1 Revisiting salary discrimination @ Acme Bank: Background A bank is facing a discrimination suit in which it is accused of paying its female employees

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Revisiting salary discrimination @ Revisiting salary discrimination @ Acme Bank: BackgroundAcme Bank: Background

• A bank is facing a discrimination suit in which it is accused of paying its female employees less than their male employees

• The bank had 208 employees in 1995 --140 females and 68 males

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• Raw Data: First 6 records of the data are shown below

Females MalesAvg. Salary 37.21 45.51

Avg. Experience 9.58 9.87Avg. Prior Experience 2.55 2.01

Median Educ Level 3 5Median Job Grade 2 4

Avg. Age 40.89 39.38

• Summary statistics by Gender

3456789

A B C D E F G HEmployee Gender Salary YrsExp YrsPrior JobGrade EducLev Age

1 Male 32 3 1 1 3 262 Female 39.1 14 1 1 1 383 Female 33.2 12 0 1 1 354 Female 30.6 8 7 1 2 405 Male 29 3 0 1 3 286 Female 30.5 3 0 1 3 24

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Simple regression: Salary vs. GenderSimple regression: Salary vs. Gender

Regression StatisticsMultiple R 0.35R Square 0.12Adjusted R Square 0.12Standard Error 10.58Observations 208

ANOVAdf SS MS F Significance F

Regression 1 3149.634 3149.6 28.115 2.935E-07Residual 206 23077.47 112.03Total 207 26227.11

CoefficientsStandard

Error t Stat P-value Lower 95%Upper 95%

Lower 95.0%

Upper 95.0%

Intercept 45.50544 1.28353 35.453 1E-89 42.9749 48.036 42.975 48.036Gender -8.29551 1.564493 -5.302 3E-07 -11.379986 -5.211 -11.38 -5.211

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Difference in salariesDifference in salaries• Female salaries are lower than male salaries on average by

$8,295 (coefficient of Gender; Gender=0 for male and Gender=1 for female)

• Although r-square is low (12%), most people would agree that the difference in salaries is statistically significant based on the low p-value

• Simple regression only looks at things at the gross or surface level

• Multiple regression helps us net out effects of other important variables, such as prior experience, job grade, educational background etc. We need to include additional variables (information) in the analysis to see if salaries are different

• We expand the model by including prior experience (YrsPrior) in the banking industry, and years at this bank (YrsExp)

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MR1: Salary vs Gender, Years at the current MR1: Salary vs Gender, Years at the current bank (YrsExpr), and prior experience in bank (YrsExpr), and prior experience in

banking (YrsPrior)banking (YrsPrior)Regression Statistics

Multiple R 0.701615R Square 0.492264Adjusted R Square 0.484798Standard Error 8.079397Observations 208



CoefficientsStandard Error t Stat P-value Lower 95% Upper 95%Intercept 35.49166 1.341022 26.46614 6.8E-68 32.84762014 38.13570179Gender -8.080212 1.19817 -6.743794 1.56E-10 -10.4425983 -5.71782594YrsPrior 0.131338 0.180923 0.725933 0.468712 -0.22538094 0.488056769YrsExp 0.987994 0.080928 12.20828 4.31E-26 0.828430755 1.147556606

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QuestionsQuestions1. Is the MR1 better than the simple regression

model (ensure that the model is significant, and compare r-squared values)?

2. Interpret the coefficient of Gender in MR1

3. Interpret the coefficients of YrsExp and YrsPrior

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Further expansionFurther expansion

• Next, we add Job Grade (see slide #2) to the MR1 model

• Since Job Grade is a categorical variable with 6 levels, 5 dummy variables are created to represent these levels (Job_2 through Job_6 )

• Job_2 was set to 1 if Job Grade was 2, and zero other wise – similar approach was used in coding Job_3 through Job_6

4. Can you tell which Job Grade is represented by the default setting where Job_2, Job_3, Job_4, Job_5 and Job_6 equal zero? Please ask me to elaborate if you are not clear on this.

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MR2MR2Regression Statistics




CoefficientsStandard Error t Stat P-value Lower 95% Upper 95%Intercept 30.22959 1.173029 25.77055 2.54E-65 27.91643109 32.54275608Gender -1.962205 1.005084 -1.95228 0.052308 -3.94418833 0.019777785YrsPrior 0.148943 0.131975 1.12857 0.260438 -0.11130602 0.409192156YrsExp 0.408408 0.07776 5.25218 3.85E-07 0.255069005 0.561746428Job_2 2.575278 1.182071 2.178616 0.030535 0.244285407 4.906270924Job_3 6.294688 1.170322 5.378596 2.1E-07 3.986863271 8.602512557Job_4 10.47453 1.367644 7.658809 8.01E-13 7.777590918 13.17146325Job_5 16.01136 1.559333 10.26808 4.04E-20 12.93642212 19.08629908Job_6 27.64718 2.417629 11.43566 1.33E-23 22.87971662 32.41463752

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QuestionsQuestions5. Is this model better than the first two models (check

that the model is significant; please also look at r-squared)?

