Vuyanich v. Republic Nat. Bank of Dallas

505 F. Supp. 224 (1980)

Joan Rance VUYANICH

v.

REPUBLIC NATIONAL BANK OF DALLAS.

Ellen JOHNSON

v.

REPUBLIC NATIONAL BANK OF DALLAS.

Nos. CA-3-6982-G, CA-3-7949-G.

United States District Court, N. D. Texas, Dallas Division.

October 22, 1980.

*225 *226 *227 *228 *229 JoAnn Peters, Anderson & Peters, Dallas, Tex., for Vuyanich.

Linda N. Coffee, Palmer, Palmer & Coffee, Dallas, Tex., for Johnson.

Richard L. Arnold, Tobolowsky & Schlinger, Dallas, Tex., for intervenor Fenton.

Wayne S. Bishop, Seyfarth, Shaw, Fairweather & Geraldson, Washington, D. C., for defendant.

MEMORANDUM ORDER

PATRICK E. HIGGINBOTHAM, District Judge.

TABLE OF CONTENTS

I. History of Case 230

II. Class Reevaluation 233

A. "Across-the-Board" Suits 234

B. EEOC Charges by Intervenors 237

C. Class Redefinition 238

III. The Bank and Its History 242

IV. Bank Personnel Policy 243

A. Personnel Division and Hiring 243

B. Affirmative Action 247

C. Categorization of Employees 248

V. Statistical Evidence 251

A. Sources of the Data 251

B. Problems with the Data 255

1. Plaintiffs' Challenges 256

2. The Bank's Challenges 256

C. Technical and Institutional Competence 258

D. The "Anecdotal Evidence" 259

VI. The Theory Behind the Parties' Mathematical Modeling 261

A. Introduction 261

B. Job Relatedness and Equal Treatment 262

C. Controlling for Productivity 265

D. The Mathematics of Regression Analysis 267

1. Uses of Multiple Regression 267

2. Econometrics and the Ordinary

Least Squares Form of Multiple

Regression Analysis 268

a. Estimating Multiple Regressions 269

b. Statistical Inference 271

3. What Can Go Wrong? 273

E. Econometric Indication of Discrimination 275

*230

VII. Compensation 279

A. The Legal Standard for Compensation Discrimination 279

B. Compensation Data 285

1. Plaintiffs' Models 285

2. The Bank's Models 299

C. Applying the Law to the Data 304

D. Summary 319

VIII. Initial Placement and Promotion Analysis 319

A. The Data Presented 319

1. Plaintiffs' Data 320

2. The Bank's Data 331

B. Applying the Law to the Data 338

1. Black and Female Nonexempts 339

2. Female Exempts 342

3. Black Exempts 343

C. Summary 344

IX. Hiring 344

A. Statistical Inference 345

B. Plaintiffs' Case: The Data 350

C. Plaintiffs' Case: The Legal Standards 354

D. Plaintiffs' Case: Applying the Law to the Data 357

E. The Bank's Case: The Data 362

F. The Bank's Case: The Legal Standards 369

1. Occuptional and Educational Weights 369

2. Geographical Weights 375

G. The Bank's Case: Applying the Law to the Data 376

H. Summary 385

X. Terminations 385

XI. Terms and Conditions 386

A. Departmental Segregation 386

B. Maternity Leave 389

C. Marriage Policy 392

D. Training 392

E. Dress 393

F. Summary 393

CONCLUSION 394

This order constitutes the court's findings of fact and conclusions of law entered after a five-week trial to the court on the Phase I liability issues of a class action race and sex discrimination case under Title VII of the Civil Rights Act of 1964, 42 U.S.C. งง 2000e et seq. ("Title VII"). [1] For the sake of clarity, and so that this opinion may stand as a self-contained unit, the facts and procedural history of the case will be set forth in full.

I. History of the Case

Plaintiff Joan Rance Vuyanich, a black female, was first employed with defendant Republic National Bank ("the Bank" or "Republic") on April 28, 1969, as an agent contact clerk in the Money Order Department. She was then the only black employee of her department. Soon after her arrival, she began to have problems with two white female co-workers, whom Ms. Vuyanich believed to be shouldering less than a fair share of the workload. Complaints to her supervisor resulted (in her view) in only temporary improvement. On June 29, 1969, Ms. Vuyanich married a white male whom she had met during previous employment. Her supervisors first met her new husband *231 approximately one month later. A few days after this introduction, Ms. Vuyanich was called to her supervisor's office and told that there was a clash of personalities between herself and her co-workers, that the complaints about the workload were her fault, that she was not suitable for the job, and that she should resign. When asked about a transfer, her supervisor replied that Ms. Vuyanich probably did not need a job anymore since her husband was white. [2] Ms. Vuyanich was discharged from the Bank on July 28, 1969.

On August 13, 1969, Ms. Vuyanich filed a charge against the Bank with the Austin Regional Office of the Equal Employment Opportunity Commission ("EEOC"), setting forth the above events and charging the Bank with violation of Title VII. On May 23, 1972, the Dallas District Director of the EEOC issued his findings of fact, in which he found, inter alia, that Vuyanich's supervisor had discharged her instead of taking other steps to resolve the racially motivated personality conflict between her and her co-workers. A determination of reasonable cause to believe that a violation of Title VII had occurred was issued on November 6, 1972. Conciliation efforts were unsuccessful, and on March 6, 1973, the EEOC issued a statutory right-to-sue letter. On March 22, 1973, three years and eight months after her discharge, Ms. Vuyanich filed the first of these two consolidated actions.

Plaintiff Ellen Johnson, a black female, applied for a job at the Bank on September 23, 1971. Ms. Johnson was a 1971 graduate of the University of Texas at Arlington with a major in government. She first applied for a position as a management trainee or in personnel administration, but was told that no such positions were available. She then expressed her willingness to accept any position available; she was not offered a position of any kind. [3]

On October 15, 1971, Ms. Johnson filed a discrimination charge with the Dallas District Office of the EEOC. This charge alleged across-the-board race and sex discrimination by the Bank with respect to hiring, recruiting, job requirements, training, promotion, and personnel rules. On August 14, 1973, the District Director issued a determination of reasonable cause with respect to most of these allegations. On November 1, 1973, the EEOC issued a right-to-sue letter, and on December 3, 1973, the second of the present actions was filed.

The procedural history of these cases is complex, the Vuyanich case having been assigned at one time or another to five different judges of this court. The first significant event in the cases took place on November 7, 1974, when Judge Mahon conditionally certified the Vuyanich case as a class action on behalf of female and black employees and potential employees of the Bank who were adversely affected by the practices alleged by Ms. Vuyanich. Except for sporadic discovery activities, the cases remained dormant until March 12, 1976. On that date, Judge Mahon consolidated the Vuyanich and Johnson cases for discovery purposes. Judge Mahon also granted plaintiffs' motions to strike the Bank's jury demand, basing his decision on Curtis v. Loether, 415 U.S. 189 , 94 S. Ct. 1005 , 39 L. Ed. 2d 260 (1974), and Johnson v. Georgia Highway Express, Inc., 417 F.2d 1122 (5th Cir. 1969). At the same time, he denied the Bank's motion to strike Ms. Vuyanich's allegations of sex discrimination, holding her EEOC charge sufficient under Sanchez v. Standard Brands, Inc., 431 F.2d 455 (5th Cir. 1970), and Gamble v. Birmingham Southern Railroad, 514 F.2d 678 (5th Cir. 1975), to support claims both of race and sex discrimination. Vuyanich, supra n.1, 409 F. Supp. at 1086-90 .

On March 15, 1978, the cases having been transferred to the present judge, the class status of the cases was updated and redefined. *232 Based on statistical evidence presented at a two-day hearing, the court certified a class consisting of:

All females of all races and all blacks of either sex; 1) who are or have been employed by the Republic National Bank on or after February 16, 1969, and 2) who applied for employment but were not hired at the Republic National Bank on or after February 16, 1969 to date.

Vuyanich, supra n.1, 78 F.R.D. at 354 . [4] At the same time, the court consolidated the cases for all purposes. Trial of the so-called "Phase I liability" issues, see Swint v. Pullman-Standard, Inc., 539 F.2d 77 , 94 (5th Cir. 1976); Baxter v. Savannah Sugar Refining Corp., 495 F.2d 437 , 443-44 (5th Cir. 1974), cert. denied, 419 U.S. 1033 , 95 S. Ct. 515 , 42 L. Ed. 2d 308 (1975), was severed from trial of individual damage issues.

In early 1979, the court conducted a further hearing for the purpose of refining the class definition. Both sides presented sophisticated statistical analyses in support of their respective positions. On April 25, 1979, the court reaffirmed its earlier class certification order, dividing the original class into five subclasses. [5] The court simultaneously approved requests for designation of three class members as additional class representatives. These three new class representatives have been referred to as intervenors. [6]

*233 The five certified subclasses are as follows:

Subclass Subclass

Subclass Representative(s) Attorney(s)

black and female exempt Ellen Johnson Linda Coffee with

employees Marjorie Lee Jackson Joann Peters

female nonexempt employees [7] Marisu Fenton Richard Arnold

black nonexempt employees [7] Joan Vuyanich Joann Peters with

Dorothy Hooks Linda Coffee

unsuccessful black and female Ellen Johnson Linda Coffee with

applicants for exempt positions Joann Peters

unsuccessful black applicants Ellen Johnson Linda Coffee with

for nonexempt positions Joann Peters

The employee subclasses include employees who have worked for the Bank during the period from February 16, 1969, to the date of trial. The applicant subclasses include those who applied for positions at the Bank during the period from February 16, 1969, to the date of trial. [8]

The Phase I trial commenced on October 15, 1979. In the course of 24 days of testimony, the court heard from over three dozen witnesses and received thousands of exhibits into evidence. Ten of these witnesses were experts in the fields of computer science, statistics, business, or economics. [9] The testimony of these experts, together with their written reports, forms the evidentiary base for the statistical analyses at the heart of plaintiffs' case and the Bank's rebuttal of that case.

II. Class Reevaluation

Before turning to substantive issues, we deal with posttrial challenges by the Bank to the status of this case as a class action. Although the issues of class certification have already been considered repeatedly and exhaustively, see n.1, supra, the court has a continuing duty to reevaluate class status on the basis of events which transpire and circumstances which develop as the litigation unfolds. E. g., Guerine v. J & W Investment, Inc., 544 F.2d 863 , 864 (5th Cir. 1977); Cooper v. University of *234 Texas, 482 F. Supp. 187 , 190 (N.D.Tex.1979), appeal docketed, No. 80-1412 (5th Cir. Apr. 15, 1980). This reevaluation process is complex, and will be discussed in detail in due course. The spokes of all the Bank's challenges to the class as it is now structured run in the final analysis to a central hub. That hub is an attack on the maintainability of so-called "across-the-board" suits. We turn then to their role in the evolving framework of Title VII class action litigation.

A. "Across-the-Board" Suits.

The concept of an "across-the-board" class action, i. e., an action challenging a wide range of employment practices alleged to result from a common discriminatory animus, owes its origin to Johnson v. Georgia Highway Express, Inc., 417 F.2d 1122 (5th Cir. 1969). The Fifth Circuit in Johnson reversed a ruling by the trial court that a discharged black employee could only represent other discharged black employees. The court found such a limitation to be error "as it is clear from the pleadings that the scope of appellant's suit is an `across-the-board' attack on unequal employment practices alleged to have been committed by the appellee pursuant to its policy of racial discrimination." 417 F.2d at 1124 . Noting the "Damoclean threat of a racially discriminatory policy," Hall v. Werthan Bag Corp., 251 F. Supp. 184 , 186 (N.D.Tenn.1966), hanging over the entire racial group, the court held that the plaintiff could represent all black employees of the defendant in their claims of alleged discrimination in hiring, firing, promotion, and maintenance of facilities. 417 F.2d at 1124 .

The Johnson decision received continued support in Fifth Circuit Title VII jurisprudence. Thus in Carr v. Conoco Plastics, Inc., 423 F.2d 57 (5th Cir.), cert. denied, 400 U.S. 951 , 91 S. Ct. 241 , 27 L. Ed. 2d 257 (1970), a group of black employees was permitted to maintain an "across-the-board" suit challenging hiring and internal personnel policies. In Jack v. American Linen Supply Co., 498 F.2d 122 (5th Cir. 1974), the court held that a former employee could properly represent a class of present and future employees. Similarly, in Long v. Sapp, 502 F.2d 34 (5th Cir. 1974), the court, reaffirming Johnson, held that a terminated employee could challenge racially discriminatory policies allegedly "pervad[ing] all aspects of the employment practices of" her former employer:

Having shown herself to be a black and a former employee ... she occupies the position of one she says is suffering from the alleged discrimination. She has demonstrated the necessary nexus with the proposed class for membership therein. As a person aggrieved, she can represent other victims of the same policies, whether or not all have experienced discrimination in the same way.

502 F.2d at 43.

The Fifth Circuit approach, characterized as "pioneering," Worley v. Western Electric Co., 22 Empl.Prac.Dec. ถ 30,600, at 14,224-25 (N.D.Ga.1979), received widespread but not universal support in other circuits. The "across-the-board" approach was recognized and approved by the Third Circuit, Wetzel v. Liberty Mutual Insurance Co., 508 F.2d 239 , 247 (3d Cir.), cert. denied, 421 U.S. 1011 , 95 S. Ct. 2415 , 44 L. Ed. 2d 679 (1975), the Fourth Circuit, Barnett v. W. T. Grant Co., 518 F.2d 543 , 547-48 (4th Cir. 1975), the Sixth Circuit, Senter v. General Motors Corp., 532 F.2d 511 , 523-24 (6th Cir.), cert. denied, 429 U.S. 870 , 97 S. Ct. 182 , 50 L. Ed. 2d 150 (1976); Tipler v. E. I. duPont deNemours & Co., 443 F.2d 125 , 130 (6th Cir. 1971), the Eighth Circuit, Donaldson v. Pillsbury Co., 554 F.2d 825 , 829-32 (8th Cir.), cert. denied, 434 U.S. 856 , 98 S. Ct. 177 , 54 L. Ed. 2d 128 (1976); Reed v. Arlington Hotel Co., 476 F.2d 721 , 722-23 (8th Cir.), cert. denied, 414 U.S. 854 , 94 S. Ct. 153 , 38 L. Ed. 2d 103 (1973); Parham v. Southwestern Bell Telephone Co., 433 F.2d 421 , 425 (8th Cir. 1970), and a variety of district courts, e. g., Mack v. General Electric Co., 329 F. Supp. 72 , 75-76 (E.D.Pa.1971); Wilson v. Monsanto Co., 315 F. Supp. 977 , 979 (E.D.La.1970); see Hall v. Werthan Bag Corp., supra ("across-the-board" action as to injunctive relief but not as to back pay). See generally Developments in the Law-Employment *235 Discrimination and Title VII of the Civil Rights Act of 1964, 84 Harv.L. Rev. 1109, 1219-21 (1971). The "across-the-board" approach was rejected in the Tenth Circuit, Taylor v. Safeway Stores, Inc., 524 F.2d 263 , 270-71 (10th Cir. 1975), and by a number of district courts, e. g., Tolbert v. Daniel Construction Co., 332 F. Supp. 772 , 775 (D.S.C.1971); White v. Gates Rubber Co., 53 F.R.D. 412 , 413 (D.Colo.1971); Hyatt v. United Aircraft Corp., 50 F.R.D. 242 , 245-47 (D.Conn.1970); Burney v. North American Rockwell Corp., 302 F. Supp. 86 , 90-91 (C.D.Cal.1969); Colbert v. H-K Corp., 295 F. Supp. 1091 , 1093 (N.D.Ga.1968), vacated on other grounds, 444 F.2d 1381 (5th Cir. 1971).

