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Re. Trends in Control of Unobserved Confounding

Ertefaie, Ashkan; Small, Dylan S.; Leonard, Charles E.; Ji, Xinyao; Hennessy, Sean

doi: 10.1097/EDE.0000000000000890
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Department of Biostatistics and Computational Biology, University of Rochester, Rochester, NY, ashkan_ertefaie@urmc.rochester.edu

Department of Statistics, University of Pennsylvania, Philadelphia, PA

Center for Pharmacoepidemiology Research and Training, Center for Clinical Epidemiology and Biostatistics, Perelman School of Medicine at the University of Pennsylvania, Philadelphia, PA

Department of Statistics, University of Pennsylvania, Philadelphia, PA

>Center for Pharmacoepidemiology Research and Training, Center for Clinical Epidemiology and Biostatistics, Perelman School of Medicine at the University of Pennsylvania, Philadelphia, PA

Code for replication: The computer code is available as an R package “TrendInTrend” on CRAN.

Supported by National Science Foundation (NSF grant SES-1260782).

The authors report no conflicts of interest.

Supplemental digital content is available through direct URL citations in the HTML and PDF versions of this article (www.epidem.com).

We appreciate thoughtful commentary by Shahn1 on the use of methods such as the newly developed trend-in-trend design2 to control for unmeasured confounding. We would like to clarify two of the assumptions that Shahn enumerated as underlying this research design.

Assumption (b) enumerated by Shahn is that “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.”1 While the trend-in-trend design does require the outcome to be logistic with respect to some specified function of covariates, that function does not need to be linear, even though that was the functional form used in the original paper.2 Any specified function will suffice to derive the population-average model that is obtained by integrating out the set of measured and unmeasured covariates in the individual-level outcome model.

Assumption (g) enumerated by Shahn is that “there are no calendar time trends in confounders within strata.”1 This is stated slightly more strictly than is actually needed. In truth, the design is unbiased as long as any trends in the prevalence of measured or unmeasured causes of the outcome are equal across strata defined by the cumulative probability of exposure, and unmeasured confounders over time can be modeled as depending on time-invariant latent variables and independent, identically distributed time-varying variables. In the eAppendix; http://links.lww.com/EDE/B380, we rigorously justify this relaxation and prove the unbiasedness of the trend-in-trend design under this less restrictive assumption. Moreover, Ji et al2 presented simulated scenarios (Table 3) in which covariates were serially correlated, and the results remained unbiased.

We would therefore propose a friendly amendment to the list of assumptions underlying the trend-in-trend design, as follows: (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 logistic regression with respect to some specified function exposure, calendar time, and the set of measured and unmeasured factors that influence the exposure and/or outcome; (c) the outcome model given exposure, calendar time, and stratum is a logistic regression that is 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 at baseline and calendar time have a multiplicative effect on probability of exposure; (f) the outcome is rare; and (g) any time trends in the prevalence of confounders are equal across strata of the cumulative probability of exposure. As noted by Shahn, assumptions (c), (d), and (f) can be assessed empirically for any given application of the method.

Ashkan Ertefaie

Department of Biostatistics

and Computational Biology

University of Rochester

Rochester, NY

ashkan_ertefaie@urmc.rochester.edu

Dylan S. Small

Department of Statistics

University of Pennsylvania

Philadelphia, PA

Charles E. Leonard

Center for Pharmacoepidemiology Research

and Training

Center for Clinical Epidemiology and Biostatistics

Perelman School of Medicine at the University

of Pennsylvania, Philadelphia, PA

Xinyao Ji

Department of Statistics, University of Pennsylvania

Philadelphia, PA

Sean Hennessy

Center for Pharmacoepidemiology Research

and Training

Center for Clinical Epidemiology and Biostatistics

Perelman School of Medicine at the University

of Pennsylvania

Philadelphia, PA

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REFERENCES

1. Shahn Z. Trends in control of unobserved confounding. Epidemiology. 2017;28:537–539.
2. Ji X, Small DS, Leonard CE, Hennessy S. The trend-in-trend research design for causal inference. Epidemiology. 2017;28:529–536.

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