The appropriateness of survival analysis for determining lost pay in discrimination cases: application of the ‘Lost Chance’ doctrine to Alexander v. Milwaukee¹

Abstract

1. Introduction

While damages for a lost opportunity have been available in contract law² and tort law for many years,³ the economic justification and appropriateness of the approach for determining damages in equal employment cases concerning competitive promotions was recently provided by Judge Posner in Doll v. Brown.⁴ In Biondo v. City of Chicago⁵ and Alexander v. Milwaukee, the Seventh Circuit required that damages due to plaintiffs who prevail in discrimination cases be determined in accordance with the lost chance doctrine.⁶ This article shows how survival analysis can be used to estimate both the probability a plaintiff would have been promoted during the period in question and the expected damages a plaintiff is due. In Biondo the juries were asked to estimate the probabilities each plaintiff would have been promoted during the period and their lost pay. That opinion by Judge Easterbrook noted that their findings were inconsistent with the fact that only about one-third would have been promoted.⁷ An advantage of the survival analysis framework is that at each promotion, all eligible employees are considered and their probability of being selected is estimated. The calculation incorporates relevant factors, e.g. length of time served in the current position, special skill or education. Because the method considers the relative qualifications of the eligible pool at the time of each promotion, it can incorporate changes in relevant characteristics of the eligible employees over time.⁸ Unlike Judge Easterbrook’s decision in Biondo the method does not assume that an individual who is promoted after the discriminatory practices ended would have been certain to have received one of the promotions during the time when discrimination was practiced. The lost salary is estimated by multiplying the estimated probability of being a captain times the salary differential.

As data from the Alexander et al. v. Milwaukee case will be used to illustrate the methodology, Section 2 will summarize the relevant issues and background information from it. Section 3 describes the basic concepts underlying the statistical approach and the methods are illustrated on the data from Alexander in Section 4. Section 5 compares some of the results obtained from the survival analytic method with those introduced in the case, including the success probabilities estimated by the jury. Sections 6 and 7 discuss other issues arising in compensatory damage calculations.

2. Background

2.1 The Alexander v. Milwaukee reverse discrimination case⁹

In 1996, the City of Milwaukee appointed Arthur Jones to head the police force. Seventeen white males who were lieutenants in the city’s police force and were not promoted to captain during the 18 November 1996 through 18 November 2003 time period when Jones was in charge, claimed that the promotion practices discriminated against them. According to Wisconsin law, when a captain’s position opened up, the police chief nominated a candidate, who was already in the police force and was fit for promotion. A five-member board of commissioners reviewed the files and interviewed each nominee. During Chief Jones’s tenure, he submitted 41 nominees, all of whom were approved by the Board. All of the parties having a role in the promotion process were defendants in the case and their different legal responsibilities and defences are discussed in the opinion.¹⁰ This aspect of the case will not be discussed here as our focus concerns the statistical methodology appropriate for calculating damages under the ‘lost chance’ approach. The survival analysis of the promotion data described here can also be used in determining liability.¹¹

The Department kept records of the racial and gender diversity of the police force, in part in response to discrimination suits filed by minorities in the 1970s. Indeed, a consent decree governing affirmative action in hiring, but not promotion, was still in force during the early part of the relevant time frame. A researcher on the staff of the Fire and Police Commission periodically produced statistical reports on diversity. They contained affirmative action goals for recruiting and hiring but stated there were no ‘affirmative action goals for promotion’. In 2001, an updated report included data on the command-staff. It noted that white men were under-represented at the rank of captain and higher as they held only 44% of these positions, noticeably lower than their 53% share in all jobs in the Department.¹² The plaintiffs also submitted statistics comparing their 44% share of captains to their nearly 80% share of lieutenants.¹³

The City did not have written procedures and vacancies at the captain level were not announced or posted. At the trial Chief Jones said that he considered the skills, abilities, knowledge and, to a lesser extent, seniority of the promotable lieutenants. He denied that race or gender was a factor, however, several statistical studies introduced by the plaintiffs’ expert demonstrated highly significant differences in promotion rates and time to promotion disadvantaging white males.¹⁴ The jury found that the City and Chief Jones had intentionally discriminated in favour of women and minorities and that the Commissioners had participated in the discriminatory practices. The trial judge rejected the defendants’ request for a new trial because the jury’s decision was reasonably supported by the evidence and the appellate court affirmed liability. The appeals court, however, disapproved of the approach the district court took to determining the compensation due each plaintiff under the ‘lost chance’ doctrine and remanded the case for further proceedings to re-calculate the damages.

2.2 Calculation of compensatory damages under the Lost Chance approach

When there is discrimination in a competitive process, the 7th Circuit had previously noted that the ‘loss of chance’ doctrine, often used in tort cases, is appropriate since during the period when a discriminatory policy was in effect, members of the disadvantaged group only had a probability of being selected for each promotion. As Judge Posner explained in Doll v. Brown¹⁵ in order to avoid under-compensation and consequently under-deterrence of harm to the public in medical malpractice cases, the trier of fact estimates the probability the patient would have survived had the proper treatment been given and awards the corresponding percentage of damages to the plaintiff. In both Herskovits v. Group Health Cooperative¹⁶ and McKellips v. Saint Francis Hospital, Inc.¹⁷ the failure to properly diagnose patients with cancer and heart disease, respectively, reduced their probabilities of surviving by about 15–20%. Both courts rejected the suggestion that the probability of the ‘lost chance’ be at least 50% in order for the defendants to be liable. The plaintiff’s award was their ‘lost chance’ percent of full damages. The Doll opinion notes that the lost chance doctrine is well suited to the context of competitive promotions. For example, if there were four equally qualified minority candidates for a position who were discriminated against when an employer appointed a less qualified majority member, only one of the four could have received the position. Thus, it makes sense to award each of them one-fourth of the lost pay. Otherwise, ‘the employer would get off scot-free’.¹⁸ The idea of a ‘lost chance’ can be explained to the jury in the context of a lottery, where due to a computer problem a sizeable fraction of the tickets were not entered. If say, 20% of the tickets were omitted, collectively the purchasers lost a 20% chance of winning.

