An Application of Inverse Probability Weighting Estimation of Marginal Structural Models of a Continuous Exposure

Abstract

To the Editor: Inverse probability weighting of marginal structural models has been proposed as a method to control for time-dependent confounding in settings where confounders are affected by previous exposure, but has generally been limited to settings where exposure is binary.1 When exposure is continuous, the correct distributional form for the exposure model needs to be specified and possible nonconstant exposure variance (i.e., heteroscedasticity) and outliers need to be considered.2 Recently, simulations have demonstrated methods to construct inverse probability weights for continuous exposure using a normal distribution, a gamma distribution, or a quantile binning approach, but empirical examples are scarce.2 We applied each of these three methods to estimate the effect of (continuous) benzodiazepine exposure on delirium in a large cohort of critically ill adults.3 Benzodiazepine exposure, presence of delirium, mortality or discharge, and 10 time-varying confounders were recorded daily and seven confounders were measured at intensive care unit admission. We adopted the normal, gamma, and an adaptation of the quantile binning exposure models based on Naimi et al.,2 to estimate stabilized inverse probability weights. Benzodiazepine exposure was modeled using the zero-inflated Poisson distribution. The stabilized inverse probability weights generated were used to estimate marginal structural models of the average causal effect of benzodiazepine exposure on delirium risk. For comparison, “ordinary” confounding adjustment was performed by including time-fixed and time-varying confounders as covariates in a regression model. Benzodiazepines exposure was highly skewed (median daily midazolam equivalent dose = 6.6 mg [range: 0.1–858.7, IQR: 57.5]). Differences between inverse probability weighting methods (i.e., mean and range of the weights) were most apparent in the untruncated weight distributions (Table). For example, inverse probability weights from quantile binning with eight bins (mean 1.06, min 0.05, max 89.3) had a smaller range compared with the normal inverse probability weights (mean 2.58, min 0.01, max 1488.13). Despite the variation in untruncated weights, application of these weights yielded similar effect estimates for each method of which all were closer to the null (odds ratio = 1) compared with “ordinary” adjustment.TABLE: Estimates of the Relation Between Benzodiazepine Use and Delirium for Different ModelsIn inverse probability weighting of marginal structural models, a trade-off exists between confounding bias reduction and increased bias and variance due to nonpositivity.4 To investigate this trade-off weights were progressively truncated for each method, and as levels of weight truncation increased, the marginal structural model effect estimates became more similar to the estimate from ordinary adjustment for time-varying covariates (Table). The detailed analysis is presented in the eAppendix (https://links.lww.com/EDE/A939). To assess the relation between weight truncation and confounder imbalance, the inverse probability weights were used to fit a weighted exposure model with the R2 expected to be approximately zero.5 To find the level of truncation leading to a minimal confounder imbalance (lowest R2), this analysis was repeated for different levels of weight truncation. The normal exposure model (before weighting) had a R2 of 0.51, while refitting this exposure model with untruncated weights resulted in an increased R2 of 0.54. This indicates that untruncated weights from the normal model increased confounder imbalance. For the normal inverse probability weighted model, the minimum imbalance was observed at 4% truncation (R2 = 0.124). This indicates that there is still an association between confounders and exposure after weighting which may be due to exposure model misspecification. Noting the level of imbalance estimated by the weighted exposure model only quantifies imbalance with respect to that particular specification and distributional form of the exposure model, methods to assess balance with respect to separate confounders could be used.6 Despite large differences in weight distributions between methods for obtaining inverse probability weighting for continuous exposures, different methods yielded similar exposure–outcome effect estimates. However, these results should be interpreted cautiously since large untruncated weights, and remaining imbalance after truncation, suggest that nonpositivity (possibly with respect to a combination of confounders) may invalidate estimates. ACKNOWLEDGMENTS The authors thank P. M. C. Klein Klouwenberg, MD, PharmD, and W. Pasma, DVM, Department of Intensive Care Medicine, University Medical Centre Utrecht, Utrecht, The Netherlands, for their support and assistance in data acquisition and data management. C. Marijn Hazelbag Julius Center for Health Sciences and Primary Care UMC Utrecht Utrecht, The Netherlands [email protected] Irene J. Zaal Intensive Care UMC Utrecht Utrecht, The Netherlands John W. Devlin School of pharmacy Northeastern University Boston, MA Nicolle M. Gatto WSR Epidemiology Pfizer Inc New York, NY Department of Epidemiology Mailman School of Public Health Columbia University New York, NY Arno W. Hoes Julius Center for Health Sciences and Primary Care UMC Utrecht Utrecht, The Netherlands Arjen J. C. Slooter Intensive Care UMC Utrecht Utrecht, The Netherlands Rolf H. H. Groenwold Julius Center for Health Sciences and Primary Care UMC Utrecht Utrecht, The Netherlands