Marginal Risk Differences and Risk Ratios

From Regime-Specific Risks to Contrasts

A simulated risk under a treatment regime is an absolute quantity: it describes the probability of the outcome by a specified follow-up time if the population followed that regime. Comparing two such risks produces familiar measures of effect. The risk difference expresses an absolute change in probability; the risk ratio expresses a multiplicative change. Both are marginal contrasts because each risk averages over the target population’s baseline characteristics and the covariate histories that arise under the regime.

Let \(Y_K\) indicate whether the event has occurred by fixed follow-up time \(K\), and let \(g_1\) and \(g_0\) denote two specified treatment regimes. Write \(\psi_j=P(Y_K^{g_j}=1)\) for the regime-specific risk under \(g_j\), for \(j=0,1\). In practice, the parametric g-formula estimates these risks by fitting models for covariate evolution and the outcome, then averaging predicted outcomes over simulated histories and the target baseline distribution.

These are contrasts of risks, not contrasts of conditional outcome-model coefficients. In particular, a coefficient in a logistic model is generally a conditional log odds ratio; it is not the marginal risk difference or risk ratio. The g-formula obtains the marginal risks first, then forms the contrast.

Deriving the Risks from Longitudinal Histories

Suppose \(L_0\) is baseline covariate information and \(L_1\) is a covariate measured later in follow-up. Under a regime \(g\), the law of total probability gives the endpoint risk by averaging the outcome probability over the baseline and the covariate distribution generated under that regime:

For continuous covariates, the corresponding sums are integrals. This expression makes the marginal nature of the estimand explicit: the distribution of \(L_1\) can differ between regimes, and the outcome risk is averaged over each regime’s own covariate histories. The regime-specific risk estimates \(\hat\psi_1\) and \(\hat\psi_0\) then give the plug-in contrasts \(\hat\Delta=\hat\psi_1-\hat\psi_0\) and \(\hat R=\hat\psi_1/\hat\psi_0\), when the denominator is positive.

Worked Longitudinal Example

Consider a target population in which \(P(L_0=1)=0.4\) and \(P(L_0=0)=0.6\). A parametric g-computation analysis simulates \(L_1\) under two static treatment regimes, \(g_1\) and \(g_0\), and predicts the probability of the event by time \(K\) for each history. The following probabilities are stipulated outputs of the fitted transition and outcome models:

For \(g_1\), when \(L_0=0\), the probability of \(L_1=0\) is \(1-0.30=0.70\). The conditional event risk averaged over \(L_1\) is therefore \(0.70(0.08)+0.30(0.20)=0.056+0.060=0.116\). When \(L_0=1\), that risk is \(0.30(0.18)+0.70(0.35)=0.054+0.245=0.299\). Averaging over baseline \(L_0\) gives

For \(g_0\), when \(L_0=0\), the average is \(0.80(0.04)+0.20(0.12)=0.032+0.024=0.056\). When \(L_0=1\), it is \(0.40(0.10)+0.60(0.25)=0.040+0.150=0.190\). Thus

The resulting contrasts are

By time \(K\), the estimated risk is 7.96 percentage points higher under \(g_1\), and approximately 1.73 times the risk under \(g_0\). The calculations explicitly average over the regime-specific distribution of \(L_1\); using one common distribution for \(L_1\) would generally estimate different quantities.

Uncertainty for the Contrasts

The two risk estimates are usually dependent: they may be computed from the same fitted models and the same observed sample. If their estimated variances and covariance are available, the variance of their difference follows directly from the variance of a linear combination.

To verify the result, write \(U=\hat\psi_1\) and \(V=\hat\psi_0\). Since \(E[U-V]=E[U]-E[V]\),

The covariance term is not optional. Omitting it treats the risk estimates as independent, which they often are not and can either overstate or understate uncertainty.

