The FDA recently published revised guidance on statistical methods for adjusting for baseline covariates in trials. Overall I like the guidance and think it will prove useful. In this post I’ll give a few thoughts on aspects of the revised guidance, organised according to the sections of the guidance document.
Robust standard errors
Robust sandwich standard errors are advocated rather than ‘nominal standard errors’ (model based ones), on the basis that model based SEs may be biased when the outcome regression is misspecified. Actually, for linear models, the model based SE is valid even under misspecification (Wang et al 2000) when randomisation is 1:1. When it is not 1:1 however, sandwich SEs are indeed needed (Bartlett 2020) even in the linear model case.
Stratified randomisation and standard errors
Often randomisation is stratified on some of the prognostic factors. If one correctly specifies the outcome regression and adjusts for the variables used in the stratified randomisation, valid inferences are obtained. If one however views the outcome model as a possibly misspecified working model, then how can we obtain valid standard errors and tests if one does not necessarily assumed the working model is correctly specified?
In the linear model case, the FDA guidance recommends using the methods of Bugni et al 2018. They show (as was already known earlier) that if one uses stratified randomisation but performs a t-test comparing groups ignoring the stratification factors, inferences are conservative. Bugni et al describe a modification to this standard t-test which ensures the type 1 error control is exactly the desired level (i.e. removing the conservativeness).
Bugni et al also consider ‘t-test with strata fixed effects’, which is a linear model with treatment and indicators for each stratum as covariates, using the usual heteroscedastic robust SEs. It’s important to note the model here is not a model where the baseline variables used to perform the stratified randomisation are entered themselves as covariates, but rather indicators for membership of strata defined by these variables. Thus if one stratifies randomisation on two binary covariates, there are four strata, and this method includes indicators for each of these as covariates. They show that the test of treatment here has correct type 1 error if randomisation is 1:1. In fact they also show that the usual model based SE provides exact (asymptotically) type 1 error control, provided randomisation is 1:1.
Conditional vs unconditional effects
For linear models one of the things stated is that:
Covariate adjustment through a linear model (without treatment by covariate interactions) also estimates a conditional treatment effect, which is a treatment effect assumed to be approximately constant across subgroups defined by baseline covariates in the model.
This is saying that for linear models which don’t include treatment by covariate interactions, the treatment effect estimate has both an interpretation as a marginal treatment effect (how does the population mean outcome change if you switch from treatment A to B) and as a conditional effect (how does the mean outcome change in subpopulations defined by levels of the covariates adjusted for). Of course for the latter interpretation to be correct, as the quoted text says, it must be the case that in truth these conditional/subpopulation effects are identical across these subpopulations. In reality there is no reason why this will necessarily be true, certainly not exactly.
The guidance notes that
The linear model may include treatment by covariate interaction terms. However, when using this approach, the primary analysis should still be based on an estimate from the model of the average treatment effect.
I don’t think it is immediately obvious how a model which includes interactions between treatment and baseline covariates can be used to obtain an estimate of the unconditional/marginal effect. To do so, one can use equation 5 of Tsiatis et al 2008, where h0(Xi) and h1(Xi) are the model predicted outcome means under control and active treatment from the fitted outcome model
The guidance helpfully explains the issue of non-collapsibility, which affects odds ratios and hazard ratios. The simple table demonstrating non-collapsibility is particularly appealing.
Conditional effects estimated by outcome regression
The guidance notes that regression (e.g. logistic) models adjusting for covariates estimated conditional effects. The difficulty, as per my comments above in relation to linear models, is that there is no reason why say the conditional odds ratio should be the same across subpopulations defined by the baseline covariates. The guidance states:
Sponsors should discuss with the relevant review divisions specific proposals in a protocol or statistical analysis plan containing nonlinear regression to estimate conditional treatment effects for the primary analysis. When estimating a conditional treatment effect through nonlinear regression, the model will generally not be exactly correct, and results can be difficult to interpret if the model is misspecified and treatment effects substantially differ across subgroups. Interpretability increases with the quality of model specification.
This acknowledges the issue, but it is not clear to me how one can operationalise this given requirements for prespecification of a primary analysis model. At the very least, it indicates I think one should perform diagnostics to detect whether the conditional effects are constant, but what would one do if these reveal heterogeneity? Moreover, the power to detect model misspecification could be low, such that one may conclude there is no evidence against a null hypothesis of common conditional effects even in cases where there may be moderate heterogeneity of these effects.
Covariate adjusted estimation of unconditional effects
The guidance indicates trials could use methods which exploit baseline covariates for improved power but still target the marginal or unconditional treatment effect (e.g. Moore and van der Laan 2009). The guidance even helpfully gives a recipe for how to construct such estimators, and by doing so demonstrates they are pretty straightforward to implement. One thing the guidance does not mention is how to obtain valid inferences using such methods when using stratified randomisation and when working outcome models may possibly misspecified. Fortunately, Wang et al 2019 recently extended the aforementioned results of Bugni et al 2018 to show this can be achieved.
Marginal effects and transporting
One criticism of marginal effects, or at least their estimation of them from randomised trials, is that their estimation using standard methods implicitly relies on an assumption that the patients in the trial are a representative sample from the population of interest. As has been helpfully pointed out to me in the past (e.g. by Stephen Senn and Frank Harrell), this is never really the case. Given this, and the fact that the magnitude of marginal effects can change if you were to modify the population definition, this raises some doubts about the interpretation of estimates of marginal effects from randomised trials. In this regard, there has been quite a lot of work (which so far I am only somewhat familiar with) which looks at how to combine data from a trial with external information about the target population of interest, in order to estimate the effect in the target population. Recent papers on these developments include Ackerman et al 2020 and Dahabreh et al 2020.