Comment on "Using the Peters-Belson method in equal employment opportunity personnel evaluations" by Sinclair and Pan

The Peters-Belson (P-B) method (also called the Blinder–Oaxaca approach method in the economics literature; Blinder, 1973; Oaxaca, 1973) is a linear regression–based method that is used in salary discrimination cases. The underlying idea is to create a ‘statistical match’ to each member of the protected or unfavoured group by predicting the wage that an individual from that group would have received had that individual been a member of the majority or favoured group (see Altonji and Blank, 1999, for a review). The paper of Sinclair and Pan (2009) applies an extension of the P-B method to logistic regression (Gastwirth and Greenhouse, 1995; Nayak and Gastwirth, 1997) to personnel actions where the outcome is a binary variable, e.g. hired or not, instead of a continuous variable such as salary. As in the case for linear regression, they use logistic regression to adjust for differences in the distribution of covariates between the favoured versus unfavoured groups in the prediction of fairness of a personnel action. In logistic regression, the P-B method follows the same basic recipe as it does for linear regression. A logistic regression is fitted to the favoured group and then applied to the individuals in the unfavoured group to estimate their predicted probability of success. The difference between an individual’s observed value (0 or 1) and predicted values is their ‘shortfall’. Here, the predicted value is the probability of not being hired. The total short falls for all members of the unfavoured group estimates the number of positions (hires or promotions) they would have received had their qualifications been assessed by the same formula the employers used in evaluating the candidates from the favoured group. This average short fall is the P-B estimate of disparity. Sinclair and Pan (2009) give an excellent presentation of the properties of the P-B method and how it compares with the standard method where a regression is estimated using the combined data from both groups and an indicator variable is included in the regression model to distinguish the two groups. An estimate of the disparity for standard method comes directly from this additional term. In this Comment, I will provide some graphical illustrations of several important issues related to using the two methods for estimating disparity and augment some of the excellent statistical discussion provided by Sinclair and Pan (2009). This discussion will focus on linear regression instead of logistic regression because it is easier to illustrate the properties of both the P-B and the standard method graphically for linear regression in this situation, as these properties extend to the logistic regression setting. To focus on the basic concepts, I consider simple linear regression where there are two variables, dependent variable wages (W ) and a covariate X that is a predictor of wages such as years of experience, and there are only two groups to be compared, favoured and unfavoured. I will assume that the slopes of lines for the