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STATISTICAL MODELING TO SUPPORT A CLAIM OF INTENTIONAL DISCRIMINATION James W. L. Cole, Pittsburgh, PA David C. Baldus, University of Iowa In the context of discrimination cases the courts have acknowledged

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STATISTICAL MODELING TO SUPPORT A CLAIM OF INTENTIONAL DISCRIMINATION James W. L. Cole, Pittsburgh, PA David C. Baldus, University of Iowa In the context of discrimination cases the courts have acknowledged that Figures speak, and when they do, courts listen. ' Indeed, conclusions drawn from statistical analyses often form an important and valued element of the evidence supporting claims of discrimination against minorities. But when one reads the cases in which statistics have been applied, one observes that the courts have left unresolved a number of methodological issues. Our purpose here is to examine some issues that arise in a particular class of discrimination cases and to pursue their resolution. The cases to be considered have the following defining properties in common: 1. The plaintiffs belong to a group (say, Group A) which is constitutionally or statutorily protected. 2. The practice under challenge is a selection process which assigns to each candidate for selection one of two outcomes, namely selection or rejection. Thus candidates are effectively winnowed out in a process represented schematically as in Figure 1 which highlights a) the pre -selection pool of applicants containing Group A (protected minority) and Group B (majority or other) components. b) the similarly constituted post -selection pool of those selected, and c) the decision maker, the individual or group actually making the choices. Pre -selection Pool FIGURE 1 \ Selection by \ Decision -maker Post -selection pool Group A NA Group B NB 3. The essence of the challenge is that the decision -maker was covertly applying a policy that used group membership to influence the applicants' selection chances and hence put Group A applicants at a disadvantage relative to others. However, those elements of the policy governing the selection process that are open to public view do not explicitly refer to group membership or any equivalent criterion as a factor influencing chances of selection. 4. The policy governing the selection process has left the decision maker ample room to use personal discretion in making choices, in that choices are not determined substantially by qualification criteria that are open to public view. Selection processes susceptible to challenges with these properties will include those found in the substantive areas of employment selection, promotion, school admissions, some instances of criminal sentencing, and jury selection. The very early landmark case Yick Wo v. Hopkins2 involved such a claim, and a long and rich line of challenges against jury selection processes provide numerous other examples.3 What is being claimed in cases having these properties is intentional discrimination of the type prohibited in particular under the due process and equal protection clauses of the 14 -th Amendment of the U. S. Constitution. In two recent opinions, namely Washington v. Davis4 and Arlington Heights v. Metropolitan Housing Corporation, the Supreme Court has made it clear that, to be successful, such challenges raised on constitutional grounds require proof of two facts. These are the existence of discriminatory impact, relative disadvantage falling to the plaintiff as a result of the suspect practice, and the existence of an intent to discriminate underlying the practice. The dual nature of this requirement has been unclear in many decisions against jury selection systems because in these decisions a single piece of evidence has been found to establish a prima facie case of intentional discrimination. Specifically, the plaintiffs have prevailed on showing that over a suitably long period of time citizens with the same group membership have been substantially or consistently under -represented on the jury lists. However, in its discussion of requirements for proof of intentional discrimination in Castaneda v. Partida, the Supreme Court observed that the sufficiency of a single piece of evidence in such cases simply reflects the fact that the one fact can have two implications: the substantial under -representation speaks directly to the question of discriminatory impact while at the same time suggesting by its magnitude the presence of illicit motive.6 Selection processes can be classified usefully according as the decision -maker's choices are or are not guided by consideration of overt and verifiable qualification criteria. If the decision- 465 maker's choices are influenced in part by such qualifications we will call the process a 'guided' discretionary process. In such processes, because equal protection doctrine is concerned with the equal treatment of equally situated candidates, such qualification variables that legitimately divide the candidates into equivalence classes of 'equal situation' must be taken into account before either a relevant measure of impact can be obtained or a valid inference of motive can be drawn. If no such qualifications are cited, we will call the process 'purely' discretionary. With respect to those aspects of the governing policy that are oren to public view, and hence for purposes of measuring the discriminatory impact of the discretionary aspects of the policy, each candidate who exceeds a basic threshold level of eligibility can be considered to be as qualified for selection as any other such candidate. Because purely discretionary processes will be easier to model, we will consider them first. Fortunately, they do provide a suitable context for discussing interesting questions. Modeling in Purely Discretionary Processes For challenges of purely discretionary selection processes the four frequencies, NA, NB, na, and nb found in the accompanying four -fold table contain the information usually considered necessary to establish a prima facie case of either discriminatory impact or discriminatory intent. Group Group A Group B Numbers Selected nb Table 1 Numbers Rejected NA NB Totals n N - n N Pre -selection Pool Totals Usually the information in these quantities is summarized by a number or statement that compares one measure that reflects how the minority group was actually treated with a corresponding measure constructed to show how the group should have or would have been treated absent any discriminatory behavior. The following measures of actual treatment are frequently used: NA NB a) the selection rate (or pass rate), PA = na /NA, b) the rejection rate (or fail rate), 1 - c) the inverse selection rate, 1 /PA, d) the minority