The continuing propriety of "across-the-board" class actions under Title VII was called into question in 1977 by East Texas Motor Freight Systems, Inc. v. Rodriguez, 431 U.S. 395 , 97 S. Ct. 1891 , 52 L. Ed. 2d 453 (1977). In that case, the district court had denied certification of an "across-the-board" suit, and had rejected the named plaintiffs' claims on their merits after trial. The Fifth Circuit reversed, certifying a class, and itself found classwide liability on the basis of the trial record. The Supreme Court in turn reversed the court of appeals, holding, based upon the failure of the named plaintiffs' claims, their failure to move for class certification, and their conflicts of interest with members of the class, that the named plaintiffs were not proper class representatives. In this regard, the Court stated:

We are not unaware that suits alleging racial or ethnic discrimination are often by their very nature class suits, involving classwide wrongs. Common questions of law or fact are typically present. But careful attention to the requirements of Fed.Rule Civ.Proc. 23 remain nonetheless indispensable. The mere fact that a complaint alleges racial or ethnic discrimination does not in itself ensure the party who has brought the lawsuit will be an adequate representative of those who may have been the real victims of that discrimination.

431 U.S. at 405-06, 97 S. Ct. at 1897-1898 . Citing Schlesinger v. Reservists Committee to Stop the War, 418 U.S. 208 , 216, 94 S. Ct. 2925 , 2929, 41 L. Ed. 2d 706 (1974), the Court noted that "a class representative must be part of the class and `possess the same interests and suffer the same injury' as the class members." 431 U.S. at 403 , 97 S.Ct. at 1896.

Mirroring the pre- Rodriguez dissension among the circuits, courts are divided on the question of whether "across-the-board" suits survive Rodriguez. In Payne v. Travenol Laboratories, Inc., 565 F.2d 895 , 900 (5th Cir.), cert. denied, 439 U.S. 835 , 99 S. Ct. 118 , 58 L. Ed. 2d 131 (1978), the Fifth Circuit, without discussing Rodriguez, stated:

Plaintiffs' action is an "across-the-board" attack on unequal employment practices alleged to have been committed by Travenol pursuant to a policy of racial discrimination. As parties who have allegedly been aggrieved by some of those discriminatory practices, plaintiffs have demonstrated a sufficient nexus to enable them to represent other class members suffering from different practices motivated by the same policies.

In Satterwhite v. City of Greenville, supra n.7, the court again stated:

Nor is Rodriguez or this opinion contrary to the policy favoring "across-the-board" Title VII class actions. See Johnson v. Ga. Highway Express, supra . It is not necessary that the representative suffer discrimination in the same way as other class members, but it is necessary that she suffer from the discrimination in some respect.

578 F.2d at 993-94 n.8. Finally, in Falcon v. General Telephone Co., 626 F.2d 369 (5th Cir. 1980), the court settled any remaining doubt, holding squarely that "an employee complaining of one employment practice [may] represent another complaining of another practice, if the plaintiff and the members of the class suffer from essentially the same injury." The Fourth Circuit has reached a contrary result, Hill v. Western Electric Co., 596 F.2d 99 (4th Cir.), cert. denied, 444 U.S. 929 , 100 S. Ct. 271 , 62 *236 L.Ed.2d 186 (1979); see Vuyanich, supra n.1, 82 F.R.D. at 433 n.8, and district courts are split on the issue. [10]

The Payne, Satterwhite and Falcon cases are controlling precedent in this circuit for the proposition that "across-the-board" suits remain appropriate in proper circumstances notwithstanding Rodriguez. Even were the slate clean, however, the factors favoring "across-the-board" suits identified by this court in its earlier order, see Vuyanich, supra n.1, 82 F.R.D. at 432-33 , would remain cogent today. The fundamental thesis of "across-the-board" actions remains the existence of a discriminatory animus cutting across a variety of employment practices; where there is substantial evidence that such an animus exists, the class representative "possesses the same interests" in its elimination and "suffers the same injury" from its presence, within the meaning of Rodriguez, as the class members she represents. [11] Whether the direct approach typical of the Fifth Circuit, in which a broad class is certified, or the two-step approach of the Third Circuit, where a narrower class is certified but broader issues examined, see Alexander v. Gino's, Inc., supra n.10, is employed, an "across-the-board" suit serves to vindicate classwide rights on a classwide basis.

Notwithstanding the propriety of "across-the-board" actions in appropriate cases, it must be kept in mind that not every Title VII suit, nor even every Title VII class action, is an "across-the-board" suit. Only where the plaintiff alleges that the particular discrimination she has suffered is due to a racist or sexist animus pervading the defendant's practices, and only where the plaintiff presents sufficient proof of that allegation at the class certification stage, should an action be certified as an "across-the-board" class action. See Falcon v. General Telephone Co., supra , at 376 n.9. The thrust of this court's earlier opinion was that the plaintiffs in this case have met that burden. At the class certification stage, the plaintiffs presented statistical evidence showing discrimination in hiring, pay, promotion, and termination. These statistics constituted substantial evidence that discrimination at the Bank, if indeed such discrimination were provable at trial, was the product of a common discriminatory *237 animus characteristic of "across-the-board" actions. It was on that basis that this action was so certified. [12]

One additional factor, not fully present at the time of the court's 1979 certification order, counsels in favor of "across-the-board" treatment for this case. The court noted at that time that challenged practices such as promotion, pay, training, testing, transfer, job assignment and classification, job content, and constructive discharge were intertwined. Vuyanich, supra n.1, 82 F.R.D. at 433 . What was not then apparent, but which now reinforces the decision to treat this as an "across-the-board" suit, was the fact that much of the statistical proof presented at trial would mirror this intertwined relationship. The introduction of multiple regression econometric studies, discussed more fully in sections VI through VIII, infra, which can in certain circumstances measure discrimination in a wide variety of forms through related statistical tests, [13] suggests that "across-the-board" treatment, far from fragmenting the proof adduced at trial, may in fact be a preferred method for testing for discrimination by an employer with a large diverse work force.

B. EEOC Charges by Intervenors.

The Bank next challenges this court's earlier holding, see Vuyanich, supra n.1, 82 F.R.D. at 437-38 , that Marisu Fenton, Dorothy Hooks, and Marjorie Jackson could intervene as the representatives of certain subclasses despite having failed to file EEOC charges. Citing Hodge v. McLean Trucking Co., 607 F.2d 1118 (5th Cir. 1979), it asserts that the case of Wheeler v. American Home Products Corp., 563 F.2d 1233 (5th Cir. 1977), on which this court's holding was based, has been overruled. This argument must fail for two reasons. First, one panel of the Fifth Circuit has no power to overrule a decision of a previous panel. E. g., Ford v. United States, 618 F.2d 357 , 361 (5th Cir. 1980); Gates v. Collier, 616 F.2d 1268 , 1272 (5th Cir. 1980). Second, even were a panel possessed of such a power, examination of Hodge reveals it to be fully consistent with, if not dictated by, the Wheeler doctrine.

The Fifth Circuit in Oatis v. Crown Zellerbach Corp., 398 F.2d 496 (5th Cir. 1968), first established the rule that unnamed plaintiffs in a Title VII class action need not have exhausted administrative remedies by filing a discrimination charge with the EEOC. The court in that case reasoned that it would "be wasteful, if not vain, for numerous employees, all with the same grievance, to have to process many identical complaints with the EEOC." 398 F.2d at 498 . Thus it held that the EEOC complaint of the named class representative was sufficient to support an action on behalf of the entire class.

This rule was extended to intervenors in Wheeler v. American Home Products Corp., supra , an action in which class status had been denied. See Romasanta v. United Airlines, Inc., 537 F.2d 915 , 919 n.7 (7th Cir. 1976), aff'd sub nom. United Airlines, Inc. v. McDonald, 432 U.S. 385 , 97 S. Ct. 24 , 64, 52 L. Ed. 2d 423 (1977). The Oatis and Wheeler cases both rely on the EEOC complaint of the original plaintiff to stand in lieu of EEOC complaints by others. Hence it is not surprising that intervention was denied *238 in Hodge, where the original plaintiff had failed to file an effective EEOC complaint, since in that case the intervenors could not rely on the original plaintiff's charge. This being the case, Hodge cannot be interpreted as altering the long-standing rule in the Fifth Circuit that intervenors need not exhaust EEOC remedies.

Finally, the Bank argues that there must be an independent basis of jurisdiction for each subclass, i. e., that for each subclass there must be a representative who has filed an EEOC charge. [14] The Bank argues that this requirement is a necessary concomitant of an asserted requirement under Fed.R.Civ.P. 23(c)(4)(B) that each subclass meet all the requirements of Rule 23. See Monarch Asphalt Sales Co. v. Wilshire Oil Co., 511 F.2d 1073 , 1077 (10th Cir. 1975); Weathers v. Peters Realty Corp., 499 F.2d 1197 , 1200 (6th Cir. 1974); cf. Chmieleski v. City Products Corp., 71 F.R.D. 118 , 150 (W.D.Mo.1976) (numerosity). Assuming arguendo that subclasses must always satisfy each requirement of Rule 23, it does not follow that an independent jurisdictional base is required for each subclass. The ratio decidendi of Oatis and Wheeler was that the named plaintiff's EEOC charge will be typical of those which would have been filed by those she represents. Subclassing was undertaken in this case for the dual purposes of facilitating presentation of issues, Vuyanich, supra n.1, 82 F.R.D. at 433 , and eliminating potential conflicts of interest, id. at 435. Since the court found plaintiffs' claims to be an "across-the-board" attack on discriminatory practices, typicality, while serving as a convenient basis for separation of subclasses, was not a factor compelling subclassing. Otherwise stated, the entire class, notwithstanding subclassing, continues to complain of "across-the-board" discrimination. Hence, just as absent class members in an "across-the-board" suit may rely on the named plaintiff's charge, Oatis, supra , and just as intervenors who assert claims common to those asserted by existing plaintiffs may travel on those plaintiffs' complaints, Wheeler, supra , the subclass representatives may rely on the EEOC charges of plaintiffs Vuyanich and Johnson. [15] See Vuyanich, supra n.1, 82 F.R.D. at 437-38 .

C. Class Redefinition.

This court has recognized its "continuing duty to monitor and modify the class according to the facts that develop." Cooper v. University of Texas, supra , at 190. Behind this seemingly simple phrase, however, lies considerable complexity. Throughout the evaluation process, the legal standards remain the same: the case must satisfy the four prerequisites of Fed.R.Civ.P. 23(a), and must fit into one or more of the categories of Fed.R.Civ.P. 23(b). Nonetheless, the extent to which a determination that these requirements have been satisfied may be reexamined during the course of the litigation must vary from requirement to requirement and must depend on the procedural posture of the case. Just as class certification is not a ritual exercise which once done may be laid to one side, class reevaluation cannot depend on a formalistic invocation of the five class action requirements.

The four prerequisites of Fed.R.Civ.P. 23(a) serve to protect the differing and *239 occasionally antagonistic interests of the named parties, the unnamed class members, and the court. The numerosity requirement of Fed.R.Civ.P. 23(a)(1), for example, protects the interests of absent class members, who might otherwise be unnecessarily deprived of the right to control their own litigation, and those of the court, "in assuring a full and fair exposition of views by all affected parties when it is practicable to join them in a single proceeding." Scott v. University of Delaware, 601 F.2d 76 , 88 (3d Cir.), cert. denied, 444 U.S. 931 , 100 S. Ct. 275 , 62 L. Ed. 2d 189 (1979). The commonality requirement of Fed.R.Civ.P. 23(a)(2) serves to focus the issues, protecting both the court and the defendant by ensuring that resources are not wasted through inquiry into a multitude of diverse controversies. The adequacy of representation test of Fed.R.Civ.P. 23(a)(4) is for the primary benefit of absent class members, whose rights might otherwise be adjudicated in a binding fashion without a complete presentation of the facts and legal arguments supporting their contentions. See generally C. Wright & A. Miller, Federal Practice and Procedure ง 1765 (1972). The typicality requirement of Fed.R.Civ.P. 23(a)(3) has been characterized as a "double schizophrenic." 3B Moore's Federal Practice ถ 23.06-2, at 23-192 (2d ed. 1980). It serves both the class and the court, complementing the commonality requirement as well as providing an intuitive check on the court's determination that the class representative will adequately protect the rights of absent class members. The differing nature of these purposes suggests that differing levels of inquiry apply to each of the requirements as the litigation evolves.

At the outset of the case, all four requirements must be satisfied. [16] Fed.R. Civ.P. 23(c)(1) requires the court, "[a]s soon as practicable after the commencement of" the action, to determine whether it may be maintained as a class action. The standard of proof for such a determination is, however, ill-defined. As the court stated in its earlier order with regard to commonality:

Of course a plaintiff must do more to demonstrate the existence of the question than simply assert its existence. Bare bones conclusions are insufficient. At the same time, a plaintiff need not make out a prima facie case of liability. The higher courts have not yet articulated where between these marks a plaintiff must place his proof.

Vuyanich, supra n.1, 82 F.R.D. at 431 . Similar uncertainty exists as to how far a plaintiff must go in proving numerosity, adequacy, and typicality. We do know that a plaintiff need not show at the class certification stage that she has a winning individual claim as a sine qua non to the typicality of her claim or the adequacy of her representation. Huff v. N. D. Cass Co., 485 F.2d 710 (5th Cir. 1973) ( en banc ). At this early stage, the typicality requirement will commonly be met by a showing that the issues which will likely be raised at trial by the named plaintiff are typical of those of the class, and the adequacy requirement by a demonstration that plaintiff will likely be an adequate representative at trial due to her incentive and ability to represent the class and the absence of conflicts of interest between herself and the class.

Fed.R.Civ.P. 23(c)(1) also authorizes the court to alter or amend a class certification order as the litigation progresses. Hence if pretrial activity demonstrates that a case is not appropriate for class treatment, the district court should decertify the action. Lamphere v. Brown University, 553 F.2d 714 , 720 (1st Cir. 1977). Considerations such as this led this court to decertify the portion of the class composed of unsuccessful female applicants for non-exempt positions. Vuyanich, supra n.1, 82 F.R.D. at 438 . As discovery progresses, it is reasonable to hold the plaintiff to a higher *240 standard of satisfaction of such requirements as numerosity and commonality, as to which more precise proof will become available. At the same time, reliance on incentive, ability, lack of conflict, and typicality as measures of adequate representation, while still present, will give way to an evaluation of the extent to which the plaintiff's litigation performance demonstrates adequacy. Thus if it becomes apparent that adequate representation is not being provided, the court must withdraw class status from the suit. Guerine v. J & W Investments, supra, at 864-65.

The duty to continually reevaluate class status does not end with the commencement of trial. Because, however, by the end of trial the court's interests in efficiency and manageability have for better or for worse been realized or frustrated, the focus of reevaluation at this stage must be on the factors protecting the parties, named and unnamed. Principal among these is adequacy of representation, and the court must not hesitate to decertify a class in whole or in part if the plaintiff has failed to present at least minimal evidence, or has otherwise demonstrated that her representation of all or part of the class is less than adequate. Cooper v. University of Texas, supra , at 198. Likewise, if the court finds that the numerosity requirement, which is based in part on protection of the absent class members' right of autonomy, is not satisfied, the court must decertify. Scott v. University of Delaware, supra, at 88-89. By this stage of the proceedings, the emphasis is on actual rather than predictive measures of adequacy and numerosity: the proof has presumably been fully developed, and the court may evaluate plaintiff's actual adequacy at trial.

