Performance of the marginal structural cox model for estimating individual and joined effects of treatments given in combination

Abstract

Background: The Marginal Structural Cox Model (Cox-MSM), an alternative approach to handle time-dependent confounder, was introduced for survival analysis and applied to estimate the joint causal effect of two time-dependent nonrandomized treatments on survival among HIV-positive subjects. Nevertheless, Cox-MSM performance in the case of multiple treatments has not been fully explored under different degree of time-dependent confounding for treatments or in case of interaction between treatments. We aimed to evaluate and compare the performance of the marginal structural Cox model (Cox-MSM) to the standard Cox model in estimating the treatment effect in the case of multiple treatments under different scenarios of time-dependent confounding and when an interaction between treatment effects is present.

Methods: We specified a Cox-MSM with two treatments including an interaction term for situations where an adverse event might be caused by two treatments taken simultaneously but not by each treatment taken alone. We simulated longitudinal data with two treatments and a time-dependent confounder affected by one or the two treatments. To fit the Cox-MSM, we used the inverse probability weighting method. We illustrated the method to evaluate the specific effect of protease inhibitors combined (or not) to other antiretroviral medications on the anal cancer risk in HIV-infected individuals, with CD4 cell count as time-dependent confounder.

Results: Overall, Cox-MSM performed better than the standard Cox model. Furthermore, we showed that estimates were unbiased when an interaction term was included in the model.

Conclusion: Cox-MSM may be used for accurately estimating causal individual and joined treatment effects from a combination therapy in presence of time-dependent confounding provided that an interaction term is estimated.

Keywords: Causal inference; Longitudinal data; Marginal structural models; Multitherapy; Time-dependent confounding.

Fig. 1

Causal directed acyclic graphs corresponding to the structure of simulated data. A₁ and A₂ are the treatments, L is the time-dependent confounder and Y is the outcome. Case 1 and 2 considered all relationship between A₁, A₂ and L. The time-dependent Confounder was strongly associated to the treatments A₁ and A₂ in the case 1 whereas it was weakly associated to the treatments A₂ in the case 2. Coefficients of the time-dependent confounder in the functions of treatment prediction were set to 0.004 and 0.001, respectively. Case 3: relationship between A₂ and L were not considered. Data were simulated from a marginal structural model as the confounding in the exposures-outcome relationship arises via T₀ as follows: Y (m + 1) ← T₀ → L (m) → A₁ (m), Y (m + 1) ← T₀ → L (m) → A₂ (m)

Fig. 2

Bias and coverage rate of treatment effects estimates for the sub-cases 1A, 1B, 1C, 1D and 1E

Fig. 3

Bias and coverage rate of treatment effects estimates for the sub-cases 2A, 2B, 2C, 2D and 2E

Fig. 4

Bias and coverage rate of treatment effects estimates for the sub-cases 3A, 3B, 3C, 3D and 3E

Fig. 5

Bias of treatment effects estimates for the sub-cases 1B and 1D according to whether interaction was estimated in the model

Fig. 6

Distribution of stabilized weights related to PIs and other ARVs