Chapter5p2lecture

Chap. 5, part II, page 1

Chapter 5, part II: Comparisons among several samples Testing Equality in a Subset of Groups: The Spock Conspiracy Trial Data. In 1968, Dr. Benjamin Spock was tried in the United States District Court of Boston on charges of conspiring to violate the Selective Service Act by encouraging young men to resist the draft for Vietnam. The defense challenged the jury selection, claiming that women had been systematically under-represented. The Spock jury had no women. Juries in Boston are selected in 3 stages. From the city directory, the Clerk of the Court selects 300 names at random. Before the trial, a venire of 30 or more jurors is selected from the 300 names, according to law, at random. The final jury is selected from the venire in a nonrandom process allowing each side to exclude certain jurors for a variety of reasons. The Spock defense pointed to the venire for their trial, which contained only one woman. That woman was released by the prosecution, resulting in an all male jury. The defense argued that the judge in the trial had a history of venires in which women were under-represented, contrary to law. The data consist of the percent women in the venires of Spock’s judge and six other Boston area District Court judges. Here are the summary statistics: Descriptives percent

N SPOCK'S A B C D E F Total

9 5 6 9 2 6 9 46

Mean 14.6222 34.1200 33.6167 29.1000 27.0000 26.9667 26.8000 26.5826

Std. Deviation 5.03879 11.94182 6.58222 4.59293 3.81838 9.01014 5.96888 9.17911

Std. Error 1.67960 5.34054 2.68718 1.53098 2.70000 3.67838 1.98963 1.35339

95% Confidence Interval for Mean Lower Bound Upper Bound 10.7491 18.4954 19.2923 48.9477 26.7090 40.5243 25.5696 32.6304 -7.3068 61.3068 17.5111 36.4222 22.2119 31.3881 23.8567 29.3085

Minimum 6.40 16.80 27.00 21.00 24.30 17.70 16.50 6.40

Maximum 23.10 48.90 45.60 33.80 29.70 40.20 36.20 48.90

Here is the ANOVA table comparing the equal means model to the separate means model: ANOVA percent

Between Groups Within Groups Total

Sum of Squares 1927.081 1864.445 3791.526

df 6 39 45

Mean Square 321.180 47.806

F 6.718

Sig. .000

Chap. 5, part II, page 2 So, we see that the separate means model is significantly better than the equal means model. Now, the question becomes: Is the venire from Spock’s judge significantly different from the other 6, and are the other six basically the same? Here is the ANOVA table comparing the two means model to the equal means model, where Spock’s venire has parameter µ1 and the other venires have mean µ0. ANOVA percent

Between Groups Within Groups Total

Sum of Squares 1600.623 2190.903 3791.526

df 1 44 45

Mean Square 1600.623 49.793

F 32.145

Sig. .000

Now, finally, we want to compare the two-means model to the separate means model. How do we do this? Remember: SPSS will always use the equal means model as the reduced model. We need to calculate the Extra Sum of Squares where the two means model serves as the reduced model. ESS = SSR2 − SSR7 =

Then, our F statistic is:

F

=

Our rejection decision is then made by comparing this value to the 95th quantile of the appropriate F distribution, or the value closest to this in the table. From the table we find,

F

=

What is our decision?

Chap. 5, part II, page 3 Kruskal Wallis Nonparametric Analysis of Variance When outliers are present, but the data are thought to be meaningful, one alternative to ANOVA is the Kruskal-Wallis nonparametric analysis of variance. Here, we use the rank transform and then conduct the F test on the ranks. The test statistic is:

KW =

1

σ R2

SSBR

where σ R2 is the sample variance of all n ranks, where n is the number of observations in all groups.

Results of Kruskal-Wallis Test for the Spock Data Ranks percent

Judge SPOCK'S A B C D E F Total

N 9 5 6 9 2 6 9 46

Mean Rank 6.39 33.20 34.00 28.06 23.50 24.08 23.28

Test Statisticsa,b Chi-Square df Asymp. Sig.

percent 21.965 6 .001

a. Kruskal Wallis Test b. Grouping Variable: Judge

To perform the KW test in SPSS, go to Analyze Æ NonparametricÆ K independent samples, and then choose the “Kruskal-Wallis test” box.