Time-fixed vs time-varying causal diagrams for immortal time bias

Abstract

We thank Shrier and Suissa1 for their interest in our editorial2 on causal diagrams3–7 for immortal time bias (ITB) and for their detailed analysis of our causal diagrams. They proposed a time-varying approach for these diagrams and concluded that the structure of ITB for both exposure misclassification and exclusion scenarios is selection (collider-stratification) bias. We chose to use simpler causal diagrams in our editorial in order to explain the key concepts without confusing the readers who might be less familiar with these diagrams, but we welcome the chance now to consider the more detailed analysis of ITB using time-varying diagrams. Here, we first explain why we believe that our time-fixed causal diagrams are correct. Then we will demonstrate that Shrier and Suissa’s presentation and interpretation of time-varying causal diagrams for ITB are incorrect, and also provide the correct time-varying diagrams that are actually equivalent to the time-fixed diagrams originally presented in our editorial. The outcome variable for time-to-event analysis is (T, Y), referring to time at risk and failure indicator, respectively. In practice, T is measured discretely (e.g. daily) and so an equivalent representation of the outcome would be (Y1, Y2,…, Yt) where the subscript t should be as fine as the actual intervals of time measurement. In our example of a sequentially randomized trial without censoring and noncompliance, the sequential representation of the exposure/outcome was unnecessary. Therefore, we used the simpler and more compact alternative with (Y, T) as the outcome. We considered each participant with time-varying exposure, switching from A = 0 to A = 1, as two pseudo-participants with appropriate splitting of T at the time of switching. For the first pseudo-participant, A = 0, Y = 0 and T equals the immortal time for the actual participant. Under this time-fixed conceptualization, ITB refers to either exclusion or exposure misclassification of the first pseudo-participants who are unexposed and immortal. For simplicity, we have omitted T from the diagram. Thus, the time-fixed causal diagrams in our editorial, Figure 1a and b in the letter, correctly represent the setting in which ITB occurs. Also there is no confounding bias in our figures representing correct handling of ITB (Figure 1c and d in the letter) if exposure affects outcome as claimed by Shrier and Suissa. Immortal time bias occurs under the causal null hypothesis of no effect of exposure on the outcome. Suppose NE and NU are the number of events in the exposed and unexposed groups in a sequentially randomized trial, and TE and TU are follow-up person-times in these groups, respectively. Let TI (<TE) denote the sum of the immortal times, i.e. person-times not under exposure in the exposed group. Under the assumption of constant hazard, the consistent estimators of the causal rate ratio (RRc), exclusion rate ratio (RRe) and misclassification rate ratio (RRm) will be NE/(TE-TI)NU/(TU+TI)⁠, NE/(TE-TI)NU/TU and NE/TENU/TU⁠, respectively. We assume a very large sample so that these estimators well approximate corresponding parameters. The causal null hypothesis implies RRc=1⇒RRe=RReRRc=NE/(TE-TI)NU/TUNE/(TE-TI)NU/(TU+TI)=TUTU+TI⇒0<RRe<1 as TU, TI > 0. RRm=RRmRRc=NE/TENU/TUNE/(TE-TI)NU/(TU+TI)=TUTU+TI×TE-TITE=RRe×TE-TITE⇒0<RRm<RRe as TE > TI > 0. In sum, RRc=1 implies 0<RRm< RRe<1, i.e. ITB is present, in favour of the exposed group and more severe bias in the misclassification than exclusion, even under the causal null hypothesis. We believe that the time-varying causal diagrams presented by Shrier and Suissa suffer from several important errors. First, their Figure 2a and b incorrectly suggest that ITB would not occur under the causal null hypothesis because exclusion of the arrow from A0 to Y0+ will make Y0+ a non-collider; hence collider-stratification bias will no longer exist. As mentioned in our editorial and mathematically shown in Figure 1, exclusion or misclassification of the immortal times will introduce bias even under the causal null hypothesis.2 Second, the bias arisen as a result of conditioning on Y0+ in Figure 2a mentioned in the letter (but not shown in the figure) is unrelated to ITB and in fact represents selection bias inherent in the hazard ratio,8 which remains even in the correct handling of ITB, time-dependent analysis and sequential approach (Figure 2c and d in our editorial). Third, selection variable S in their Figure 2b should be a function of A0 and A1 as the immortal times in the participants with A0 = 0 and A1 = 1 are excluded; Y0+ affects selection through A1. Fourth, the same Figure 2b represents both misclassification (by including A*) and exclusion (by including S with a square around), which is impossible as the immortal times are mistakenly either misclassified or excluded. Fifth, the variable treatment assignment (not needed to be on the figure) as well as variable A* should also be time-varying. Time-fixed and time-varying causal diagrams representing two types of immortal time bias under the causal null hypothesis in a sequentially randomized trial: misclassification [(a) and (c)] and exclusion [(b) and (d)]. In time-fixed causal diagrams, the variables A and Y denote exposure and outcome, U is an unmeasured risk factor for the outcome, and A* and E represent misclassification and exclusion of immortal times. Under these diagrams, each exposed person effectively contributes as two persons: unexposed (A = 0) before receiving the exposure, and exposed (A = 1) after receiving. In time-varying causal diagrams showing only two visits, the variables A0 and A1 denote the exposure in Visits 0 and 1 (assumed to be constant until the next visit), Y0+ and Y1+ denote the outcome between Visit 0 and Visit 1, and Visit 1 and the next one, and A0* and E0 represent misclassification and exclusion of the immortal times Tvisit 1 − Tvisit 0. For simplicity, the time variables accompanying Y, Y0+ and Y1+ have been omitted. The time-fixed causal diagrams for ITB in our editorial have been reproduced in Figure 2a and b. The correct time-varying causal diagrams for ITB under the causal null hypothesis (where A0 is independent of Y0+ and Y1+, and also conditional on Y0+, A1 is independent of Y1+) are presented in Figure 2c and d. The variables A0* and E0, representing misclassification and exclusion of the immortal times Tvisit 1 − Tvisit 0, are functions of A0 and A1, hence the arrows from these variables to A0* and E0. ITB is measurement bias in Figure 2c representing the misclassification scenario as A0* and Y0+ are associated via A1. ITB is selection bias9 in Figure 2d, representing the exclusion scenario as conditioning on E0, the descendant of the collider A1 on the path A0→A1←Y0+, introduces a non-causal association between A0 and Y0+. The time-fixed Figure 2a and b are in fact equivalent to the time-varying Figure 2c and d with the following changes: the variables A0/A1 and Y0+/Y1+ were collapsed into A and Y, and Y0+ is now replaced by the variable U (not included in the time-varying diagrams), the cause of Y0+ and Y1+. The equivalence of the time-fixed (Figure 2a and b) and time-varying diagrams (Figure 2c and d) provides further credence to the accuracy of our work.2,10 M.A.M. produced the first draft, and M.N. and M.E. proposed critical revisions. All authors approved the final version of the manuscript. There was no conflict of interest.

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