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Adaptive designs and optimal subgroups

This post is part of our Q&A series.

A question from graduate students in our Fall 2018 offering of “Special Topics in Biostatistics – Adaptive Designs” at Berkeley:

Question:

Hi Mark,

We were interested in your opinion on few topics that have come up in class a few times.

If we isolate an optimal subgroup, we can, perhaps, answer interesting questions about, say, drug efficacy (as in, does this drug work for anybody as opposed to on average?). Having an adaptive design that then preferentially samples from this optimal group could be a good way to increase power and limit resource constraints. This imposes interesting statistical questions- how do we deal with such bias sampling within an adaptive sequential design, and how to generalize to a superpopulation? (it seems that, by design, we converge towards the mean for the optimal subgroup)?

On the other hand, perhaps ethical concerns regarding who we identified as an optimal subgroup could be alleviated by enforcing fairness constraints, or more elaborate “resource constraint” rules that limit how much of the resources are allocated to the optimal subgroup (wealthier families are most responsive, but perhaps not the best solution practically).

Along these lines, it seems that these type of ideas can be further extended to, for example, location (adaptability deciding where to sample next), or maybe even time?

We look forward to your insight!

Best,

I.M., N.H., and R.P.


Answer:

Hi I.M., N.H., and R.P.,

We still start out with a well defined full data random variable X=(W,Y0,Y1) with some probability distribution PX,0. In this case, the design under our control might involve sampling from a marginal distribution gW and a conditional distribution gA of A, given W.

gW is now not equal to the population distribution QW,0, but might be a biased sample version of that. Our observed data structure of (W,A,Y)=(W,A,YA) has a probability distribution determined by q0, conditional distribution of Ya, given W, and g=(gW,gA), and is given by

  1. Sample W from gW,
  2. sample {Y0,Y1} from q,
  3. sample A from ga, and
  4. define O=(W,A,Y=YA).

We could now define an oracle design for g0, some gq0 that might be defined by qY.

For example, it might define gW as sampling from the conditional distribution of W, given E(Y1Y0W)>0, which corresponds with only sampling subjects for which there is a positive treatment effect, and one might define the oracle design gA as setting A=1.

A target parameter of interest could be EYg0, i.e., the mean outcome of Y if we would sample O from Pq0,g0. I would probably first want to know how to estimate this quantity, maybe EYg0EYgW,A=0, causal effect of treatment for optimal subgroup) – based on iid. sampling from a fixed design Pq0,g0, where g0 could be the actual distribution of W combined with a standard RCT for gA. This now resembles estimation of the treatment effect among the optimal subgroup for a fixed design, and Alex Luedtke and I have a paper on that.

Such an iid estimator, e.g., an iid TMLE of EYg for a given design g based on sampling from a fixed design Pq0,g0 would now be a basis for estimation in an adaptive design where OiPq0,g0,i is sampled using design g0,i, changing with i, based on the available data at the time point at which the unit for Oi needs to be sampled.

Such an iid TMLE of EYg for an iid design Pq0,g0 would involve targeting an initial estimator of q0 with a TMLE step using weighting (e.g., in the loss or in the clever covariate) by gg0=(gWgA)(gW,0gA,0).

We might now simply replace that weight by ggi=(gWgA)gW,igA,i to obtain an IPTW-TMLE for the adaptive design sampling Oi from Pq0,gi. Generally, I do not see any reason why the whole theory for sequential adaptive design would not apply to this setting as well. It is an example of the general sequential adaptive design (see 2008 tech report) in which now $(W,A)$ are the design variables that are generated by the distribution set by the experimenter. In fact, in that technical report, I present as one example an adaptive design on the sampling of $X$ in the regression context.

Indeed, the choice of oracle design is in our hands, and, in particular, we can refine what we view as the target subgroup who need to be treated. For example, if treatment is always beneficial, we might define a subgroup by a resource constraint (e.g., only 30% can be treated). One could imagine that a ranking of the conditional treatment effect would result in the top 30% of covariate values that are not fair w.r.t. subgroups, and one could then decide to define a rule for treating that also takes into account fairness w.r.t. certain subgroups, while, under both a fairness and resource constraint, optimizing the mean outcome. That will then result in a different oracle design and the resulting adaptive design will then learn that particular oracle design…

Best Wishes,

Mark

P.S., remember to write in to our blog at vanderlaan (DOT) blog [AT] berkeley (DOT) edu. Interesting questions will be answered on our blog!