There exists a comprehensive framework for estimating various causal effects of time-varying treatments from observational cohort data assuming that all confounders are measured.1,2 Of course, this is an untestable assumption, and in many observational studies at least some important confounders are unobserved. In the trend-in-trend design,3 the authors provide a clever method for eliminating bias from a broad category of unmeasured confounders (including all time-invariant baseline confounders such as place of birth) as long as there is a calendar time trend in treatment prevalence and certain additional assumptions hold. Other designs attempt to strike a similar bargain, that is, they also alter the standard cohort assumptions to achieve control of unmeasured confounding. The authors mention that the trend-in-trend is related to the difference-in-difference method and calendar time as an instrumental variable. There is also a class of “self-controlled” methods (including the case-crossover,4 case-time-control,5 and self-controlled case series6) that seek to adjust for unobserved baseline confounding by essentially using subjects as their own controls. Maclure et al.7 provide a useful overview of the use of self-controlled designs in pharmacoepidemiology. Here I survey the assumptions required by the trend-in-trend design and the various self-controlled methods. Each method’s specific combination of strengths and assumptions defines unique terrain where it is most suitable, even as all methods make some similar general tradeoffs in pursuit of similar goals.
The trend-in-trend design makes the following assumptions (in no particular order): (a) there is a constant instantaneous subject-specific treatment effect, which is the estimand; (b) the individual level outcome model at each person-time is a linear logistic regression in exposure, calendar time, and the set of measured and unmeasured intrinsic covariates that influence the exposure and/or outcome; (c) the outcome model given exposure, calendar time, and stratum is a logistic regression linear in exposure, calendar time, and an exposure–stratum interaction; (d) there is a strong population level calendar time trend in treatment prevalence; (e) intrinsic covariates and calendar time have a multiplicative effect on probability of exposure; (f) the outcome is rare; and (g) there are no calendar time trends in confounders within strata. [Note that assumptions (a), (d), and (g) together imply that confounders cannot be affected by past exposure.] The length of this list suggests that eliminating bias from unmeasured confounding does not come cheaply. Unlike the usual cohort assumptions, these assumptions impose homogeneity and transience on the exposure effect, place restrictions on the type of measured confounding that might exist, and specify functional forms of outcome models. However, assumptions (c), (d), and (f) can be checked, and, in a given context, the other assumptions might easily be more palatable than the extremely strong assumption of no unobserved confounding. The question then becomes, in a context where unobserved confounding is of primary concern, how do the assumptions made by the trend-in-trend design compare to the assumptions of other methods that seek to eliminate bias from unobserved baseline confounding?
The case-crossover4 is one such method. It is a case-only design, meaning it only incorporates data from subjects who experience the outcome at some time (the cases). In its simplest incarnation, for each person-time at which an outcome occurs, a preceding “control” time is chosen from the same subject. The case–control pairs are then used to compute the Mantel–Haenzsel estimator of a common odds ratio. By matching person-times on subject, confounding by baseline variables (which are of course invariant within subjects) is eliminated.
The case-crossover and trend-in-trend designs make a lot of similar assumptions. The case-crossover estimator targets the assumed constant instantaneous subject-specific hazard ratio,8 which, in a rare outcome survival context, is the same estimand as in the trend-in-trend design (see assumption (a)). Both methods appear to depend on an assumption that exposure at time t has no direct effect on future outcomes at time t + 1 or later. In the case-crossover literature, this assumption is called treatment transience. In the trend-in-trend design, the assumption appears to be implicit in the omission of past treatment history from the person-time level outcome model. Measured time-varying confounders can be adjusted for in the case-crossover design, but as in the trend-in-trend, there are restrictions on these confounders. In particular, time-varying confounders must have transient effects on the outcome and either not be affected by past exposure or not be affected by any unobserved causes of the outcome.8 The case-crossover further assumes that there is no unmeasured time-varying confounding, whereas the trend-in-trend design allows for unmeasured time-varying confounding as long as the distribution of confounders does not vary with calendar time within strata.
Unlike the trend-in-trend, the case-crossover estimator does not make any assumptions about functional forms of causal or distributional models. However, when the effect of exposure is nonzero, an additional necessary assumption in the case-crossover design (with no apparent counterpart in the trend-in-trend) is that there are no unobserved time-varying common causes of the outcome at different person-times within the same subject.8 And where the trend-in-trend requires an ecologic time trend in exposure, the case-crossover prohibits time trends in exposure within subjects. As within subject time trends will usually accompany ecologic time trends, the case-crossover can rarely be applied in any situation where the trend-in-trend might be applied.
The case-time-control design5 is a modification of the case-crossover method intended to accommodate situations in which there is a time trend in exposure. In the simplest version, subjects are sampled as in a standard case–control design. Within each subject, two person-times are sampled—one from the “current” time period, which is when the outcome occurs in the cases, and one from a preceding “reference” time period. The estimand remains the constant instantaneous subject-specific rate ratio, and it is still assumed that there are no unmeasured time-varying confounders. But unlike the case-crossover, the case-time-control design makes modeling assumptions. Specifically, let i denote group (1 = case, 0 = control), j denote period (1 = current period, 0 = reference period), k denote outcome (1 = event, 0 = no event), l index subjects within group, and Eijkl denote the exposure status of subject l in group i with outcome k at time j. Then the case-time-control method assumes that
This model is very similar to the subject-specific model underlying the trend-in-trend design. μ here maps to β0 there; θ1−θ0 here is the exposure effect of interest, which is β1 there; π1−π0 here is the period effect of time on exposure, whereas β2 there is the effect of time on the outcome; and sil is a subject level random effect capturing all time-invariant confounding as γTXti captures measured and unmeasured confounding in the trend-in-trend. From the cases alone it is possible to identify (θ1 − θ0) + (π1 − π0), and from the controls alone it is possible to identify π1 − π0. Thus, it is possible to identify θ1 − θ0 by a difference in differences approach. An important restriction imposed by (1) is that the period effect on exposure is the same among cases and controls. If baseline variables that influence the probability of a subject becoming a case also influence subject-specific exposure trends, then (1) will be misspecified. This restriction is similar in substance to assumption (e) of the trend-in-trend design, which also excludes the possibility that the (multiplicative) effect of time on exposure varies with covariates that influence the outcome. (A modification of the case time control called the “case-case-time-control”9 uses control-time from future cases to mitigate concerns that exposure trend might be different in cases than controls.)