The role of typicality in the post-trial class reevaluation scenario is uncertain. To the extent that typicality mirrors the commonality requirement, whose role is diminished once the trial is concluded, its role will also be diminished. To the extent that typicality serves as a predictor of adequacy of representation, its importance is overshadowed by the more objective measures of adequacy which become available after prolonged observation by the court of the conduct of plaintiff and her counsel. While the typicality requirement does serve a residual role of providing a check on the accuracy of that evaluation, that role is limited: unless the named plaintiff's claim appears at trial to be so atypical of the those of the class that the adequacy of her representation is drawn into question, the court should not decertify the class merely because the individual plaintiff's proof differs from that presented on behalf of the class. This is especially the case where, as here, consideration of the named plaintiff's individual claim has been severed from trial of class issues. [17]

The Bank argues that there must be a "continuing nexus" at all stages of the litigation between the named class representation and the class members she represents. While this assertion is undoubtedly correct in the abstract, the "continuing nexus" test cannot be mechanically applied without regard to the procedural posture of the case. In each of the cases cited by the Bank, an appellate court held that a class representative whose individual claim has failed cannot continue to represent the class, either on appeal or on remand following appeal. East Texas Motor Freight Systems, Inc. v. Rodriguez, supra ; Armour v. City of Anniston, 597 F.2d 46 (5th Cir. 1979), vacated, 445 U.S. 940 , 100 S. Ct. 1334 , 63 L. Ed. 2d 774 , remanded, 622 F.2d 1226 (5th Cir. 1980); Davis v. Roadway Express, Inc., 590 F.2d 140 (5th Cir. 1979), on rehearing, 621 F.2d 775 (5th Cir. 1980) (reaffirmed, but on other grounds than in earlier opinion); Camper v. *241 Calumet Petrochemicals, Inc., 584 F.2d 70 (5th Cir. 1978); Satterwhite v. City of Greenville, supra n.8. [18] It is a far different matter, however, to hold that the failure of a named plaintiff's claim at trial requires the retroactive decertification of the class and consequent failure of the class claims. [19]

This distinction is recognized by the "continuing nexus" cases themselves. In Rodriguez, the progenitor of this line of cases, the Court stated:

Obviously, a different case would be presented if the District Court had certified a class and only later had it appeared that the named plaintiffs were not class members or were otherwise inappropriate class representatives. In such a case, the class claims would have already been tried, and, provided the initial certification was proper and decertification not appropriate, the claims of the class members would not need to be mooted or destroyed because subsequent events or the proof at trial had undermined the named plaintiffs' individual claims. [Citations]. Where no class has been certified, however, and the class claims remain to be tried, the decision whether the named plaintiff should represent a class is appropriately made on the full record, including the facts developed at the trial of the plaintiffs' individual claims.

431 U.S. at 406 n.12, 97 S. Ct. at 1898 n.12. [20] The Fifth Circuit in Satterwhite outlined the reasons underlying this distinction:

Where a class is certified, and class claims tried, before the lack of merit or mootness of the representative's claim is discovered, the class representative has already assiduously asserted the claims of the constituents. The conservation of both litigants' and judicial resources makes it desirable not only to avoid abortion of the litigation but also to prevent prejudice to the members of a certified class who, in the midst of a law suit, suddenly discover that their representative's claim is no longer viable.

578 F.2d at 994. Accord, Drayton v. City of St. Petersburg, 477 F. Supp. 846 , 857 n.19 (M.D.Fla.1979). Indeed, in addition to wasting the resources of the parties and the court and frustrating the reasonable reliance of absent class members, decertification on the basis of failure of the individual plaintiff's claim would in many cases disserve the defendant, by depriving it of an adjudication of nonliability binding on the class.

It remains only to apply these standards of class reevaluation to the present case. The Bank makes no argument that the numerosity and commonality requirements are no longer satisfied, and indeed no such argument could be made given the state of the record. The course of the trial has revealed no continuing pattern of inadequacy of representation warranting decertification. Decertification of particular subgroups within the class and subclasses involves an evaluation of whether the plaintiffs have produced at least minimal evidence on the issues applicable to those subgroups, and is best deferred until consideration and evaluation of those parts of the *242 plaintiffs' case. As will be seen, no decertification is warranted.

Finally, the testimony of the named class representatives reveals their claims to be sufficiently typical that no substantial doubts are raised as to the adequacy of their representation. The testimony of plaintiffs Vuyanich and Johnson has already been described in section I, supra, and will not be repeated here. The testimony of intervenors Jackson, Fenton, and Hooks at trial mirrored that given by them during the class certification hearing and summarized in n.6, supra. The wide variety of discriminatory practices testified to by these individuals reinforces the court's earlier determination that the essence of their claims was the presence of a racially and sexually discriminatory animus pervading the Bank's personnel practices. Whether this animus in fact existed is a question which must await more detailed evaluation of the statistical and other evidence presented at trial. The court is convinced at this juncture, however, that the claims of the named plaintiffs and intervenors, whether meritorious or not, are typical of those asserted on behalf of the class.

III. The Bank and Its History [21]

With 2,400 full-time employees and assets exceeding $8 billion, Republic National Bank is among the 25 largest banks in the United States and is the largest bank in the South. Through its sister companies, its corporate influence is further extended, as it is the "flagship" bank of the Republic of Texas Corporation, a bank holding company incorporated under the laws of Texas with some 30 wholly-owned subsidiaries. The Bank's principal business is commercial lending. Republic lends to consumers as well as businesses, but consumer lending accounts for not more than 5% of its loan portfolio. Indeed, Republic is the third largest nonretail unit bank in the United States.

The "line" function of commercial lending is carried out by the Bank's five commercial banking departments. The Banking Department, consisting of the Cash Management, Commodity, Metropolitan, Southwestern, and National Divisions, provides business loans and financial consultation to the customers of those divisions' respective geographic regions. Loans to individuals are provided by the Consumer Lending Group of the Metropolitan Division. Loans to other banks are generated by the Correspondent Banking Department, while the Real Estate Department offers single mortgage lending services and interim financing for the building and construction industry. The Petroleum and Minerals Department makes loans to companies in the petroleum and related minerals industry, and the International Department provides financial assistance to United States based companies who wish to deal in foreign markets, in addition to financial service to those markets.

Other departments in the Bank include the Trust and Investment Department, the Finance and Credit Administration Department, the Operations Department, and the Funds Management Department. The Trust and Investment Department is responsible for managing money and other assets for individuals and organizations, according to specified conditions set out in trust agreements. The department consists of the following divisions: Securities Management Services, Personal Services, Operations Services, Corporate Services, Legal Counsel, Taxes, and Business in Trust.

The Funds Management Department is responsible for making investment judgments affecting the flow of funds in and out of the Bank. The objective of this department is to ensure that surplus funds are properly invested to obtain maximum return, and that funds needed to meet the Bank's money commitments are secured at the lowest cost. This department also houses the Bank's municipal bonds and sales and *243 trading operation, as well as bond portfolio management.

The Finance and Credit Administration Department is responsible for the day-to-day business of the Bank. This department includes the following divisions: Controller, Personnel, Credit and Corporate Finance, and Loan Review and Special Loans. The Operations Department is responsible for all of the Bank's banking customer services functions and data processing, programming, production, and development. Other staff divisions at the Bank include Corporate Planning, Legal Counsel, Marketing and Public Affairs, and Economic Research, all of which report directly to the President, and the Audit Division, which reports to the Chairman of the Board.

The Bank's Personnel Division is responsible for coordinating and administering the Bank's equal employment and personnel policies, including hiring, promotion, compensation and benefits, counseling, discipline, and termination. These responsibilities are carried out by three group managers. The Personnel Group, headed by Vice President Thomas E. Barksdale, has responsibility for the salary administration section and the payroll and benefits section. The Personal Development Group, led by Vice President Dan White, coordinates Bank employee participation in external employee development programs, and the development and presentation of internal programs leading to personal development. The Employment Group is the responsibility of Jerry M. Watson, Vice President and Manager of Interview and Selection. This group carries out the Bank's staffing and recruiting functions. Watson also has responsibilities relating to affirmative action, internal employee transfers, terminations, counseling, and record keeping. Each group manager reports to Thomas Croft, Senior Vice President and Director of Personnel, who in turn reports to the Executive Vice President and Manager of the Bank's Finance and Administration Department.

The Bank's structure thus reflects independent staff divisions serving the core lending function, including economic research, legal counsel, marketing, and public affairs. Over the decade spanned by this lawsuit, that structure has been staffed with a work force that has ranged in size from a low of 1,450 employees to a high of approximately 2,400. Over this time the internal lines of the structure have occasionally shifted, with resulting changes in departmental or division categories. Today the work force falls into nine separate organizational departments, with the staff divisions earlier mentioned.

Roughly speaking, the Bank's work force may be sliced horizontally according to the Fair Labor Standards Act categorization of exempt and nonexempt employees. During the period from 1969 to 1972, 71% of the work force was nonexempt and 29% was exempt (Plaintiffs' Exhibit 641). The same approximate percentage existed throughout.

Before 1965 there was a virtual absence of black employees from the Bank's work force. From 1968 through 1972, there were no blacks in the exempt category. The first black officer of Republic Bank did not arrive until 1973. As late as 1972, females were significantly better represented than blacks in their employment in the exempt category. Like blacks, however, and throughout the 10-year period at issue, there was a concentration of females in the nonexempt categories, and in turn in the lower grades of the nonexempt range. The numbers of females hired into exempt positions, and the movement of females upward within the nonexempt category, have risen gradually over the 10-year period. Today there are approximately 15 black employees in the exempt category, all of whom are below the vice president level. Of 570 Bank officers, 132 are female, 13 of whom are at the level of vice president.

IV. Bank Personnel Policy

A. Personnel Division and Hiring.

The Personnel Division of the Bank is headed by a Vice President. The Vice President is responsible for the Bank's personnel activities, including day-to-day staffing, campus recruiting, EEO activities, the infirmary, counseling, and career advancement. *244 Eight exempt employees work under his direct supervision. Hiring is done in many ways. Nonexempt employees are recruited largely through word of mouth and applicants for nonexempt positions are usually "walk-ins." The experience of the Personnel Division has been that applicants for exempt positions are attracted by advertisements, referred by other companies to the Bank, or recruited at college campuses. The use of employment agencies and search firms is principally confined to the filling of secretarial positions or hiring of accountants and data processing specialists. Recruiting is also done through governmental and civic agencies, including the Texas Employment Commission, National Association of Bankers, the Dallas Interracial Council, Minority Women's Employment Council, the Black Chamber of Commerce, and the Mexican-American Chamber of Commerce. In exempt job offerings, the Personnel Division has throughout the period covered by this suit placed great emphasis on the interview process. The Bank has used from time to time both black and female interviewers. The ultimate hiring decisions for exempt employment have, however, been made by white males.

Throughout the 10 years covered by this suit, the Bank has had a large number of job titles and positions. The number of such titles has ranged to upwards of 3,500, with as many as 700 to 950 titles in use at a given time. In mid-1979, as part of the defense of this suit and, according to the Bank, for other business purposes as well, the entire job structure was regrouped by occupational codes and families. The purpose of this effort was said to be to achieve horizontal symmetry in job functions by looking past job titles to job functions. Employees with similar job functions albeit with different descriptive titles, were grouped together in the same job family. This division of the work force, whose results are the benchmarks for labor availability in the statistical hiring studies and other studies, is reviewed in depth where those issues are discussed. The point here is that in an overview of the Bank structure, we find a shifting organization as the Bank grew in size followed by a dramatic adjustment accomplished in one sweep. The validity of that precipitate change must be examined. See section IX(G), infra.

With recruiting for entry positions leading to "line" jobs, the effort begins on college campuses, and has historically been concentrated in the Southwest. Recruiting at college campuses is confined with little exception to recruitment of exempt employees entering the credit analyst program. [22] The credit analyst program is the main entry channel for future bank executives. As stated by the Bank, "the Credit Training Program is the means by which [the Bank] seeks out and develops persons with the potential to assume positions of managerial responsibility with the bank" (Defendant's Post Trial Brief at 286). In hiring into this program, the Bank places great emphasis upon a business-related degree with emphasis in accounting and finance, a preference that will be examined later in the context of hiring and availability of potential employees. See section IX(G), infra. In the time period 1970-1978, 181 persons were hired into the credit training program. Of this group, 34 were hired for their experience. Of the remaining 147, only two did not have a business-related degree. These two held degrees in mathematics and law, respectively. This is consistent with the overall hiring pattern for the years 1970-1978, when there were a total of 997 exempt hires, of which 860 (86%) had at least a bachelor's degree. Of those who did not, virtually all had experience related to the jobs for which they were hired (Defendant's Exhibits 508, 566).

Credit analysts assist loan officers while they are being "trained" and provide a pool from which the loan "floor" draws. The credit training program is said by the Bank to be the breeding ground for its loan officers. And except for occasional lateral hiring of an individual with equivalent training, *245 it is the only means for staffing loan officer positions. An examination of the pattern of Bank hires and track of progression followed by incumbent management of the Bank reveals that as a practical matter, it is the main track for higher level executives within the Bank. This is not surprising in view of the Bank's core function of commercial lending and its insistence upon college degrees in business fields for persons in commercial lending. Indeed, throughout, the majority of all Bank officers have been in lending and marketing jobs.

The credit training program at the Bank has existed throughout the time period. In fact, the current president of the Bank is a product of that program. The program's changes have been primarily in increased intensity in curriculum. In 1969, the Bank employed a professor at the Harvard Business School to design a more intensive program and the Bank has implemented that program. In a nutshell, there is no other internal training or internship program of a formal nature at the Bank for exempt employees.

The Credit Department has approximately 36 desks. A new employee historically spends approximately nine months in the department before entering one of the respective loan divisions. The new employee is expected to progress through levels of credit analyst, unit manager, and supervisor. In this nine month period, the analyst receives increasing responsibility for the credit analysis work of prospective borrowers or existing bank customers. Part of the program consists of a "live-in" system. Under this program, analysts, after a few months at the Bank, are assigned to a particular division and work for that division alone for approximately a month. This rotating assignment of live-ins allows the analyst to become familiar with the somewhat differing emphases of the respective areas of the Bank while allowing the lending officers in those areas to observe the work of the credit analyst. Credit analysts are not officers of the Bank. It is anticipated, however, that upon completion of the credit program and acceptance of a position in one of the respective lending divisions the new employee will be elected to officer status.

When we turn to the filling of vacancies for nonexempt positions, the emphasis shifts from the campus to the Bank's internal staff and to more local labor pools. Moreover, there is far more lateral hiring for nonexempt jobs than for exempt jobs. There is also some movement from nonexempt into the lower exempt jobs as well as some overlap of exempt and nonexempt personnel hiring responsibility. With nonexempt employees, the Bank's promotional and training policies have changed over the years. Before 1975, the movement of employees from job to job within the Bank lacked formal structure. There were no predetermined paths, and no specific prohibitions or established channels. Movement was usually accomplished through one or a combination of three methods: supervisory referral; employee request to appropriate managers; and employee-initiated contact with interviewers in the Personnel Division. Not surprisingly, this loosely structured process caused personnel decisionmaking to be more subjective than the system which came into effect in 1975.

In February of 1975, the Bank created a "career advancement program." The career advancement program consists of formalized job posting and bidding, enabling employees to learn of the existence of other positions within the Bank. The program contemplates that job openings will be posted outside the employee cafeteria, at the drive-in motor bank, and at the Spring Valley Commuter center, with listings updated daily. In April of 1977, the program was expanded to include exempt job openings except those with the title of officer, manager, or administrator. Under the program, an employee interested in a transfer completes a career advancement interview request form and forwards it to the Personnel *246 Division. As applications for transfers are received in the Personnel Division, the personnel files of the applicants are reviewed for eligibility for transfer by a personnel representative. An employee is not eligible for transfer consideration until he has been in his current job for six months. Eligible applicants are then scheduled for screening interviews in the Personnel Division. Interviews are conducted by the supervisor in the section to which the transfer is sought. Bank policy is that effort is to be made to fill vacancies by internal transfer or promotion before outside hiring is attempted, although this policy does not appear to be uniformly followed.

When the Bank turns to outside employees, authorization must be granted by the Personnel Division upon receipt of a personnel requisition by a line manager or a request for addition to staff. [23] The Personnel Division's employment group interviewers begin efforts to fill the job. Applicants are screened by the Personnel Division and those applicants considered to possess the qualifications for the requested job are referred to management in the target division.

