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Group comparisons and hierarchical modeling

Group comparisons and hierarchical modeling - 8 Group...

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8 Group comparisons and hierarchical modeling In this chapter we discuss models for the comparison of means across groups. In the two-group case, we parameterize the two population means by their average and their difference. This type of parameterization is extended to the multigroup case, where the average group mean and the differences across group means are described by a normal sampling model. This model, to- gether with a normal sampling model for variability among units within a group, make up a hierarchical normal model that describes both within-group and between-group variability. We also discuss an extension to this normal hierarchical model which allows for across-group heterogeneity in variances in addition to heterogeneity in means. 8.1 Comparing two groups The first panel of Figure 8.1 shows math scores from a sample of 10th grade students from two public U.S. high schools. Thirty-one students from school 1 and 28 students from school 2 were randomly selected to participate in a math test. Both schools have a total enrollment of around 600 10th graders each, and both are in urban neighborhoods. Suppose we are interested in estimating θ 1 , the average score we would obtain if all 10th graders in school 1 were tested, and possibly comparing it to θ 2 , the corresponding average from school 2. The results from the sample data are ¯ y 1 = 50 . 81 and ¯ y 2 = 46 . 15, suggesting that θ 1 is larger than θ 2 . However, if different students had been sampled from each of the two schools, then perhaps ¯ y 2 would have been larger than ¯ y 1 . To assess whether or not the observed mean difference of ¯ y 1 - ¯ y 2 = 4 . 66 is large compared to the sampling variability it is standard practice to compute the t -statistic, which is the ratio of the observed difference to an estimate of its standard deviation: P.D. Hoff, A First Course in Bayesian Statistical Methods , Springer Texts in Statistics, DOI 10.1007/978-0-387-92407-6 8, c Springer Science+Business Media, LLC 2009

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126 8 Group comparisons and hierarchical modeling t ( y 1 , y 2 ) = ¯ y 1 - ¯ y 2 s p 1 /n 1 + 1 /n 2 = 50 . 81 - 46 . 15 10 . 44 1 / 31 + 1 / 28 = 1 . 74 , where s 2 p = [( n 1 - 1) s 2 1 + ( n 2 - 1) s 2 2 ] / ( n 1 + n 2 - 2), the pooled estimate of the population variance of the two groups. Is this value of 1.74 large? From introductory statistics, we know that if the population of scores from the two schools are both normally distributed with the same mean and variance, then the sampling distribution of the t -statistic t ( Y 1 , Y 2 ) is a t -distribution with n 1 + n 2 - 2 = 57 degrees of freedom. The density of this distribution is plot- ted in the second panel of Figure 8.1, along with the observed value of the t -statistic. If the two populations indeed follow the same normal population, then the pre-experimental probability of sampling a dataset that would gener- ate a value of t ( Y 1 , Y 2 ) greater in absolute value than 1.74 is p = 0 . 087. You may recall that this latter number is called the (two-sided) p -value . While a small p -value is generally considered as indicating evidence that θ 1 and θ 2 are different, the p
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