The self-controlled case series6 is another case-only design like the case-crossover. It makes the modeling assumption that the data are generated by a nonhomogeneous Poisson process with rate
where i indexes subjects, j indexes exposures within subjects, and k indexes time period from subject baseline. Φi is a subject level random effect capturing the multiplicative effects of all baseline covariates, αk is the time period effect, and βj are the exposure effects of interest. Model (2) has similar parallels to the individual level trend-in-trend outcome model as those enumerated for model (1). Under model (2), the effect of the jth exposure period is constant and instantaneous across subjects and time, but because the effect can depend on j the exposure effects are not quite transient in the same sense as the other methods considered here. The self-controlled case series estimates αk and βj by maximizing the likelihood conditional on total number of events and exposure history in each subject under the assumption that exposures and observation periods are independent of past outcomes. Under this “no outcome-dependent exposure” assumption, only subjects with at least one event contribute to the conditional likelihood, making the self-controlled case series estimator a case-only estimator. And Φi drops out of the conditional likelihood contribution of subject i, which eliminates unmeasured baseline confounding. When considering serious outcomes (e.g., myocardial infarctions as in an application of the trend-in-trend design presented in ), exposures are very likely to be influenced by past occurrences of the outcome. Farrington et al.10 present a modified self-controlled case series estimator that retains the properties of the original under event-dependent exposures as long as exposure periods are short. In cases where a drug is taken for a long time, the short exposure period assumption would fail. Also, while observed time-varying confounders can be added to model (2) to adjust for measured time-varying confounding, unobserved time-varying confounders would lead to bias.
To summarize this accounting of assumptions, the trend-in-trend works under largely different circumstances than the self-controlled case series (which fails if there are long exposure periods and major outcomes) or case-crossover (which fails when there are time trends in exposure), but similar circumstances to the case time control. The case time control imposes fewer modeling assumptions. However, like the other self-controlled methods, case time control only adjusts for unmeasured baseline confounding. Perhaps the most important advantage of the trend-in-trend design is that it can adjust for unobserved time-varying confounders (as long as they do not have calendar time trends within strata).
Zooming out, another take-away is the generally similar flavor of the assumptions made by these designs. In particular, it is interesting that all of them target a constant, transient (or “semitransient” in the case of self-controlled case series), instantaneous, and subject-specific effect. In the trend-in-trend, case time control, and self-controlled case series, this is a natural consequence of defining the effect as the coefficient of instantaneous exposure in a linear model containing time and essentially all baseline variables as covariates. The case-crossover does not specify any model, however, and yet still estimates this quantity. It might be useful to identify the precise source of this estimand’s magnetic pull in methods that allow for unobserved confounding, because the estimand is often not of primary public health interest.11
ABOUT THE AUTHOR
ZACH SHAHN is a postdoctoral fellow working on problems in causal inference for time-varying treatments at the Harvard School of Public Health. He is the author of a forthcoming paper that places the case-crossover design in a formal counterfactual framework.
1. Robins J. A new approach to causal inference in mortality studies with a sustained exposure period—application to control of the healthy worker survivor effect. Math Model. 1986;7:1393–1512.
2. Robins JM, Hernán MA. Fitzmaurice G, Davidian M, Verbeke G, Molenberghs G. Estimation of the Causal Effects of Time-Varying Exposures. In:Longitudinal Data Analysis. 2009.New York, NY: Chapman and Hall/CRC Press.
3. Ji X, Small DS, Leonard CE, Hennessy S. The trend-in-trend research design for causal inference. Epidemiology 2016;28:529–536.
4. Maclure M. The case-crossover design: a method for studying transient effects on the risk of acute events. Am J Epidemiol. 1991;133:144–153.
5. Suissa S. The case-time-control design. Epidemiology. 1995;6:248–253.
6. Farrington P, Pugh S, Colville A, et al. A new method for active surveillance of adverse events from diphtheria/tetanus/pertussis and measles/mumps/rubella vaccines. Lancet. 1995;345:567–569.
7. Maclure M, Fireman B, Nelson JC, et al. When should case-only designs be used for safety monitoring of medical products? Pharmacoepidemiol Drug Saf. 2012;21(Suppl 1):50–61.
8. Shahn Z, Hernan M. A formal counterfactual treatment of the case crossover design. Technical Report. 2016.
9. Wang S, Linkletter C, Maclure M, et al. Future cases as present controls to adjust for exposure trend bias in case-only studies. Epidemiology. 2011;22:568–574.
10. Farrington CP, Whitaker HJ, Hocine MN. Case series analysis for censored, perturbed, or curtailed post-event exposures. Biostatistics. 2009;10:3–16.
11. Robins JM, Greenland S, Hu FC. Estimation of the causal effect of a time-varying exposure on the marginal mean of a repeated binary outcome. J Am Stat Assoc. 1999;94:687–700 (See section 7).