Hello everyone,
If I may, I would like discuss the application of ANCOVA.
Say we are considering ANCOVA in an experimental setting i.e. in a RCT scenario. We randomly allocate n1 patients to group A and n2 patients to group B and take a measure (Y) at baseline, we then administer the treatment and placebo to group A and B respectively and measure Y again six months later .
We wish to evaluate the effect of treatment on end of trial outcome.
An ANCOVA model is essentially a regression model and, in our simple case above, we have "6th month Y value"(i.e. end-of-trial outcome) as the dependent variable, "treatment type" as the independent variable and "baseline Y" as the covariate.
We can think of ANCOVA in two ways:
(i)As baseline Y is likely to be associated with end-of-trial outcome, we can view it as acting like a "blocking factor". Thus, after conducting an ANCOVA (i.e. including baseline Y as a covariate), error variance is reduced and the test on "treatment" is more powerful (smaller P-value and narrower confidence interval).
(ii) Randomisation is expected to balance treatment groups as regards baseline Y but, in practice, it is not unusual to observe imbalances after randomisation. Thus we can also view ANCOVA as "adjusting" our treatment comparisons to a common value of baseline Y i.e. holding baseline Y constant across our two treatment groups and comparing the two "adjusted" means.
Now it is often said that there should be independence of the covariate and treatment effect. To use an example I have seen previously, say QoL is our dependent variable, "treatment type" is the independent variable and "anxiety level" is the covariate. We will take it that anxiety is associated with QoL. If anxiety is measured after administering treatment and the treatment affects anxiety levels then the adjustment for anxiety may hide or exaggerate the treatment effect. It will therefore make the treatment effect difficult to interpret.....My question is....doesn't this problem occur in situation (ii) above when we have imbalance (albeit slight imbalance) as regards baseline Y in our two treatment groups? i.e. where there is dependence between the covariate and the treatment. I guess the reason why (ii) is acceptable is that *prior to randomisation* we are saying that there is an *assumption* of independence of the covariate (baseline Y) and treatment (but, of course, after randomisation we usually find that baseline values of the covariate are not identically distributed in all treatment groups)?
Say we consider another hypothetical example (non RCT this time) where we have "achievement score" as the dependent variable, "educational establishment (primary school, secondary school) " as the independent variable and "age" as the covariate. Here, the assumption of "independence of the covariate and treatment effect (independent variable)" is clearly violated, yet I have still seen many examples which use ANCOVA in this sort of scenario where (in this case) they would use ANCOVA to compare the two educational establishments as regards (average) achievement score at a common value of age.
I would greatly appreciate your views on these issues.
Kindest Regards,
Kim
Dr Kim Pearce PhD, CStat, Fellow HEA
Senior Statistician
Faculty of Medical Sciences Graduate School
Room 3.14
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Newcastle University
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