For a risk ratio, inference is often carried out on the log scale. Suppose the vector of risk estimates is jointly asymptotically normal, with covariance matrix consistently estimated, and the true risks satisfy \(\psi_1>0\) and \(\psi_0>0\). The function \(h(x,y)=\log(x/y)\) is differentiable in a neighbourhood of \((\psi_1,\psi_0)\), with gradient \((1/\psi_1,-1/\psi_0)\). The multivariate delta method therefore gives the following approximate variance:

For justification, the delta method approximates the variance of \(h(\hat\psi_1,\hat\psi_0)\) by the gradient at the true risks multiplied on both sides of the covariance matrix. Expanding that quadratic form yields the two variance terms and the covariance term displayed above. Replacing the unknown risks and covariance matrix with consistent estimates gives the plug-in formula. Strict positivity of both true risks is necessary for differentiability of the log-ratio function at the parameter value, and positivity of both estimated risks is necessary for the displayed plug-in expression to be defined. \(\square\)

For the longitudinal example, suppose the estimated sampling variances are \(0.000400\) and \(0.000225\), and the estimated covariance is \(0.000120\). These are supplied here as sampling-uncertainty quantities; they are not Monte Carlo simulation variances. For the difference,

A normal-approximation 95% interval for the risk difference is \(0.0796\pm1.96(0.01962)\), or approximately \((0.0411,0.1181)\). For the log risk ratio, the three terms are approximately \(0.000400/(0.1892)^2=0.01117\), \(0.000225/(0.1096)^2=0.01873\), and \(2(0.000120)/(0.1892\cdot0.1096)=0.01157\). Hence the estimated log-ratio variance is \(0.01117+0.01873-0.01157=0.01833\), with standard error \(\sqrt{0.01833}\approx0.1354\). Since \(\log(1.7263)\approx0.546\), the log-scale 95% interval is \(0.546\pm1.96(0.1354)=(0.281,0.811)\). Exponentiating its endpoints gives an approximate risk-ratio interval of \((1.32,2.25)\).

These calculations concern sampling uncertainty in estimated risks. A finite number of simulated histories can add Monte Carlo error; increasing the simulation size reduces that component but does not remove sampling uncertainty or model misspecification. The uncertainty calculation must match the source of uncertainty being reported.

Scale, Diagnostics, and Limitations

The risk difference is often useful for communicating absolute impact and planning services; the risk ratio summarizes relative change. Their numerical values need not suggest the same practical importance. A 50% relative increase from a risk of 2% is an increase of 1 percentage point, while the same relative increase from 40% is 20 percentage points. Report the risks as well as the contrast, identify the comparator and follow-up time, and state the target population.

A common error is to calculate uncertainty for the ratio while assuming the two risk estimates are independent. The correction is to use their joint covariance, as in the delta-method formula. Another important boundary case occurs when \(\psi_1=0\) and \(\psi_0>0\): the risk ratio itself is defined as zero, but its logarithm is not, and the log-ratio function is not differentiable at that parameter value. The log-scale delta-method interval is not justified there. If \(\psi_0=0\), the ratio is undefined. Very small positive risks can also make ratio estimates unstable and normal approximations poor.

Useful checks include verifying that each estimated risk lies between 0 and 1, confirming the contrast’s direction matches the stated regime ordering, and checking that the covariance matrix is plausible (in particular, its implied variances cannot be negative). Compare each risk with the simulated history distributions and inspect whether a small number of histories or baseline strata dominate the estimate. These checks can reveal coding or numerical problems, but they cannot establish correct model specification or the causal identification assumptions.

The main alternative is to report each regime-specific risk without reducing them to a single contrast. This preserves absolute levels but makes direct comparison less concise. Odds ratios are another familiar measure, but they are not risk ratios and can differ substantially from them when risks are not small. No scale is universally best: the choice should follow the scientific question, and reporting both a difference and a ratio can clarify absolute and relative impact. Neither choice repairs violations of exchangeability, positivity, consistency, or model adequacy.