6. Are female salaries significantly lower than male salaries at the 10% level?

7. Are you clear on the interpretation of coefficients for the Job Grade dummy variables? Can you determine the difference in salaries for a person moving from Job Grade 2 to 3, all else equal?

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Full ModelFull Model• More information was added to further

expand the model

• Four dummy variables (Ed_2 through Ed_5) were added to represent the 5 education levels

• Age for each employee was included

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MR3MR3SUMMARY OUTPUT




Coefficients Std Error t Stat P-value Lower 95% Upper 95%Intercept 29.1566 2.5491 11.4380 0.0000 24.1291 34.1842Gender -1.7468 1.0081 -1.7328 0.0847 -3.7349 0.2414YrsExp 0.4919 0.1002 4.9067 0.0000 0.2942 0.6896YrsPrior 0.2435 0.1422 1.7128 0.0884 -0.0369 0.5239Age -0.0020 0.0592 -0.0345 0.9725 -0.1187 0.1146Job_2 2.0095 1.2087 1.6625 0.0980 -0.3744 4.3934Job_3 5.1901 1.2950 4.0079 0.0001 2.6361 7.7442Job_4 8.7694 1.5337 5.7177 0.0000 5.7445 11.7944Job_5 13.3194 1.9199 6.9377 0.0000 9.5329 17.1059Job_6 24.1773 2.8703 8.4233 0.0000 18.5163 29.8382Ed_2 -1.1320 1.4210 -0.7966 0.4266 -3.9346 1.6706Ed_3 0.9064 1.3877 0.6532 0.5144 -1.8306 3.6434Ed_4 0.6349 2.4645 0.2576 0.7970 -4.2258 5.4956Ed_5 2.9852 1.6603 1.7980 0.0737 -0.2894 6.2598

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QuestionsQuestions8. Is the most recent model in MR3 (slide #11)

better than the MR2 (slide #8)?

9. Are female salaries significantly lower than male salaries at 10%?

10.Using p-values, identify variables that do not appear to contribute significantly to MR3

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Refining the regression Refining the regression model – removing variablesmodel – removing variables

• Age does not appear to contribute significantly to the MR3 model

• Most education levels (with the exception of Ed_5) also do not appear to contribute significantly to the MR3 model

• Thus, we should exclude Age and Education Level variables from further analysis (this gives us the same model as MR2, but it is shown again as MR4 on the next slide for easy reference)

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MR4 (Same as MR2)MR4 (Same as MR2)Regression Statistics




CoefficientsStandard Error t Stat P-value Lower 95% Upper 95%Intercept 30.22959 1.173029 25.77055 2.54E-65 27.91643109 32.54275608Gender -1.962205 1.005084 -1.95228 0.052308 -3.94418833 0.019777785YrsPrior 0.148943 0.131975 1.12857 0.260438 -0.11130602 0.409192156YrsExp 0.408408 0.07776 5.25218 3.85E-07 0.255069005 0.561746428Job_2 2.575278 1.182071 2.178616 0.030535 0.244285407 4.906270924Job_3 6.294688 1.170322 5.378596 2.1E-07 3.986863271 8.602512557Job_4 10.47453 1.367644 7.658809 8.01E-13 7.777590918 13.17146325Job_5 16.01136 1.559333 10.26808 4.04E-20 12.93642212 19.08629908Job_6 27.64718 2.417629 11.43566 1.33E-23 22.87971662 32.41463752

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Are further refinements Are further refinements possible?possible?

• Seems like YrsPrior is not adding to MR4 – let’s do the analysis again by excluding this variable

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MR5MR5SUMMARY OUTPUT




Coefficients Std Error t Stat P-value Lower 95% Upper 95%Intercept 30.5319 1.1428 26.7161 0.0000 28.2783 32.7854Gender -1.9265 1.0053 -1.9164 0.0567 -3.9088 0.0558YrsExp 0.4062 0.0778 5.2221 0.0000 0.2528 0.5596Job_2 2.6677 1.1800 2.2607 0.0249 0.3408 4.9946Job_3 6.4204 1.1658 5.5072 0.0000 4.1215 8.7192Job_4 10.5247 1.3679 7.6943 0.0000 7.8274 13.2219Job_5 16.1153 1.5577 10.3457 0.0000 13.0437 19.1869Job_6 27.4503 2.4130 11.3761 0.0000 22.6922 32.2084

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Gender and experience – is there an interaction?Gender and experience – is there an interaction?