As earlier mentioned, hiring for most nonexempt positions at the Bank begins with an applicant's visit to the Personnel Division. Walk-ins are required to sign an applicant log at one end of the Personnel Division offices. The log is maintained by the interview coordinator, but no information has been maintained regarding qualifications of particular applicants and information regarding race and sex was not kept before 1974. Sources of Bank hires also include unsolicited resumes mailed to the Bank and college recruiting programs. While exempt and nonexempt hiring have some common sources, these resumes and the college recruiting program have been the richest source for exempt applicants. Despite this small overlap in sources, the hiring approaches were different. Before 1974, the Personnel Division used a "clerical" application form and a "professional" application form. These forms were the same except that the professional form contained places for professional and technical references while the clerical form contained a space for office machine operation, typing speed, and shorthand speed. The Bank claims that the professional form was given to those who expressed an interest in exempt positions or those who had a college degree. The form, utilized by the Bank since 1974, instructs walk-in applicants to indicate specific areas of banking in which they are interested and for which they are qualified. Upon receipt of the completed application form, the interview coordinator arranges an interview with one of the Personnel Division interviewers if the applicant has the "appropriate" qualifications. That is, there is some pre-interview screening: while the Personnel Division interviews many persons, not all applicants are interviewed. But the failure to interview was not the subject of separate evidentiary focus, and we do not know, apart from inferences from mere general hiring and placement models, whether the failure to interview itself cut along racial or sexual lines. In the interviewing process there is a channeling effect, because applicants have historically been directed to interviewers who specialize in either exempt or clerical or nonexempt positions, and the route to the exempt positions not surprisingly results in a more rigorous interview process. If the *247 interviewer determines that the applicant is qualified for an open position in which she has expressed interest, the applicant is referred to the appropriate supervisor or manager. If the manager determines that the applicant is the one desired, attempts are then made to verify the applicant's history. The supervisor/manager makes the final decision to hire or reject the applicant upon receipt of verified work histories. With exempt employees, more than one manager may participate in the decision. The salary level of the incoming employee is chosen by the manager by application of salary guidelines issued by the Salary Administration Section of the Personnel Division. An offer of employment is then extended by the interviewer.

B. Affirmative Action.

Beginning in the late 1960's, and continuing with increasing emphasis in 1974 and later years, the Bank has engaged in various affirmative action efforts. In 1974, the Bank came under new and younger upperlevel management in the person of President Charles Pistor. At approximately the same time, the Bank began a more intensified effort to engage in affirmative action efforts. At least since Pistor assumed the presidency, there is little question that it has been the announced policy of the Bank's senior management not to discriminate. [24]

In 1964, the first written EEO policy was adopted and circulated within the Bank. The first Affirmative Action Plan was drawn in 1970 and was distributed to managers in 1971. The plan was submitted to and approved by the Treasury Department in early 1971. Through 1975, the Treasury Department monitored the Affirmative Action Plan. Under the Bank's Affirmative Action Plan, the Vice President for Personnel receives monthly status reports revealing the percentage of hires that are minorities. The Bank's newspaper advertisements, at least since 1971, have contained the usual addendum that the Bank is an equal opportunity employer. In the 1968-1973 period, five charges of discrimination were filed with the EEOC by employees, as part of a total of 43 charges for the 10-year period of this suit (Defendant's Exhibit 177). The EEOC in July, 1974, charged the Bank with discrimination, and a conciliation agreement was reached in December, 1976 (Plaintiffs' Exhibit 733). The agreement settled four employee charges and set various hiring and distributive goals. In 1975, the Treasury Department presented a list of "deficiencies" which resulted in a conciliation agreement in March, 1977.

Beginning in the latter part of 1968, of 19 colleges visited by the Bank, eight were predominately black and one was predominately female. As Jerry Watson advised his supervisors in April, 1975:

The selection of colleges and universities ... was strongly influenced by: (1) the Treasury Department's compliance review (May, 1974); (2) the class action discrimination charge filed by Mr. John Powell, Commissioner of the Equal Employment Opportunity Commission (July, 1974); and (3) Republic National Bank's commitment, as expressed in the Affirmative Action Plan, to increase minority and female representation in officer and exempt positions.

Defendant's Exhibit 68. Watson also commented at the same time about the Bank's hiring success, observing that:

... a very large number of minority students must be screened in order to find a disproportionately few acceptable candidates ..... *248 Id. In December, 1976, Watson commented to his supervisors about the small number of invitations the Bank was extending as a result of colleges visited, 61% of which had predominate black enrollments: "... the small number of invitations ... is an indication of our continued `selective' approach to invitations and hiring." Defendant's Exhibit 174. He further observed as of the end of the third quarter of 1976, 25.8% of hires in 1976 were minority: "... the majority of minority hiring is accounted for in nonexempt positions." Id.

C. Categorization of Employees.

As earlier mentioned, before 1973, the Bank's structure viewed vertically consisted of exempt and nonexempt employees. In turn, the nonexempt range was divided into 17 grades. There were five practical levels in the exempt categories. The first four ran through the vice president level with senior vice president and above categorized as number five. In the 1969-70 period, the Personnel Division decided that the top management of the Bank should occupy a separate structure in that their pay values were disproportionate with the lower ranks. Except for that separation of the few officers at the level of top management, the levels of bank officers were not changed. In 1973, as part of a reevaluation of pay structures and personnel assignments, the Personnel Division reexamined the 17 nonexempt grade divisions and decided that a 10-grade structure would be more effective. Among other things, it found that although there were 17 grades, less than 14% of the total of nonexempt employees actually fell in grades 11 through 17. In an effort to avoid this bunching and to correlate the nonexempt jobs with a sound pay structure, the reduction to 10 grades was made.

Beginning around 1970 and continuing until 1973, the Bank undertook a transition to a system of job evaluation known as the "Hay System." It is not necessary to set out in detail this relatively complex technique, because the court finds that nothing intrinsic to the Hay System itself creates Title VII concerns. Of course if a wage or hiring system is otherwise skewed, the Hay System is no guarantee against liability. A general appreciation of its operation, however, is important to a comprehension of the Bank's personnel policy, its employee structure and the system's usefulness in the multiple regressions in section VII(A), infra. We will return to the system's utility in the statistical measurement efforts.

The reduction of the number of nonexempt grades from 17 to 10 was part of the implementation of the Hay System. This system, in use presently at the Bank, is intended to be a technique for comparing a variety of jobs within a firm in an orderly way. The job evaluation phase of the system is claimed to be premised on the know-how needed, the problem solving involved, and the accountability of the job being analyzed. More specifically:

As Harriette Weiss, a Senior Principal with Hay and Associates, testified, the "Hay System" is basically a euphemism for the Hay Guide Chart-Profile Method of job evaluation, a proprietary method utilized by the management consulting firm for evaluating and ranking jobs by job content within an organization. With the aid of job evaluation, a firm is able to establish a rational and consistent method of calculating the relative worth of jobs, which in turn can be used as a basis for payment of compensation for the performance of those jobs.

Defendant's Post-Trial Brief at 353 (emphasis supplied). The system is summarized by Hay Associates in Figure 1.

The Bank utilized the Hay method from January 1, 1970, forward with its exempt *249 employees, the method being set up in consultation with Hay Associates. See Defendant's Exhibit 202. The first phase of the Hay study involved an effort to establish equitable internal relationships among all positions. This was attempted through a job evaluation process conducted by an in-house committee of Republic executives, usually one or two levels above the job being evaluated. The result, after review, was the assignment of "points," based on job content (using know-how, problem-solving, and accountability as factors to be taken into account) to each position being evaluated. These "points" have been referred to as Hay points throughout this litigation, though perhaps they are more accurately referred to as "client points"; comparisons with other firms are made possible by conversion of client points to "Hay points." See Defendant's Post-Trial Brief at 354 n.100.

The second phase, an effort to develop a competitive salary structure within the Bank, depends on the point values assigned to the exempt jobs in the first phase, and what other employers pay for jobs of equivalent point value:

Compensation practices are compared among [Hay Compensation Comparison] survey participants utilizing "Hay points" as the standard of reference for comparing jobs of equivalent value. A conversion formula enables each Hay client to convert Hay points to a dollar figure representing the average salary paid by survey participants or the midpoint of a salary range for a job at any given Hay point level.

Defendant's Post-Trial Brief at 359. *250

The salary range contains a minimum figure that is set as a hiring level rate of pay for applicants meeting position specifications. The maximum figure is said to be one that is only exceeded rarely by an employee with a highest rating, that of "path *251 finder." The salary range has a 50% spread. That is, the maximum is 150% of its minimum. For example, a salary range with a $8,000 minimum would have a maximum of $12,000. Under the plan the midpoint of a salary range (for example, $10,000) would be the area within which the Bank would pay for "consistently satisfactory performance on a sustained basis of the functions assigned to the incumbent." Defendant's Exhibit 202.

Under the plan, employees sometimes can find themselves below minimum if, for example, they are rapidly promoted. The plan allows the Bank to give inequity increases to reduce any such inequities. The minimum has attracted the attention of the Department of Treasury. Since 1977, as a result of a reconciliation agreement with the Treasury Department, the total males, the total females, and the total minorities below minimum are monitored.

The Bank has throughout required performance appraisals by the immediate supervisors of employees. The Hay system defines certain ranges of performance. In the exempt position category, the ranges included inadequate, marginal, beginning, fair, competent, commendable, distinguished, and pathfinder.

The nonexempt structure, to which the Hay System was applied in 1973, differs somewhat from the exempt structure in that nonexempt salaries are quoted in monthly sums whereas exempt salaries are quoted in annual sums. Of course there is a difference in this salary spread. With a grade one position, the minimum to maximum spread is 25% while with a grade ten it is 60%. There is also a difference when an employee becomes eligible for changes, as well as difference in allowable percentage salary increases. For example, for nonexempts, higher percentage increases keyed to performance, are allowed. In addition, different survey data was used for exempts and nonexempts: at the exempt level the competitive salary inquiry was national, while for nonexempts the metroplex area was the primary data source.

V. Statistical Evidence

All statistical studies in this case share a common goal: to compare the personnel decisions which have actually been made at the Bank since 1969 with the decisions which hypothetically would occur in the absence of discrimination. In the areas of compensation, initial placement, promotion, and termination, this comparison relates one segment of the Bank population to another, determining whether females or blacks at the Bank are treated differently from males or whites. In the hiring area, the Bank's hiring decisions are compared with data external to the Bank-the availability of blacks and females as measured by Census Bureau data. Each of these comparisons depends upon the existence of an accurate and complete data base reflecting the multitude of individual personnel decisions made by the Bank during the relevant time period. While the statistical models employed by the parties possess varying degrees of "robustness," i. e., ability to withstand the effect of errors in the data or violation of the assumptions on which the models rely, it is generally true that the utility of a statistical comparison is directly dependent on the accuracy and completeness of the data being compared. For this reason, we must, before turning to the statistical models and the results generated by them, carefully examine the sources of data on which those models rely. We must then evaluate the parties' challenges to those data. Having examined the input to the statistical models, we next must determine the extent to which the court is confined to the output of the studies in its presented form, and whether it may or must perform statistical calculations beyond those provided by the parties. Finally, we discuss the role of nonstatistical evidence and its relationship to the statistical studies.

A. Sources of the Data.

With some exceptions, all data used by the parties in this case were ultimately derived from employment records maintained by the Bank as its permanent business records. For each present and past employee, the Bank maintains a traditional "personnel *252 jacket," containing such items as the employee's vital statistics and salary, promotion, and performance data. The Bank also maintains a variety of personnel summaries and payroll and other accounting records. In addition, the Bank has since 1973 maintained computerized personnel records. Each of these types of personnel record has played a role in the development of the data bases used by the parties.

The data bases used by the parties, either for statistical analysis or for the generation of other data bases, are summarized in the margin. [25] The data bases used for analysis *253 were derived in principal part from a series of tapes maintained by the Bank and designated by it as the "A020 tapes." These tapes represent semi-monthly "snapshots" of the Bank's work force, and were used by the Bank to generate employee paychecks and day-to-day personnel records. The tapes contain a variety of information on each employee, including the items of information listed in n.25, supra, and pension, insurance, and actuarial information.

The Bank also maintains a personnel history tape which it designated as the "A030 tape." This tape, which has not been used on a day-to-day basis, was produced in an attempt to avoid the use of individual personnel jackets, and contains personal data on each employee, together with work histories and information about preemployment education. The A030 tape is the Bank's only computerized record of information about education.

In August of 1978, the Bank developed a tape designated as "A035," which it used for the bulk of its statistical studies. The A035 tape represents a continuous record of the employment history of all Bank employees from January 1, 1973, forward. The *254 tape was constructed by extracting information from the A020 tapes and merging that information with employee names and educational data from the A030 tape. Unlike the A020 and A030 tapes, the A035 tape contained a variable length record for each employee, with one to 95 fields per record. [26] A total of 6,395 present or past employees were represented. Eighteen of these employees, who had accidently been omitted from the A030 tape, were manually added to the A035 tape. Based on a belief that many Equal Employment Opportunity codes on the A020 tapes were in error, [27] EEO codes were changed on the A035 tape to reflect "appropriate" values.

After assembling the A035 tape, the Bank audited the tape for accuracy. A random sample of 375 employee records was chosen, and computerized information for each record was compared with the same information as recorded in the employee's personnel jacket, which the Bank believed to constitute the most reliable source of information on its employees. The audit demonstrated, to the Bank's satisfaction, that most fields contained accurate information. [28] Some fields, however, were found to be, in the Bank's opinion, too unreliable for use. These included salary ratings (24% missing and 6% erroneous), annual performance appraisal ratings (26% missing and 4% erroneous), and, most importantly, information about education. [29]

Before 1973, the Bank did not maintain any computerized personnel records, and for the pre-1973 period it was necessary to manually assemble personnel data from paper records. From information contained in personnel jackets, salary change sheets, and payroll earnings journals, the Bank assembled a computer tape, designated the "A500 tape" (sometimes designated as the "A040 tape"). This tape contained information for 1969-72 similar to that contained on the A035 tape for 1973-78.

For certain studies performed by Drs. Stoikov, Stolzenberg, and Snyder, it was necessary to go beyond the Bank's personnel records. To this end, a questionnaire was prepared and submitted to a randomly selected sample of 286 current Bank employees on June 28, 1979. Employees were asked to verify information obtained from personnel jackets and to provide responses to a variety of questions about their work habits, working conditions, motivations, and expectations. Ninety-two percent of the employees surveyed fully completed the questionnaire.

For their regression study on involuntary terminations, Drs. Stolzenberg and Snyder prepared their own ad hoc data base. This data base consisted of information from the A035 and A500 tapes, together with a variety of data on education, skills, and experience.

*255 For the purpose of developing weighted average availability figures for use in hiring studies, see section IX(F), infra, Dr. Stoikov used a variety of data on the education, experience, and geographical origin of those hired by the Bank between 1970 and 1978. These data were obtained from personnel jackets.

While plaintiffs were furnished with most of the computerized data bases developed by the Bank, including the A030, A035, and A500 tapes and selected A020 tapes, they, unlike the Bank, chose not to place primary reliance on the A035 tape. Instead, plaintiffs' computer expert, Dr. David Morgan, prepared a tape for 1973-78 by merging data from the A020 tapes with date of birth and highest grade completed from the A035 tape. For 1969-72, Dr. Morgan merged data from the A500 tape with date of birth, highest grade completed, and salary taken from excerpts from personnel jackets supplied by the Bank. Dr. Morgan's primary purpose in developing these separate tapes was to facilitate use of the Statistical Package for the Social Sciences, a group of standardized statistical computer programs used in a variety of social science applications. See generally N. Nie, C. Hull, J. Jenkins, K. Steinbrenner, & D. Bent, Statistical Package for the Social Sciences (2d ed. 1975).

B. Problems with the Data.

The parties mount a variety of challenges to the accuracy and reliability of each other's data. Each such challenge is made on two levels: First, the challenging party argues that the data are so infected with inaccuracy and omission that they are unfit for use, and that statistical studies relying on the data should be disregarded. As a fallback position, the parties argue that flaws in the data should be kept in mind in evaluating the strengths and weaknesses of the respective statistical presentations. The purpose of this section is to identify the challenges, and to evaluate whether any challenges have sufficient merit to warrant the total disregard of studies relying on the challenged data. Any inaccuracies not rising to this level will be considered in the evaluation of the statistical presentations which rely on the challenged data.

We begin by noting that the Bank's personnel records from an unusually rich source of data for statistical analysis of employment discrimination. Cf. Testimony of Dr. Francine Blau at 207 (Bank's data pool a "really large impressive sample"). Only in the case of a large employer with substantial resources is a comprehensive computerized personnel record spanning many years typically available. To this must be added the fact that a certain measure of inaccuracy in data is a daily fact of life for the social scientist. As Dr. Blau put it,

I think if social scientists in general waited for perfect data that was completely error free, no social science research would be done so the real question for the researcher is determining when any of these bias[es] or errors are serious enough to precluded [sic] analysis....

