(d) Confirm that the variable ‘b’ holds the mean value of ‘treatment’ in each centre, when the treatments are coded as 0 and 1. Confirm that for each patient w = treatment − b.

Table 8.6 Recovery time (minutes, log-transformed) after anaesthesia in a multi-centre study to compare two anaesthetics.

Centre Treatment A Treatment B No. of patients Mean SD No. of patients Mean SD 1 4 1.141 0.967 5 0.277 0.620 2 10 2.165 0.269 10 1.519 0.913 3 17 1.790 0.795 17 1.518 0.849 4 8 2.105 0.387 9 1.189 1.061 5 7 1.324 0.470 10 0.456 0.619 6 11 2.369 0.401 10 1.550 0.558 7 10 1.074 0.670 12 0.265 0.502 8 5 2.583 0.409 4 1.370 0.934 9 14 1.844 0.848 19 2.118 0.749 Table 8.7 Simulated individual-patient data from the clinical studies to compare two anaesthetics, summarized in Table 8.6.
A B C D E F 1 centre! patient treatment! b w logrt 2 1 1 0 0.555556 −0.555556 0.4973 3 1 2 1 0.555556 0.444444 0.5399 4 1 3 1 0.555556 0.444444 −0.0413 5 1 4 1 0.555556 0.444444 0.5306 . .
. .
. .
181 9 31 1 0.575758 0.424242 1.3515 182 9 32 0 0.575758 −0.575758 1.4634 183 9 33 1 0.575758 0.424242 1.6574 Treatment 0=Treatment A; 1=Treatment B.
‘b’ represents the between-study component of the treatment effect, and ‘w’ the within-centre component. Change the model fitted in Part c by removing the term ‘treatment’ and adding the terms ‘b’, ‘w’ and ‘w.centre’, allocating these terms to the fixed- and random-effect models as appropriate. Fit the resulting model, and comment on the results.