Joint assessment of modifiers and/confounders

1. Joint assessment of modifiers and/confounders Maybe one-at-time assessment is not enough The merits of regression analysis start to kick in!

2. Joint effect modification • Now consider a model like: • (Yikes!) • …but this is just 4 lines: E(Y) versus age; • For the 4 groups: • Females receiving placebo: • Males receiving placebo: • Females receiving active:

3. …and lastly • Males receiving active: • How do we interpret this ..er mess? • Look at the drug effect for each gender as a function of age

4. Drug effect is: active - placebo • For females: • For males: • How does drug effect depend on gender? • Take the difference again: • So we can see that measures the extent to which gender as an effect modifier depends on age

5. But also measures: • The extent to which age as an effect modifier depends on gender • You can spin the interpretation in either way here (as is the case with most ‘interaction’ measures)

6. Now lets look at the ‘3 factor’ example from Rabe-Hesketh • Define indicator variables for each of the 3 drug groups: x, y and z • Decide on a ‘baseline’ drug group: say x. • The model’s estimates/predcitions do not depend on this choice but do provide interpretations for the coefficients • Since there are 3 groups, we need 2 coefficients to display the 2 degrees of freedom associated with the differences among the 3 groups

7. Then we can build a ‘saturated’ model • This model will give estimates that reproduce the 12 cell averages • The model separates into a number of 2 df sets:

8. Notice that this last set of 2 df: • Can be expressed in 3 ways (each way means the same thing!): • How does the diet/biofeed interaction depend on drug group? • How does the drug/diet interaction depend on biofeed group? • How does the drug/biofeed interaction depend on diet group?

9. The decision to whether to (and how to) separate up the 2 df sets should be made ‘in advance’ • Individual one df tests can be made using the usual t-tests. It is always to good idea to check out ‘what such tests mean’ (what a concept!) • Of course, it may be that 2(or more) df tests cover the issues at hand. In such cases, one should offer the appropriate F test. • For example, if one fits: • regr sbp x y d b db dx dy bx by dbx dby • …and then tries: • test dbx=dby=0 • One receives the 2 df F test for ‘whether or not the db interaction depends on drug group’