Discussion of “A formal causal interpretation of the case‐crossover design”

Discussion on “A formal causal interpretation of the case‐crossover design” by Zach Shahn, Miguel A. Hernan, and James M. Robins


INTRODUCTION
We congratulate Shahn, Hernán, and Robins for giving a formal counterfactual treatment of the case-crossover design. In their interesting piece, they clarify assumptions and interpretation, and also identify a potential bias of a classical estimator when the effect is nonnull.
Our main points are as follows: (i) Adding control-crossover to eliminate the need for "no time trends in treatment." The case-crossover design is classically seen as evaluating the exposure odds at the case time ( ) relative to that at an earlier control period ( − 1 , for example, Hernández-Díaz et al., 2003). Table 1 (left  panel) shows the resulting data where, in the notation of the present article, = 1, = ∑ ℎ ( 1 ℎ = 1, 0 ℎ1 = 0), and = ∑ ℎ ( 1 ℎ = 0, 0 ℎ1 = 1). The casecrossover odds ratio estimator is thenˆ1 = ∕ and may be obtained using conditional logistic regression. This estimator is sensitive to time trends in exposure and a time-adjusted estimate for the odds ratio may be obtained using additional data on noncases ascertained at the same times ( and − 1 ) as the cases (the case-time-control design), see Table 1, right panel. Here, the time trend in exposure may be estimated byˆ2 = * ∕ * and an adjusted estimator for the parameter of interest, that is, the odds of failure This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2022 The Authors. Biometrics published by Wiley Periodicals LLC on behalf of International Biometric Society.
given exposure relative to that under nonexposure, iŝ =ˆ1∕ˆ2 or, preferably, this parameter may be estimated using conditional logistic regression based on both cases and time-matched noncases. A crucial assumption in the present paper is the "No Time Trends in Treatment" assumption (9). A natural question is now whether this assumption could be avoided if additional data on noncases were sampled from the same full cohort in the obvious way, that is, as follows. Select, for each case, one subject ℎ with ℎ = 0 and ℎ( −1) = 0 and ascertain the exposure 1 ℎ , 0 ℎ at times , = − 1 , … , − . The equivalent ofˆ2 would then be the ratiô and the question is then whether a ratio likê ∕ˆ2 can be shown to estimate the same causal parameter as in the present paper without assumption (11)? (ii) Do other and more efficient estimators exist for the considered estimand? This paper gives a formal framework for a wellknown estimator showing, under appropriate conditions, that it converges to a certain target parameter, which is a (constant) causal hazard ratio.
Once an estimand has been defined, it is interesting TA B L E 1 Classical exposure data from cases and time-matched noncases ascertained at the case period ( ) and at an earlier control period ( − 1 ) to speculate whether more efficient estimators can be developed and, ultimately, whether an overall most efficient estimator exists. In this search, it is of interest to seek the so-called efficient influence function.

Cases Time-matched noncases
Various approaches exist to find this function with some being easier to apply than others depending on the given context. In the specific setting where an estimator already exists, we wonder whether one could first derive its corresponding influence function and next project it into the tangent space? (iii) Collapsibility.
An effect measure is said to be collapsible when the marginal exposure effect, as described by this effect measure, is identical to the corresponding conditional measure when the exposure and the variable that is conditioned on are independent. This is, for instance, the case for the risk difference and the risk ratio, whereas the odds ratio does not have this property. It is also well known that the (discrete time) rate difference and rate ratio are generally noncollapsible, but that the hazard difference is collapsible (Martinussen & Vansteelandt, 2013;Sjölander et al., 2016). In this respect, it makes a difference when moving from discrete time to continuous time and when working with rate/hazard differences as pointed out by Sjölander et al. (2016). For the rate ratio, the situation is different as it remains noncollapsible in continuous time. It is therefore surprising that the hazard ratio considered in the current paper is claimed to be collapsible.
Maybe the authors wish to expand on this? Also, we wonder what happens if time intervals are made smaller and smaller eventually progressing to the continuous time case. (iv) This brings to our final point. We wonder whether or not it is possible to frame the problem in continuous time?