Marginal Mean Specifications for Binary Outcomes

Binary Outcomes and Marginal Event Probabilities

For a repeated binary response, each observation records whether an event occurred. Examples include whether a patient was readmitted at a visit, whether a participant reports a symptom, or whether a treatment succeeds. The central quantity in a marginal mean model is the event probability for a specified covariate profile. It describes the mean response in the target population at that profile, not a latent or individual-specific probability.

Let $Y_{ij}$ be the binary response for unit $j$ in cluster $i$, with $Y_{ij}=1$ indicating the event and $Y_{ij}=0$ otherwise. Let $\mathbf X_i$ denote the covariates for the cluster. The marginal mean for response $j$ is

\mu_{ij}=E[Y_{ij}\mid \mathbf X_i] =P(Y_{ij}=1\mid \mathbf X_i).

The equality holds because a binary variable has expectation equal to its probability of being one. “Marginal” here means that the mean is specified without conditioning on a subject-specific random effect. Responses within a cluster may be associated; the mean specification by itself does not assert independence.

Definition: A binary marginal mean model specifies the event probability $\mu_{ij}=P(Y_{ij}=1\mid\mathbf X_i)$ as a function of observed covariates. It does so by relating a transformation of $\mu_{ij}$ to a linear predictor, $g(\mu_{ij})=\eta_{ij}=\mathbf x_{ij}^{\mathsf T}\boldsymbol\beta$. Here $0<\mu_{ij}<1$, $\mathbf x_{ij}$ is a specified covariate vector, and $\boldsymbol\beta$ is a vector of regression coefficients.

For binary responses, the two common choices considered here are the logit and probit links. Both map probabilities between zero and one to an unrestricted real-valued scale, and both have inverse links that turn a linear predictor into a valid event probability.

Logit and Probit Specifications

Definition: The marginal logit model is $$ \operatorname{logit}(\mu_{ij}) =\log\!\left(\frac{\mu_{ij}}{1-\mu_{ij}}\right) =\mathbf x_{ij}^{\mathsf T}\boldsymbol\beta, \qquad \mu_{ij}=\frac{\exp(\mathbf x_{ij}^{\mathsf T}\boldsymbol\beta)} {1+\exp(\mathbf x_{ij}^{\mathsf T}\boldsymbol\beta)}. $$ The marginal probit model is $$ \Phi^{-1}(\mu_{ij})=\mathbf x_{ij}^{\mathsf T}\boldsymbol\beta, \qquad \mu_{ij}=\Phi(\mathbf x_{ij}^{\mathsf T}\boldsymbol\beta), $$ where $\Phi$ is the cumulative distribution function of $N(0,1)$.

In either model, the linear predictor is a real number, while its inverse-link transformation is a probability. Given an estimated coefficient vector $\hat{\boldsymbol\beta}$ and a covariate profile $\mathbf x$, the predicted event probability is

\hat{\mu}(\mathbf x)= \begin{cases} \displaystyle\frac{\exp(\mathbf x^{\mathsf T}\hat{\boldsymbol\beta})} {1+\exp(\mathbf x^{\mathsf T}\hat{\boldsymbol\beta})}, &\text{logit},\\[1.2em] \Phi(\mathbf x^{\mathsf T}\hat{\boldsymbol\beta}), &\text{probit}. \end{cases}

This is a prediction for the specified covariates. If the profile includes treatment, visit, or their interaction, each of those terms must be included in the linear predictor before applying the inverse link. A coefficient is not automatically a change in probability: it is a change on the scale defined by the link.

Worked Example: Logit Model for a Repeated Binary Outcome

Suppose a marginal logit model for event status at follow-up is $\operatorname{logit}(\mu)=-1.2+0.7A+0.3T$, where $A=1$ denotes treatment and $A=0$ control, and $T=1$ denotes follow-up while $T=0$ denotes baseline. Calculate the four fitted event probabilities and interpret the treatment coefficient.

For control at baseline, $A=0,T=0$, so $\eta=-1.2$. The probability is $\exp(-1.2)/(1+\exp(-1.2))=0.3010/1.3010\approx0.2315$.

