Conditional versus marginal effect plot

Non-collapsibility plot

STC
ML-NMR
ML-UMR
MAIC
How the marginal odds ratio drifts away from a fixed conditional odds ratio as a prognostic covariate becomes stronger.
STCML-NMRML-UMRMAICProof of concept

Conditional versus marginal effect plot example

Conditional (fixed at 0.45) and marginal treatment odds ratios as the prognostic strength of a normally distributed covariate increases, with no effect modification and no confounding. Computed by Monte Carlo integration.
Family
Outcome regression and transportability
Purpose
Explain why conventional STC can target the wrong estimand on non-collapsible scales.
Inputs
An outcome model and a covariate distribution.
Software
Custom simulation and ggplot2 (shown)

What it shows

Odds ratios and hazard ratios are non-collapsible: the population-average (marginal) effect differs from the within-stratum (conditional) effect even when the covariate is not an effect modifier and not a confounder. Conventional STC plugs covariate means into a conditional regression and compares the result with a published marginal effect, mixing the two. This proof-of-concept plot, of the kind used in the methodological literature, shows how large the gap can become as a covariate’s prognostic effect grows.

How to read it

  • Horizontal axis: strength of the covariate’s prognostic effect.
  • Blue line: the conditional odds ratio, constant by construction.
  • Red line: the marginal odds ratio in the population.
  • Gap between them: the estimand mismatch.

Interpretation

With no prognostic covariate the two coincide at 0.45. As the covariate’s effect grows to 3 log-odds units per SD, the marginal odds ratio attenuates to about 0.69, although nothing about the treatment has changed. An STC that reports the conditional value against a published marginal comparator would overstate the treatment benefit.

Pitfalls

  • The gap is not bias in either quantity; each is correct for its own estimand. The error lies in comparing them.
  • Risk differences and mean differences on the identity link are collapsible and do not show this pattern.
  • G-computation, ML-NMR, and ML-UMR marginalize over the covariate distribution to produce a coherent marginal effect.

Code

library(ggplot2)

# Non-collapsibility of the odds ratio: with the conditional log OR fixed at
# -0.8, the marginal log OR moves toward zero as the prognostic effect of a
# covariate (not an effect modifier) grows
set.seed(1)
x <- rnorm(2e5)
strength <- seq(0, 3, by = 0.1)
marg <- sapply(strength, function(b) {
  p1 <- mean(plogis(-0.5 - 0.8 + b * x))
  p0 <- mean(plogis(-0.5 + b * x))
  qlogis(p1) - qlogis(p0)
})
d <- rbind(data.frame(strength, logor = -0.8, type = "Conditional OR (within covariate levels)"),
           data.frame(strength, logor = marg, type = "Marginal OR (population average)"))

ggplot(d, aes(strength, exp(logor), colour = type)) +
  geom_hline(yintercept = 1, colour = "#7a828c") +
  geom_line(linewidth = 1.2) +
  scale_colour_manual(values = c("#1d4e89", "#b5452b"), name = NULL) +
  scale_y_log10(limits = c(0.4, 1.05)) +
  labs(x = "Prognostic strength of the covariate (log OR per SD)",
       y = "Treatment odds ratio (log scale)",
       title = "Conditional and marginal effects differ on the odds ratio scale",
       subtitle = "No effect modification and no confounding: the gap is purely non-collapsibility") +
  guides(colour = guide_legend(ncol = 1))

References

  • Remiro-Azócar A, Heath A, Baio G. Parametric G-computation for compatible indirect treatment comparisons with limited individual patient data. Res Synth Methods. 2022;13:716-744. doi:10.1002/jrsm.1565
  • Daniel R, Zhang J, Farewell D. Making apples from oranges: comparing noncollapsible effect estimators and their standard errors after adjustment for different covariate sets. Biom J. 2021;63:528-557. doi:10.1002/bimj.201900297