Confidence intervals ANCOVA

Allows assessment of prognostic factors. Fitting the ANCOVA model provides coefficients for the covariates and although this is not the primary focus of the analysis, these coefficients and associated confidence intervals provide information on the effect of the baseline covariates on outcome. 

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. 

Suppose that we have just two centres (and two treatment groups). If we define binary indicators, say z and x, to denote treatment group and centre, respectively, then including z and x in an ANCOVA is identical mathematically to the corresponding two-way ANOVA. This connection is true more generally. If we were now to add more centres (say four in total) to the ANOVA then defining binary indicators to uniquely define these centres Xj = 1 for a patient in centre 1, Xj = 1 for a patient in centre 2, X3 = 1 for a patient in centre 3 with 0 values otherwise, then ANCOVA with terms z, Xj, Xj and X3 would be mathematically the same as ANOVA. We would obtain the same p-values, (adjusted) estimates of treatment effect, confidence intervals etc. 

This link applies also to the p-value from the unpaired t-test and the confidence interval for p, the mean difference between the treatments, and in addition extends to adjusted analyses including ANOVA and ANCOVA and similarly for regression. For example, if the test for the slope b of the regression line gives a significant p-value (at the 5 per cent level) then the 95 per cent confidence interval for the slope will not contain zero and vice versa. 

The t-tests and their extensions ANOVA, ANCOVA and regression all make assumptions about the distribution of the data in the background populations. If these assumptions are not appropriate then strictly speaking the p-values coming out of those tests together with the associated confidence intervals are not valid. 

Recall that, in ANCOVA, the model has an ANOVA portion and a regression portiOTi. The covariant is the regression portion. Hence, we have a b or slope for the covariate, which is b = 0.9733 (Table 11.8). This, in itself, can be used to determine if the covariate is significant in reducing overall error. If the b value is zero, then the use of a covariate is not of value in reducing error, and ANOVA would probably be a better application. A 95% confidence interval for the (3 value can be determined. 

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