For treatment at baseline, $A=1,T=0$, so $\eta=-1.2+0.7=-0.5$, and $\mu=\exp(-0.5)/(1+\exp(-0.5))=0.6065/1.6065\approx0.3775$. The fitted risk difference at baseline is $0.3775-0.2315=0.1460$.

For control at follow-up, $A=0,T=1$, so $\eta=-1.2+0.3=-0.9$, and $\mu=\exp(-0.9)/(1+\exp(-0.9))=0.4066/1.4066\approx0.2891$. For treatment at follow-up, $A=1,T=1$, so $\eta=-1.2+0.7+0.3=-0.2$, and $\mu=\exp(-0.2)/(1+\exp(-0.2))=0.8187/1.8187\approx0.4502$. The follow-up risk difference is $0.4502-0.2891=0.1611$.

At either time, the treatment coefficient $0.7$ is a treatment-versus-control contrast in log odds, since the model has no treatment-by-time interaction. The corresponding odds ratio is $\exp(0.7)\approx2.014$. It does not mean the event probability rises by 0.7, nor does it imply a risk difference of 0.7.

Interpreting Coefficients and Contrasts

A coefficient’s interpretation depends on the link and on the model terms. For a binary logit model, a coefficient is a change in log odds for a one-unit covariate change, holding the other terms in the linear predictor fixed. Exponentiating gives an odds ratio. For a probit model, a coefficient is a change in the normal-score scale $\Phi^{-1}(\mu)$; it is not an odds ratio or a constant probability change.

A contrast compares two specified covariate profiles. If their linear predictors are $\eta_a$ and $\eta_b$, their fitted probabilities are obtained separately through the inverse link, and the probability-scale contrast is $\mu_a-\mu_b$. Thus a complete interpretation should identify both profiles and the scale of the reported contrast.

Theorem: In a binary logit model with no interaction between a binary covariate $A$ and the other terms being compared, the odds ratio for changing $A$ from 0 to 1 is $\exp(\beta_A)$, at every fixed value of those other covariates.

Proof. Fix the other covariates and write their contribution to the linear predictor as $c$. The odds at $A=1$ are $\mu_1/(1-\mu_1)=\exp(c+\beta_A)$, while the odds at $A=0$ are $\mu_0/(1-\mu_0)=\exp(c)$. Their ratio is

\frac{\mu_1/(1-\mu_1)}{\mu_0/(1-\mu_0)} =\frac{\exp(c+\beta_A)}{\exp(c)} =\exp(\beta_A).

The fixed-covariate contribution cancels, proving the result. If the model includes an interaction involving $A$, the coefficient for $A$ alone is generally the log odds ratio only at the reference value of the interacting covariate. $\square$

Worked Example: A Treatment-by-Time Interaction

Consider the marginal logit model $\operatorname{logit}(\mu)=-2+0.8A+0.5T+0.4AT$, with $A$ and $T$ coded as in the previous example. Find the treatment odds ratio and the fitted risk difference at baseline and follow-up.

At baseline, $T=0$, the treatment contrast in the linear predictor is $0.8$, so the treatment odds ratio is $\exp(0.8)\approx2.2255$. The control probability is $\operatorname{logit}^{-1}(-2)\approx0.1192$, and the treatment probability is $\operatorname{logit}^{-1}(-2+0.8)=\operatorname{logit}^{-1}(-1.2)\approx0.2315$. The baseline risk difference is $0.2315-0.1192=0.1123$.

At follow-up, $T=1$, the treatment contrast is $0.8+0.4=1.2$, so the treatment odds ratio is $\exp(1.2)\approx3.3201$. The control probability is $\operatorname{logit}^{-1}(-2+0.5)=\operatorname{logit}^{-1}(-1.5)\approx0.1824$. The treatment probability is $\operatorname{logit}^{-1}(-2+0.8+0.5+0.4)=\operatorname{logit}^{-1}(-0.3)\approx0.4256$. The follow-up risk difference is $0.4256-0.1824=0.2432$.

The interaction changes the treatment contrast on the log-odds scale from $0.8$ at baseline to $1.2$ at follow-up. The risk differences also differ, but they are obtained by applying the inverse logit at each profile and subtracting; they are not the interaction coefficient.

