Artsy Case Solution

Artsy Case1

The Artsy Corporation has been sued in the United States Federal Court on charges of employment discrimination under Title VII of the Civil Rights Act of 1964. (Artsy is an actual corporation and the data given in the case is real, but the name has been changed to protect the firm's true identity.) The litigation at contention here is a "class action" lawsuit brought on behalf of all females whom the company employed, or who had applied for work with the company, between 1979 and 1987. Artsy operates in several states, runs four quite distinct businesses, and has many different types of employees. The allegations against Artsy include issues of hiring, pay, promotions, and other "conditions of employment." In such large class action employment discrimination lawsuits statistical evidence commonly plays a central role in the determination of guilt or damages. In an interesting twist on traditional legal procedures, the precedent in these cases is that plaintiffs may make a "prima-facie" case purely in terms of circumstantial statistical evidence. If that statistical evidence is reasonably strong, the burden of proof shifts to the defendants to rebut the plaintiff's statistics with other statistical data, other statistical analyses of the same data, or by non-statistical testimony. In practice, statistical arguments often dominate the proceedings of such EEO cases. Indeed, in this case the statistical data used filled numerous computer tapes and the supporting statistical analysis comprised thousands of pages of computer printouts and reports. We work here with a small subset of the voluminous data that pertain to one of the several contested issues in one of the company's locations.

1 Peter J. Kolesar, 2001.

Specifically, the data in Table 1 relate to the pay of 256 employees on the bi-weekly payroll at one of the Artsy Company’s Pocahontas, Maine production facilities. The data include:

an identification number (IDNUMBER) that would permit us to identify the person by name or social security number,

the person's sex (SEX) where a 0 denotes female and a 1 denotes a male,

the person's job grade in 1986 (GRADE), the length of time (in years) the person had been in that job grade as

of 12/31/86 (TING), and the person's weekly pay rate as of 12/31/86 (RATE). The issue of

concern is fair pay for female employees.The plaintiff's attorneys have proposed settling the pay issues for this group of female employees for a "back pay" lump payment of 25% of their pay during the period 1979 to 1987. It is our task to examine the data in the table for evidence in favor of, or against the charges of pay discrimination against the females. To make our mission explicit suppose that we are to advise the lawyers for the Artsy Company on how to proceed. (An alternative mission would be to assist the plaintiffs.)

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Please consider the following issues:1) Overall, how different is pay by sex? Are the differences in pay

statistically significant? Is a statistical hypothesis test appropriate in an issue like this? If so, how should it be done? How could it be explained to a judge? What arguments do you anticipate the plaintiffs will be making with these data?AnswerBox plot can be used to compare the average values of pay graphically.

The box plot suggest that median pay for feamles are less compared to that of males. They are a few number of outliers also.Hypothesis testingH0: There is no signifiacnt difference in the mean pay of males and females.H1: The mean pay of females are significantly lower than that of males.Test Statistics used is independnet sample t test. The test statistic used is

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where

Rejection criteria: Reject the null hypothesis, if the calculated value of t is greater than the critical value of t at 0.05 significance level.

Details

Group Statistics

SEX N Mean Std. Deviation Std. Error Mean

RATE Female 171 832.77 158.529 12.123

Male 85 1128.18 223.338 24.224

Independent Samples Test

RATE

Equal variances

assumed

Equal variances

not assumed

Levene's Test for Equality of

Variances

F 18.431

Sig. .000

t-test for Equality of Means t -12.195 -10.905

df 254 127.396

Sig. (2-tailed) .000 .000

Mean Difference -295.405 -295.405

Std. Error Difference 24.224 27.089

95% Confidence Interval of

the Difference

Lower -343.109 -349.006

Upper -247.700 -241.803

Conclusion: Reject the null hypothesis. The sample provides enough evidence to support the claim that The mean pay of females are significantly lower than that of males.Plaintiffs may argue that this difference in average salary is due to gender discrimination in the Artsy Corporation.