representation rate in the post -selection pool, ra na /n (to be compared with RA NA /N, the representation rate in the pre -selection pool), or even e) the actual number of minority candidates chosen, These rates or numbers can be compared in a variety of ways. Usually comparisons are made in terms of arithmetic differences or ratios, but other formulas have been suggested.? The courts have had some problems with the variety of measures available, the most obvious being a lack of consistency and direction in the choice of form- ulas for summarizing the numerical information bearing on the issue at hand. However, this lack of direction is not in itself the most troublesome aspect. As the courts move away from cases with clear -cut factual bases and encounter those that are closer, they are more inclined to compare the numbers in the case at hand with those of precedent cases.8 To do this without adopting some function of the tetrad (na, NA, NB) that evaluates the evidence in terms of a single number is to trust the reliability of subjective judgment. On the other hand, any function chosen to evaluate the tetrad in the context of a particular question of fact should have an essentially monotonic (strictly) increasing relationship with the actual legal significance of the tetrad as it relates to the question of fact. For example, if in a given situation an arithmetic difference of 10 percentage points between RA and ra is much more strongly suggestive of unlawful motive when RA = 11% than when RA = 91 %, the blind dependence on RA -ra to support an inference of motive will invite the drawing of erroneous conclusions. We propose that a primary determinant in choosing a measure should be the purpose of the measure -- whether it is intended to measure discriminatory impact or to support an inference of motive. In measuring discriminatory impact, we are inquiring as to the relative harm done to members of the minority group as they go through the selection process, and this suggests that the underlying modeling be motivated by a útility theoretic approach. On the other hand, in inferring motive, we are seeking to identify an aspect of the decision - maker's behavior and principles of behavioral modeling should dominate. Measuring the Discriminatory Impact. In a purely discretionary selection system, each Group A applicant entering the process can be said to have the same probability of being selected, say PA, as any other such applicant. If we assume further that each candidate has the same (positive or negative) utility, say u, of being selected and utility 0 (zero) of being rejected9, then the expected utility for a minority candidate is clearly EA(U) u PA. Finally, assuming that in the absence of discrimination minority candidates would have the same probability of selection as that which the majority candidates have, say PB, the expected utility for a minority candidate would be E* (U) = u PB Therefore a measurement of harm should clearly be based on some comparison between upa and upb. However, to be consistent with the principles of utility theory, a given arithmetic shortfall in the expected utility must represent the same degree of harm to the applicant regardless of the value of the expected utility that would obtain in a non -discriminatory process. That is to say, the appropriate measure of harm is the arithmetic difference E *(U) - EA(U) = u(pb - PA). Finally, since the minority and majority probabilities of selection are estimated without statistical bias by the corresponding observed selection rates, these results clearly suggest that a measure of discriminatory impact based on the difference between selection rates is preferable A significant exception to this argument applies when the selection process under challenge is that of selecting the venire from which a jury will be chosen when that jury is to decide the fate of a minority criminal defendant. For in this situation, the courts must be concerned primarily with the impact on the rights of the defendant, not those of the prospective veniremen. Moreover, in this situation, the composition of the post- selection pool (that is, the venire from which the jury will ultimately be chosen) is the only aspect of the venire -selection process that bears on the defendant's rights. Consequently, while the difficulty of modeling the subsequent jury selection may make the application of the utility- theoretic argument impractical and its results of doubtful acceptability to the courts, it is clear that the minority representation rates that result from the venire selection process are the pivotal quantities in establishing the impact on the defendant's rights. Modeling to Infer Discriminatory Motive. When we turn to the goal of modeling to infer motive, the focus of the model shifts from the treatment of the applicant to the behavior of the decision - maker. The first step in this modeling process is to pin -point as nearly as possible the kind of non -discriminatory selection mechanism the decision -maker might have used or claims to have used. The second is to adopt a model that appears to simulate that mechanism adequately. The third is to consider how discriminatory behavior might manifest itself in the context of the given selection mechanism. The fourth is to decide how that form of discriminatory behavior is to be incorporated into the model. Finally, we apply the conclusions drawn in the first four steps to the pursuit of our current objective, namely to choose a measure best suited to reflecting behavior suggestive of a discriminatory motive in the eyes of the court. For example, the selection of prospective jurors from the citizens in a district is often supposedly done by application of an explicitly random mechanism. The appropriate model for such a process will be more or less obvious depending on the complexity of the mechanism and the care taken to adhere to it. For example, in straightforward situations the model of simple random sampling from the pool of all citizens meeting specified eligibility requirements may suffice, but when allowances are made for various forms of hardship, the model may require modification. In the context of such a process, two forms of discrimination would seem to be most likely. The first is the ever -present possibility that the number of Group A applicants was held below some tacit quota. The second is the exclusion from consideration of some fixed portion, consisting say of individuals, from the pool of eligible Group A candidates. If the process is being influenced by a desire to limit Group A selections to a quota, it is more likely that the decision -maker is concerned with controlling the size of the Group A representation rate than the Group A selection