Inverse Probability Weighting of Count Exposures in the Presence of Missing Data: A Simulation Study

Abstract

Inverse probability of treatment weighting (IPTW) is widely used to estimate causal effects, but guidance is limited for count exposures. It is also unclear how IPTW performs when combined with multiple imputation in this context. In this study, we evaluated five IPTW methods applied to count exposures: multinomial binning, parametric and non-parametric covariate balancing propensity scores (CBPS, npCBPS), generalised boosted models (GBM), and energy balancing. Our simulations were informed by an example using data from the 1970 British Cohort Study, aiming to estimate the effect of psychological distress, measured as a count of symptoms at age 34, on self-reported longstanding illness at age 42. We compared these approaches on bias, coverage, effective sample size, and other metrics under truncated negative binomial and Poisson exposure distributions. We also assessed the performance of Rubin's rules under different missingness mechanisms. Under complete data, multinomial, CBPS, GBM, and energy weights produced low bias and near-nominal coverage, whereas npCBPS resulted in bias and poor coverage due to extreme weights. When data were missing completely at random, similar performance patterns were observed for IPTW with multiple imputation. Under missing at random, bias increased with higher missingness, but this was present for both IPTW and covariate-adjusted regression, possibly reflecting a limitation of the imputation model rather than a failure of IPTW. Overall, these findings support the use of multinomial, CBPS, GBMs, and energy weights for count exposures in similar settings while highlighting trade-offs between these methods and the need for imputation models accommodating right-truncated overdispersed counts.

Figures (5)

• Figure 1: Histograms of the Malaise score and the simulated exposures. A, Observed Malaise score at age 34 in the 1970 British Cohort Study (N = 9 596; range 0--9). B, Simulated exposure (N = 1,000,000) from a conditional negative binomial distribution, truncated by resampling values greater than 10. C, Simulated exposure (N = 1,000,000) from a conditional Poisson distribution, truncated by resampling values greater than 10. BCS70 -- 1970 British Cohort Study; NegBin -- Negative Binomial.
• Figure 2: Balance plot for the motivating example. Points show weighted or unadjusted Pearson and point-biserial correlations between the exposure and each covariate; horizontal whiskers show the range of the correlations across 94.0 multiply imputed datasets. The npCBPS method failed to achieve balance for several covariates and produced wide correlation ranges, so its results are omitted here for visual clarity; a corresponding balance plot including all methods is provided in Figure \ref{['fig:bcs70_balance_npcbps']} of the Supplementary Information. Data are from the 1970 British Cohort Study (N = 16638.0). (np)CBPS -- (Non-Parametric) Covariate Balancing Propensity Score; GBM -- Generalised Boosted Models.
• Figure 3: The forest plot shows risk ratios and 95% confidence intervals for longstanding illness per 4-point higher Malaise Inventory score from unadjusted, inverse probability of treatment weighted, and covariate-adjusted Poisson regression models, pooled across 94.0 multiply imputed datasets (sample N = 16638.0). The unadjusted model gives the largest estimate (risk ratio 1.79); all weighted and covariate-adjusted analyses give similar risk ratios (ranging from 1.50--1.53 across the methods). Winsorisation of weights at the 99th percentile yielded slightly higher estimates than the corresponding raw weights. CI widths are similar across methods, except for npCBPS, where the wide CI (1.08--2.10) reflects extreme weights and is substantially reduced after winsorisation. (np)CBPS -- (Non-Parametric) Covariate Balancing Propensity Scores; GBM -- Generalised Boosted Models.
• Figure S1: A heatmap comparing relative computation times of the five weight estimation methods. Pairwise relative times (ratios of CPU times) were obtained for each of the 20 simulated datasets, each containing N = 5,000 observations. Geometric means of these ratios were taken across all datasets for each pairwise comparison. A higher relative time indicates that the numerator method was slower against the method in the denominator. All methods were implemented using the WeightIt R package. The CPU time of the multinomial approach included preprocessing of the exposure by collapsing categories with prevalences below 1%. (np)CBPS, (Non-Parametric) Covariate Balancing Propensity Score; GBM, Generalised Boosted Models.
• Figure S2: Balance plot for the motivating example including npCBPS alongside other approaches. Points show weighted or unadjusted Pearson and point-biserial correlations between the exposure and each covariate; horizontal whiskers show the range of the correlations across 94.0 multiply imputed datasets. Data are from the 1970 British Cohort Study (N = 16638.0). (np)CBPS -- (Non-Parametric) Covariate Balancing Propensity Score; GBM -- Generalised Boosted Models.