Id. at 37-38. Nor are courts immune to this pervasive imprecision: the latitude allowed a district court in computing Title VII back pay awards, see, e. g., United States v. Allegheny-Ludlum Industries, Inc., 517 F.2d 826 , 852 n.29 (5th Cir. 1975), cert. denied, 425 U.S. 944 , 96 S. Ct. 1684 , 48 L. Ed. 2d 187 (1976), is but one example of the judicial response to limitations of employer personnel data.

In light of these factors, together with the fact that the parties' approaches to the data assembly problem are facially reasonable and the fact that the parties' respective data bases are at their core quite similar, a heavy burden must be met before a party can justify the rejection in toto of any statistical analyses on grounds of errors or omissions in the data. Identification of flaws is helpful for later use in evaluating the results, but identification is not alone sufficient to warrant rejection. Instead, the challenging party bears the burden of showing that errors or omissions bias the data, i. e., that erroneous or omitted items are not distributed in the same way as items which are present and correct. That *256 party must then show, at the least, that this bias alters the result of the statistical analyses in a systematic way, i. e., that errors or omissions make the models more (in the case of challenges) or less (in the case of the plaintiffs' challenges) likely to show discrimination than would correct and complete data. See section VII(C) (introduction). Having established these ground rules, we turn to the specific challenges made by the parties.

1. Plaintiffs' Challenges.

The plaintiffs eschewed use of the A035 tape because, in their words, it was "prepared for litigation." That the tape was indeed prepared with this litigation in mind is not seriously questioned by the Bank. Nevertheless, Bank officials have testified that it is their intention to use the A035 tape as a master personnel history tape in day-to-day operations. In particular, they expect the tape to play a role in the preparation and monitoring of future Affirmative Action Plans. Moreover, to say that the tape was "prepared for litigation" says nothing about its accuracy absent some indication that that fact has resulted in erroneous, omitted, or biased data.

Plaintiffs' expert Dr. Janice Madden questioned the validity of the hiring qualifications data used for availability calculations, arguing that "recall" ex post facto of the qualifications for which employees were hired renders those data suspect. While it is true that the primary reason for which an employee was hired was determined after the fact for the purpose of the availability analyses, this classification was not accomplished without objective guidelines. Messrs. Stotts and Barksdale testified that in preparing the breakdown (Defendant's Exhibit 549) they used as a rule of thumb the presence or absence of a degree conferred within the preceding twelve months: those with such a degree were presumed to have been hired for their educational qualifications, and those without such a degree for their experience. Apart from the allegedly subjective nature of the classification process, plaintiffs offer no reason to believe that the classifications were systematically biased in any way.

The plaintiffs finally attack the data obtained through the questionnaire which was submitted to a sample of current Republic employees. They argue that employees who did complete the questionnaire "may not have understood what they were considering," and that those who did not complete it may not have wanted to cooperate in the Bank's defense of this lawsuit. The high participation rates in the questionnaire survey discredit these assertions. Of 313 current employees selected for the questionnaire, 286 (91%) were actually given the questionnaire, the remainder being on vacation, maternity leave, or foreign assignment. Of these 286, 285 (99.7%) completed the questionnaire at least partially, and 268 (93.7%) completed the questionnaire fully. There is no indication that responses from absent or uncooperative employees would have been differently distributed from those of other employees. While it is impossible to know whether the answering employees understood the questionnaire, the questionnaire is not difficult to understand and those who answered must be presumed to have known what they were doing. [30]

2. The Bank's Challenges.

The Bank first challenges the plaintiffs' use of year-end A020 tapes as the source for most of their data. It argues that errors in the year-end tapes will be perpetuated until the next year's tape in the plaintiffs' analyses, while such errors will be corrected in the Bank's analyses by information from the A020 tape for the next payroll period. While this is indeed a flaw in the plaintiffs' data, the court is convinced that it is a minor one: the errors (presumably few in number to begin with) will be propagated only if they happen to appear in *257 the year-end tape, and then only until the next year-end tape. Moreover, no systematic bias has been shown.

The Bank also attacks the plaintiffs' inclusion of hourly employees in the data for some but not all years. The short median tenure of such employees (about one year; Defendant's Exhibit 481) and the small proportion of Bank employees they represent (averaging 160 hourly employees; Defendant's Exhibit 480) suggest that any error caused by their inclusion is small. Again, no systematic bias has been shown.

The Bank's most substantial challenge to the plaintiffs' data focuses on the use of the "highest grade completed" field. For 1973-78, this challenge relies on the results of the A035 audit, see n.28, supra, which shows an omission rate of 31% and an error rate of 10% for the highest grade completed field. The plaintiffs performed an analysis (Plaintiffs' Exhibit 1308, Tables 11 & 12) which purports to show that errors and omissions in this field display no systematic pattern and hence do not affect the regression results. This attempt must fail, however, since there is no mathematical way to determine the extent or effect of measurement error. D. Baldus & J. Cole, Statistical Proof of Discrimination ง 9.11, at 298 (1980). In common sense terms, if correct data were present with which to compare the data which were used, those correct data could have been used in the first place and the problem would never have arisen. The problem with plaintiffs' error/omission analysis is that there is error and omission in both groups under comparison: the total Bank population, as to which there is a 31% omission rate and 10% error rate in the highest grade completed field, is compared with the regression sample (total Bank population less those missing one or more critical data items). Not surprisingly, the comparison shows little difference.

For 1969-72, the Bank performed an audit of the manually added fields from the plaintiffs' tape. Results of this audit are shown in the margin. [31] For these years, there is a 12.8% omission rate and a 24.5% error rate in the highest grade completed field. The Bank attributes these flaws to the use of excerpts from personnel jackets rather than the entire jackets, and to imprecision in the translation of high school, college, and graduate degrees into years of education by Dr. Morgan and his coders.

As with other challenges, the Bank has failed to show that these flaws in the data produce any systematic bias. This court does not accept the Bank's assertion that an error rate of, e. g., 10% is alone sufficient to warrant rejection of the data. See Defendant's Post-Trial Brief at 65. There is no indiction that any errors or omissions are not randomly distributed among sexes and races [32] , or that the errors and omissions *258 caused an overestimate or underestimate of education for any particular racial or sexual group. No guidance is given as to what was treated as an "error": a discrepancy of one year between education as recorded in a personnel jacket and education as recorded on the plaintiffs' tapes is likely recorded as an "error," yet there is no indication of the proportion of such small errors.

The flaws in the plaintiffs' educational data are indeed serious enough to warrant care in evaluating the studies which rely on those data. Those flaws do not, however, warrant the extreme sanction of disregarding the studies as irrelevant. The high proportions of the data which are both present and correct (59% for 1973-78 and 63% for 1969-72) and the even higher proportions of correct data among all included data (86% for 1973-78 and 72% for 1969-72) render the data at least marginally probative of the existence vel non of discrimination. Cf. Fed.R.Evid. 401.

C. Technical and Institutional Competence.

An important question which must be resolved before turning to the analysis of the parties' statistical presentations relates to the proper role of the court in evaluating those studies. Of course, the record on which the court must base its decision consists of the exhibits and testimony, principally the reports and other exhibits prepared by the experts and their live testimony concerning those reports and exhibits. But a nagging question remains: To what extent may (and should) the court, using data contained in the record and mathematical and statistical techniques explained by the experts in their reports and testimony, perform its own calculations? Two limitations on such activity must be examined: the technical competence of the court to perform such calculations; and the extent to which voluminous calculations can be required by a party which did not perform such calculations as part of its trial presentation.

The role of the court as factfinder requires that it bring to its task its own fund of common knowledge and general reasoning ability. To go beyond these sources of decisionmaking power, however, presents three related dangers. First and foremost among these is the risk of error. Many judges (including this court) have little or no formal training in statistics, econometrics, or higher-level mathematics. True to the aphorism that a little knowledge is a dangerous thing, such a court may be led into the temptation to apply its limited fund of such knowledge to the solution of mathematical problems in the case before it, without a full appreciation of the risks and limitations of the techniques it is employing and without assurance of their suitability for the task to which they are put. Thus the explanation by an expert of a statistical technique or formula and its applicability to a particular problem does not warrant the application of that technique or formula to an unrelated problem, no matter how closely related the two may seem to the uninitiated. Only where the expert has justified the use of an approach for a particular problem, evaluating the risks and limitations of the approach in the context of the problem, ought the court to apply the technique in performing calculations not performed in the evidentiary presentation.

An overwillingness to undertake computational efforts also creates the risk of decisional disparity. Courts with varying degrees of inclination and ability in the application *259 of complex technical approaches to the solution of social problems will reach varying results, and the outcome of cases requiring such application will turn more on the plaintiffs' choice of forum and the luck of the draw in the District Clerk's office than on the merits of the cases. Indeed, absent some neutral and consistently applied approach to computation by the court, a court must risk the accusation that it has picked and chosen among the data, performing or not performing additional computations to suit a preconceived result.

Going too far beyond the calculations performed for and explained at trial is also unfair to the parties. In a typical complex Title VII class action, the parties have by the commencement of trial deposed each other's experts, discovered those experts' reports, counseled with their own experts, explored possible cross-examination, developed rebuttal evidence, and otherwise prepared for the evidentiary onslaught they expect to meet. For the court to appoint itself as an additional statistical expert, ex parte, after trial, and without opportunity for challenge by the parties, frustrates the expectations of the parties and diminishes the reliability which the adversary system was designed to insure. Similarly, complex calculations performed for trial but not explained will not receive the benefit of sustained expert scrutiny of their validity and implications, and so cannot be relied upon.

Beyond problems of technical competence are those of what may be referred to as institutional competence. Obviously, a court cannot be called upon to perform multiple regression analyses requiring a computer merely because the raw data which would support such analyses have been introduced into evidence. More generally, the introduction of raw data and the explanation of a statistical technique ought not, at least where the data are voluminous or the technique complex, place the burden on the court to perform the calculations. While the court cannot shirk its duty to review the evidence and decide the case, to require a myriad of calculations based on file cabinets full of statistical exhibits would exact not diligence but fanatical devotion to detail. See n.117, infra.

To say that the court should be reluctant to go far afield from the parties' own calculations does not mean, however, that in all cases the court is hamstrung by a party's failure to perform calculations necessary or helpful to the decision of the case. Where a simple statistical technique has been explained through expert testimony in terms understandable by one with no advanced statistical background, and where the use of that technique has been justified for the problem in question, the performance of simple calculations using the technique is not the creation of new evidence but only a more detailed examination of existing evidence. Cf. L.C.L. Theatres, Inc. v. Columbia Pictures Industries, 421 F. Supp. 1090 , 1103 & n.9 (N.D.Tex.1976), rev'd in part on other grounds, 566 F.2d 494 (5th Cir. 1978) (post-trial summaries permissible where underlying data are in evidence); 75 Am. Jur.2d Trial ง 991 (1974) (jury experimentation permissible if effect is not to introduce extraneous evidence); Annot., 95 A.L.R. 2d 351 (1964) (same). This type of calculation aids rather than harms the parties, enabling the court to fill gaps created when minor flaws not recognized by the parties are discovered in the data. Limited calculations may alternatively enable the court to transform the parties' statistical data into a form which gives greater insight into the existence of discrimination, a result which likewise ought to be welcomed.

D. The "Anecdotal Evidence."

The Second Circuit recently stated that "statistics showing a significantly disparate racial impact have consistently been held to create a presumption of Title VII discrimination." Guardians Association of the New York City Police Department, Inc. v. Civil Service Commission, 630 F.2d 79 , at 88, 23 Empl.Prac.Dec. ถ 31,153, at 16,973-74 (2d Cir. 1980). In Hazelwood School District v. United States, 433 U.S. 299 , 307-08, 97 S. Ct. 2736 , 2741, 53 L. Ed. 2d 768 (1977), the Court held that "[w]hen gross statistical disparities can be shown, they alone may in a proper case constitute prima facie proof of *260 a pattern or practice of discrimination." See also Johnson v. Uncle Ben's, Inc., 628 F.2d 419 , (5th Cir. 1980).

Like those courts, we make inferences from statistical patterns in this litigation. Indeed, the court must rely on such statistical evidence: neither the testimony of individual applicants and employees, nor that of Bank executives, [33] provides in this case a sufficient independent basis for a finding of the presence or absence of class-wide discrimination of the sorts here alleged. See section VIII(A), infra. Anecdotal evidence does, however, serve a useful function in this case, in light of the principle that tests of statistical correlation cannot by themselves identify the causative factors which produce the observed results. See D. Baldus & J. Cole, supra, ง 9.42, at 320; R. Wonnacott & T. Wonnacott, Econometrics 173 (2d ed. 1979). See generally F. Mosteller & J. Tukey, Data Analysis and Regression-A Second Course in Statistics 260-62 (1977). It serves a useful function as one of the three ways in which the plaintiffs have bolstered the thesis that it is real-world behavior rather than random statistical patterns which causes there to be correlations and modelling results damaging to the Bank. We leave discussion of two of the ways-statistical significance and the use of theory in designing the models-to later sections. See section VI(D) nn.58 & 54, infra; sections VI(C) and IX(A), infra.

The anecdotal evidence-evidence of individual experiences, specific employment practices, and testimony of bank officials-which in itself is not weighty enough to alter the results obtained from the statistical evidence, lends support to the idea that any discrimination found in the statistical analyses is due to discriminatory behavior rather than to chance. Cf. Waintroob, The Developing Law of Equal Employment Opportunity at the White Collar and Professional Level, 21 Wm. & Mary L.Rev. 45, 96-98 & 103-105 (1979).

While we do not here detail such anecdotal evidence, we have considered all such evidence carefully. That there is such evidence can hardly be challenged. For example, the following exchange took place at the deposition of Thomas G. Croft, Senior Vice President and Director of Personnel:

Q When do you believe that the bank began to be in compliance or began to offer equal employment opportunity to females?

A When? I don't understand your question.

Q Well, my previous questions concerning being in compliance and offering equal employment opportunity to blacks and females, you said something about historically, and then we narrowed it to presently, and you said yes, you believed that you were or that the bank was. Therefore, I presume that you intended that historically perhaps it was not, and I'm saying when did the bank begin, as far as you're concerned, to offer equal employment opportunity to women and to blacks.

A I think we probably began in the early '70's.

Q When you began in the early '70's-excuse me just a minute.

(Off-the-record discussion.)

MS. PETERS: (To the reporter.) What was our last question and answer?

(Thereupon, the last question and answer were read by the reporter.)

A I would, if I may, I'd like to add to that. At least there's been a spiritual change, if you will, in the thinking of the management of the bank which started in the early '70's, and historically these laws and the executive orders are relatively new. So it's difficult to say when it came about or when it started or where we are now, but I must say, to the best of my knowledge, backed up by what the government agencies have to say, *261 that we are in compliance with the law of the land. Again, we've had people like Sharon Jobe, who is vice president and general counsel for the holding company. Sharon has done a lot of things to change people's minds about the capability of a female executive. She's doing some work with the-or she's preparing to do some work with the affiliates of the holding company, again working with Bob Stoller. So I guess maybe spirtually [sic] is not the best word, but it's the one that comes to mind, that change did begin to take place in the early '70's.

Deposition of Thomas G. Croft at 126-27 (December 7 and 8, 1977). [34] See also Deposition of Tom Croft at 106-07 (December 9, 1977) and his trial testimony concerning the December 9 answers. Croft's candid remarks form a backdrop to the statistical evidence, to which at long last we now turn.

VI. The Theory Behind the Parties' Mathematical Modeling

A. Introduction.

Both the plaintiffs and the Bank rely on econometric models rooted in principles of labor economics in order to perform one task: to represent in mathematical terms the Bank's employment practices. Neither side posits that every time a Bank administrator is faced with a personnel decision he "evaluates each applicant according to a predetermined rule for accounting and weighting key characteristics [using a system where] [t]he relevant characteristics are specified in advance, and so is the rule for combining them to produce a score for each applicant." [35] Both parties instead offer models claimed to represent the Bank's behavior on the whole.

The mathematical models draw heavily on quantitative techniques long used by social scientists and more recently used in judicial resolution of antitrust, securities, and employment discrimination disputes. The use of quantitative techniques in this case is unique only in that the techniques used here, especially those involving multiple regressions, are more sophisticated and are more heavily relied on by counsel for both sides than in most litigation. By pushing the techniques to the center of the fray, the parties have compelled the court to examine the techniques' theoretical under-pinning-the "human capital" theory of labor economics and general productivity theory-as well as the operational mechanics of multiple regression itself. Only by doing so will we understand the strengths and limitations of the models, and thus determine whether any particular model offered is sufficient to establish or rebut a prima facie case. Once it is determined that a particular model generates probative evidence, the court must determine whether a model represents a behavioral pattern violative of Title VII. For this second step, an understanding of both the mathematical techniques and Title VII are necessary.