In order to discuss the damage calculation in Alexander v. Milwaukee, relevant aspects of the precedential case, Biondo need to be reviewed. In Biondo nineteen white male firefighters in the Chicago Fire Department, who had higher scores on the 1986 promotion exam for lieutenant than minority members who had lower scores but were promoted when the City created two racially segregated lists and made appointments from them in rank-order, sued. The Court found that the plaintiffs were discriminated against but questioned the damage calculations. The juries had assigned probabilities for their subsequent promotions to captain and later to battalion chief for each plaintiff as well as an estimated date for these promotions. The opinion noted that the probability estimates of the juries were overly optimistic, e.g. of 15 plaintiffs who were assigned a 100% chance of promotion, only 9 subsequently were promoted in one of two later promotion exams. Judge Easterbrook realized that the best evidence of the chance the plaintiffs’ lost was missing because they did not present comparative evidence, e.g. showing that firefighters who had slightly higher scores than they did on the 1986 exam and became captains, had very high promotion rates. The opinion vacated the judgment and remanded the case for a new trial focused only on the calculation of back pay and damages for emotional distress. Using aggregate data on the promotion rates from subsequent exams, the court noted that each of the 13 plaintiffs, who had not yet become captains, would have received a timely promotion to lieutenant and had about a 33% chance of promotion to captain by 2002. The court also discussed the front pay calculation, stating that it should be the time period a reasonable person needs to achieve the same position in the absence of discrimination. While the lingering effects of the discrimination may have hindered the plaintiffs’ opportunity for promotion until the trial in 2001, the opinion stated that front pay cannot logically continue after the next unimpeded promotional opportunity, i.e., the first post-2002 captain exam (for plaintiffs who were lieutenants in 2002) or battalion chief (for those who had become captain by 2002).¹⁹

2.3 The damage calculations in Alexander v. Milwaukee²⁰

At the damages phase of the trial, the defendants requested that the jury be instructed to consider all the lieutenants who were eligible for promotion at each of the dates the jury found that a plaintiff was wrongfully passed over. This would have required the jury to consider the entire set of officers who were qualified to be promoted to captain at the time of each promotion. The jury, however, was instructed to calculate the probability that each individual plaintiff would be promoted at the time a vacancy was filled, keeping in mind the other qualified plaintiffs. Thus, jurors were not told to consider the probability of promotion of all the eligible lieutenants at the time. The Special Master then used the dates of promotion and probabilities of promotion the jury determined for each plaintiff in calculating their economic damages.

It is clear that an estimate of the probability of promotion obtained by considering only a subset of eligible competitors at each promotion will overestimate the promotion probabilities of this smaller set. As Judge Ripple’s opinion observed only if the plaintiffs provided evidence that in the absence of discrimination they would have been promoted over all non-plaintiff candidates, e.g. they had longer seniority and were more qualified, would the district court have been justified in telling the jury to only consider the plaintiffs as the totality of eligible officers.

In the subsequent proceedings the plaintiffs followed Judge Easterbrook’s approach in Biondo and assumed that the four plaintiffs, out of the original 17, who were promoted after Jones was no longer the chief of police would have received a promotion during the discriminatory period. Hence, probability estimates were developed for the 13 plaintiffs who had not yet been promoted. This was accomplished by estimating the number of promotions they would have received.²¹ Several modifications needed to be made to account for retirements, which occurred during the tenure of Jones or after he left but before the trial. Ultimately, the case settled and the final awards are not available. The next section presents the survival analysis approach, which incorporates seniority in predicting the success probability of each member of the eligible set at the time of each promotion. The method does not adopt Judge Easterbrook’s assumption that individuals who received a promotion after Jones left had a 100% chance of receiving a promotion during the period. Not only does the set of potential selectees change during the time period, an individual officer’s relative seniority increases over time, so someone who receives a promotion after being eligible for say, seven years, might well have had a lower probability of promotion at an earlier time, when they had less seniority.

3. The compensatory damages obtained from a survival analysis

According to the lost chance doctrine, economic loss estimates have two major components: the promotion probabilities the plaintiffs would have had they been treated the same as the unprotected group members, and the increased wages and pension benefits they would have received had they been promoted to captain. The first component, the probability of promotion, is an uncertain event. The second part of the calculation is conceptually simpler as it is just the increased compensation the individual would have as a captain. This section explains appropriate statistical methods for estimating the hypothetical promotion probabilities, i.e. the probabilities of promotion that the plaintiffs would have had in a discrimination-free environment.

