# Including a baseline covariate Model B

Model A was a very simple model for assessing the effect of treatment on DBP. It is usually reasonable to assume that there may be some relationship between pre- and post-treatment values on individual patients. Patients with relatively high DBP before treatment are likely to have higher values after treatment, and likewise for patients with relatively low DBPs. We can utilise this information in

Table 1.1 Number of patients included in analyses of final visits by treatment and centre.

Treatment

Table 1.1 Number of patients included in analyses of final visits by treatment and centre.

Treatment

 Centre A B C Total 1 13 14 12 39 2 3 4 3 10 3 3 3 2 8 4 4 4 4 12 5 4 5 2 11 6 2 1 2 5 7 6 6 6 18 8 2 2 2 6 9 0 0 1 1 11 4 4 4 12 12 4 3 4 11 13 1 1 2 4 14 8 8 8 24 15 4 4 3 11 18 2 2 2 6 23 1 0 2 3 24 0 0 1 1 25 3 2 2 7 26 3 4 3 10 27 0 1 1 2 29 1 0 2 3 30 1 2 2 5 31 12 12 12 36 32 2 1 1 4 35 2 1 1 4 36 9 6 8 23 37 3 1 2 6 40 1 1 0 2 41 2 1 1 4 Total 100 91 94 288

Note: Several additional centres were numbered but did not eventually participate in the study.

the model by fitting the baseline (pre-treatment) DBP as an additional effect in Model A:

DBPi = n + b • pre + tk + et, where b = baseline covariate effect, pre = baseline (pre-treatment) DBP.

Here, we will take the values recorded at visit 2 as the baseline values. We could, of course, have considered using either the visit 1 value, or the average of the visit 1 and visit 2 values, instead. The visit 2 value was chosen because it measured the DBP immediately prior to randomisation, after one week during which all patients received the same placebo medication. The baseline DBP is measured on a quantitative scale (unlike treatments). Such quantitative variables are commonly described as covariate effects and an analysis based on the above model is often referred to as analysis of covariance. The term b is a constant which has to be estimated from our data. There is an implicit assumption in our model that the relationship between the final DBP and the baseline value is linear. Also, that within each treatment group, an increase of 1 unit in the baseline DBP is associated with an average increase of b units in the final DBP. Figure 1.1 shows the results from fitting this model to the data (only a sample of data points is shown, for clarity).

This demonstrates that performing an analysis of covariance is equivalent to fitting separate parallel lines for each treatment to the relationship between post-treatment DBP and baseline DBP. The separation between the lines represents the magnitude of the treatment effects. The analysis will be considered in much greater detail in Section 2.5, but we note for now that two of the treatments appear to be similar to one another, while the lowest post-treatment blood pressures occur with treatment C.

Figure 1.1 Plot to illustrate the analysis of covariance. Treatment:_A;---------B;

Figure 1.1 Plot to illustrate the analysis of covariance. Treatment:_A;---------B;

The use of a baseline covariate will usually improve the precision of the estimates of the treatment effects. It will also compensate for any differences between the mean levels of the covariate in the treatment groups prior to treatment being received. Of course, our assumption that there is a linear relationship between pre-and post-treatment values may not be true. If this were the case, fitting a baseline covariate could lead to less precise results. However, in practice the assumption is very frequently justified in medicine and it has become almost standard to take baseline values into account in the model if they are available.

An alternative way of using baseline values (which we do not recommend) is to analyse the differences between pre- and post-treatment values. However, this generally leads to less accurate results than the 'covariate' approach, particularly when the relationship between pre- and post-treatment values is weak.