Despite the quantitative social science nature of its analysis, the court fashions no new policy based on its own concept of the public good. In attempting to evaluate the models against the background of an understanding of both the mathematics and the law, we seek to ensure that any hidden value choices are not introduced by the quantitative techniques. We confess to a determination not to reject the unfamiliar for its unfamiliarity, and to a hope that the translation of Title VII equal opportunity doctrine to technocratic terms will make possible more sensitive and accurate detection of the presence or absence of discrimination, as defined not by the court's or the economist's view of public policy but by the court's view of policy set by Congress in the form of Title VII.

*262 Stripped of jargon, the parties are agreed as to the fundamental theory behind the Bank's employment decisions as to compensation, initial placement, and promotion: the Bank treats people differently according to their productivity. [36] The plaintiffs argue that the Bank considers, in addition, the racial and sexual group status of the individual. The Bank counters that it does not. In section VI(B), infra we show that Title VII does not proscribe differentiation according to productivity.

In Section VI(C), infra, we describe the human capital theory, which here provides the intellectual justification for inclusion of productivity-related factors in the mathematical constructs. In section VI(D), infra, we describe the mechanics of modeling, with focus on "single equation, ordinary least squares regression" models. Finally, in section VI(E), infra, we discuss how particular econometric results can determine whether employees are being treated according to their group-status as well as their productivity.

B. Job Relatedness and Equal Treatment.

There are two concepts of discrimination employed in interpreting fair employment laws. The first is "equal treatment," a doctrine wherein all persons are to be treated by employers without regard to their racial or sexual group status. It seeks to assure that in competing for jobs a person is not handicapped by his group status. The focus of this concept is upon individuals and acknowledges their varying skills. The second concept is that of "equal achievement." Its focus is not upon individuals with varying skill levels, but is upon the actual distribution of employment among groups. That is, it looks to the results of the contest, not to whether the rules are the same for everyone. Equal achievement subscribers contend that equality of treatment of minorities is spurious rhetoric because minorities remain hobbled by the effects of past discrimination. [37] The choice between these two interpretations is not an academic exercise. Equal achievement subscribers would place on employers the burden of correcting for the disabling effect of societal discrimination such as inadequate education and training. This court is persuaded that in doing so, they would frustrate the stated purpose of Congress.

The construction of "job relatedness," to be discussed more fully herein, turns directly upon the choice between equal treatment and equal achievement. An employer does not discriminate by paying a white more for a higher productivity level in a job than a black otherwise similarly situated. Judicial decisions, although not always true to an equal treatment subscription in their analysis of Title VII cases, generally do not hold to the contrary. The difficulties stem from failure to give to courts accurate measures of productivity of individuals. Left at sea, we simultaneously have recognized that productivity is a valid differentiating factor but have indulged in a tendency to hedge asserted indicators of such productivity with high standards of proof because we recognize that absent accurate measure they contain great potential for masking differentials actually based on race or sex. Excessively high standards of proof have sometimes resulted in adoption of the equal achievement approach by evidentiary default. [38]

*263 The choice between the equal treatment and equal achievement approaches is not for the courts because Congress made that choice when it passed Title VII; and it left an unequivocal record that its choice was critical to the very enactment of the legislation. Congress had equal treatment in mind and the courts cannot, if they accept institutional limitations of a tripartite government, do otherwise โ€” either directly or by unspoken decisions that erode the premise, whether done intentionally or unwittingly.

Title VII was directed solely at unequal treatment of individuals; [39] it was not intended to infringe on independent decision-making based on reasons of business efficiency as long as the employer did not take the group status of the individual into consideration. [40] Senators Clark and Case, floor managers of Title VII, stated in an interpretative memorandum:

It has been suggested that the concept of discrimination is vague. In fact it is clear and simple and has no hidden meanings. To discriminate is to make a distinction, to make a difference in treatment or favor, and those distinctions or differences in treatment or favor, which are prohibited by section 704 are those which are based on any of the five forbidden criteria: race, color, religion, sex and national origin ....

110 Cong.Rec. 7213 (1964), quoted in No-Alternative Approach, supra n.37, at 103 n.29. Responding to opponents of Title VII, who were concerned that the bill would lead to quotas and other equal achievement oriented reactions, the supporters stated:

The language of ... [Title VII] simply states that race is not a qualification for employment. Every man must be judged according to his ability. In that respect, all men are to have an equal opportunity to be considered for a particular job ....

. . . . .

... It is possible that although a ... particular business will contain no Negroes, no charge of discrimination will be made. But businesses ... may not systematically exclude Negroes, when the only ground for exclusion is the color of a man's skin.

110 Cong.Rec. 8921 (1964), quoted in Disparate-Impact Liability, supra n.37, at 927. Senator Case also explained:

Whatever its merit as a socially desirable objective, title VII would not require, and *264 no court should read title VII as requiring, an employer to lower or change the occupational qualifications he sets for his employees simply because proportionately fewer Negroes than whites are able to meet them ... [nor would it require lower standards] because prior cultural or educational deprivation of Negroes prevented them from qualifying ....

[T]he very purpose of title VII is to promote hiring on the basis of job qualifications, rather than on the basis of race or color.

Id. at 928.

The concept of job relatedness, born in Griggs v. Duke Power Co., supra n.38, is an application of the equal treatment philosophy. In Griggs, the Supreme Court held that an employment practice nondiscriminatory on its face, but which has a disparate impact on a protected class, is illegal under Title VII unless the employer can show the practice is a "business necessity." Griggs struck down the requirement of a high school diploma and satisfactory aptitude test scores for hiring into supervisory jobs because the employer showed "no demonstrable relationship to successful performance" of the requirements in the jobs for which they were used. 401 U.S. at 431 , 91 S.Ct. at 853. As the Court stated, Title VII prohibits "practices that are fair in form, but discriminatory in operation. The touchstone is business necessity. If an employment practice which operates to exclude Negroes cannot be shown to be related to job performance, the practice is prohibited." Id. at 431, 91 S. Ct. at 853 . Duke Power Company's qualifications were unrelated to efficiency, being surrogates for a direct racial criterion. See Disparate-Impact Liability, supra n.37, at 929. The Court's chief concern was not equal achievement, but equality of treatment. Id. Later Supreme Court cases confirm the equal treatment interpretation of Title VII. Id. at 930-33. [41]

The bare words "business necessity" have been interpreted as involving more than "business purpose," under which a practice can be justified by showing that any benefit accrues to the employer through the use of the practice. No-Alternative Approach, supra n.37, at 100. The typical formulation of the rule is that the practice must be one that is essential to the safe and efficient operation of the business. Disparate-Impact Liability, supra n.37, at 916, 918-19.

While some commentators assert that the requirement of more than business purpose to justify the use of a race-correlated practice is more consistent with equal achievement than with the equal treatment interpretation of Title VII, [42] this is not always true for two reasons. First, while a particular race- or sex-correlated predictor [43] may indeed be indicative of productivity (and thus one that would have business purpose), the racist or sexist employer may be placing a heavier emphasis on that predictor than would a non-racist or non-sexist employer. [44] Thus, if the possessor of a particular degree is marginally more productive-if, for example, he is worth twenty cents an hour more-but if the racist or sexist employer pays not twenty cents an hour more, but fifty cents an hour more or even refuses to hire those without the degree, then the inference that the employer is using the predictor as an excuse for paying one group more is both rational and fair. Second, if there is an alternative predictor available which would serve equally well with lesser disparate impact, then to use the initial predictor would have undue *265 impact on minorities not justified by productivity considerations. Cf. No-Alternative Approach, supra n.37, at 115-16; Parson v. Kaiser Aluminum & Chemical Corp., 575 F.2d 1374 , 1389 (5th Cir. 1978), cert. denied, 441 U.S. 968 , 99 S. Ct. 2417 , 60 L. Ed. 2d 1073 (1979).

On the other hand, a "balancing" approach, wherein a court balances the impact on an employer of disallowing a practice and the impact of blacks or females of allowing the practice, is not consistent with the equal treatment construction of Title VII. Cf. No-Alternative Approach, supra n.37, at 101 & 119. For instance, in Bing v. Roadway Express, Inc., 444 F.2d 687 , 690 (5th Cir. 1971), the court appears to have compared the benefits accruing to the employer from the use of a practice and the extent of the disparate impact-a mixed equal treatment-equal achievement approach. See Swint v. Pullman-Standard, 624 F.2d 525 (5th Cir. 1980); Note, 12 Ga.L. Rev. 104, 106 n.12 (1977). Often, these courts are looking at equality of results not because they fail to adhere to the equal treatment interpretation of Title VII, but because they were not presented with sophisticated quantitative indicators of equality of treatment itself, and so had to make do with looking at results. This court, however, having direct quantitative evidence of whether there has been equality of treatment, need not deviate from such an interpretation. This court will, and must, adhere to the equal treatment philosophy intended by Congress.

We defer discussion to section VI(E), infra, of the specifics of how econometric modeling may indicate whether differential treatment is due entirely to productivity differences or to group status as well. We can, however, describe intuitively three ways in which such non-productivity-based behavior may manifest itself. First, the mathematical relationship found may indicate that a non-job-related or otherwise improper predictor is being used as a basis for differentiation. Second, the mathematical relationship found may indicate that even though a black and white are identical in all productivity-characteristics, they are being treated differently ("unequal treatment of twins"). Third, though such identical blacks and whites may be treated equally, the productivity characteristics disproportionately possessed by the favored group may be rewarded more than they would be rewarded by a nonracist (or nonsexist) employer (an aspect of "improper treatment across twins"). See section VI(E), infra.

C. Controlling for Productivity.

As earlier stated, the fundamental behavioral assumption of both the plaintiffs and the Bank is that the Bank treats people with different productivities (or true indicators of productivity) differently. Thus the mathematical models are designed to determine if there is any differential treatment not entirely attributable to such productivity differences. The productivity of an individual is gauged, in part, by those observable characteristics, such as schooling, thought to affect productivity. Cf. Gwartney, Asher, Haworth, & Haworth, supra n.40, at 636-37.

The plaintiffs' experts explicitly rely on what is called "human capital theory" to determine what observable characteristics out to be controlled for in their multiple regression models. See Plaintiffs' Exhibit 504, at 1; Testimony of Dr. Janice Madden at 11. See generally Sahota, Theories of Personal Income Distribution: A Survey, 16 J.Econ.Lit. 1 (1978); Blaug, The Empirical Status of Human Capital Theory: A Slightly Jaundiced Survey, 14 J.Econ.Lit. 827 (1976); P. Samuelson, Economics 751-52 (11th ed. 1980). Dr. Blau explained the theory behind her modeling as follows:

Economists define wage (or salary) discrimination as occurring when pay differences between groups exist that cannot fully be explained by productivity differences between those groups. (Becker, 1957). This definition matches our intuitive or common sense view that if one individual is more productive than another, he or she is entitled to higher pay. However, pay differences that cannot be explained by productivity differences are *266 suspect, especially if they are associated with race and/or sex differences.

One problem with applying this definition of salary discrimination is that firms generally do not keep records of employees' productivity (e. g., output per work hour). Indeed in some industries, like banking, employee output would be very hard to measure. For this reason, economists generally focus upon the characteristics of individuals that make one more or less productive than another, rather than upon productivity itself. The human capital theory (Becker, 1964; Mincer, 1973) is a widely accepted analysis of the determinants of earnings differences among individuals. The theory focuses upon the investment that individual workers and their employers make that increase the workers' knowledge and skills and thus make them more productive. The following factors are particularly important:

(1) Increases in knowledge and skills through formal education (generally measured by years of schooling).

(2) Increases in knowledge and skills through informal training acquired on-the-job. This is of two types:

(a) General training which is useful in a variety of firms (generally measured by years of labor market experience).

(b) Specific training which is useful only in the firm in which the training is acquired (generally measured by years of experience with the firm).

Thus, human capital theory also leads us to some common sense conclusions. If one individual has more education and experience than another, he or she is probably entitled to higher pay (with the size of the pay difference depending on the size of the difference in education and experience). However, pay differences in qualifications are suspect, especially if they are associated with race and/or sex differences. Economists generally attribute such unexplained pay differences to discrimination.

How does such salary discrimination arise? Two major mechanisms have been identified:

(1) UNEQUAL PAY FOR EQUAL WORK. Pay differentials among individuals with similar qualifications that are due to pay differences within occupational categories.

(2) UNEQUAL WORK. Pay differentials among individuals with similar qualifications that are due to differences in occupational distributions (i. e., differences in access to occupations on the basis of race and/or sex). Such discrimination in access to occupations may be due to discriminatory hiring and/or promotion policies of the firm.

Plaintiffs' Exhibit 504, at 1-2.

The Bank's principal expert, Dr. Stoikov, similarly offers multiple regressions which control for characteristics a productivity-minded employer might wish to use as predictors. For instance, as to her compensation analysis, Dr. Stoikov stated:

Sex- or race-related pay disparities are estimated using a model which relates an employee's pay to those characteristics which are thought to influence the quantity and quality of his/her work. These characteristics include the skills an individual brings to his/her employment at the time of hire that have been acquired through education and previous employment experience, the skills an individual has acquired since the time of hire, and an individual's motivation to be productive at his/her present job assignment and to assume future assignments which entail larger amounts of responsibility, skill and effort.

Individuals bring different quantities and types of skills to their employment, as well as different degrees of willingness to acquire new skills or to assume additional responsibilities. Moreover, when these factors are averaged separately for men and women (nonblacks and blacks) differences usually are observed. Therefore, we can anticipate that there will be a difference in the average pay of men and women (nonblacks and blacks) that reflects the differences in the quantity and quality of work-related characteristics *267 they possess. This difference is not attributable to their employer's discriminatory behavior, but rather is an expected pay disparity attributable to differences in employee-determined, work-related characteristics. Remaining sex- or race-related pay disparities are pay differences that cannot be related to male/female (nonblack/black) differences in these employee-determined, work-related characteristics. They may or may not stem from an employer's discriminatory behavior.

Defendant's Exhibit 32, at 16-18 (footnote omitted).

D. The Mathematics of Regression Analysis.

In today's society, among those who claim special insight denied the common run of men, the "esoteric language is mathematics; the special means of inspiration, the computer, the forbidden path of truth, science." B. Ackerman, et al., The Uncertain Search for Environmental Quality 1 (1974). Econometricians-who with multiple regression analysis can provide an important addition to the judicial toolkit necessary for reconstructing from bits and pieces of data the framework of past events-are no exception.

The practical use of multiple regression has grown markedly over the past 25 years due to the development of statistical methodology itself, increasing availability of statistical data, and most importantly, the development of the computer. Fisher, Multiple Regression in Legal Proceedings, 80 Colum.L.Rev. 702, 702 (1980). Regression analysis is increasingly being used in legal proceedings and commentary. Id.; Statistical Evidence on the Deterrent Effect of Capital Punishment: Editors' Introduction, 85 Yale L.J. 164, 167 n.15 (1975) (Editor's Introduction). To record this court's understanding, correct or not, of the regression analyses presented by the experts in this case, and the limitations of those analyses-an understanding made necessary by technically complex trial challenges to their validity-we here describe how they are used, how they work, and when they do not. Cf. id. at 167 n.15 & 169; Fisher, supra, at 702. [45]

1. Uses of Multiple Regression.

The two primary uses of multiple regression analysis can be illustrated through the following examples where such analyses have actually been used:

(i) For years after the disappearance of coal-burning locomotives, there was dispute on the preservation of the jobs of railroad firemen. One issue was whether the presence of a fireman on trains contributed to railroad safety.

(ii) Cable television systems (CATVs) have been involved in administrative proceedings where one issue is the magnitude of the effect of the entry and activity of CATVs upon the profits and growth of broadcast television stations. This issue presents such questions as the influence of CATVs on the viewing audience of particular stations and the effect of changes in a station's audience on its revenues. Of course some claim that such effects are small while others insist they are large.