Suppose one has data for n individuals who are eligible for promotion at some time during the relevant time period.²² The event that any individual receives a promotion is a random variable that can happen any time between the date the individual becomes eligible and the date they retire or get promoted. We model the risk of promotion as a time-varying process through the Cox proportional hazards model

(1)

The hazard rate for subject i at time point t, can be understood as the event rate at time t given that the event of interest, the ith individual is promoted, has not occurred before t. The subject-specific hazard rate at each time point is assumed to be the product of an unspecified baseline hazard rate and the exponential of a linear combination of relevant covariates.²³ The baseline hazard function, is shared by all subjects. At any time t, the covariates, increase or decrease hazard rate of the ith employee by a multiplicative factor. Notice the covariate values can be either fixed, such as race, gender, or varying over time, e.g. seniority in the lieutenant position. For each unit increase in the pth covariate, the hazard rate at every time point is multiplied by e^βp. For example, if β_whitemale = −1, then the probability of a white male eligible at time t, would receive that promotion is reduced by a factor e⁻¹ = .37, implying that the probability of this employee receiving the promotion was 37% of that of a comparably qualified non-white-male. Similarly, if the value of the coefficient for seniority is 1.25, then an employee’s hazard of promotion increases by e^1.25 or 3.49 for each additional year of seniority.²⁴

At the calendar date of each promotion, the Cox proportional hazards model considers all lieutenants eligible for the particular promotion as the risk set, and each candidate’s chance of being selected is proportional to the exponential of a linear combination of the covariates, reflecting their job-related qualifications. The regression coefficients estimates are the values which maximize the probability of the promoted individual getting selected out of the risk set. A description of the procedure for estimating the regression coefficients β are explained in  Appendix A.1. It is shown there that the cumulative baseline function, for a subject with covariate an be estimated by

(2)

which is the integral of the hazard rate estimates for subject i, over time interval (0, t). The hazard rate for one subject at any time point is the product of the baseline hazard rate and the subject-specific qualifications ⁠. At each event time, s, the baseline hazard rate is estimated by the total number of events, divided by the sum of the qualification measures of the risk set, Then the probabilities of being a captain and being a lieutenant at time point t for a subject i are respectively estimated by

(3)

SAS codes for obtaining the estimates of and (captain; t) are also provided in the  Appendix A.2.

The estimated probabilities of being a captain at any time point are used to weight the difference between the captain and lieutenant salaries in calculating the expected economic loss. Detailed examples of this calculation are given in Section 4.2.

4. Analysis of the data from Alexander v. Milwaukee

4.1 Estimates of lost chance

Information on all promotions and retirements between 18 November 1996 and 31 May 2003 when Mr Jones was Police Chief was examined for evidence of race-conscious promotion rates and later used to estimate damages. Two Cox proportional hazards models are applied to the promotion and retirement processes respectively. The promotion process focuses on the employees who are working, i.e. those who had not retired at the time of each promotion. If the retirement process is affected by the discriminatory practices, one needs to appropriately modify the compensation calculation (see  Appendix A.3). The time axis for both models is calendar time. Both sets of adjustment covariates include a 0/1 variable indicating membership in the protected group (1 for the protected group), another indicating position (detective versus police) and the number of years the subject served before becoming a lieutenant, which are time-invariant, i.e. their values do not change with the dates promotions were made during the period. The proportional hazards assumption for the binary covariates was validated by log-log-survival plots and the functional form of continuous variables was verified by martingale residuals plot. Besides the three time-invariant covariates, the promotion model adjusts for the number of years the subject had served as a lieutenant at each promotion time. The retirement risk of an employee is related to the number of years they have been eligible. The two seniority measures (number of years as a lieutenant and number of years since eligibility) are time-varying; they increase over time. Several functional forms were explored to model the relationship between the time-varying covariates and the two outcomes (promotion, retirement) and the forms with smaller AIC and a direct interpretation were chosen. For the promotion process, a quadratic relationship with the promotion risk was identified, which means the promotion risk first increases as the seniority of the lieutenant accumulates, however, once the risk reaches its peak, waiting longer leads to a lower chance of promotion. Wage growth follows a similar pattern. Here, most promotions occurred by the time one served five to six years as a lieutenant and the chance of promotion declined afterwards. The relationship between the length of time since becoming eligible and retirement risk was adequately modelled as a linear one. Finally, the Cox–Snell residuals indicated satisfying performance of the Cox models (plots available from the authors).

In Table 1, the terms Exp and Exp ⁠, the exponentials of the regression coefficients equal the ratio of the hazards corresponding to one unit increase in the covariate. A small p-value, say <0.05, indicates that the regression coefficient is significantly different from zero, i.e. the covariate has a significant effect on the outcome as the hazards ratio for the covariate is significantly different from exp(0) = 1.

