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# Handout15 - Lecture 15 1 Bootstrap hypothesis tests...

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Lecture 15 1. Bootstrap hypothesis tests: 2-sample t -test 2. Types of missing values 3. Multiple imputation 4. Making missing-value datasets: MCAR and MNAR 5. Indicator variables 6. Proc MI and Proc MIanalyze 1 Bootstrap tests Our bootstrap example (confidence interval for a correlation): 1. draw B samples with replacement from the data, 2. calculate the statistic r § b from each bootstrap sample, b = 1,..., B , 3. use the histogram of the bootstrap statistics © r § b to find confidence interval or standard error. Hypothesis test differs from confidence intervals and standard errors: test has a null hypothesis H 0 , p -value for the test is calculated assuming H 0 . To bootstrap a test, we need to draw the bootstrap samples from the null hypothesis distribution. 2

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The two sample t -test compares two population means μ Y and μ Z by comparing estimates of these means from independent samples y = © y 1 , y 2 ,..., y n from population Y and z = { z 1 , z 2 ,..., z m } from population Z . The standard assumptions are: 1. The data are simple random samples 2. The two populations are Normal 3. The two populations have the same variance 4. The observations are correctly labeled with their population: no misclassification. The null hypothesis is H 0 : μ Y = μ Z . 3 Brain glucose example Magnetic resonance imaging gives researchers a non-invasive way to measure chemicals in the brain minute by minute. One study measured blood sugar (glucose) in the brains of 14 people with diabetes and 14 healthy people. 4
The TTEST Procedure Lower CL Upper CL Variable diabetic N Mean Mean Mean Std Dev Std Err brain_ 14 4.6832 5.3214 5.9596 1.1054 0.2954 glucose 0 brain_ 14 4.1943 4.6857 5.1771 0.8511 0.2275 glucose 1 brain_ Diff (1-2) -0.131 0.6357 1.4021 0.9865 0.3728 glucose Variable Method Variances DF t Value Pr > |t| brain_glucose Pooled Equal 26 1.71 0.1001 brain_glucose Satterthwaite Unequal 24.4 1.71 0.1009 From SAS Help, test statistic for Variances Unequal is t = ¯ y ° ¯ z q SE( ¯ y ) 2 + SE(¯ z ) 2 observed test statistic is t obs = 1.71. 5 The test indicates no difference. But these are small samples that looked skewed in opposite directions—perhaps the t -test is missing something? Bootstrap the sampling distribution of unequal-variances t , assuming H 0 . 6

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Bootstrap 2-sample t-test We have two independent samples: y = © y 1 , y 2 ,..., y n from population Y and z = { z 1 , z 2 ,..., z m } from population Z . 1. Calculate the observed test statistic from the samples: t obs = ¯ y ° ¯ z p SE( ¯ y ) 2 + SE(¯ z ) 2 This version of the t -statistic does not assume equal population variances. 2. Create two transformed data sets y 0 and z 0 with equal means to satisfy the null hypothesis. We want equal means, but don’t want to change the standard deviations. How can we do that? 7 Let ¯ y be the mean of sample y , and ¯ z be the mean of sample z . Subtract ¯ y from each observation in sample y : y 0 = © y 0 1 , y 0 2 ,..., y 0 n = © ( y 1 ° ¯ y ), ( y 2 ° ¯ y ),...,( y n ° ¯ y ) Subtract ¯ z from each observation in sample z : z 0 = © z 0 1 , z 0 2 ,..., z 0 n = {( z 1 ° ¯ z ), ( z 2 ° ¯ z ),...,( z m ° ¯ z )} Easy to show y 0 and z 0 have equal means (zero), which is the null hypothesis.
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Handout15 - Lecture 15 1 Bootstrap hypothesis tests...

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