Inverse Probability Weights for Quasicontinuous Ordinal Exposures With a Binary Outcome: Method Comparison and Case Study

Abstract

Inverse probability weighting (IPW), a well-established method of controlling for confounding in observational studies with binary exposures, has been extended to analyses with continuous exposures. Methods developed for continuous exposures may not apply when the exposure is quasicontinuous because of irregular exposure distributions that violate key assumptions. We used simulations and cluster-randomized clinical trial data to assess 4 approaches developed for continuous exposures-ordinary least squares (OLS), covariate balancing generalized propensity scores (CBGPS), nonparametric covariate balancing generalized propensity scores (npCBGPS), and quantile binning (QB)-and a novel method, a cumulative probability model (CPM), in quasicontinuous exposure settings. We compared IPW stability, covariate balance, bias, mean squared error, and standard error estimation across 3,000 simulations with 6 different quasicontinuous exposures, varying in skewness and granularity. In general, CBGPS and npCBGPS resulted in excellent covariate balance, and npCBGPS was the least biased but the most variable. The QB and CPM approaches had the lowest mean squared error, particularly with marginally skewed exposures. We then successfully applied the IPW approaches, together with missing-data techniques, to assess how session attendance (out of a possible 15) in a partners-based clustered intervention among pregnant couples living with human immunodeficiency virus in Mozambique (2017-2022) influenced postpartum contraceptive uptake.

Keywords: HIV; causal inference; epidemiologic methods; inverse probability weighting; nonparametric statistics; propensity score.

Figure 1

Marginal and conditional exposure distributions across 3,000 simulated data sets based on singleton live births in Quebec, Canada, 1995--2005. Panels on the left (panel A, X₁; panel C, X₂; panel E, X₃, panel G, X₄; panel I, X₅; panel K, X₆), show the marginal distributions of 6 exposures. Panels on the right (panel B, X₁; panel D, X₂; panel F, X₃, panel H, X₄; panel J, X₅; panel L, X₆) show residuals from multiple linear regressions of the 6 exposures on the confounding covariates. Panels A, B, E, and F suggest that we successfully recreated the marginal and conditional exposure distributions from the study by Naimi et al. (3) for X₁ and X₃. For X₂, panel C shows that introducing a higher correlation between left-skewed maternal age and the exposure shifted the exposure mean upwards (from 16.96 to 23.74), increased the variance (from 2.16 to 3.37), and made the exposure marginally left-skewed. Panels G, I, and K show that with rounding to integers instead of tenths, X₄, X₅, and X₆ recreate the distributions of X₁, X₂, and X₃. The appearance of heteroskedasticity in panels D and J is due to the higher numbers of individuals with higher exposure levels. X₁: conditionally normal, marginally normal exposure rounded to the nearest tenth; X₂: conditionally normal, marginally skewed exposure rounded to the nearest tenth; X₃: conditionally skewed, marginally skewed exposure rounded to the nearest tenth; X₄: conditionally normal, marginally normal exposure rounded to the nearest integer; X₅: conditionally normal, marginally skewed exposure rounded to the nearest integer; X₆: conditionally skewed, marginally skewed exposure rounded to the nearest integer.

Figure 2

Bias and mean squared error (MSE) of marginal log odds ratio estimates across 3,000 simulated data sets based on singleton live births in Quebec, Canada, 1995--2005. In each row, the gray shaded area shows a probability density function of the biases; the black circle shows the median bias; the thicker black bar shows the interquartile range of the biases; and the thinner black bar shows the range of 95% of biases across the simulated data sets. The right side of each panel shows the MSE, the mean of the squared bias (“true” β₁ minus estimated β₁) across all simulated data sets, for each stabilized inverse probability weighting (sIPW) approach. A) Conditionally normal, marginally normal exposure rounded to the nearest tenth (X₁); B) conditionally normal, marginally normal exposure rounded to the nearest integer (X₄); C) conditionally normal, marginally skewed exposure rounded to the nearest tenth (X₂); D) conditionally normal, marginally skewed exposure rounded to the nearest integer (X₅); E) conditionally skewed, marginally skewed exposure rounded to the nearest tenth (X₃); F) conditionally skewed, marginally skewed exposure rounded to the nearest integer (X₆). UW, unweighted; OLS, ordinary least squares; CBGPS, covariate balancing generalized propensity score; npCBGPS, nonparametric covariate balancing generalized propensity score; QB10, quantile binning with 10 bins; QB15, quantile binning with 15 bins; QB20, quantile binning with 20 bins; CPM, cumulative probability model.

Figure 3

Intervention session attendance in the Homens para Saúde Mais (HoPS+) Trial, Zambezia Province, Mozambique, 2017--2022. A) HoPS+ session attendance among female participants (n = 315); B) HoPS+ session attendance among male participants (n = 315).