Table 1

Open in new tab

Estimated regression coefficients of the Cox PH models for the promotion and retirement processes

Covariate .	.	Exp(⁠⁠) .	p-value .	.	Exp(⁠⁠) .	p-value .
White male	−2.13	0.12	<0.001	0.18	1.20	0.767
Detective	−0.17	0.84	0.611	−0.23	0.79	0.425
Years before lieutenant	−0.01	0.99	0.856	−0.10	0.90	0.007
Years since lieutenant	0.41	1.51	0.012	–	–	–
Years since lieutenant2	−0.02	0.98	0.089	–	–	–
Years eligible for retire	–	–	–	0.12	1.13	.017

Covariate .	.	Exp(⁠⁠) .	p-value .	.	Exp(⁠⁠) .	p-value .
White male	−2.13	0.12	<0.001	0.18	1.20	0.767
Detective	−0.17	0.84	0.611	−0.23	0.79	0.425
Years before lieutenant	−0.01	0.99	0.856	−0.10	0.90	0.007
Years since lieutenant	0.41	1.51	0.012	–	–	–
Years since lieutenant2	−0.02	0.98	0.089	–	–	–
Years eligible for retire	–	–	–	0.12	1.13	.017

Table 1

Open in new tab

Estimated regression coefficients of the Cox PH models for the promotion and retirement processes

Covariate .	.	Exp(⁠⁠) .	p-value .	.	Exp(⁠⁠) .	p-value .
White male	−2.13	0.12	<0.001	0.18	1.20	0.767
Detective	−0.17	0.84	0.611	−0.23	0.79	0.425
Years before lieutenant	−0.01	0.99	0.856	−0.10	0.90	0.007
Years since lieutenant	0.41	1.51	0.012	–	–	–
Years since lieutenant2	−0.02	0.98	0.089	–	–	–
Years eligible for retire	–	–	–	0.12	1.13	.017

Covariate .	.	Exp(⁠⁠) .	p-value .	.	Exp(⁠⁠) .	p-value .
White male	−2.13	0.12	<0.001	0.18	1.20	0.767
Detective	−0.17	0.84	0.611	−0.23	0.79	0.425
Years before lieutenant	−0.01	0.99	0.856	−0.10	0.90	0.007
Years since lieutenant	0.41	1.51	0.012	–	–	–
Years since lieutenant2	−0.02	0.98	0.089	–	–	–
Years eligible for retire	–	–	–	0.12	1.13	.017

Coefficient estimates from both models are listed in Table 1. The survival analysis confirms the soundness of the original decision that white males were discriminated against during the tenure of Chief Jones. Notice that the white male coefficient in the promotion model (see column 2) is −2.13 and is statistically significantly less than zero (p < 0.001). This indicates a substantially decreased promotion chance for white males compared to non-white-males with the same seniority and position. Indeed, the hazards ratio (Exp⁠) in the third column is 0.12, which means that the risk of a white male being promoted to captain is 12% that of a non-white-male with equivalent seniority and position. Besides membership in the protected group, the single most important predictor for promotion risk is number of years as a lieutenant. Both the linear and quadratic order terms of the number of years since lieutenant are statistically significantly different from zero in predicting the promotion risk.

Although we expected that members of the protected group might have higher chance of retiring after they became eligible (25 years of service) because of their decreased chance of promotion, the data do not provide evidence of a statistically significant impact of being a white male. Indeed, the estimated coefficient is 0.18 (see column 5 of Table 1), which is not statistically significantly different from zero. Therefore, the retirement process under a nondiscriminatory environment would be similar to the observed retirement process, so the actual retirement dates will be used in calculating compensatory damages. The retirement process estimates individual’s probability of being on the job, which equals one minus the probability of being retired.²⁵ When the retirement process is not affected by protected group status one uses the actual retirement dates.

The promotion probabilities for the plaintiffs under the hypothetical scenario in which the protected and unprotected group members had an equal opportunity for promotion can be estimated by changing their white male indicator from 1 to 0. Thus, their promotion probability is estimated by that of a non-white-male in the same position with the same seniority before being appointed a lieutenant who served the same length of time as a lieutenant as the plaintiff. Therefore, we substitute a modified set of covariates into formula (2) to estimate the chance of promotion had there been no discrimination during the period. The only difference between and is the value of the white male indicator. It should be mentioned that the data collection ended on 31 May 2003 so promotion probabilities beyond this date cannot be estimated from the Cox model. The baseline promotion rate is assumed to be constant after 31 May 2003. The constant baseline hazard rate is estimated using the data from 18 November 1996 to 31 May 2003. The covariate effects are also extended into the period beyond 31 May 2003. With these assumptions for baseline hazard rate and covariate effects, the promotion probabilities after 31 May 2003 are estimated using formula (3).

In the following subsection, the damage calculations for three plaintiffs, each having one of the three most common job histories of the seventeen plaintiffs will be discussed. Plaintiff 1 retired during the period, on 20 July 2002, as a lieutenant. Plaintiff 2 was not promoted and was still working, i.e. had nor retired, by the time of compensation calculation (31 August 2005). Plaintiff 3 was promoted on 6 December 2003, soon after Jones stepped down on 18 November 2003. The probabilities that each plaintiff would have been promoted, in a non-discriminatory environment, over time are plotted in Fig. 1. Plaintiff 1 was promoted to lieutenant on 20 April 1997. His probability of being a captain increased to 90% by 19 July 2002 when he retired. Plaintiffs 2 and 3 became lieutenant on 16 July 1995 and 8 October 1995, respectively. Fig. 1B and Fig. 1C show that their probabilities of being a captain increased dramatically after about five years of serving as a lieutenant and essentially reached 1 by the end of 2001. The yearly average probabilities of being a captain for the three plaintiffs are presented in Fig. 2. Because the probabilities are time varying, the yearly average weights each probability by the proportion of time in the year that probability was applicable.²⁶

Fig. 1.

Estimated probabilities of being captain for (A) Plaintiff 1, (B) Plaintiff 2 and (C) Plaintiff 3.