Males FemalesYrsExp 0.8877 0.2354

JobGrade 0.8210 0.7147

• The above table shows correlations between Salary and the Yrs of Experience and Job Grade for Males and Females

• The correlation between Yrs of Experience and Salary appears to be much stronger for males than females – in other words, male employees are moving up the salary ladder faster than female employees – the analyst felt this may be the source of salary discrimination at Acme Bank

• Thus, there appears to be an interaction: The effect that Yrs of Experience has on Salary depends on whether the employee is male or female

• Regression analysis can be improved by adding an interaction term in the model – the method is described in the next slide

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Variables for the modelVariables for the model

• To capture the interaction between Yrs Experience and Gender, a new variable called Gen*YrsExp was created by multiplying the value of Gender (0 or 1) by the employee’s experience (YrsExp) at Acme

• Thus the MR model with interaction is: Salary against Gender, YrsExp, and Gen*YrsExp

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Regression model with Regression model with interactioninteraction




Coefficients Std Error t Stat P-value Lower 95% Upper 95%Intercept 30.43 1.216574 25.01288 4.6E-64 28.03135469 32.82870079Gender 4.10 1.665842 2.460168 0.014719 0.813774918 7.382728839YrsExp 1.53 0.09046 16.88875 1.3E-40 1.349404533 1.706118905Gen*YrsExp -1.25 0.136676 -9.129628 6.83E-17 -1.51727662 -0.97832011

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QuestionsQuestions11.What is the regression equation for

male employees?

12.What is the regression equation for female employees?

13.How do we interpret the regression coefficients in the slide #19?

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QuestionsQuestions14. According to the regression model, what is the

salary for a male (Gender=0) who has 1 year of experience at this bank?

15. What is the predicted salary for a male (Gender=0) who has 6 years of experience at this bank?

16. Answer the above questions for a female (Gender=1) at this bank

Contd.

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QuestionsQuestions17.Looking at your answers, can you tell

if there is an interaction?

18.Can you explain the interaction?

19.Is the interaction significant (=10%)?

Contd.

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How many interaction variables?How many interaction variables?• Suppose we want to test the interaction between

Gender and Job-Grade

20. How many interaction variables would we need?

21. Let’s add the interaction variable to our best MR model so far (MR5 on slide #16) to see if further improvements are possible …. The new model (MR6) is shown in the following slide – do you think the model with interaction is better (is MR6 better than MR5 -- why)?

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MR6: Full model with interactionMR6: Full model with interactionRegression Statistics




Coefficients Std Error t Stat P-value Lower 95% Upper 95%Intercept 26.1042 1.1054 23.6143 0.0000 23.9243 28.2841Gender 6.0633 1.2663 4.7881 0.0000 3.5662 8.5605YrsExp 1.0709 0.1020 10.4975 0.0000 0.8697 1.2720Gen*YrsExp -1.0211 0.1187 -8.6001 0.0000 -1.2552 -0.7869Job_2 2.5965 1.0101 2.5705 0.0109 0.6046 4.5884Job_3 6.2214 0.9982 6.2328 0.0000 4.2530 8.1898Job_4 11.0720 1.1726 9.4423 0.0000 8.7597 13.3842Job_5 14.9466 1.3402 11.1521 0.0000 12.3037 17.5895Job_6 17.0974 2.3907 7.1517 0.0000 12.3831 21.8117

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But your data has an outlier!But your data has an outlier!• Before we accept that there are significant

differences between male and female salaries, we’d like to address the issue of outliers

• Specifically, there is a female employee in the highest job grade (Job Grade 6), has 33 years of experience at Acme, but whose salary is only $30,000 – this could a major source of discrimination at Acme Bank

• To see if this is the case, we remove this employee from our data and redo the regression analysis

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MR7: Regression with outlier removedMR7: Regression with outlier removedRegression Statistics



Regression 8 21781 2722.625 124.0064 7.74677E-73Residual 198 4347.191 21.95551Total 206 26128.19

Coefficients Std Error t Stat P-value Lower 95% Upper 95%Intercept 26.7103 1.0440 25.5840 0.0000 24.6515 28.7691Gender 4.3531 1.2321 3.5331 0.0005 1.9234 6.7828YrsExp 0.8977 0.1012 8.8675 0.0000 0.6980 1.0973Gen*YrsExp -0.7206 0.1252 -5.7578 0.0000 -0.9674 -0.4738Job_2 2.7179 0.9485 2.8655 0.0046 0.8474 4.5883Job_3 6.2572 0.9370 6.6777 0.0000 4.4093 8.1050Job_4 10.9838 1.1008 9.9777 0.0000 8.8129 13.1547Job_5 15.4645 1.2619 12.2547 0.0000 12.9759 17.9530Job_6 22.3234 2.4530 9.1004 0.0000 17.4861 27.1608

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QuestionsQuestions22.Does MR7 support the argument

that male and female salaries are different? Does this make the case stronger or weaker for those accusing Acme of gender discrimination?

23.Is MR7 better than MR6?

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Outcome of the Case….Outcome of the Case….

•So what was the outcome of the case .. Any guesses?

Documents

1 Revisiting salary discrimination @ Acme Bank: Background A bank is facing a discrimination suit in which it is accused of paying its female employees