2) The Artsy Company wishes to argue that a legitimate explanation of any pay rate difference is the difference in job grades by sex. (In this analysis we will tacitly assume that each person's job grade is, in fact, appropriate for them, even though the plaintiff's attorneys have charged that females have been unfairly kept in the lower grades. Other statistical data, not available here, are used in the analysis of the job placement issue.) The company’s lawyers ask, "Is there a relatively easy way to understand, to

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analyze and display the pay differences by job grade? Easy enough that it could be presented to an average jury without confusing them?” Again, try to anticipate the possible arguments of the plaintiffs. To what extent does job grade appear to explain the pay rate differences between the sexes? Propose and carry out appropriate hypothesis tests or confidence intervals to check whether the difference in pay between sexes is statistically significant within each of the grades.

AnswerHere Two ANOVA with interaction term can be adopted to answer the question. H01:There is no signifiacnt difference in the mean pay of males and females.H11: There is signifiacnt difference in the mean pay of males and females.

H01:There is no signifiacnt difference in the mean pay in differnet grades .H12: There is signifiacnt difference in the mean pay in differnet grades.H03: There is no significant interaction effect between sex and grades for pay rate.H13: There is significant interaction effect between sex and grades for pay rate.

Here we are mainly interested in the 3rd hypotheis about interaction effect. Test statistic used is F test (ANOVA)Rejection criteria: Reject the null hypothesis, if the calculated value of t is greater than the critical value of t at 0.05 significance level.

Details

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GRD SEX

Female Male

Mean Std. Deviation N Mean Std. Deviation N

dimension2

1 664.68 81.492 22 804.00 . 1

2 725.47 56.311 51

3 830.09 57.085 22 835.33 36.776 9

4 833.67 87.505 18 824.20 87.688 5

5 887.00 67.578 24 918.64 161.421 11

6 1006.20 99.692 15 1130.80 113.927 10

7 1093.24 122.897 17 1212.85 133.423 33

8 1274.00 128.693 2 1375.94 103.461 16

Total 832.77 158.529 171 1128.18 223.338 85

Tests of Between-Subjects Effects

Dependent Variable:RATE

SourceType III Sum of

Squaresdf Mean Square F Sig.

Corrected Model 1.127E7 14 804906.055 90.301 .000

Intercept 8.378E7 1 8.378E7 9398.924 .000

SEX 109218.580 1 109218.580 12.253 .001

GRD 4244536.124 7 606362.303 68.027 .000

SEX * GRD 114509.511 6 19084.919 2.141 .050

Error 2148174.881 241 8913.589

Total 2.352E8 256

Corrected Total 1.342E7 255

a. R Squared = .840 (Adjusted R Squared = .831)

Conclusion: Fails to reject the null hypothesis about the interaction . The sample provides enough evidence to support the claim that there is no discrimination in pay rate at different grades by sex. The other two hypothesis are significant and suggest that there is significant difference in pay rate with respect to gender and grade.

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The interaction plot also supports the above arguments. The model adequacy measure R2 =0.831. Thus 83.1% variability in pay rate can be explained by the two way ANOVA.

3) In the actual case, the analysis carried out in (2) above suggested to the attorneys that differences in pay rates are due, at least in part, to differences in job grades. They had heard that in another EEO case the dependence of pay rate on job grade had been investigated with regression analysis. Perform a simple linear regression of pay rate on job grade. Interpret the results fully. Is the regression significant? How much of the variability in pay does job grade account for? What light does this analysis shed on the pay fairness issue? Does it help or hurt the Artsy company?

AnswerHere scatter diagram can be adopted to graphically represent the relationship between pay rate and grade.

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The scatter diagram suggest that there is a positive correlation between grade and pay rate. The estimated regression equation is

Rate =533.937 +90.001*Grade

Coefficientsa

Model

Unstandardized Coefficients

Standardized

Coefficients

t Sig.B Std. Error Beta

1 (Constant) 533.937 15.341 34.804 .000

GRD 90.001 3.105 .876 28.989 .000

a. Dependent Variable: RATE

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Thus for a unit increase in grade, the pay rate increase by 90.001 units. The t test for the significance of regression coefficient is highly significant with t statistic =28.989, p-value =0.000. Thus we can conclude that grade have a significant effect on the rate.

Model SummaryModel

R R SquareAdjusted R

SquareStd. Error of the

Estimatedimensio 1 .876a .768 .767 110.724

a. Predictors: (Constant), GRD

The model adequacy measure R2 suggests that 76.8% variability in pay rate can be explained by the simple regression model with grade as the explanatory variable.