Open in new tabDownload slide

Fig. 2.

Estimated yearly average probabilities of being promoted for (A) Plaintiff 1, (B) Plaintiff 2 and (C) Plaintiff 3.

Open in new tabDownload slide

4.2 Estimates of compensatory damages

The calculation of the lost wages and pension benefits is described for the three representative plaintiffs.²⁷ The calculation for Plaintiff 1 is given in Table 2. The first column lists the years when the plaintiff received lower wages and pension benefits because they lost a fair opportunity for a promotion due to discrimination. Plaintiff 1 became a lieutenant on 20 April 1997 and has zero probability of being a captain before that date. At the time Milwaukee formally required a minimum of one year of service as a lieutenant before one is eligible for promotion, however, Jones promoted four non-white-male lieutenants within one year of joining the lieutenant rank. In a nondiscriminatory setting the white males should be treated the same was as non-white males, so for the compensation calculation it will be assumed that all lieutenants became eligible for promotion to captain when they became lieutenants. The end of the compensation period, also called the cut-off time, theoretically would be the date when the actual promotion probabilities of the plaintiffs equal their hypothetical ones in a fair employment environment. Usually, this occurs sometime after the period of discriminatory practices ended. Because Plaintiff 1 retired as a lieutenant during the tenure of Chief Jones, he deserves back pay for the time until he retired as well as the additional pension benefits for as long as he lives.²⁸

Columns 2–4 in Table 2 list the differences between the base wages received by the plaintiffs and the wages they would receive, had they been promoted. In the Milwaukee Police system, lieutenants and captains receive biweekly wages, and the listed base wages also refer to the biweekly payment amount. Thus, there are roughly 26 pay periods in a year. As there are three levels within each rank, the differences between biweekly wages at each level of captain and the biweekly wage the plaintiff earned as a lieutenant are given. During their first year of being a captain, the individual is paid the salary of captain level 1, this base salary increases to level 2 in the second year and reaches level 3 at the beginning of his third year and remains there. Had discrimination not occurred, a plaintiff had a non-zero probability of being each level of captain, so three differences between the ‘hypothetical’ wage as captain and the lieutenant’s salary they earned are listed for each year. The nondiscriminatory probabilities of being at each of the three captain levels, which was calculated using formula (3) and described in Section 4.1 are given in columns 5–7. The values of the lost wages for each year in column 8 are the weighted averages of the wage differences for the three levels where the weights are the probabilities of being at each level that year.

The pension for Plaintiff 1 is decided by his wages in the last 26 pay periods before retirement as his yearly pension is set at 74% of the sum of the last 26 payments. The pension compensations are also weighted by the probabilities of being at each level of captaincy in these last 26 pay periods. Furthermore, the pensions are assumed to increase 2.5% annually as the court appointed expert adopted that figure. Finally, because the compensation damages were calculated in August 2005, an inflation rate of 4.2% is applied to the back pay due from before August 2005 and to front pay after that. The amounts of compensatory damages after accounting for inflation are given in column 9.

The calculation for Plaintiff 2 is described in Table 3. Although Plaintiff 2 became a lieutenant on 16 July 1995, the discriminatory practices did not start until 18 November 1996 when Jones became the Police Chief. Before that date, the plaintiff was not discriminated against. Therefore, his promotion probabilities in a race-neutral environment were the same as the observed promotion probabilities, which were zero before 18 November 1996.²⁹ For promotions after 18 November 1996, even though the plaintiff was not selected in real life, he would have a nonzero chance in the hypothetical fair employment scenario as predicted by the Cox model and shown in Figs 1 and 2.

Since the lieutenant had not retired by the time of compensation calculation (31 August 2005), the expert used the court suggested retirement date, which was the date he became eligible for retirement. Thus, the date 20 July 2006 was used for Plaintiff 2 and we will do the same in our calculation. The compensations in wages and pensions are calculated in the same way as that for Plaintiff 1.

Table 4 lists the calculation for Plaintiff 3, who was promoted on 6 December 2003 soon after Chief Jones stepped down. A zero difference in columns 2–4 indicates that the plaintiff was receiving a captain’s wage at that particular level and a negative difference indicates that the plaintiff was at a higher level, receiving a higher wage that year. Plaintiff 3 reached the highest level of captain in 2005 and needs no further compensation after that year.

5. Comparison of the proposed compensation calculation and the method the jury was instructed to follow

Although the appellate court remanded the case for a recalculation of damages using the lost chance doctrine, it is useful to compare the results obtained from the survival analytic approach to those calculated by the court appointed expert based on the jury’s determination of the date and probability of each plaintiff’s promotion. The calculations differ only with respect to the promotion probabilities. The same wage rates for the various levels of captain that increased over time along with the corresponding pension increases, retirement dates and the inflation rate assumed by the court’s expert are used. It will be seen that the survival analytic approach yielded noticeably different damage estimates than the original ones.

For Plaintiff 1, the survival approach estimates a total compensation amount of $118,819 in contrast to $160,381 obtained following the jury’s determinations. Similarly, the damages (total compensation) calculated using survival analysis are $200,510 and $42,128 for Plaintiffs 2 and 3, respectively, while the court expert’s calculation gave $87,612 and $24,570. Notice that the first plaintiff would be awarded less in our calculation, while the other two plaintiffs would receive more. The major source of these differences stems from the fact that the survival method yields a distribution of possible times for a plaintiff to reach the different levels of captain, while the original method determined a fixed probability for each plaintiff’s success on the date of one particular promotion. It will be seen that the two methods can lead to a much different length of time in the captain position for which a plaintiff receives the pay and pension differential.

