Quick thread meandering into a weird DAG for trials. Here are three DAGs. The first two are uncontroversial but I've never seen the third depicted as a DAG.
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I recently read a description of a 'conditionally-randomised experiment', purposefully introducing different probabilities of treatment Z conditional on the value of binary covariate X.
*No-one does this but the idea was to introduce the idea of conditional exchangeability
*No-one does this but the idea was to introduce the idea of conditional exchangeability
No-one does this but people do randomise Z conditional on X all the time in actual RCTs. Rather than doing this to *imbalance* Z, it's done to *perfectly* balance Z by X.
This balancing is most frequently done by stratification.
bmj.com/content/345/bm…
This balancing is most frequently done by stratification.
bmj.com/content/345/bm…
If you've never had an introduction to clinical trials, here's an example of stratified randomisation:
Take a categorical covariate. For a pair (or 'block') of people with a given value of that covariate, one will be assigned to control arm and the other to research arm.
Take a categorical covariate. For a pair (or 'block') of people with a given value of that covariate, one will be assigned to control arm and the other to research arm.
Take a 'block' (pair) of people:
For the first, P(Z=1)=0.5
For the second, P(Z=1)=0 or 1, depending on what the first person got. Note that this probability was caused by their covariate value X.
For the first, P(Z=1)=0.5
For the second, P(Z=1)=0 or 1, depending on what the first person got. Note that this probability was caused by their covariate value X.
Back to my three causal DAGs. The third depicts X->Z for stratified randomisation (see above).
Stratification leads to greater balance than we would get from simple randomisation (where there is no path from X->Z) – the very opposite of confounding!
Stratification leads to greater balance than we would get from simple randomisation (where there is no path from X->Z) – the very opposite of confounding!
If you read the third causal DAG, you realise that you need to condition on X to identify Z->Y. This is wrong; there is no bias for the estimated treatment effect here.
But do read on, things are about to look up.
But do read on, things are about to look up.
Something that's well-known in the trials world is that, if you use stratified randomisation, you have to account for this in your analysis to get an unbiased estimate of the *variance* of the treatment effect (@stephensenn will now plug Genstat).
The causal DAG with the blue correctly tells you that you need to account for X in the analysis, and you do! I guess if you think about it in terms of exchangeability this makes sense: stratification leads to exchangeability conditional on Strata.
I'm out of time, practical session, so I'm posting this thread now.
Thoughts?
Thoughts?
