Chapter 8: Linear Combinations and Multiple Comparisons of Comparing pairs of means within the ANOVA ModelSimultaneous Inferences: multiple comparisonsPlanned and unplanned comparisonsExample: mice diet experimentChap. 8, multiple comparison Chapter 8: Linear Combinations and Multiple Comparisons of Means Comparing pairs of means within the ANOVA Model The two-sample pooled t procedure for comparing any pairs of means, say 1µ and 2µ uses 21YY − and SE(21YY − ) = 2111nnsp+ where 2)1()1(21222211−+−+−=nnsnsnsp. The only change in adapting this to several groups is to use the pooled standard deviation from all of the groups if the assumption of equal standard deviations seems reasonable. DescriptivesMonths survived49 27.40 6.134 6.4 35.557 32.69 5.125 17.9 42.371 42.30 7.768 18.6 51.956 42.89 6.683 24.2 50.756 39.69 6.992 23.4 49.760 45.12 6.703 19.6 54.6NPN/N85N/R50R/R50N/R loproN/R40N Mean Std. Deviation Minimum Maximum The equal variance assumption seems reasonable for this experiment so we will use the pooled standard deviation from all 6 treatments. )1()1()1()1()1()1(212222211−++−+−−++−+−=IIIpnnnsnsnsns…… = 678.6599.44595648)703.6(59)125.5(56)134.6(48222==++++++…… The degrees of freedom for the t distribution when you use this pooled standard deviation is the denominator in the above expression which is In−, where n is the total sample size (349 in our example) and I is the number of groups or treatments (6 in our example). So we use a t with 343 degrees of freedom for the mice experiment. • One desired comparison is between groups 1 and 2: the unrestricted non-purified diet (NP) to a standard 85 calorie diet (N/N85). The result is summarized in part e) on p. 116. First, note that SE(21YY − ) = 571491678.61121+=+nnsp = 1.301. A 95% confidence interval for 21µµ−:Chap. 8, multiple comparison )(SE)975(.2134321YYtYY −±−= 35.5 – 42.3 ± 1.967 (1.301) = -6.8 ± 2.56 ≈ -9.4 months to -4.2 months Conclusion: It is estimated that the 85 calorie standard diet increases mean life expectancy by 6.8 months over an unrestricted diet with a 95% confidence interval of 4.2 to 9.4 months. • A test of the null hypothesis that 21µµ= against a one-sided alternative that 21µµ< (we would have to decide before collecting the data that we were only interested in detecting an increase in mean life expectancy with the 85 calorie diet): Test statistic = 301.18.6)(SE2121−=−−YYYY = -5.23 Compare to t distribution with 343 d.f. P-value = area to left of –5.22 < .0001 Conclusion: The data provide very strong evidence that the 85-calorie diet increases life expectancy over the unrestricted diet. Note: if the equal standard deviations assumption did not appear reasonable, then we could have done the confidence interval and hypothesis test the usual way using the pooled standard deviation from the two groups or the unpooled Welch’s t procedures. The advantage of pooling all 6 groups is a better estimate with increased degrees of freedom. The overall F-test is only one step in the comparison of several groups. We’ve seen how the pooled standard deviation could be used in confidence intervals and hypothesis tests for the difference of any pair of group means. Simultaneous Inferences: multiple comparisons After we have a significant F test that there is a least one difference in the treatment means, we might like to perform some tests to see which group means are different. In order to control our type I error for the entire experiment, we need to distinguish between individual and simultaneous confidence intervals and hypothesis tests. If we form individual 95% confidence intervals for a set of linear combinations of means, then we cannot be 95% confident that they all include the true parameters they’re estimating. The actual confidence that a family of confidence intervals are simultaneously correct is called the familywise confidence level. The Bonferroni inequality creates simultaneous confidence intervals with any desired familywise confidence level. To create 100(1-α)% simultaneous confidence intervals for k parameters, we make each confidence interval an individual 100(1-α/k)% confidence interval. . Example: Simultaneous 95% confidence intervals for 10 parameters:Chap. 8, multiple comparison Bonferroni guarantees that the familywise confidence level is at least 1-α, but it can be overkill, especially when k is large. There are several ways that have been developed for creating simultaneous confidence intervals among means that can be less drastic. Planned and unplanned comparisons • Planned comparisons: contrasts which are the researcher decides are of interest before the data are collected. We can control the familywise confidence level using the Bonferroni inequality or one of the other methods listed below. • Unplanned comparisons: contrasts which the researcher decides are of interest after examining the data. These may be chosen from a larger set of contrasts which have been examined or may be chosen after looking at the data to suggest contrasts of interest. The confidence intervals must take into account that you actually (in the first case) or essentially (in the second case) examined a large number of contrasts and picked out the most “significant” one or ones. When we ‘data snoop’ and decide to test only the group means that appear to be the most different, our family-wise error correction should be that for all possible pair-wise comparisons. In all cases of family-wise error correction, the confidence interval for a contrast always has the form: Estimate ± (multiplier) x (standard error) The specific method used determines only the multiplier. If you have a legitimate choice between two or more procedures, you can choose the one with the smaller multiplier. In SPSS, JMP and other packages, the standard errors of one or more contrasts can usually be calculated automatically. Pairwise comparisons between all pairs of means can usually be obtained by clicking an appropriate button or with a couple of lines of code. Test Multiplier . Fisher’s LSD for planned comparisons only after the F test has shown significance. ,(1 / 2)nItα−−Bonferroni where k = is the number of all possible. pairwise comparisons of means,
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