Treatment-by-covariate interactions

We call these equations (or models), main effects models. In the next subsection we will be adding to the main effects (of treatment and the covariates), treatment-by-covariate interaction terms. 

Figure 6.5 Scatter plot showing treatment-by-covariate interaction...

We have previously discussed treatment-by-centre interactions, where the treatment effect is seen to be different in the different centres. We now have a treatment-by-covariate interaction, where the treatment effect depends on the value of the covariate. We can investigate this by considering a more general model where we allow the lines to have different slopes as well as different intercepts ... 

If a significant treatment-by-covariate interaction is found then it would be useful to divide the patients into subgroups in terms of the size of the primary tumour, say small, medium and large and look at the treatment difference within those subgroups to try to better understand the nature of the treatment-by-covariate interaction. 

Assessing the treatment-by-covariate interactions is then based on comparing the bs bii with b2i> with b22 and with I723 in separate hypothesis tests. 

For several covariates we simply introduce a cross-product term for each covariate with corresponding coefficients d, 2 and dj. The presence of treatment-by-covariate interactions can then be investigated through these coefficients. 

Provides a convenient framework for the evaluation of treatment-by-covariate interactions in some cases such interactions are anticipated, in other cases such analyses are exploratory. 

Should treatment-by-covariate interactions be found, either through a test of homogeneity in an adjusted analysis or through ANCOVA, then analysis usually proceeds by looking at treatment differences within subgroups. Plots of treatment effects with associated confidence intervals within these subgroups are useful in this regard. 

We can also investigate the presence of treatment-by-covariate interactions by including cross-product terms ... 

Questions relating to those interaction terms are addressed through the d coefficient as before. In the above example, looking for treatment-by-covariate interactions would be asking whether the treatment benefit, in terms of a reduction in the likelihood of the baby suffering respiratory distress, was the same for babies delivered at 37, 38 and 39 weeks. 

Recall from Section 6.5.2 that, when assessing interactions, we use a significance level of 0.10 rather than 0.05 due to a lack of power. In Figure 10.1 most of the interactions, except that involving baseline LDL cholesterol, give p-values well above 0.10 and so there is no evidence of treatment-by-covariate interactions for... 

Non-parametric procedures tend to be simple two group comparisons. In particular, a general non-parametric version of analysis of covariance does not exist. So the advantages of ANCOVA, correcting for baseline imbalances, increasing precision, looking for treatment-by-covariate interactions, are essentially lost within a non-parametric framework. 

Model explorations, if conducted at all, should be justified with suitable a priori power and should be consistent with the principle of BE assessment. For example, effects of formulation by covariate interaction should not be tested, for the same reason that, in standard BE analysis, treatment by subgroup interaction is typically not tested. This implies that the number of explorations should be kept very small, and preferably avoided in general. 

An example is shown in Table 16.14. A two-factor analysis of variance for the covariate, as shown in Table 16.15, clearly indicates that the two sexes started with approximately the same means (p = 0.5598). Moreover, there were no differences between the group means in either sex as indicated by the large tail probabilities for treatment (p = 0.8823) and sexxtreatment interaction (p = 0.6532). These facts justify using sex as a factor in the analysis, as was done here. 

The above analysis establishes that there was no significant sex difference, as indicated by the tail probabilities for sex (p = 0.2667) and sexxtreatment interaction (p = 0.9784). There was also some indication that there may have been some treatment effect across the treatment groups in both sexes (p = 0.0559). Examination of the variate means indicated that both sexes seemed to have lower means than their respective controls. The picture was clouded by the fact that there was a similar slightly lower tendency, though not very consistent, in the covariate means as well. Under this circumstance, it is more appropriate to take both the covariate and the variate into any optimal analysis. Table 16.19 shows an analysis of covariance for the factorial model. 

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