Fisher, supra, at 703. [46]

In the first case, multiple regression is being used to "test hypotheses"-does a particular variable [47] (presence or absence of *268 firemen) have any effect on some other variable (railroad safety). In the second case, multiple regression is being used for "parameter [48] estimation"-there being little doubt that audience size affects television revenue, and the real question being how much. Fisher, supra, at 704.

Both the firemen and CATV cases above involve "conditional forecasting"-a prediction of what will happen to the "dependent variable" [49] (such as railroad safety) if an "independent variable" [50] (such as the number of firemen) is changed or, looking retrospectively, what would have happened to the dependent variable had the value of an independent variable been different. Fisher, supra.

Determining whether firemen do affect railroad safety faces two difficulties in the absence of multiple regression analyses. First, the factor whose influence one wishes to test or measure is usually not the only major factor affecting the dependent variable. Thus, for instance, the amount of traffic on the railroads affects accidents as well. If we could make controlled experiments, it would be easy to quantify the relationship. A controlled experiment here would involve varying number of firemen, traffic on railroads, and the other variables expected to affect the number of accidents one at a time, holding everything else constant and observing the resulting number of accidents. This would be difficult and costly. We are left then with analyzing nature's experiments. See id. at 705; cf. R. Wonnacott & T. Wonnacott, Econometrics 7 (2d ed. 1979). Second, even if the effects of other systematic factors can be accounted for, there typically remain elements of chance. Id. Falling objects follow a physical law, but the behavior of individuals does not. Cf. R. Wonnacott & T. Wonnacott, supra at 6.

2. Econometrics and the Ordinary Least Squares Form of Multiple Regression Analysis.

Put in more formal terms, econometrics may be viewed as the science of model building, using quantitative tools used to construct and test mathematical representations of parts of the real world. R. Pindyck & D. Rubinfeld, Econometric Models and Economic Forecasts xi (1976). The fundamental underpinning of econometrics is the basic idea of relationships among economic variables. Relationships are grouped to form a model, the number of relationships included in an economic model depending on the objectives for which the model is constructed and the degree of explanation that is being sought. [51]

The models presented in this case involve, for the most part, one type of econometric modeling. They are "single-equation regression models." R. Pindyck & D. Rubinfeld, supra, at 1. Most of the single equation regression models are of a common variety: the behavior of the "endogenous" variable (a variable determined within the economic system under study) is assumed to be a linear function [52] of a set of "exogenous" *269 variables (those determined outside the system), [53] and the variables are assumed to possess certain other properties such that the convenient "ordinary least squares" method of estimating the relationships among those variables can be used. R. Pindyck & D. Rubinfeld, supra, at 1, 225; Editors' Introduction, supra ; G. Maddala, supra n.49, at 5; R. Wonnacott & T. Wonnacott, supra, at 334-35.

Multiple regression begins by specifying the major variables believed to affect the dependent variable. Fisher, supra, at 705. For instance, in our railroad example, we may wish to include as explanatory variables the number of firemen and the amount of railroad traffic, using as the dependent variable the number of railroad accidents. This involves using independent variables which reflect the important or systematic influences that may affect railroad safety. The "minor influences" are placed in a "random disturbance term," treating their effects as due to chance. Id. at 705-06. The relationship between the dependent variable and the independent variable of interest-for example, the relationship between the number of accidents and the number of firemen-is then estimated by culling the effects of the other major variables. Multiple regression is thus a substitute for controlled experimentation. Id. at 706. The results of multiple regressions-such as what we will call "coefficients" in the ordinary least square methodology-can be read as showing the effect of each independent variable on the dependent variable, holding the other independent variables constant. Moreover, relying on statistical inference, one can make statements about the probability that the effect described is due only to a chance fluctuation. Cf. id.

Central to the validity of any multiple regression model and resulting statistical inferences is the use of a proper procedure for determining what explanatory variables should be included and what mathematical form the equation should follow. The model devised must be based on theory, prior to looking at the data and running the model on the data. If one does the reverse, the usual tests of statistical inference do not apply. And proceeding in the direction of data to model is perceived as illegitimate. [54] Indeed it is important in reviewing the final numerical product of the regression studies that we recall the model's dependence upon this relatively intuitive step.

a. Estimating Multiple Regressions.

If the relationship of interest is to include only one independent variable ("x1") to explain the behavior of a dependent variable ("Y"), and it is believed that the relationship of Y to x1 is a straight line, then the relationship is expressed mathematically as:

(1) Y = a + b1x1

where a and b1 are constants, Y is the dependent variable, and x1 is the independent variable. Diagrammatically, the relationship is illustrated by the straight line in Figure 1. The econometrician thus seeks to determine the value of "a" (called the "intercept" or "constant") and the value of "b1" (the "coefficient" or "slope" of x1). Once he obtains these two numbers, for any value of x1, he would know the exact value of Y. See G. Maddala, supra n.49, at 74; J. Johnston, supra n.51, at 122. Because there are random influences in life, it is unlikely that the relationship between Y and x1 will be so *270 exact. Instead, plotting values of Y against values of x1 will likely produce a scatter of points as in Figure 2. Thus the correct relationship is not described by equation (1) but instead by:

(2) Y = a + b1x1 + u

where u represents random influences and is called the "residual" or "error." See G. Maddala, supra n.49, at 74. The econometrician attempts to cut through the noise generated by these random disturbances and extract the "signal"-that is, the line around which the points are scattered. He passes the line through the points so that it is as close as possible to the scatter of points, in the sense that the sum of the squared deviations between the predicted and actual Y values is minimized. Not inappropriately, this is called "least squares regression." G. Maddala, supra n.49, at 75. See Figure 3.

In general, numerical estimates of a and b1 are obtained by entering the scatter of points in a computer programmed to perform "least squares" calculations. These empirical estimates, based on the relatively crude "least squares" method of processing empirical observations, will be "good"-in that they are likely to be quite close to the true values-if and only if certain assumptions hold as to the nature of and relationship among the dependent variable, the independent variable(s), and the error term. See, e. g., R. Pindyck & D. Rubinfeld, supra, at 20-24, 55; R. Wonnacott & T. Wonnacott, supra, at 55-69; J. Johnston, supra n.51, at 126; J. Kmenta, Elements of Econometrics 9-14, 161 (1971). [55]

Regressions involving more than one explanatory variable ( e. g. number of firemen as well as railroad traffic) are more frequently used and are known as "multiple regressions." Id.; R. Wonnacott & T. Wonnacott, supra, at 71. To illustrate, we may assume that there is a suspected relationship between Y, the dependent variable, and the explanatory variables, x1, x2, x3, ... xk, such that:

(3) Y = a + b1x1 + b2x2 + b3x3 + ... + bkxk + u

where a, b1, b2, b3, ... bk are constants. As with a model with only one explanatory variable, a is called the intercept or constant; b1, the coefficient of x1; b2, the coefficient of x2; and so forth; and u, the "error term" or "residual." [56] When empirical observations are placed in the computer programmed to do least squares manipulations (just as we input the "points" in the model with one explanatory variable), it will generate numerical estimates of a, b1, b2, ... bk based on how the dependent variable changes when the independent variables move in a variety of ways. Cf. Fisher, supra, at 712. These numerical estimates of b1, b2, b3 ... bk, are analogous to the b1 in Figure 3. When only one explanatory variable is used, b1 is the slope of the line: that is, it tells us how much Y will change for a unit change in the value of x1. With a multiple regression, b1 tells how much Y will change for a unit change in the value *271 of x1, holding the other explanatory variables constant; b2 is the change in Y corresponding to a unit change in the value of x2, holding the other explanatory variables constant; and so forth. Thus if the proper model for railroad safety were described by:

(4) Y = a + b1x1 + b2x2 + u

where Y is the number of railroad accidents, x1 is the number of firemen, and x2 the miles of railroad traffic, we can "run" this model on the appropriate data, and obtain, for example, Y = 50 + ฝ x1 + พ x2.

This equation tells us, among other things, that with each additional fireman, the number of railroad accidents is reduced by ฝ of an accident, while with each additional mile of railroad traffic, the number of railroad accidents is increased by พ of an accident. [57]

b. Statistical Inference.

With certain assumptions about the particular probability distribution of the error term, one can go beyond estimating effects of independent variables on the dependent variable to gauging the certainty or accuracy of those effects. Fisher, supra, at 716; G. Maddala, supra n.49, at 79. For instance, one can perform calculations that support statements about the range of values likely to contain the true coefficient of any of the explanatory variables.

In particular, one can determine the size of the interval on either side of the estimated coefficient which has a given probability ( e. g., .95) of containing the true parameter. This range of variables is referred to as a "confidence interval." R. Pindyck & D. Rubinfeld, supra, at 31. For example, after performing the appropriate calculations, we might find that there is a 95% chance that the true coefficient of firemen in our railroad example is larger than ผ and smaller than พ. In other words, the 95% confidence interval for the firemen coefficient is ฝ ฑ ผ.

One can also perform calculations to test the hypothesis, at a given level of statistical significance, [58] that the true coefficient is actually zero; that is, that the independent variable to which it corresponds has no effect on the dependent variable. Fisher, supra, at 717. If what is known as the "t-statistic" for the particular coefficient is large enough, then we can reject the hypothesis that the true coefficient of the variable in *272 question is equal to zero. See R. Pindyck & D. Rubinfeld, supra, at 31; R. Wonnacott & T. Wonnacott, supra, at 86. For example, if a 5% level of significance is used, a sufficiently large t-statistic for the coefficient indicates that the chances are less than one in 20 that the true coefficient is actually zero. Fisher, supra, at 717. The magnitude of the t-statistic necessary to reject such hypothesis varies with the desired level of significance. Thus the t-statistic would need to be larger with a 5% level of significance than with a 10% level of significance.

The magnitude of the t-statistic is also dependent on the particular type of hypothesis to be tested. When an analyst uses a "two-tailed hypothesis"-that is, where he wishes to determine whether there is any relationship, positive or negative, between the independent and the dependent variable-a t-statistic of approximately two means (in the case of large samples) that the chances are less than one in 20 that the true coefficient is actually zero. Cf. id. An explanatory variable is, however, usually included in the equation because of a prior theoretical reason for expecting it to affect the dependent variable in a specific direction. R. Wonnacott & T. Wonnacott, supra, at 86. For instance, we expect additional firemen to decrease, not to increase, the number of accidents. In such situations, we test a more specific hypothesis-a "one-tailed hypothesis"-not whether or not a particular coefficient is positive or negative as with the two-tailed test, but whether or not, in our railroad example, it is negative or zero. In this circumstance, the "one-tailed test" is one in which 5% would be the probability of observing some negative coefficient if the true value were zero. Cf. Fisher, supra, at 717 n.26. See also n.148, infra. The t-statistic required for significance at the 5% level on a one-tailed test is only approximately 1.6. Fisher, supra, at 717 n.26.

Speaking purely from a statistical point of view, this does not mean that only results significant at the 5% level should be considered; less significant results may be suggestive. Id. at 718-19. Thus even if the t-statistic is small for a particular dependent variable, this does not mean there is no relationship between that dependent variable and the independent variable. R. Wonnacott & T. Wonnacott, supra, at 88. This is so because the size of the t-statistic corresponding to statistical significance is ultimately dependent on the level at which statistical significance is arbitrarily set. "The most commonly used significance level is five percent, but this is purely arbitrary." G. Maddala, supra n.49, at 45. It follows that in many instances, reporting only whether a particular coefficient is significant or not should be avoided; it is more informative to report confidence intervals, test statistics, or other quantitative measures of significance. Id. at 45-46.

Moreover, significance tests and confidence intervals are controversial when the test data includes an entire universe of decisions. For example, in a promotion case, data on every promotion decision may be available, rather than a mere random sample of such decisions. When all such data are available, some statisticians argue that statistical tests are either meaningless or misleading, while others say that they can be useful. D. Baldus & J. Cole, supra n.55, ง 9.32, at 316. See also Testimony of Dr. John Spaulding, Jr., at 1094-95; Johnson Post-Trial Brief at 4 n.4.

Considering the arbitrary nature of the adoption of the 5% level of significance, it is not surprising that courts show flexibility in determining what level of significance to be required in a legal context. Cf., e. g., United States v. Georgia Power Co., 474 F.2d 906 , 915 n.11 (5th Cir. 1973); D. Baldus & J. Cole, supra n.55, ง 9.221, at 308 n.36 & ง 9.41, at 318. See generally section IX(A), infra. Indeed, the Fifth Circuit has specifically stated that a 10% level of significance rather than the statisticians' more conventional and more stringent 5% level " might be acceptable" in the context of the validity of job tests under Title VII. Watkins v. Scott Paper Co., 530 F.2d 1159 , 1187 n.40 (5th Cir.), cert. denied, 429 U.S. 861 , 97 S.Ct. *273 163, 50 L. Ed. 2d 139 (1976) (emphasis in original). [59]

Statistical tests more elaborate than t-statistics and confidence intervals for individual coefficients are also available. For instance, "R2" is often calculated; this is a statistic which provides an overall index of how well Y can be explained by all the independent variables, that is, now well a multiple regression fits the data. R. Wonnacott & T. Wonnacott, supra, at 180-81. The higher the R2, the greater the association between movements in the dependent and independent variables. Fisher, supra, at 720. There are problems, however, associated with the use of R2. A high R2 does not necessarily indicate model quality. See R. Pindyck & D. Rubinfeld, supra, at 58; G. Maddala, supra n.49, at 122-24. Cf. Fisher, supra; D. Baldus & J. Cole, supra n.55, ง 8.22, at 266-67. For instance, the addition of more explanatory variables to the regression equation can never lower R2 and is likely to raise it. Thus one could increase R2 by simply adding more variables, even though, because of "over-inclusion" and "multicollinearity" (terms we will later describe) it may be improper econometrically to do so. Cf. R. Pindyck & D. Rubinfeld, supra.

3. What Can Go Wrong?

A "perfect" model would explain completely the process under study. While such models are found in the physical sciences, they are rare in the social sciences. D. Baldus & J. Cole, supra n.55, ง 8.21, at 264. Indeed, it has been argued that no model in the social sciences ever meets the requirements for a perfect regression analysis. Id., ง 8.22, at 266. But this does not mean that because a model is subject to challenge, its results are valueless. Cf. Editors' Introduction, supra , at 169. Small departures from assumptions necessary for a perfect regression may have small deleterious effects. Cf. Fisher, supra, at 711.

The value of a regression would obviously be affected by problems in the underlying data or by mismeasurements of the explanatory variables. Cf., e. g., R. Pindyck & D. Rubinfeld, supra, at 194-202; Kmenta, supra at 336-41; G. Maddala, supra n.49 at 201-07; J. Johnston, supra n.51, at 281-91; Beyond the Prima Facie Case, supra , n.57, at 417-21. See generally section V, supra, and section VII(C), infra (introduction). And, as discussed earlier, the model must be one justified by theory. We turn here to a few of the less intuitively obvious ways in which ordinary least squares can provide unreliable results important to this case.

In a perfect regression model using the ordinary least squares technique, [60] three major assumptions would hold:

(a) that the effects of the random disturbance term are independent of the effects of the independent variable; (b) that the values of the random term for different observations are not systematically related and that the average squared size of the random effect has no systematic tendency to change over observations; and (c) that the sum of random effects embodied in the disturbance term is distributed normally, in the "bell curve" generally characteristic of the distribution *274 of the sum of independent random effects.

Fisher, supra, at 708.

One way the first assumption may be violated is if some relevant explanatory variable has been left out of the analysis. This is one type of "misspecification" or "specification error." For instance, if yield of a crop is dependent on both amount of fertilizer and rainfall, and if in our regression we include only fertilizer, there will be improper omission of an explanatory variable. The problem caused by omission of variables is that the regression coefficient(s) of the included explanatory variable(s) ( e. g., of fertilizer) would be "biased" (that is, not likely to be correct "on the average"), [61] and the usual tests of significance concerning the included regression coefficient(s) (such as calculation of a confidence interval) will be invalid. Kmenta, supra, at 392-95. See J. Johnston, supra n.51, at 169. Certain statistical tests are available to suggest whether this sin of omission has occurred. See, e. g., Kmenta, supra, at 405.