4) It is argued that seniority within a job grade should be taken into account since the Artsy Company's written pay policy explicitly calls for the consideration of this factor. How different are times in grade by sex? Enough to matter?AnswerHere independent sample t test can be applied.H0: There is no significant difference in the mean times in grade among males and females.H1: There is significant difference in the mean times in grade among males and females.Test Statistics used is independnet sample t test.

Group Statistics

SEX N Mean Std. Deviation Std. Error Mean

TinG Female 171 1.286 1.0602 .0811

Male 85 2.628 1.8322 .1987

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Independent Samples Test

TinG

Equal variances

assumed

Equal variances

not assumed

Levene's Test for Equality of

Variances

F 84.526

Sig. .000

t-test for Equality of Means t -7.411 -6.254

df 254 112.747

Sig. (2-tailed) .000 .000

Mean Difference -1.3423 -1.3423

Std. Error Difference .1811 .2146

95% Confidence Interval of

the Difference

Lower -1.6989 -1.7675

Upper -.9856 -.9170

Conclusion: Reject the null hypothesis. The sample provides enough evidence to support the claim that there is significant difference in the mean times in grade among males and females. Clearly mean times in grade for males is higher than that of females. The box plot also support this argument

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5) The Artsy legal team wants an analysis of the simultaneous influence of grade and time in grade on pay. Perform a multiple linear regression of pay rate versus grade and time in grade. Is the regression significant? How much of the variability in pay rates does this model explain? Will this analysis help your clients? Could the plaintiffs effectively attack it? Utilize residuals in your analysis of these issues. AnswerHere the pay rate can be analyzed using a multiple regression model with interaction term for grade and time in grade .

Coefficientsa

ModelUnstandardized Coefficients

Standardized Coefficients

t Sig.B Std. Error Beta1 (Constant) 537.622 22.454 23.944 .000

TinG 18.943 12.838 .124 1.476 .141GRD 75.867 4.659 .739 16.284 .000TinG*GRD 2.891 2.114 .138 1.368 .173

a. Dependent Variable: RATE

The estimated regression model isRate =537.622+18.943*TinG+75.867*GRD+2.891*TinG*GRDThe t test for the significance of regression coefficients suggest that only grade have significant effect on the rate. The main effect of TinG and Interaction effect of TinG*GRD has only insignificant effect on Rate.The assumptions of regression model are validated using the residual analysis. The Histogram and PP plot of residuals suggest that the errors have a normal distribution.

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The homogeneity of variance assumption is validated using the plot of residuals against the predicted value.

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Thus with the support of residual analysis, we can claim that the major factor that influence the rate is the grade. The model adequacy measure R2 indicates that 81.5% variability can be explained by the multiple regression models.

Model Summaryb

ModelR R Square

Adjusted R Square

Std. Error of the Estimate

dimensio 1 .903a .815 .813 99.296a. Predictors: (Constant), TinG*GRD, GRD, TinGb. Dependent Variable: RATE

The results from the above analysis give a solid statistical evidence to claim that there is no significant level of discrimination based on gender. All the assumptions of regression analysis are also valid. Thus it is difficult for the plaintiffs to effectively attack it .

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6) The attorneys ask: “Is it possible to do a regression analysis that simultaneously considers the effect on pay of grade, time-in–grade and sex?” If so, carry one out.Answer Here a multiple regression analysis with dummy coded variable for sex can be used to answer the question. The estimated regression model is Rate = 526.882+75.019*Grade +59.667*Sex+30.79*TinG

Coefficientsa

Model

Unstandardized Coefficients

Standardized

Coefficients

t Sig.B Std. Error Beta

1 (Constant) 526.882 14.131 37.285 .000

GRD 75.019 3.325 .730 22.562 .000

SEX 59.667 15.980 .123 3.734 .000

TinG 30.790 4.562 .202 6.749 .000

a. Dependent Variable: RATE

The regression coefficients can be interpreted as For a unit increase in grade, the pay increase by 75.019 units .For males, the pay rate is 59.667 units higher than that of females.For a unit increase in time in grade, the pay increase by 30.790 units.The t test for the significance of regression coefficients are highly significant with p values less than 0.05. Thus we can conclude that all explanatory variables have significant effect on pay rate. Here model adequacy measure R2=0.823 . Thus 82.3% variability pay rate can be explained by the regression model.

