This can dilute the significance of a departure from the null hypothesis. We can focus the test on departures of a particular form.

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1 One-Degree-of-Freedom Tests Test for group occasion interactions has (number of groups 1) number of occasions 1) degrees of freedom. This can dilute the significance of a departure from the null hypothesis. We can focus the test on departures of a particular form. 1

2 For example: if we expect all responses after the baseline to respond similarly: µ after baseline = µ baseline + shift, where shift depends on treatment group, we estimate the difference of (µ 1 + µ 4 + µ 6 )/3 µ 0 between treated and placebo groups: [(µ S,1 + µ S,4 + µ S,6 )/3 µ S,0 ] [(µ P,1 + µ P,4 + µ P,6 )/3 µ P,0 ] or we can focus on the area under the curve 2 (µ 1 µ 0 ) (µ 4 µ 0 ) + 1 (µ 6 µ 0 ) and again estimate the difference between the groups. 2

3 Test these hypotheses in PROC MIXED by adding the statements contrast response vs baseline treatment * week / chisq; contrast AUC treatment * week / chisq; The additional output is Contrasts Num Den Label DF DF Chi-Square F Value Pr > ChiSq Pr > F response vs baseline <.0001 <.0001 AUC <.0001 <

4 Exploiting the Baseline Profile analysis tests for any differences in change over time. With baseline data and randomized groups, we know that the baseline responses can differ between groups only by sampling variation. We can exploit this knowledge by converting the later responses into differences from baseline, or treating the baseline measurement as a covariate instead of a response. 4

5 Convert to differences:... data tlc_uni; /* Convert the data from multivariate form (one record per subject, with * 4 responses), to univariate form (one record per response, with a new * variable week to identify the occasion; differences from baseline */ set tlc; drop y0 -- y6; week = 1; y = y1 - y0; output; week = 4; y = y4 - y0; output; week = 6; y = y6 - y0; output; run; proc sort data = tlc_uni; by treatment; proc mixed data = tlc_uni order = data; class child treatment week; model y = treatment week treatment * week / solution chisq; repeated week / subject = child type = un; run; 5

6 The key output is Mixed Model Analysis of TLC data 8 Differences from baseline Type 3 Tests of Fixed Effects Num Den Effect DF DF Chi-Square F Value Pr > ChiSq Pr > F treatment <.0001 <.0001 week <.0001 <.0001 treatment*week <.0001 <.0001 Note that the main effect of treatment is now of substantive interest. In fact, it is the same as the simpler one-degree-offreedom test described above. These treatment and treatment week Chi-Squares sum to the interaction term in the previous analysis. 6

7 Covariate approach:... data tlc_uni; /* Convert the data from multivariate form (one record per subject, with * 4 responses), to univariate form (one record per response, with a new * variable week to identify the occasion; keep y0 as a covariate */ set tlc; drop y1 -- y6; week = 1; y = y1; output; week = 4; y = y4; output; week = 6; y = y6; output; run; proc sort data = tlc_uni; by treatment; proc mixed data = tlc_uni order = data; class child treatment week; model y = treatment week treatment * week y0 / solution chisq; repeated week / subject = child type = un; run; 7

8 The key output is Mixed Model Analysis of TLC data 8 Use baseline data as a covariate The Mixed Procedure Solution for Fixed Effects Standard Effect treatment week Estimate Error DF t Value Pr > t y <.0001 Type 3 Tests of Fixed Effects Num Den Effect DF DF Chi-Square F Value Pr > ChiSq Pr > F treatment <.0001 <.0001 week <.0001 <.0001 treatment*week <.0001 <.0001 y <.0001 <

9 The week and treatment week tests are the same as in the deviation from baseline analysis, but the main effect of treatment is slightly more significant. The y0 coefficient of is more than 2 standard deviations less than 1 (significant at the 5% level). 9

10 Constraining baseline means to be equal:... data tlc_uni; set tlc; drop y0 -- y6; week = 1; y = y1; output; week = 4; y = y4; output; week = 6; y = y6; output; /* put baseline in its own group: */ week = 0; y = y0; treatment = Z ; output; run; proc sort data = tlc_uni; by treatment; proc mixed data = tlc_uni order = data; class child treatment week; model y = week treatment * week /* no main effect of treatment */ / solution chisq; repeated week / subject = child type = un; run; 10

11 The key output is Mixed Model Analysis of TLC data, baseline means equal 8 The Mixed Procedure Solution for Fixed Effects Standard Effect treatment week Estimate Error DF t Value Pr > t Intercept <.0001 week week week week treatment*week A <.0001 treatment*week A <.0001 treatment*week A treatment*week P treatment*week P treatment*week P treatment*week Z

12 Type 3 Tests of Fixed Effects Num Den Effect DF DF Chi-Square F Value Pr > ChiSq Pr > F week <.0001 <.0001 treatment*week <.0001 <.0001 This approach: exploits the information about baseline data; is as powerful as any other; uses data for a subject even when the baseline is missing. 12

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