However, in at least one circumstance, this problem may not be a serious one in cases where the issue is whether discrimination exists or does not exist. Where it is possible to use as proxy for the presence (or absence) of discrimination against a particular group a "dummy" or "group status" explanatory variable, [62] such an omission will not threaten the validity of the group status coefficient (and hence, the validity of the model's suggestion of the existence or nonexistence of discrimination) unless the omitted variable is related to the group status variable. D. Baldus & J. Cole, supra n.55, ง 8A.1, at 273. Thus, here, where the plaintiffs' model had compensation as its dependent variable, and various explanatory variables (including dummy variables for race and sex), Dr. Madden stated:

Q. Do you agree that your report would have been more valid if you measured all potential productivity?

A. Well, since my purpose was to analyze sex and race my report would have been more valid had there-to the extent that there are any omitted productivity variables that are correlated with race or sex. To the extent they're not correlated with race or sex [it] makes no difference whatsoever.

Testimony of Dr. Janice Madden at 104. Cf. Testimony of Dr. Judith Stoikov at 91.

Another type of specification error occurs when one or more irrelevant variables are included in the model. This overinclusion by itself causes fewer problems than under-inclusion. See R. Wonnacott & T. Wonnacott, supra, at 413; R. Pindyck & D. Rubinfeld, supra, at 190; Kmenta, supra, at 399; J. Johnston, supra n.55, at 169. Overinclusion of variables, however, increases the risk of "multicollinearity."

Multicollinearity refers to a situation where due to the high (but not perfect) correlation of two or more variables (or combination of variables), it becomes difficult to disentangle their separate effects on the dependent variable. R. Pindyck & D. Rubinfeld, supra, at 67; G. Maddala, supra n.49, at 183. Multicollinearity creates broad confidence intervals, and estimates of coefficients become sensitive to particular sets of sample data: a multicollinear model makes it difficult to establish that an individual explanatory variable influences the dependent variable. R. Wonnacott & T. Wonnacott, supra, at 353; J. Johnston, supra n.51, at 160. Thus, even if two explanatory variables should be included in the regression, if multicollinearity is serious it may be necessary to drop one of them. This, in turn, may cause problems associated *275 with omission of variables, but those problems might in certain circumstances be acceptable in the face of more serious problems of multicollinearity. See R. Pindyck & D. Rubinfeld, supra, at 68; G. Maddala, supra n.49, at 190. Cf., e. g., Kmenta, supra, at 390-91 (an alternate solution to multicollinearity is acquisition of more data). There are some rough rules of thumb to judge whether multicollinearity is serious or not. G. Maddala, supra n.49, at 186; Kmenta, supra, at 389-91. Thus, in a discrimination model, when too many qualification variables are included, the patterns of correlation among the explanatory variables will be such that the confidence interval of the group status coefficient is inflated. Cf. D. Baldus & J. Cole, supra n.55, ง 8A.1, at 274. Hence, if multicollinearity exists, the probability will be increased that the net impact of group status will be judged statistically nonsignificant, even in cases where there are actual differences in the treatment. Id. at 275.

Another form of specification error, occurs in the case where the analyst chooses to use a regression equation that is linear in the explanatory variables when the true regression model is nonlinear in the explanatory variables. For instance, the analyst may think the relationship is best described by, and thus runs the regression on:

(5) Y = a + b1x1 + u

while the true relationship is:

(6) Y = a + b1x1 + b2(x1)2 + b3(x1)3 + u

Specification of a linear model when the model is nonlinear-an error in the "form" of the specification-can lead to biased estimates. R. Pindyck & D. Rubinfeld, supra, at 190-91.

We have discussed the first major assumption underlying ordinary least squares with reference to specification errors. When the second assumption is violated, i.e., when the scatter or variance of the error terms about zero (the point of perfect prediction) is not approximately the same for all values of each independent variable, "heteroscedasticity" is said to exist. R. Pindyck & D. Rubinfeld, supra, at 17; R. Wonnacott & T. Wonnacott, supra, at 194-95; D. Baldus & J. Cole, supra n.55, ง 8A.42, at 284. Heteroscedasticity can produce errors such as errors in confidence intervals. Id. at 285.

The third major assumption underlying ordinary least squares is that the error term follows the "normal distribution." [63] With respect to this assumption, basic least squares regression models are "quite `robust' in that they will tolerate substantial deviations without affecting the validity of the results." D. Baldus & J. Cole, supra n.55, ง 8A.41, at 284. Nonnormality of errors can be detected, through the use of such techniques as the Kolmorgorov-Smirnov test. G. Maddala, supra n.49, at 306.

E. Econometric Indication of Discrimination.

In econometric terms, one way used (here and in the economic literature) to study differences among individuals' wages is to estimate a regression such as:

(7) Y = a + b1x + b2x2 + ... + bkxk + u

where Y is the level (or natural logarithm) [64] of earnings, income or wage rate, and x1,x2 ..., xk are observable productivity characteristics. Cf. Blinder, Wage Discrimination: Reduced Form and Structural Estimates, 8 J. Human Res. 436, 437 (1973). To illustrate, we could assume that the employee's wages are a function of his years of *276 schooling and years of relevant experience, both factors being postulated by human capital theory to be productivity-enhancing. In such a model, Y would be earnings, x1 would be the number of years of schooling, and x2 would be the number of years of relevant experience. If the assumptions necessary for ordinary least squares hold, and if this model is a valid one, then by running the model on actual data for the firm, we would obtain actual numbers for a, b1, and b2. To determine the predicted earnings for any employee, we would plug his particular values of x1 and x2 into equation 7, using the estimates of a, b1, and b2.

Recalling from our discussion in section VI(B), supra, there are three ways the econometric models, if true mathematical representations of real-world behavior, may indicate that the employer is discriminatory: (1) use of non-job-related criteria with disparate impact; (2) unequal treatment of twins; and (3) improper treatment across twins.

The first type of indication of discrimination may occur, for instance, where the employer states that he behaves according to a certain model, and that model includes explanatory variables which do not meet the legal standard of business necessity. The process of mathematically representing an employer's behavior should at this stage be external to the legal framework, such as those concerned with the legality of the employer's use of various predictors. In determining whether the employer is violating Title VII through the use of non-job-related criteria with disparate impact, one must follow a three-step procedure: [65]

(i) On purely econometric grounds, derive a mathematical model which best represents how the employer behaves. At this stage, it is not necessary for there to be more than business purpose to justify inclusion of a variable. Indeed, to require such a higher standard would be flawed methodologically. This is not to say that all relevant productivity-related variables should always be included: the problems which may be introduced by overinclusion and multicollinearity, described in section VI(D), supra, must always be considered.

(ii) Examine the variables with disparate impact and see if they are adequately justified under the "business necessity" standard.

(iii) For those variables justified under (ii), see if those variables have been subject to group status-based manipulation by the employer.

Step (ii) above is in part a restatement of the job-relatedness requirement:

Once the racially adverse impact of an examination is demonstrated by statistical or other evidence, the burden of proof shifts to the employer to prove that the exam is job-related. Albemarle Paper Co. v. Moody, 1975, 422 U.S. 405 , 426, 95 S. Ct. 2362 , 2375, 45 L. Ed. 2d 280 , 301. See Griggs v. Duke Power Co., supra , 401 U.S. at 432, 91 S. Ct. at 854 , 28 L.Ed.2d at 164. Employment practices having a racially adverse effect are subject to the same scrutiny: if a prima facie case of discriminatory practice is shown, it becomes the employer's burden to demonstrate the job performance validity of its practices. Washington v. Davis, [ 426 U.S. 229 , 96 S. Ct. 2040 , 48 L. Ed. 2d 597 ], supra. The employer's burden is not satisfied by establishing merely a rational basis for a test; the test must be validated. Washington v. Davis, supra , 426 U.S. at 247, 96 S. Ct. at 2051 , 48 L.Ed.2d at 611. Validation, in general, requires a demonstration that "the qualifying tests are appropriate for the selection of qualified applicants for the job in question." Id. Validation by any one of a number of methods of making such a showing, see id. at 247 n.13, 96 S.Ct. at 2051, 48 L. Ed. 2d at 611 , suffices to refute statistical evidence that a test has had a disproportionate racial impact. Id. See also Albemarle Paper *277 Co. v. Moody, supra , 422 U.S. at 425-31, 95 S. Ct. at 2375-78 , 45 L.Ed.2d at 300-304. The same sort of validation will rebut the inferences drawn from statistical evidence of racial discrimination based on work practices.

Scott v. City of Anniston, 597 F.2d 897 , 901-02 (5th Cir. 1979), cert. denied, 446 U.S. 917 , 100 S. Ct. 1850 , 64 L. Ed. 2d 271 (1980). [66] See also Smith v. Olin Chemical Corp., supra (re those criteria which are "manifestly job-related"); Garcia v. Gloor, supra n.36, at 269 n.6, 270 & 272 ("there is no disparate impact if the [employment] rule is one the affected employee can readily observe and nonobservance is a matter of individual preference"); Dothard v. Rawlinson, 433 U.S. 321 , 331, 97 S. Ct. 2720 , 2727, 53 L. Ed. 2d 786 (1977) (re directness of predictors).

What is to be proven in job-relatedness is separate from the methods used in such proof. We leave description of the methods to section IX(F)(1), infra. The employer must show that he is using the predictor to the same extent as would a race-(or sex-) blind employer. [67]

Finally, predictors must be such that if "the legitimate ends of safety and efficiency could be served by a reasonably available alternative system of classification with less discriminatory effect, the classification cannot be continued." Note, 12 Ga.L.Rev. 104, 106 n.12 (1977). See United States v. Jacksonville Terminal Co., 451 F.2d 418 , 451 (5th Cir. 1971), cert. denied, 406 U.S. 906 , 92 S. Ct. 1607 , 31 L. Ed. 2d 815 (1972). More specifically, the Fifth Circuit has held that:

[F]or a practice, which is not intentionally discriminatory or neutral but perpetuates consequences of past discrimination, to be justified by business necessity, the practice must "not only foster safety and efficiency, but must be essential to that goal ... and there must not be a acceptable alternative that will accomplish that goal equally well with a lesser differential racial impact." Parson v. Kaiser Aluminum & Chemical Corp., 575 F.2d 1374 (5th Cir.), cert. denied, 441 U.S. 968 , 99 S. Ct. 2417 , 60 L. Ed. 2d 1073 (1979).

Swint v. Pullman-Standard, 624 F.2d 525 , 536 (5th Cir. 1980). But cf. Garcia v. Gloor, supra n.36, at 271.

Step (iii) of our three-step analysis relates to use of predictors which are susceptible to manipulation by the employer. If a predictor which, while not valueless as a predictor of productivity, is subject to manipulation by a racist employer, the econometric results generated may well be biased in favor of a finding of nondiscrimination. Cf., e. g., Greenspan v. Automobile Club of Michigan, 495 F. Supp. 1021 , 22 Empl.Prac. Dec. ถ 30,812, at 15,193-94 (E.D.Mich.1980); Finkelstein, The Judicial Reception of Multiple Regression Studies in Race and Sex Discrimination Cases, 80 Colum.L.Rev. 737, 738-41 (1980). Thus, for instance, in James v. Stockham Valves & Fittings Co., 559 F.2d 310 (5th Cir. 1977), cert. denied, 434 U.S. 1034 , 98 S. Ct. 767 , 54 L. Ed. 2d 781 (1978), the defendant presented an earnings regression study which included a merit rating as one of its explanatory variables. The court observed, "If there is racial bias in the subjective evaluations of white supervisors, then that bias will be injected into ... [the] earnings analysis." Id. at 332. Use of manipulable predictors in multiple regressions may cloak the employer's "personal biases in the mantle of a scientific judgment." Cf. Underwood, supra n.35, at 1443.

*278 Since to include manipulable predictors as explanatory variables may be to bias the econometric results in favor of the employer, if despite such inclusion the plaintiffs are able to show non-productivity-related pay differentials, the results are all the more impressive. But in the defendant's hands, while models which include manipulable predictors among the explanatory variables are not valueless, especially if those predictors are shown not to have been manipulated, they may be suspect.

The second way in which econometrics may indicate discrimination occurs when the coefficients indicate that a white and a black are being rewarded differently for the possession of the same productivity characteristic(s): there is not equality of treatment of twins. See section VII(C), infra. Referring to the econometric model described in the beginning of this section (wherein x1 is the number of years of schooling and x2 is the number of years of relevant experience), we can see that there are at least two possible econometric indications of this sort of inequality: [68]

1. First, one can run the same equation on data twice, the initial time running it on data solely for one group, and second time on data solely for the other group. After doing this, one compares the coefficients obtained for the two groups. Cf. Testimony of Dr. Francine Blau at 66.

So if we obtain for males Y = 10000 + 500x1 + 350x2, and for females Y = 9000 + 400x1 + 300x2 at the same job, it is clear that females are being discriminated against. That is, any female regard less of her particular combination of schooling and experience is receivingless than a corresponding male: since the regression coefficients (a, b1, and b2) are each higher for males than for females, males are being rewarded more for each productivity-related characteristic. This is the simple situation, where it is clear that females are being discriminated against-where a (female) is less than a (male), and b1 (female) is less than b1 (male), b2 (female) is less than b2 (male), and so on.

A more difficult case would be that where males are rewarded more for one characteristic, while females are rewarded more for another: a "mixed" case. Cf., e. g., Defendant's Exhibit 32, at Table 5. Here some men would be paid more than women with identical portfolios of productivity characteristics, while some women would be paid more than men with identical portfolios of productivity characteristics. For instance, suppose that Y = 10000 + 500x1 + 350x2 for males and Y = 10000 + 400x1 + 1000x2 for females.

(a) A woman with no schooling and two years of experience will be paid 10000 + 0 + 2000 or $12,000; a similar man will be paid 10000 + 0 + 700 or $10,700. Here a woman earns more than a man with the identical portfolio of productivity characteristics.

(b) A woman with three years of schooling and no experience will be paid 10000 + 1200 or $11,200; a similar man will be paid 10000 + 1500 + 0 or $11,500. Here a woman earns less than a man with the identical portfolio of productivity characteristics.

In such a "mixed" case, one might, for example, offer statistical evidence to show that as a result of different regression coefficients for the two groups, because of the empirical distribution patterns of productivity characteristics in the two groups one group is favored at the expense of the others. Cf., e. g., Oaxaca, Male-Female Wage Differentials in Urban Labor Markets, 14 Int'l Econ.Rev. 693, 694-97 (1973); Defendant's Exhibit 32, at 24-25. [69]

*279 One manner of comparing the regression coefficients for the two groups without actually running two separate regressions as described above is to use what are called "interaction" terms. By adding such terms to the regression equation to be estimated, one can produce estimates of the difference in the weights placed on each qualification. Thus, if we added an interaction term for race and experience, and one for race and schooling, the coefficient of the race-experience interaction term indicates the difference between the coefficient for experience for whites and blacks, and the coefficient of the race-schooling interaction term indicates the difference between the coefficients for schooling for whites and blacks. See D. Baldus & J. Cole, supra n.55, ง 8.123[2], at 262-64.

2. The other major way of detecting discrimination between two groups is to use "dummy variables." This method, in contrast to the more general method outlined above, can only be used if it is assumed that all the coefficients for the explanatory variables are exactly the same for the two groups being compared; that the only difference with being female (or black) is represented by one overall effect (that is, the coefficient of the "dummy variable"). Cf. Testimony of Dr. Francine Blau at 66.

Thus, suppose that x1 is the "dummy variable" for race ("1" is input for being black, and "0" for being white) and x2 and x3 are the two productivity predictors. If the multiple regression is run and we obtain Y = $5000 - 700x1 + 300x2 + 400x3, then this means that a black will earn $700 a year less than an equivalent white. Cf. D. Baldus & J. Cole, supra n.55, ง 8.02, at 242; R. Wonnacott & T. Wonnacott, supra, at 100-103.

The third way an employer may not be treating people solely according to productivity differences is what we have termed improper treatment across twins: (i) econometrically, this may mean that a particular race- (or sex-) correlative personal productivity characteristic may be rewarded by a racist (or sexist) employer more than it would be by a nonracist (or nonsexist) employer or (ii) the relative pay of various jobs may not correspond to their "worth". As for (i), suppose that

This text is long and has been trimmed here. Open the source document for the complete record.