Multivariate Techniques
- 1. 1 Terry Chaney PA 5033 Multivariate Techniques 1. Terminating employees based solely upon an individual’s age is illegal under the Age Discrimination Act. However, it is often difficult to separate the effects of an employee’s job performance from that person’s age when dealing with age discrimination cases. In order to determine if employees have been wrongfully terminated based on age, a number of binary logistic regressions were done to test if age alone factored into the firing decisions. A limited dependent variable model was used because the dependent variable (job termination) is a dummy variable with only two values (0 = not fired, 1 = fired). This model was initially used to test job termination as a function of age; the same model was used to regress job termination on age and performance. N = 319; 53 were fired (P = .166) Fired = f(+age): ageFired 146.958.8 )1( PP = .146(.166).834 = .020 = 2% R2 p = .85 Wald = 37.998 Fired = f(+age, -perf): perfageFired 635.130.843.5 R2 p = .868 Wald = 28.679; 6.432 Age = )1( PP = .130(.166).834 = .0179 = 1.8% Perf = )1( PP = -.635(.166).834 = -.088 = -8.8% Omitted variable bias occurs when there is a relevant independent variable omitted from the specification of the equation. This will cause bias in the estimates of the coefficients on the variables that are included in the equation. Omitted variable bias is shown in the difference between our first two limited dependent variable models. The first model is simply a regression of termination as a function of age. The second model adds performance as an additional
- 2. 2 explanatory variable. The addition of performance results in a reduced value for the coefficient on age (from a 2% change per year to a 1.8% change). This is because age and performance share explanatory “space” and without performance in the equation, the coefficient on age absorbs some of the impact of performance. By including performance in the second model we are eliminating some of the omitted variable bias in the original equation. Age and performance are also statistically linked. In a regression of performance as a function of age, the relationship between age and performance was statistically significant at the 1% level. The high F-statistic suggests that we are creating a good instrument. ageperf 055.552.6 adjusted-R2 = .112 t = -6.414 F-statistic = 41.14 This result means that with a model of termination as a function of age and performance it will be difficult to separate out the effects of increasing age and performance. In order to separate the effects of age and performance on termination, we need to “cleanse” the performance variable of the effects of age. Using the residuals of the above equation we were able to create a new variable representing performance without the effect of age, or real performance. Using this new variable we estimated termination as a function of age and real performance. We then used a binomial logit model in order to determine the effects of age and real performance. We chose this model because a linear probability model in OLS would result in estimates outside the 0 and 1 bounds of the expected values, producing nonsensical results. The binomial logit model will provide results within the bounds of the probability interval. The binomial logit model also addresses the problem of heteroskedasticity in the linear probability model. The most significant disadvantages to the binomial logit model is that it requires a large sample size (the 319 observations in our sample may not be enough) and also requires high representation of both groups (those fired and those not fired) in the sample. Weighing these
- 3. 3 issues, we chose a two-stage binomial logit model as our best model to explain the effect of age and real performance on the termination of an employee. The model we used is as follows: Fired = f(age, real_perf) 2. Running the model above yields the following results: Fired = f(age, real_perf) = -10.002 + 0.165 age – 0.635 real_perf SE: 0.026 0.250 N = 319 R2 p = 0.868 Both age and real performance (performance cleansed of the effects of age) are highly significant in predicting whether an individual is fired. Calculating the change in probability of each (as *P(1-P)) yields an increase in the probability of being fired of 23% for each additional ten years of age, and a decrease of 8.8% for each one-unit increase in worker performance. Overall, this model accurately predicts whether a worker will be fired 86.8% of the time, compared to a random sample accuracy of 83.4% (that is, the prediction that no individual will be fired, giving no consideration to age or performance, is correct 83.4% of the time). The model, however, continues to do a poor job of predicting correctly for those that were fired and does a much better job of predicting those that were not fired. Based on the strength of these results, it is clear that both age and performance are important determiners of whether an individual is fired. What is not clear is whether discrimination is occurring. The improvement in prediction of only 3.5 percentage points suggests that there may still be other considerations that are not captured here – for example, union membership. In addition, this model accurately predicts firings (as opposed to all outcomes) only 30% of the time – 37 of the 53 terminations were not predicted by the model. On the other hand, the fact that age is a highly significant predictor even after controlling for workers’ performance strongly suggests that something shady is going on.