Table 5 presents the estimated probabilities of each plaintiff reaching each of the three captain levels obtained from the survival approach alongside those made by court’s expert based on the jury’s estimated date and probability of promotion. The jury first estimated the probability a plaintiff would have received a promotion, which was either 0.80 or 0.50. Then the jury picked a date during the time period, on which they felt the plaintiff would have been promoted had the discrimination not occurred. The plaintiff would then be a level 1 captain for the first year, at level 2, the next year and level 3 the following year and afterwards. A fixed promotion date is not consistent with the opportunities each plaintiff had for advancement during the time period when discriminatory practices were in effect because a lieutenant who was not promoted on the date chosen by the jury remained eligible for later promotions. Furthermore, in a system where seniority is a legitimate consideration, a fixed probability ignores the increase in promotion probability over time. Thus, if an eligible lieutenant does not receive a particular promotion, their qualifications would have increased when the next one is made. This is reflected in the estimates obtained from the survival model but not in the procedure the jury was instructed to employ as they had to select single date for the possible promotion for each plaintiff. The survival estimates of the lost chance consider the relative qualifications of the eligible pool at the time of each promotion, i.e. the method considers the promotion process as a sequence of selections, and incorporates the positive effect of increasing seniority over time on promotion. Furthermore, other covariates can be incorporated in the basic equation (1), so that other changes, e.g. an employee’s completing a relevant degree program or special training can also be accounted for.

The wage and pension loss estimates following the method the jury was told to use depend critically on the determination of the date of each plaintiff’s likely promotion. For plaintiff 1, the jury picked a relatively early date (7 March 1999), just less than two years after the plaintiff became a lieutenant (20 April 1997), for his promotion. The survival approach estimated his promotion probability at 3% at the end of one year and 14% after two years of service. Only after serving four years on the lieutenant rank (2001), does the survival approach indicate that plaintiff 1 had a substantial (73%) chance of being a captain. By giving Plaintiff 1 an 80% chance of a promotion to captain in 1999, far larger than the 14% obtained by the survival approach, and the consequent higher wage in subsequent years, the method used by the jury leads to a much higher damage estimate than the one obtained from the survival analysis. For Plaintiff 2, who became a lieutenant on 16 July 1995, the jury picked 3 March 2002 as the promotion date. As he had already served nearly six years, the survival analysis estimated that the plaintiff already had a total probability of 96% of being promoted in a discrimination-free environment by that time. Indeed, he had a 4% chance of being a level 1 captain, a 15% chance a level 2 and 78% chance a level 3. The jury’s assignment of a 50% chance for Plaintiff 2 of being a level 1 captain in March, 2002 and 0 for previous times, led to a noticeable underestimate of the lost wages. For Plaintiff 3, who became a lieutenant on 8 October 1995, the jury picked 1 October 2000 as his hypothetical promotion date. Since many lieutenants had been promoted to captain by the time they served 5 years, the survival approach estimated he had a 10%+14%+36%=60% of being a captain. By placing 80% probability of promotion to level 1 at that day, instead of spreading his promotion probability over three captain levels, which would account for the opportunities plaintiff 3 had of an earlier promotion and reaching a level 2 or a level 3 captain by the date the jury chose, the method in the jury instructions leads to an underestimate of the lost wages.

6. Adjustment for the fixed number of promotions during the tenure of Chief Jones

The survival method predicts the ‘success’ of members of the protected group by assuming they should have been treated as a member of the majority group. The expected number of promotions they should have received, which is the sum of their probabilities of promotion by the end of the period or the time they retired, plus the number of promotions obtained by members of the unprotected group typically exceeds the number of promotions given during the period.³⁰ To illustrate the issue, consider a number, and say 20, of promotions made at one time from a set of similarly qualified eligible employees, 20 in the protected group and 80 from the majority group. Suppose all 20 promotions are awarded to the majority group, so they had a promotion rate of 25%. Had the same rate (1/4) applied to the minority employees 5 would have received a promotion. This would imply a total of 25 promotions; however, only 20 were actually made. The minority group is only entitled to their fair share of those 20, which would be one-fifth (their fraction of the qualified pool) or four. The same result is obtained by adjusting their predicted number (5) by the ratio of the actual number of promotions to the ‘theoretical total’ of 25. In order to be budget neutral, one should multiply the total number of promotions (40) by the fraction of the expected number of promotions in the ‘hypothetical’ fair employment setting that white males would have received.³¹ In the Alexander v. Milwaukee case the expected number of promotions due white males in a ‘fair environment’ was 72.3 whereas 19.1 captaincies went to non-white-males. This leads to a total of 91.4, so one needs to reduce each damage calculation by 40/91.4 = 0.438. Thus, plaintiff 1 should receive $52,043, Plaintiff 2, $87,823 and plaintiff 3, $18,452. The reason for this relatively large correction is that most of the white males had served longer as lieutenants than those of other race-gender background before Chief Jones was appointed so had they been treated like the others many would have been promoted.

