After ANOVA If your F < F critical: Null not rejected, stop right now!! If your F > F critical: Null rejected, now figure out which of the multiple means differ from which others Use multiple comparison test (many available) 2 kinds of MCT  comparisons between all pairs of means  comparisons between a control and all other means

For all possible pair-wise comparisons Multiple Comparison Tests Tukey Test H0: A = B For all possible pair-wise comparisons Number of comparisons of K, 2 at a time = k (k-1) 2

--Arrange the means in order of increasing magnitude Conclusions depend on order in which pairwise comparisons considered Largest to smallest, then largest to next smallest….. Then second largest to smallest……… --Calculate q for each pair of means XbarB- XbarA q = SE s2 Within groups MS from ANOVA = n # data in A & B See modification for unequal sample size pg 212 Zar

Other MCT available (see hand out) Tukey’s: --controls for experiment-wise error rate --is conservative (less likely to find difference) --rarely criticized Dunnett’s test: --designed to compare control to all other groups --not interested in all comparisons --SAS assumes first treatment level is control Scheffe’s test: --best for “multiple contrasts” --ex. want to compare level 1, 2, & 3 to level 4

Normally pick one data start; infile 'C:\Documents and Settings\cmayer3\My Documents\teaching\Biostatistics\Lectures\exam 1 anova.csv' dlm=',' DSD; input trtmt $ nampeat; options ls=80; proc print; data two; set start; proc glm; class trtmt; model nampeat=trtmt; means trtmt/tukey bon scheffe duncan snk; run; Normally pick one

The SAS System 39 15:11 Tuesday, October 4, 2005 The GLM Procedure Class Level Information Class Levels Values trtmt 3 dark highligh lowlight Number of Observations Read 15 Number of Observations Used 15 Dependent Variable: nampeat Sum of Source DF Squares Mean Square F Value Pr > F Model 2 804.400000 402.200000 5.83 0.0170 Error 12 827.200000 68.933333 Corrected Total 14 1631.600000 R-Square Coeff Var Root MSE nampeat Mean 0.493013 57.65701 8.302610 14.40000 Source DF Type I SS Mean Square F Value Pr > F trtmt 2 804.4000000 402.2000000 5.83 0.0170 Source DF Type III SS Mean Square F Value Pr > F Output gives type I and type III SS, use type 3 insensitive to model order (same for simple models)

Output for Tukey– all tests gave same results in this case The SAS System 45 15:11 Tuesday, October 4, 2005 = The GLM Procedure Tukey's Studentized Range (HSD) Test for nampeat NOTE: This test controls the Type I experimentwise error rate, but it generally has a higher Type II error rate than REGWQ. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square 68.93333 Critical Value of Studentized Range 3.77278 Minimum Significant Difference 14.008 Means with the same letter are not significantly different. Tukey Grouping Mean N trtmt A 22.200 5 lowlight A B A 16.400 5 highligh B B 4.600 5 dark Output for Tukey– all tests gave same results in this case Now must interpret Low differs from dark, high not differ from dark………. Conclusion, any light at all affects feeding level not important Sometimes lines over treatments Letters used to show diffs