Median Odds Ratio (MOR) for a logistic MAIHDA model

The Median Odds Ratio translates the between-stratum variance of a logistic MAIHDA model onto the odds-ratio scale: the median relative change in the odds of the outcome when comparing two individuals from randomly chosen strata (higher- vs lower-risk). MOR = exp(sqrt(2 * V_A) * qnorm(0.75)), where V_A is the between-stratum (latent, logit-scale) variance. An MOR of 1 indicates no between-stratum heterogeneity. The MOR is defined only for the logit link (it is the median odds ratio); a non-logit binomial fit such as probit is rejected, because its latent variance is on a different scale and the exp(...) above would not be an odds ratio.

For a cumulative-logit (ordinal) MAIHDA model the same formula applies to the latent logit-scale between-stratum variance and is the median cumulative odds ratio: the median relative change in the odds of being at or below any given outcome category between two randomly chosen strata (under the model's proportional-odds assumption it is the same for every category split).

Scope. V_A is the between-stratum variance. Contextual (context = ) and other non-stratum random effects are never included: the MOR compares two individuals from randomly chosen strata within the same context.

Crossed-dimensions fits use a different calculation. The closed form above assumes the two strata's random effects are independent, which is what makes their difference \(N(0, 2 V_A)\). In a crossed-dimensions fit (from maihda(decomposition = "crossed-dimensions")) a stratum's effect is the sum of its dimension effects plus the intersection effect, so two strata sharing a dimension – say two strata that are both "female" – share that dimension's random effect and are correlated. Their difference is then a mixture of normals, one component per pattern of shared dimensions, and applying the closed form to the summed variance overstates the MOR (substantially so when the variance sits mainly in the additive dimensions, and not at all when it sits entirely in the interaction).

The MOR reported for such a fit is therefore computed from that mixture, under an explicit sampling scheme: two distinct strata drawn uniformly at random from the strata present in the analytic sample. Writing \(\tau^2_d\) for dimension \(d\)'s variance and \(\tau^2_I\) for the intersection variance, a pair differing on the dimension set \(D^*\) has difference variance \(v = 2(\tau^2_I + \sum_{d \in D^*} \tau^2_d)\), and the MOR is exp(x) for the x solving \(\sum_{pairs} (2\Phi(x/\sqrt{v}) - 1) / n_{pairs} = 0.5\). For a canonical single-stratum fit the two calculations coincide, and that closed form is used.

Usage

maihda_mor(model)

Arguments

model

A maihda_model from fit_maihda fitted with a binomial (lme4), bernoulli (brms), or cumulative (ordinal) family and a logit link.

Value

A single number (the MOR, \(\ge 1\)), or NA_real_ if the between-stratum variance is unavailable – which for a crossed-dimensions fit also covers the case where the stratum grid needed for the mixture cannot be resolved (fewer than two strata, absent dimension columns, or more than 12 dimensions).

References

Larsen, K., & Merlo, J. (2005). Appropriate assessment of neighborhood effects on individual health: integrating random and fixed effects in multilevel logistic regression. American Journal of Epidemiology, 161(1), 81-88.

See also

maihda_discriminatory_accuracy