7. Discussion and additional issues

Because the survival approach considers the relative qualifications of all eligible employees at the time of each promotion, it implements the lost chance approach described in Biondo.³² Furthermore, all major relevant qualifications for which data have been collected can be incorporated into the analysis. In Alexander, seniority as lieutenant was the major predictor, although we included the length of time one served prior to becoming a lieutenant and whether or not one was a detective. While being a white male did not have an effect on the retirement process in Alexander, in Appendix 3 the appropriate modifications are described when the discrimination had an effect on retirement decisions.³³ By modelling each process separately, one can incorporate the factors that affect each process in the appropriate manner. For example, number of years since the eligibility for retirement is important for the retirement process, but only the years served as lieutenant had a significant effect on promotion to captain. Furthermore, some characteristics such as possession of a special degree could increase one’s probability of promotion but might not affect their retirement decision.

The calculation of front pay in Section 4.1 required us to make an assumption about the hazard rate during the time after the data was collected in May 2003. Since the appellate court ordered the damages to be recalculated in 2007, several years of additional data could have been used. Determining the length of time front pay should be awarded deserves more attention. In Biondo,³⁴ the court noted that front pay cannot extend past the time a reasonable person would need to achieve the position in the absence of discrimination. The promotions in question in that case were exam based, so the court felt that front pay should end when the first unhindered promotional opportunity, i.e. at the time of the first post-trial exam.³⁵ In situations where seniority and other factors have a role, possibly in addition to one’s score on an exam, estimating the time when the lingering effects of discrimination have ended is more involved. The white male lieutenants in Alexander deserved about 13 more promotions, i.e. there would be a backlog of promotions due them. Since seniority was a major factor, the first time the previous discriminatory practices would not affect a plaintiff who ranked number 12 or 13 on experience. His promotion would occur after all the more senior lieutenants had been promoted. Retirements that were not influenced by the reduced promotional opportunities during the period of discrimination, would reduce this number. Because the eligible pool also changes over time, during the post-trial period it is possible that some of the less senior eligible have some special skill or qualification justifying their promotion over a more senior plaintiff. Should these individuals be promoted before the ‘backlog’ of promotions due the protected group has been made? If they should wait, they too are affected by the earlier discriminatory practices. Some guidance from the courts is needed in order to develop the appropriate statistical model to ensure the awards are consistent with the ‘lost chance’ approach.

Defendants might argue that the front pay awarded the plaintiffs in Alexander would be reduced if the chance they would receive a promotion before they retire in the future were considered. In the three illustrative examples of Section 4, this issue would only affect Plaintiff 2 and its effect would be small as he became eligible for retirement one year after the first court compensation calculations were made. Thus, in real life he did have a non-zero probability of becoming a captain after the damage calculations, which assumed he was not promoted before retiring. If such an event has a moderately high probability, additional post-trial data would be very useful as it is difficult to estimate the number of future promotions as well as the time when they would occur. As noted above, this problem is further complicated by the changing composition of the eligible pool as the addition of some highly qualified employees would diminish the probability of promotion for members of the protected group.

While the survival method used here does not resolve all the issues involved in calculating damages, it provides a sound approach to estimating back pay and a framework for estimating front pay, especially if courts require employers to continue to preserve the relevant data until the case is fully resolved. Because job-related covariates are considered, if courts adopt the proposed method, employers should have an incentive to preserve more complete employment data as job-related factors may well explain apparent disparities.³⁶

Acknowledgement

It is a pleasure to thank Ms. Wenjing Xu for her careful review of the SAS code and helpful comments and the reviewers for many useful suggestions.

Appendix

A.1 Estimation of regression coefficient in the Cox proportional hazards model

The basic idea behind the survival model can be understood by considering the risk set at the time of one promotion. If there were n individuals, who were similarly qualified, each would have had the probability or chance 1/n of receiving the promotion. Since the employees differ in seniority and other job-related qualifications, e.g. the possession of a job-related degree, one needs to modify the probabilities of each individual accordingly. This is accomplished by weighting the probabilities according to a formula, which is estimated from the data. To convey the idea to a jury, a small numerical example could be presented and the jury could then be told that the parameters of the model are estimated using all the ‘risk sets’. Charts (Fig. 2) giving the estimated probabilities for each plaintiff would summarize the results. Jurors would see how seniority increases one’s chance of promotion in a non-discriminatory environment. The following technical details are for the interested reader.

The unknown regression coefficient β values are estimated by maximizing the product of a series of conditional probabilities

The formula above is also called profile or partial likelihood. In the profile likelihood, t₀ and τ denote the earliest and latest time points in the observed dataset respectively, that is, the begin and end of the study period. The at-risk indicator if subject j is eligible for promotion hasn’t had the event or been censored by time t, and 0 if subject j is no longer at risk. And is also a dummy variable, which takes the value 1 if the event (promotion) happened to subject i at time point t, and 0 otherwise. When ⁠, subject i contributes a conditional probability, which can be interpreted as the probability of the event happens to subject i out of all eligible subjects with In our motivating example, this is the probability of selecting a particular individual and award him the promotion given all eligible lieutenants. Furthermore, everyone’s chance is weighted by their qualifications measured by After the expression for and from the Cox proportional hazards model (1) are plugged in, the profile likelihood is equal to

Notice the common baseline hazard rate is cancelled out from the numerator and denominator. The maximum partial likelihood estimator for are usually solved by numeric methods. The parameter estimates, perform well with sufficient (>30) number of events, which is 40 in the promotion model and 48 in the retirement model.

