For Table 2.6, we fitted a logistic model, treating death penalty as the respons

Question

For Table 2.6, we fitted a logistic model, treating death penalty as the response (1- yes) and defendant's race (1 = white) and victims, race (1 = white) as indicator predictors. Table 5.16 shows results 5.15 a. Interpret parameter estimates. Which group is most likely to have the yes re- sponse? Find the estimated probability in that case. Interpret 95% confidence intervals for conditional odds ratios c. Test the effect of defendant's race, controlling for victims' race, using a (i) Wald test and (ii) likelihood-ratio test. Interpret. d. Test the goodness of fit of the model. Interpret Table 5.16 Software Output (Based on SAS) for Exercise 5.15 on the Death Penalty Criteria For Assessing Goodness of Fit Criterion Deviance Pearson Chi-Square1 Log Likelihood DF Value 0.3798 0.1978 -209.4783 Standard Error 0.5069 0.3671 0.6006 Likelihood Ratio 95t Conf Limits Chi-Square Parameter Estimate Intercept def vic -3.5961 -0.8678 2.4044 50.33 5.59 16.03 -2.7349 -1.56330.1140 -4.7754 1.3068 3.7175 LR Statistics Chi-Square Pr>Chisq Source DF def vic 0.0251 5.01 20.35

Explanation / Answer

a) There are 2 parameter estimates:

Def = -0.8678 indicates that when the defendant is White then there is a decrease in the Yes response than when the defendant is Black. This indicates that Black defendants are more prone to receive the death penalty.

The second parameter estimate is 2.4044, which is positive and hence indicates that if the victim is white there is more prob of yes response.

So combined together it indicates that a white victim and Black defendant will produce the highest number of Yes response. This is in ine with the data as that is the group which has the highest prob. of Yes response = 22.9%.

The estimated prob. = exp(-3.5961+2.4044)/1+exp(-3.5961+2.4044) = 0.23 i.e. 23% which is very close to the actual observed probability.

b) The 95% CI indicates that if we do this test with a different data each time and construct the CI each time then the true value of the odds ratios will lie within these ranges 95% of the time. Hence assuming that the sample at hand is the most likely sample, we can claim that the mean odds ratio will lie in the 95% CI almost 95% of the time..

So the coefficient of yes response for Def will be always negative i.e. Black defendant has more chance of Death penalty. The 95% CI for Victim is always positive so it indicates that the White victim is always more prone to give a verdict of death penalty.

c) In the LR Test for Defendant, the p-value is 0.0251 < 0.05 the level of significance. So the defendants race alone (i.e. even after controlling for the victim race) has an impact on the prob. of yes response. The Wald Chi-sqaure test statistic is 5.59. The critical value of Chi-sqaure(1) distribution is 3.84. So by Wald score test also, we find that Defendant race is an important variable to determine the prob of Yes response.

d) Both Deviance and pearson Chi-sqaure take value which are much smallere than the critical value of the chi-sqaure (1) distribution. which indicates that the model fit is good. Since each of Deviance and Chi-sq statistic measures the gap between observed and expectecd freq so the null hyp is that the factors have no effect. If the Deviance and chi-sq statistic is < critical value we reject Null hyp and confirm that the predictors have a good fit to the data.

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