Statistical Techniques IIEXST7015Post-ANOVAMore LSMeans24a_LSMeans_ModelComparison 1LSMeansThere is something else about the SAS LSMeans statement you should know. There are actually several "unusual" or unexpected behaviors of this statement. One we will discuss in connection with AnCova. However, there is another general behavior that we should see first. 24a_LSMeans_ModelComparison 2LSMeans (continued)What is the overall mean? Rep TmtTmt 1 2 3 4 5 Mean1 2 4 6 42 2 6 43 3 3 7 8 9 64 4 45 3 4 6 7 56 5 6 7 67 3 5 4Sum 100n 2024a_LSMeans_ModelComparison 3LSMeans (continued)Rep TmtTmt 1 2 3 4 5 Mean1 2 4 6 42 2 6 43 3 3 7 8 9 64 4 45 3 4 6 7 56 5 6 7 67 3 5 4Mean 5Sum 100 33LSMean 4.71n 20 724a_LSMeans_ModelComparison 4LSMeans (continued)LSMeans calculates means as the mean of means, not the raw mean of all observations. This is particularly important in unbalanced factorial designs. For one unbalanced 4 by 5 factorial the means and lsmeans are given below. 24a_LSMeans_ModelComparison 5LSMeans (continued)Raw dataTmt2Tmt1 1 2 3 4 51 2 3 1 2 23 4 2 4 34 5 3 342 5 6 4 8 39 6 5 363 4 6 4 3 85 8 6 7784 7 6 5 4 58 9 7 7 68 9 7 79 824a_LSMeans_ModelComparison 6LSMeans (continued)Comparison of Means & LSMeans. Tmt2Tmt1 1 2 3 4 5 LSMean Raw Mean1 3 4 2 3 33.00 3.002 7 6 5 8 35.80 5.503 6 7 5 5 86.20 6.004 8 8 6 6 6.56.90 7.00LSMean6.00 6.25 4.50 5.50 5.135.48 5.35Mean6.08 6.20 4.30 5.25 4.7324a_LSMeans_ModelComparison 7LSMeans (continued)Which is better? This depends on the situation. Suppose we caught fish in the summer and in the winter, and wanted to express the average temperature at which fish were caught. The winter mean is 15c and the summer mean is 25c. What is the mean. 24a_LSMeans_ModelComparison 8LSMeans (continued)We do the calculations on the individual catches and find the mean is equal to 24. How can that be? Well we did 180 samples in the summer and only 20 samples in the winter. So the summer temperatures dominate our samples. 24a_LSMeans_ModelComparison 9LSMeans (continued)Perhaps the average temperature would be better expressed as 20, the mean of the means. That is LSMeansI generally use LSMeans. When testing hypotheses such as H0:µ1=µ2=µ3it is best that the overall mean not be dominated by some cell that has an unusually high number of observations. 24a_LSMeans_ModelComparison 10LSMeans (continued)On the other hand, cells with more observations are better estimates of the mean than cells with fewer estimates. If the null hypothesis is true, why loose power by treating the cells equally? Traditional ANOVA will use RAW means in it's calculation. The choice is yours, except that PROC MIXED has only the LSMeans. 24a_LSMeans_ModelComparison 11Testing for differences between modelsPROC MIXED provides several tools for comparing modelsThe intent is to compare between full and reduced models. The statistics used differ from those used in regression. Reduced models may be models with some terms omitted, orReduced models may be models with a simpler variance or covariance structure24a_LSMeans_ModelComparison 12Testing for differences between models (continued)The test is called a likelihood ratio test. It produces a Chi square statistic. The degrees of freedom are the d.f. difference between the two models. 24a_LSMeans_ModelComparison 13Testing for differences between models (continued)Homogeneous variance is tested automatically with some simple modelsRecall our Typhoid strain example, we requested separate variances for each group with the statementrepeated / group=strain;The resulting output was Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq 2 14.56 0.000724a_LSMeans_ModelComparison 14Testing for differences between models (continued)Note that fitting 3 variances requires 3 d.f., while fitting a homogeneous variance model requires only 1 d.f. The 2 d.f. difference are the reason the test on the preceding page is a 2 d.f. model. This test is very similar to Bartlett's test of homogeneity of variance. 24a_LSMeans_ModelComparison 15Testing for differences between models (continued)Suppose that for the baseball example you were told that the salaries of the some positions were highly variable, while others were more stable. Perhaps we should have tested for nonhomogeneous for this example. So we add the statement, repeated / group=strain; 24a_LSMeans_ModelComparison 16Testing for differences between models (continued)SAS fits the different variances for the positions, but does not provide a test. For some cases we will not get this test automatically. In that case we can calculate it ourselves. For the original fit we got the results,Covariance Parameter Estimates Standard ZCov Parm Estimate Error Value Pr Z Alpha Lower Upperteam 3466.41 30458 0.11 0.4547 0.05 513.45 3.81E125Residual 1924296 145057 13.27 <.0001 0.05 1668871 224353424a_LSMeans_ModelComparison 17Testing for differences between models (continued)When separate variances are requested we get the following results,Covariance Parameter Estimates Standard ZCov Parm Group Estimate Error Value Pr Z Alpha Lower Upperteam 25008 35506 0.70 0.2406 0.05 4960.25 26828515Residual Position 1b 3126672 0 . . . . .Residual Position 2b 2276275 902599 2.52 0.0058 0.05 1189304 5985011Residual Position 3b 1512066 600277 2.52 0.0059 0.05 789517 3981295Residual Position c 759251 201637 3.77 <.0001 0.05 479387 1382686Residual Position if 626561 240028 2.61 0.0045 0.05 333467 1582294Residual Position of 2558744 407215 6.28 <.0001 0.05 1916409 3590143Residual Position p 1875902 208345 9.00 <.0001 0.05 1526216 2361923Residual Position ss 1384956 364052 3.80 <.0001 0.05 878092 2504484The first model estimated 2 parameters, while this model fits 9, a difference of 7.24a_LSMeans_ModelComparison 18Testing for differences between models (continued)SAS reports the number of parameters fitted in the "Dimensions" section. In order to do this 7 d.f. test we take the difference in the "-2 Res Log Likelihood" reported in the "Fit Statistics". This value was 6346.8 for the reduced model and 6323.1 for the full model. The difference is 23.7, a chi square value with 7 d.f. 24a_LSMeans_ModelComparison 19Testing for