A.2 Code to estimate probabilities of being promoted over time

/************Cox proportional hazards model for********/

proc phreg data=lawdata;

model (start_promotion, end_promotion)*c_promotion(0)=whitemale detective yrsb4lt yrsincelt yrsincelt2;

yrsincelt=max((min(end_promotion,leavelt)-startP)/365.25,0);

yrsincelt2=max((min(end_promotion,leavelt)-startP)/365.25,0)**2;

run;

/****leavelt refers to the promotion or censoring time of each subject******/

/************Proc iml to obtain********/

proc iml;

 use lawdata; /*****read in the variables as vectors*****/

 read all var {start_promotion} into startP;

 read all var {end_promotion} into endP;

 read all var {c_promotion} into cP;

 read all var {whitemale detective yrsb4lt} into XP;

num=nrow(XP);

/******input the coefficient estimates from the Cox model output************/

betaP={-2.13,-0.17,-0.01};

/******Calculate the baseline hazard rate****************************/

hazP0=j(num,1,0);cumhazP=j(num,num,0);

do s=1 to num;

Y=(startP<=endP[s])#(endP[s]<=endP);

yrsincelt=((j(num,1,endP[s])><endP)-startP)/365.25;

linP=XP*betaP+0.41*yrsincelt-0.02*yrsincelt##2;

hazP0[s]=cP[s]/sum(Y#exp(linP));

end;

/******input the hypothetical coefficient for compensation purpose: force the White Male factor effects to be zero***************************************/

betaPhyp={0,-0.17,-0.01};

/******Calculate the subject-specific hazard rate*******************/

do i=1 to num;

do j=1 to num;

yrsincelt =(((endP><j(num,1,endP[i]))-j(num,1,startP[i]))<>j(num,1,0))/365;

cumhazP[i,j]=sum(hazP0#exp(0.41* yrsincelt -0.02* yrsincelt##2) *exp(xP[i,]*betaPhyp)#(endP<=endP[j])#(endP>=startP[i,]));

end;

end;

/******Calculate the subject-specific************************/

Pcaptain=j(num,num,1)-exp(-cumhazP);

/******Output the subject-specificto a dataset*******************/

outP= cP||endP||Pcaptain`;

CREATE Pcaptain FROM outP; APPEND FROM Pcaptain; CLOSE Pcaptain;

Quit iml;

A.3 Compensations when the retirement process is impacted by the discrimination

The survival approach enables us to examine both the promotion and retirement processes to assess whether the alleged discriminatory practice affected them. As seen in Table 1, the retirement process during the period was not significantly affected in Alexander. Indeed, the estimated coefficient for the effect of being a white male on retirement was only 0.18 and the p-value of 0.767 indicates that it is not significantly different from 0 and not even near the border-line of statistical significance.³⁷ Furthermore, the martingale residuals from the promotion and retirement models are independent. Thus, the actual retirement dates during the November 1996 through March 2003 period of the data set and the court suggested retirement dates for the subsequent period were taken as fixed in the compensation calculation. In other cases, however, limiting the promotional opportunities of a protected group, could lead to those employees leaving the firm or retiring earlier than they would in a fair system. When significant differences in the retirement probabilities between the protected and unprotected groups are identified as related to discrimination, a modification of the compensation calculation is required.

Plaintiff 2 will illustrate the situation. He was hired on 20 July 1981 and became eligible for retirement with pension on 21 July 2006. We estimated the average baseline hazard rate from the observed data as 1.57. Plaintiff 2 is not a detective and he had 14 years of service before becoming a lieutenant. Therefore, Plaintiff 2’s hazard rate at any time point after 21 July 2006 is (see coefficients in Table 1), where x denotes the length of time (in years) Plaintiff 2 has been eligible to retire. For purposes of compensation, the white male coefficient 0.18 is multiplied by 0, i.e. his retirement risk are the same as those of a member of the ‘majority’ group. The corresponding probability of being retired is here t is the numbers of years since Plaintiff 2 became eligible for retirement (21 July 2006). We further calculated the compensation Plaintiff 2 would receive if he retires at different possible time points subsequent to 21 July 2006. In Table 6, the retirement probabilities and corresponding compensations for t values 1 to 6, which correspond to retirement dates 20 July 2007 to 20 July 2012, are listed.

The total probability of retiring beyond six years is 0.03 and we lump them together with the 0.04 probability in year 6. The compensation amounts in the third column are calculated the same as those in Table 3. With these probabilities for the year the plaintiff would have retired in a fair employment system, one should use the weighted average of the compensations based on different possible retirement dates, i.e.

Notice this number slightly exceeds the estimated economic loss, $200 510, in Table 3.

It is worth noting that Table 6 is an approximation since Plaintiff 2 could retire any time after becoming eligible, e.g. in 0.5 or 3.3 years. Table 6 sums up retirement probabilities in a year and uses the compensation amount corresponding to retiring at end of the year. In an actual case, the compensation amount would vary with the retirement date. A more accurate calculation would consider each possible retirement date (such as the end of every pay period), calculate the probability of retiring on that date (under a nondiscriminatory scenario) using the coefficients for the Cox proportional hazards model of the retirement process given on the right side of Table 1and the corresponding compensation amount for a person who retired on that date. Then the weighted average of the compensations at each possible retirement date, where the weights are the probability of retiring on that date, is the estimated economic loss, accounting for both promotion and retirement processes.

References