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7) Organize your analyses and conclusions in a brief report summarizing your findings for your client, the Artsy Corporation. Be complete but succinct. Be sure to advise them on the issue of the settlement. Please be as forceful as you can be in arguing "the Artsy Case" without misusing the data or statistical theory. Apprise your client of the risks they face by developing the most forceful counter argument that you believe the female plaintiffs could fairly make.

Conclusion: Statistical techniques are effectively applied here to establish that there is no discrimination in the pay rate among males and females. Inappropriate use independent sample t test suggest that there is significant difference in the pay rate with respect to gender. But regression analysis and two way ANOVA are used to disprove this argument. These statistical techniques indicates that Pay rate is determined by the grade and not the other factors. The assumption of regression analysis is also validated using the residual plot. Thus it is difficult for the female plaintiffs to raise a valid counter argument against the conclusions.

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Table 1: Artsy DataID RATE TinG SEX GRD ID RATE TinG SEX GRD ID RATE TinG SEX GRD ID RATE TinG SEX GRD1 865 1.5 0 2 65 963 2.5 0 7 129 788 1.2 0 2 193 713 1.5 0 22 820 0.5 0 4 66 747 2.5 0 2 130 808 0.5 0 2 194 952 1.5 0 63 675 1.5 0 2 67 916 0.5 0 6 131 1338 5.0 1 7 195 1376 5.0 1 64 1494 1.5 1 8 68 952 1.5 0 3 132 808 2.2 0 4 196 630 4.5 0 15 730 0.5 1 4 69 831 0.5 1 3 133 1230 3.5 1 7 197 901 1.5 0 56 710 1.5 0 2 70 854 1.5 0 3 134 1024 0.5 0 7 198 579 0.8 0 17 692 1.1 0 2 71 660 0.9 0 2 135 588 1.1 0 1 199 952 1.5 0 58 723 0.5 0 3 72 1174 0.5 1 6 136 906 0.8 0 5 200 1125 0.5 1 69 727 0.8 0 2 73 1057 3.5 1 6 137 1552 5.0 1 8 201 663 0.5 0 2