Probit Coefficients and Probability-Scale Contrasts

For the probit link, a contrast of $\delta$ on the linear-predictor scale changes the event probability from $\Phi(\eta)$ to $\Phi(\eta+\delta)$. The probability difference depends on the starting value $\eta$, even when $\delta$ is fixed.

Result: For any real $\eta$ and $\delta$, $$ \Phi(\eta+\delta)-\Phi(\eta) =\int_{\eta}^{\eta+\delta}\phi(u)\,du, $$ where $\phi$ is the standard normal density. In particular, the probability difference has the same sign as $\delta$, but its magnitude generally depends on $\eta$.

Derivation. The standard normal distribution function is differentiable with derivative $\Phi'(u)=\phi(u)$. The fundamental theorem of calculus therefore gives $\Phi(\eta+\delta)-\Phi(\eta)=\int_{\eta}^{\eta+\delta}\phi(u)\,du$. Since $\phi(u)>0$ for every real $u$, the integral is positive if $\delta>0$, negative if $\delta<0$, and zero if $\delta=0$. Its magnitude depends on the values of the density between the two endpoints, which vary with their location. $\square$

Worked Example: A Probit Treatment Contrast

Suppose the marginal probit model is $\Phi^{-1}(\mu)=-0.8+0.5A+0.4T$, where $A=1$ indicates treatment and $T=1$ follow-up. Calculate the treatment probability contrast at baseline and follow-up.

At baseline, $T=0$. For control, $\eta=-0.8$, giving $\mu=\Phi(-0.8)\approx0.2119$. For treatment, $\eta=-0.8+0.5=-0.3$, giving $\mu=\Phi(-0.3)\approx0.3821$. Thus the baseline probability contrast is $0.3821-0.2119=0.1702$.

At follow-up, $T=1$. For control, $\eta=-0.8+0.4=-0.4$, giving $\mu=\Phi(-0.4)\approx0.3446$. For treatment, $\eta=-0.8+0.5+0.4=0.1$, giving $\mu=\Phi(0.1)\approx0.5398$. Thus the follow-up probability contrast is $0.5398-0.3446=0.1952$.

The treatment coefficient $0.5$ is the same probit-scale contrast at both times because there is no interaction. The probability contrasts differ because the inverse probit transformation depends on the starting linear predictor.

Why the Marginal Interpretation Matters

The model specifies how observed covariates relate to the event probability in the population represented by the data. It does not, merely by using a logit or probit link, define a subject-specific effect. In particular, a marginal logit coefficient describes a contrast in marginal log odds at the covariate profiles being compared. A coefficient from a model that instead conditions on a subject-specific random effect would answer a different question.

The two interpretations should not be casually interchanged. When reporting a fitted marginal model, state that probabilities are conditional on the specified observed covariates and marginal with respect to unmodeled subject-specific heterogeneity. Report probability contrasts at named profiles when those are the quantities of interest, and identify whether any coefficient interpretation is on the log-odds or probit scale.

Mistake: “The logit coefficient is the change in probability for each subject.” This confuses a log-odds coefficient with a probability difference and adds an individual-level interpretation the marginal model does not supply.
Correction: The coefficient is a change in log odds for the specified marginal mean model. To obtain a probability difference, calculate the predicted probabilities at both covariate profiles and subtract them.
Mistake: Treating a probit coefficient as a constant change in event probability.
Correction: It is a shift in $\Phi^{-1}(\mu)$. Convert each profile’s predictor through $\Phi$ before interpreting a probability contrast.

Key takeaway: A binary marginal mean model specifies event probabilities at observed covariate profiles. The inverse link gives those probabilities; coefficient interpretations remain on the link scale unless a response-scale contrast is explicitly calculated.

Check Your Understanding

For each question, distinguish the linear-predictor scale from the event-probability scale.

In a marginal logit model, what is the interpretation of a coefficient for a binary covariate when there is no interaction involving that covariate?
A probit coefficient is $0.4$. Why is it incorrect to say that the event probability increases by 0.4?
For a logit model with a treatment-by-time interaction, which terms determine the treatment log odds contrast at follow-up?
What inverse-link calculation gives a predicted probability from a fitted probit linear predictor $\hat{\eta}$?
Why does a marginal mean model for repeated binary outcomes not, by itself, give a subject-specific treatment effect?