- 4. 4 The law specifies that age discrimination applies to workers over 40. Substituting the age variable for a dummy indicating whether a worker is over age 40 does not provide very satisfying results – neither performance nor being over 40 is statistically significant, and the model’s predictions are no better than the random sample. However, changing the variable to represent whether a worker is over 50 gives age much more significance and reduces the significance of performance to below the 95% level of certainty. While it is tempting to believe that this means workers over 50 are being discriminated against, it is important to note that this model also predicts firings no better than a random sample. Based solely on the predictions of the model, then, it is difficult to be certain whether age discrimination has occurred. The 86.8% accuracy of the best model considered is difficult, on its own, to draw definite conclusions from. However, there are a few additional considerations that push us towards the determination that discrimination has occurred. The first of these is the fact that no workers under age 40 were terminated. This would not be a concern if the young workers were all high performers. However, 54 of the 66 workers younger than 40 had performance scores of 5.1 or below – 5.1 being the highest score of any worker who was fired. If performance is indeed a negative function of age, these young workers’ performance could only be expected to decline in the future – but they were kept on despite their poor performance. This fact, in combination with the predictions of the model that increasing age plays a significant role in determining firings, leads us to conclude that age discrimination was indeed a factor in termination decisions. While workers’ performance was also clearly an important factor, it appears that once the poorly-performing workers were identified, the preference was to dismiss the older workers and retain younger ones.
- 5. 5 Appendix – StatisticalOutput 1. Binary Logistic Regressionof Job Termination on Age 2. Binary Logistic Regressionof Job Termination on Age and Performance 3. OLS Regression: Regressionof performance on age Classification Tablea 264 2 99.2 46 7 13.2 85.0 Observed .00 1.00 fired Overall Percentage Step 1 .00 1.00 fired Percentage Correct Predicted The cut value is .500a. Variables in the Equation .146 .024 37.998 1 .000 1.158 -8.958 1.263 50.336 1 .000 .000 age Constant Step 1 a B S.E. Wald df Sig. Exp(B) Variable(s) entered on step 1: age.a. Classification Tablea 261 5 98.1 37 16 30.2 86.8 Observed .00 1.00 fired Overall Percentage Step 1 .00 1.00 fired Percentage Correct Predicted The cut value is .500a. Variables in the Equation .130 .024 28.679 1 .000 1.139 -.635 .250 6.432 1 .011 .530 -5.843 1.653 12.492 1 .000 .003 age perf Constant Step 1 a B S.E. Wald df Sig. Exp(B) Variable(s) entered on step 1: age, perf.a. Model Summ ary .339a .115 .112 1.31881 Model 1 R R Square Adjusted R Square Std. Error of the Estimate Predictors: (Constant), agea.
- 6. 6 4. Binary Logistic Regressionof Job Termination on age and real performance 5. Binary Logistic Regressionof Job Termination on age over 40 and real performance ANOVAb 71.559 1 71.559 41.143 .000a 551.349 317 1.739 622.908 318 Regression Residual Total Model 1 Sum of Squares df Mean Square F Sig. Predictors: (Constant), agea. Dependent Variable: perfb. Coefficientsa 6.552 .411 15.924 .000 -.055 .009 -.339 -6.414 .000 (Constant) age Model 1 B Std. Error Unstandardized Coefficients Beta Standardized Coefficients t Sig. Dependent Variable: perfa. Classification Tablea 261 5 98.1 37 16 30.2 86.8 Observed .00 1.00 fired Overall Percentage Step 1 .00 1.00 fired Percentage Correct Predicted The cut value is .500a. Variables in the Equation .165 .026 39.787 1 .000 1.180 -.635 .250 6.432 1 .011 .530 -10.002 1.404 50.714 1 .000 .000 age RES_1 Constant Step 1 a B S.E. Wald df Sig. Exp(B) Variable(s) entered on step 1: age, RES_1.a. Classification Tablea 266 0 100.0 53 0 .0 83.4 Observed .00 1.00 fired Overall Percentage Step 1 .00 1.00 fired Percentage Correct Predicted The cut value is .500a.
- 7. 7 6. Binary Logistic Regressionof Job Termination on age over 50 and real performance Variables in the Equation -.238 .193 1.527 1 .217 .788 19.871 4887.791 .000 1 .997 4.3E+08 -21.227 4887.791 .000 1 .997 .000 RES_1 over40 Constant Step 1 a B S.E. Wald df Sig. Exp(B) Variable(s) entered on step 1: RES_1, over40.a. Classification Tablea 266 0 100.0 53 0 .0 83.4 Observed .00 1.00 fired Overall Percentage Step 1 .00 1.00 fired Percentage Correct Predicted The cut value is .500a. Variables in the Equation -.376 .214 3.103 1 .078 .686 1.773 .350 25.668 1 .000 5.888 -2.597 .292 79.337 1 .000 .074 RES_1 over50 Constant Step 1 a B S.E. Wald df Sig. Exp(B) Variable(s) entered on step 1: RES_1, over50.a.