Model Explanation

This visualization shows how the population-averaged log odds ratio (PA ln(OR)) changes as a function of X under a correctly specified probit random effects model.

The subject-specific model is:

Probit(P(Y = 1)) = α + a_i + βX

where a_i ~ N(0, σ²).

Under this model, population-averaged parameters are given by:

β_PA = β / √(1 + σ²)

For each value of X, the PA ln(OR)(x) is computed using:

• Probabilities at X and X+1 from the marginal probit model
• A transformation into a log odds ratio

Unlike logistic models, this quantity varies with X. To obtain a single summary measure, we evaluate ln(OR)(x) across a dense grid of X values and interpret the curve directly.

The dashed curves represent the distribution of X for two subgroups:

• Sex = Male ~ N(-1.5, 1)
• Sex = Female ~ N(1.5, 1)

Points indicate the true population-averaged values for each subgroup, computed using their respective exposure distributions.

Predictive Accuracy Visualization

Findings

It can be seen that the PA ln(OR) changes based on the Sex subgroups despite not being included in the original probit model which can cause issues with samples that do not contain all subgroups due to limitations. These subgroups can be calculated using a correctly specified Kernel Density Estimation (KDE) probit model or a misspecified stratified GEE logistic model.

Real World Impact

This project demonstrates how population-averaged effect estimates can vary depending on model specification and subgroup structure. These findings are relevant for improving the accuracy of causal interpretation in clustered or heterogeneous data settings.

Model Explanation

This comparison evaluates two approaches for estimating the population-averaged log odds ratio:

1. Kernel Density Estimation (KDE) based estimator (proposed method)
2. Stratified Generalized Estimating Equations (GEE) with a logit link (common approach)

Simulation setup:

• 200 clusters, 4 observations per cluster

• 500 simulated datasets

• Random intercept: N(0, 0.5)

• Exposure distributions differ by sex:

  • Sex = Male ~ N(-1.5, 1)

  • Sex = Female ~ N(1.5, 1)

KDE approach:

• Estimates the density of X nonparametrically using kernel density estimation

• Computes ln(OR)(x) across a grid

• Averages using density-based weights

GEE approach:

• Uses a logit link (misspecified under probit data generation)

• Requires stratification to obtain subgroup estimates

Histograms show the distribution of estimated PA ln(OR) for each method and subgroup.

Method Comparison

Findings

• The KDE-based estimator produces more precise estimates (lower variability) than GEE.
• KDE results:
  • Sex Male mean ≈ 0.801, SD ≈ 0.029
  • Sex Female mean ≈ 0.670, SD ≈ 0.021
• GEE results:
  • Sex Male mean ≈ 0.791, SD ≈ 0.047
  • Sex Female mean ≈ 0.668, SD ≈ 0.031
• KDE achieves tighter distributions and more stable subgroup estimates.
• GEE requires stratification, which reduces effective sample size and increases variability.
• Confidence interval coverage is slightly lower for KDE (~93%) than GEE (~95%), but this is offset by improved precision.
• The key advantage of KDE is that it directly targets the population-averaged parameter by adapting to the observed distribution of X.

Real World Impact

This project demonstrates how population-averaged effect estimates can vary depending on model specification and subgroup structure. These findings are relevant for improving the accuracy of causal interpretation in clustered or heterogeneous data settings.