Pooling Log Odds Ratios And Reporting Odds Ratios

Why Pool on the Log-Odds Scale?

A logistic-regression coefficient is naturally estimated on the log-odds scale, while readers often want an odds ratio. With missing data handled through multiple imputation, each completed data set gives a coefficient estimate and an estimated variance. The key is to pool those quantities on the coefficient’s scale first. Exponentiation is used afterward to express the pooled result as an odds ratio.

This order matters because exponentiation is nonlinear. The average of the odds ratios from separate imputations is generally not the odds ratio corresponding to the pooled coefficient, and it does not incorporate within- and between-imputation uncertainty according to Rubin’s rules. The target contrast must also be the same in every imputation: for example, the adjusted effect of a 10 mg increase in exposure, with the same covariate adjustment and reference category.

The estimated variances \(u_i\) are variances of the log-scale coefficient, not variances of odds ratios. If a fitted model reports a standard error \(s_i\) for the coefficient, use \(u_i=s_i^2\). The pooled standard error on the log scale is \(\sqrt{T}\). For an interval, first form the appropriate pooled interval for \(q\), using the relevant degrees of freedom and confidence level; then exponentiate its lower and upper limits. The earlier material on confidence intervals from pooled estimates supplies the general interval construction.

Conditions for a Meaningful Pooled Odds Ratio

The arithmetic is only interpretable when the estimates refer to a common quantity. The logistic model and contrast must be defined consistently across imputations: the same outcome coding, predictor scale, reference group, covariates, and coefficient. If a coefficient is for a one-unit increase in a continuous predictor in one analysis but a five-unit increase in another, those are not estimates of the same contrast until they are placed on a common scale.

The usual pooling calculation also relies on the coefficient estimates being reasonably approximated by normal sampling distributions within completed data sets, with valid variance estimates. The imputation procedure must produce plausible completed data sets under its stated missing-data assumptions, and the analysis and imputation models should be sufficiently compatible for the target coefficient. Rubin’s rules account for sampling uncertainty within completed data sets and variation between them; they do not repair a misspecified outcome model, a poorly chosen imputation model, or an incorrectly defined estimand.

Logistic coefficients are conditional on the other variables in the model. Thus, the exponentiated coefficient is an adjusted odds ratio conditional on the included covariates, not a risk ratio and not automatically a marginal population effect. An odds ratio above 1 indicates higher odds for the numerator group or higher predictor value relative to the stated comparison; it does not by itself establish a causal effect.

Why the Interval Transforms Endpoint by Endpoint

Exponentiation is strictly increasing: if \(a<b\), then \(\exp(a)<\exp(b)\). Consequently, if a log-scale interval has limits \(L\) and \(U\), the corresponding odds-ratio interval has limits \(\exp(L)\) and \(\exp(U)\), in the same order. This transformation preserves the interval’s coverage under the same model and interval procedure; it does not change the uncertainty calculation that produced \(L\) and \(U\).

The odds-ratio interval is usually asymmetric around its point estimate because exponentiation is nonlinear. That asymmetry is expected. It is not a reason to recenter the interval around the odds ratio or to calculate a new standard error by treating the transformed coefficient as though it were a normally distributed estimate.

Keeping the Predictor Scale Consistent

A reported odds ratio is incomplete unless its contrast is clear. For a continuous predictor measured in years, \(\exp(\bar{q})\) describes the odds ratio for a one-year increase. If the desired report is per 10 years, multiply the log coefficient by 10 before exponentiating. Its variance must be multiplied by \(10^2\) if the contrast itself is redefined on the 10-year scale. Equivalently, the one-year pooled result can be exponentiated to the tenth power. The estimate and uncertainty must use the same scale.

Diagnostics, Alternatives, and Common Pitfalls

Before reporting a pooled odds ratio, check that each imputation estimates the same coefficient and that each variance is on the log-odds scale. Review the coefficient and variance estimates across imputations for unusual values, large shifts, or extreme imputation-specific standard errors. Such patterns may indicate unstable model fitting, sparse outcome-by-predictor cells, separation, or inconsistent coding. Pooling can combine valid estimates; it cannot make an unstable or non-comparable set of estimates reliable.

A common mistake is to exponentiate each coefficient, average those odds ratios, and then report that average as the pooled result. This skips the Rubin pooling calculation and ignores the distinction between within- and between-imputation uncertainty. Another mistake is to exponentiate the pooled coefficient but leave confidence limits on the log scale, or to exponentiate only one limit. Transform the point estimate and both limits, and label the contrast and units in the report.

The alternative of reporting the pooled log odds ratio directly is statistically valid and can be useful when a model-scale result is desired, but it may be less intuitive for many readers. Reporting both the log coefficient and odds ratio can aid reproducibility. Neither format changes the pooling calculation. Do not interpret an odds ratio as a probability ratio: for instance, an odds ratio of 1.5 does not mean that the probability is 50% higher.