Multiple Comparisons (10.2) Assume that the null hypothesis of a single-factor ANOVA test is rejected. Ho: 1 = 2 = …. = n Ha: at least two of the i’s differ Which i’s differ? Least Significant Difference Procedure, Tukey’s Procedure, Newman-Keuls Procedure, Duncan’s Multiple Range Procedure

Tukey’s Procedure (Conservative) Use to obtain simulaneous confidence intervals for all pairwise differences i - j Each interval that does not contain zero yields the conclusion that i and j differ significantly at level  Based on the Studentized Range Distribution, Q,m, m=span,  = d.f. of MSE; for Tukey’s m = I, =I(J-1)

Tukey’s Procedure Cont’d Select  and find Q,I,I(J-1) Determine w = Q,I,I(J-1)(MSE/j)1/2 List the sample means in increasing order. Underline those pairs that differ by less than w. Any pair not underscored by the same line are judged significantly different

Example (10.11) Compare the spreading rates of (I=5) different brands of Latex paint using (J=4) gallons of each paint. The sample average spreading rates were

Example Cont’d From an ANOVA test on the equality of means, the computed value of F was found to be significant at  = 0.05 with MSE = 272.8, use Tukey’s procedure to investigate significant differences in the true average spreading rates between brands

A Caution About Interpreting  ` = experiment wise error rate. This is the confidence level for the entire set of comparisons of means = comparison wise error rate. This is the confidence level for any particular individual comparison. ` = Pr{at least 1 false rejection among the c comparisons} = 1 – Pr{no false rejections} = 1-(1-)c

Example Cont’d We used Tukey’s procedure to compare 5 different population (=0.05) means resulting in = 10 = c pairwise comparisons of means

Contrasts Elementary Contrasts: 1- 2 General Contrasts: c11+ c22+ … +cn n; where c1+c2+…+cn=0 We would like to form a CI on a general contrast, For example, construct a CI on the contrast 1+ 2 - 23

Contrasts Cont’d Let  = cii Since the Xij’s are normally distributed and the contrast is a linear combination, is normally distributed

Example Cont’d Assume that brands 2 and 5 were bought at a local paint store and 1, 3, and 4 were bought at a discount hardware store. Is there evidence that the quality of paint varies by type (classification) of store?