rate. Hence this desire is more likely to be reflected in the representation rate than the selection rate. Specifically, if the nominal model for the process is one of simple random sampling, the distribution of the representation rate would be ex- pected to differ from the nominal binomial or hypergeometric distribution in two ways. Naturally, the mean of the distribution will be reduced from the expected NA /N. But also the distribution would likely be truncated, especially at the upper end. This suggests that when results from several applications of the suspect process are available, as is often the case in jury selection challenges, the entire empirical distribution of representation rates may be relevant.11 If the Group A participation in the selection process is limited by exclusion of a fixed proportion of eligible candidates, an estimate of the excluded proportion based on the representation rates has been suggested.12 This estimate is found by simple algebra to be RA- ra RA(1 - ra) Purely random mechanisms are less common in employment selection, where typically some form of evaluation or ranking will be used. If all the evaluation criteria are strictly subjective and hence inaccessible for verification or challenge, the process will still fall in the 'purely discretionary' class. However, the five -step process of analysis and modeling should reflect the dependence on the criteria if possible. Such selection mechanisms provide more interesting modeling challenges. The particular model chosen will depend, first, on the nature of the criteria on which the informal evaluation is based; second, on what can be postulated as reasonable distributions for these criteria in the populations of candidates; and finally, on the kind of reasoning that was used to combine these criteria into a single score. In order to develop an illustration, albeit more valuable for insights produced than for realism, suppose the following. First, for each candidate there exists a vector of qualification scores (i = 1, Ni; j =A,B). Second, these vectors are p- variate normally distributed within groups- - Np(j,E) -with the same covariance matrix but possibly different means. And finally, the decision -maker looks at these criterion scores, constructs some weighted sum of them, say Yij = and selects those candidates for whom Yij exceeds a cut score y. This mode will reflect an absence of intentional discrimination only if and yg. Any difference between the weight vectors or the cut scores would indicate that group membership was being used to influence the decisions. According to this model the weighted sum Yij will be normally distributed, Yij - Thus, the probability that a candidate being randomly drawn from group j, will meet the standard for selection is y3) = - yj 467 Therefore we find that 'P (Yij the form of a linear model- - 1 P (Yi y a0 + y!) will be in --where Zij = 1 for candidates in Group A, and 0 otherwise; and the unknown coefficients have the following form: In this linear model, the parameter al contains all the evidence of discriminatory behavior. This suggests using an appropriate estimate of al as our measure from which to infer motive. From the methodology of probit analysis14 we find the maximum likelihood estimate of al to be - where and pb are the observed selection rates for the two groups. In practice, this measure would likely be resisted as unfamiliar and based on unverifiable assumptions. Thus we seek a more familiar and intuitive measure that would produce similar results in the case -to-case comparisons. We turn A first to the logit function, L(p)E Since, as can be shown, (L(PB)- L(PA)) / -1 ) (PB) - (PA) the difference L(pg) -L(pA) suggests itself as an alternative with the virtue of being easily explained in terms of betting odds. Finally, we note that if and pb are both small, the simple ratio will give case -to-case comparisons most consistent with those of among the measures now commonly used. As a last word it should be noted that al will reflect the arguably innocent effect of the group mean differences in qualification score inextricably confounded with the influences of intentional discrimination. This fact would suggest an obvious defense to the claim, and hence it raises legal and methodological questions that are serious and most interesting, but beyond the scope of this discussion. Modeling in Guided Discretionary Processes In the following discussion of guided discretionary processes we will assume that there is no challenge to the legitimacy of the overt qualification criteria guiding the decision - maker, but that in making the final selection decisions, the selector has covertly used group membership as a factor. Thus it is alleged that, within the 'equivalence classes' of candidates defined by virtue of being similarly situated with respect to the overt criteria, the selector operated to the disadvantage of the Group A members. Consequently, we shift our focus, both in measuring discriminatory impact and in detecting evidence of unlawful intent, to comparisons of treatment observed within the equivalence classes. A Y Measuring the Discriminatory Impact. Let Wij denote the vector of overt qualifications and let Pi(y) denote the probability of selection for a group j candidate whose Wij vector equals w. Then invoking the same assumptions of utility structure and the same argument as in the previous section, the suggested measure of impact for a minority candidate with qualification vector equal to w would be an appropriate estimate of the difference PB(w) This difference can be estimated in two general ways: (1) directly from selection outcomes observed for groups of candidates having Wij = w, if there exist such groups of sufficient size, or alternatively, (2) adopting models for PA(w) and PB(w) as functions of w and estimating the parameters therein by appropriate methods. If the latter approach is chosen, it will again be apparent that the model to use will again depend on the particular situation. But in this application of modeling, the final result will be judged purely on the reliability of the estimates that it provides for PB(w) - PA(w) as indicated in part by measures of goodness -of -fit of the model. If such measures indicate the need for a model in which PB(w) - varies with w, the lack of a single number summarizing the magnitude of the impact for the whole case will clearly complicate case -to -case comparisons. These complications are intrinsic to situations in which some portions of Group A may have been treated more adversely than others, and it would be unwise to confine the choice of models artificially to those which have an additive term corresponding to group

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