10 692 1.5 0 2 74 1230 1.5 1 7 138 1177 5.0 1 5 202 1390 5.0 1 711 1142 0.5 1 6 75 628 1.0 0 1 139 802 1.2 0 2 203 1038 0.7 0 712 1413 0.5 1 8 76 762 1.6 0 2 140 612 0.5 0 1 204 720 0.2 0 213 795 1.5 0 3 77 885 0.5 0 5 141 1002 1.5 0 6 205 960 4.5 1 714 825 1.5 1 3 78 865 0.1 0 4 142 932 1.5 0 4 206 756 2.3 0 215 867 0.5 0 4 79 1177 3.5 1 5 143 1191 1.5 1 7 207 597 0.9 0 116 779 0.5 0 3 80 825 1.5 0 3 144 730 0.1 0 4 208 623 0.5 0 217 1057 0.5 1 5 81 848 1.5 0 3 145 1365 0.5 0 8 209 756 2.5 0 218 706 1.5 0 1 82 682 0.8 0 1 146 810 1.5 0 3 210 804 3.5 1 119 1052 0.5 1 7 83 1240 5.0 1 7 147 856 1.5 0 1 211 1158 5.0 1 720 735 0.5 0 2 84 1519 5.0 1 7 148 1269 3.5 1 7 212 1148 2.5 0 721 780 1.5 0 2 85 730 0.1 0 4 149 624 0.8 0 2 213 1050 0.5 0 722 1255 0.5 1 7 86 1500 3.3 1 8 150 865 0.5 0 4 214 858 3.5 0 523 1264 5.0 1 7 87 806 0.4 1 5 151 698 0.9 0 1 215 1004 2.5 1 624 692 0.7 0 2 88 813 1.5 0 2 152 1238 2.0 0 7 216 1390 5.0 0 725 946 2.5 0 6 89 801 0.5 1 3 153 990 2.5 0 6 217 894 1.5 0 526 1410 5.0 1 8 90 894 0.1 0 4 154 818 1.5 0 1 218 952 0.8 1 727 747 1.5 0 2 91 825 0.5 0 4 155 687 2.5 0 2 219 1200 0.5 1 728 789 2.5 0 2 92 893 1.5 0 5 156 1067 1.5 0 7 220 842 0.5 0 329 1110 1.5 1 7 93 687 2.5 0 2 157 730 0.5 1 4 221 1131 2.5 1 730 923 0.5 0 5 94 796 0.5 0 3 158 1350 1.5 1 8 222 990 2.5 1 531 692 0.2 0 2 95 702 1.2 0 2 159 1385 5.0 1 8 223 1073 3.5 0 732 648 1.3 0 1 96 788 0.5 0 1 160 867 1.5 0 5 224 690 0.7 0 233 1067 1.5 0 7 97 1110 1.5 1 7 161 1128 3.7 1 7 225 961 5.0 0 534 870 2.5 1 5 98 779 4.5 0 1 162 1082 5.0 0 6 226 762 0.8 0 235 882 2.5 0 5 99 795 2.5 0 2 163 1396 5.0 1 8 227 1419 0.5 1 836 885 1.5 1 3 100 780 0.1 0 2 164 831 0.5 0 5 228 1258 5.0 1 737 909 0.5 1 3 101 819 2.5 1 3 165 692 0.0 0 2 229 900 1.5 0 338 1035 0.5 0 7 102 1229 4.5 1 8 166 1131 4.5 1 7 230 804 1.5 1 339 658 2.2 0 1 103 810 0.5 0 5 167 837 0.1 0 3 231 1096 0.4 0 640 860 1.5 1 4 104 630 0.2 0 1 168 735 0.5 0 2 232 932 2.5 0 541 616 0.8 0 2 105 730 0.5 0 4 169 1073 1.5 0 5 233 819 0.5 1 342 924 2.5 0 6 106 1065 0.5 1 7 170 710 0.7 0 2 234 1056 2.5 0 743 929 1.5 1 5 107 816 1.5 0 3 171 923 0.5 0 4 235 764 0.5 0 344 762 0.6 0 3 108 1172 5.0 1 7 172 1200 0.9 0 6 236 1079 1.5 0 645 1223 2.5 1 7 109 723 1.3 0 2 173 894 1.5 1 4 237 690 0.5 0 246 907 3.5 1 4 110 958 3.0 1 6 174 804 1.5 0 2 238 1183 0.5 1 647 1119 4.5 1 7 111 1275 5.0 1 8 175 590 0.8 0 1 239 837 0.2 0 548 1050 0.5 0 4 112 894 0.5 0 6 176 914 0.5 0 6 240 929 0.5 0 549 1500 4.5 1 7 113 602 1.0 0 1 177 588 1.0 0 1 241 835 1.5 0 550 740 0.7 1 5 114 1004 2.5 0 7 178 780 0.5 0 5 242 886 1.5 0 351 1183 0.5 0 8 115 1135 0.5 1 6 179 623 0.3 0 1 243 806 0.5 1 552 990 2.5 0 5 116 840 1.5 0 3 180 717 0.7 0 1 244 929 0.5 0 653 1368 5.0 1 7 117 756 2.5 0 2 181 762 0.7 0 4 245 1070 2.5 1 754 1385 0.5 1 8 118 770 1.5 0 2 182 1154 2.5 1 8 246 730 1.0 0 455 834 0.5 0 3 119 750 0.5 0 2 183 779 1.5 0 2 247 762 0.5 0 456 1263 5.0 1 7 120 687 2.5 0 2 184 771 0.5 0 3 248 1053 0.5 0 757 1154 5.0 1 6 121 900 0.5 0 4 185 1350 1.5 1 8 249 1188 3.3 0 658 1263 5.0 1 7 122 780 1.0 0 2 186 1360 0.5 0 7 250 981 1.3 0 759 814 1.5 0 5 123 1428 5.0 1 7 187 616 0.8 0 2 251 951 5.0 0 360 825 1.5 1 3 124 1275 5.0 1 8 188 1428 5.0 1 8 252 606 0.5 0 161 840 5.0 0 3 125 912 0.5 0 5 189 813 1.5 1 5 253 806 0.5 0 562 692 0.3 0 2 126 1174 0.5 1 7 190 740 0.5 1 5 254 720 1.2 0 263 837 0.9 0 3 127 710 1.8 0 2 191 635 0.2 0 2 255 981 0.5 0 664 813 0.5 0 4 128 1263 4.5 1 7 192 817 0.5 0 5 256 1038 2.5 0 7

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