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With ${\alpha }_1=0$, the marginal moment model givesThus $e^{{\alpha }_2}$ and $e^{{\alpha }_3}$ are ratios of expected visits in Lent and Easter to expected visits in Michaelmas, respectively, at the same baseline covariates. These are population mean ratios, including students with no propensity to attend; they are not odds ratios or probabilities of any attendance.

In the hierarchical model, for a fixed positive student effect $b_i$, the conditional mean ratio is also $e^{{\alpha }_j}$. It describes a multiplicative within-student change in expected visits holding the student effect fixed. For the structural-zero component all three means are zero, so no nonzero ratio is defined there. Since the distribution of $b_i$ and the mixture weight do not vary by term, integrating out $b_i$ preserves this same marginal mean ratio. This is mean-ratio preservation under a multiplicative random effect: the conditional and marginal term coefficients coincide for this multiplicative log-link model. They need not coincide in a general nonlinear mixed model, such as a logistic random-effect model. A log-mean difference ${\alpha }_j$ is not a difference of expected logarithms of counts, which can be undefined at zero.