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1、Lecture Notes Statistics 892 Mixed Models Spring 2005 223Four-way cross-over trial fitting covariance patterns -continued - dropping out the carryover termIn the previous section, we saw that the estimates of the treatment and carryover differences were dependent on whether the period or treatment s
2、tructure covariance pattern was fit. Even though the carryover effect was statistically significant with the treatment structure, it was not with the period structure. The authors of the textbook felt that the inclusion of washout periods made the carryover effect unlikely and thought the analyses s
3、hould be rerun without the carryover effect. The following tables summarize the results of reanalyzing the data without carryover in the model.Covariance Pattern -2 log likelihood param. AIC AICC BICCompound Symmetry 544.6 2 548.6 548.8 549.9Unstructured - Period 528.2 10 548.2 553.9 554.5Heterogene
4、ous Toeplitz - Period 529.1 7 543.1 545.8 547.5Unstructured - Treatment 527.4 10 547.4 553.2 553.8The heterogeneous Toeplitz and two unstructured covariance patterns all have similar fits. Because the heterogeneous Toeplitz covariance pattern has fewer parameters that need to be estimated, it is the
5、 best choice for modeling the covariance structure. Following are the variances and correlations for the four models.Variance CorrelationsPeriod Compound symmetry1 3580.29 12 0.4902 13 0.4902 0.4902 14 0.4902 0.4902 0.4902 1Period Unstructured 1 7390.17 12 2182.93 0.7505 13 3720.95 0.4449 0.6121 14
6、1275.33 0.4336 0.4316 0.5796 1Period Heterogeneous Toeplitz1 6756.49 12 2091.87 0.6515 13 4038.55 0.4477 0.6515 14 1369.32 0.4390 0.4477 0.6515 1Treatment UnstructuredA 2060.79 1B 6128.90 0.2017 1C 3441.08 0.3882 0.8188 1D 2780.88 0.1347 0.5561 0.7613 1Lecture Notes Statistics 892 Mixed Models Sprin
7、g 2005 224We see the same patterns that we saw when the crossover effect was included in the model, where in the period structures the variances in the first period are much greater than in the subsequent periods. For the treatment structure model, Treatment B, which is the placebo, again has greate
8、r variation than any of the three active treatments. Finally, we can compare the estimated treatment differences from the three models.Unstructured (Period structure):Differences of Least Squares MeansStandardEffect treata period _treata _period Estimate Error DF t Value Pr |t|treata A B 45.9105 11.
9、8378 25.8 3.88 0.0006treata A C 47.9029 11.7787 26.2 4.07 0.0004treata A D 70.5208 11.5906 25.6 6.08 |t|treata A B 43.9077 11.9391 26.4 3.68 0.0571treata A C 46.5811 12.0831 29.7 3.86 0.0006treata A D 69.2389 11.6293 26.8 5.95 |t|treata A B 43.1429 21.9679 12.6 1.96 0.0719treata A C 24.4173 15.7255
10、12.5 1.55 0.1455treata A D 68.9882 17.3586 10.7 3.97 0.0023treata B C -18.7255 12.1806 11.5 -1.54 0.1512treata B D 25.8454 17.6056 11.3 1.47 0.1693treata C D 44.5709 10.5189 11.1 4.24 0.0014First, comparing the differences in the treatments among the two period structures and the treatment structure
11、, we notice differences in the levels of significance between the treatment comparisons for the two period structures and the treatment structure. Note that the estimates for the unstructured and heterogeneous Toeplitz are similar. Although it appears that the heterogeneous Toeplitz would be the bes
12、t choice for fitting the covariance pattern, the others of the textbook did not explore this variance structure. They felt that because the unstructured for treatment structure had a smaller 2 log likelihood than the unstructured for period structure and the relatively low correlations involving tre
13、atment A, that the unstructured for treatment structure was the most plausible model. Analysis of Categorical Data from Crossover designsLecture Notes Statistics 892 Mixed Models Spring 2005 225The last area that we will briefly cover in crossover designs is when the data are not normally distribute
14、d. Many of the same issues that we saw with continuous data are encountered with categorical data. The authors of the text note that, except for the case of binary data, purely categorical response variables that dont have an underlying scale are extremely rare and the standard statistical packages (i.e. SAS) currently dont have procedures available to handle them. We will go through the example from the text of a two-treatment, two-period, two-sequence binary crossover trial.The data are ori
