Binary Biomarkers With Logistic Mixed Models

Why Use a Logistic Mixed Model?

A biomarker may be recorded repeatedly as a binary outcome: for example, whether a laboratory value is above a clinically chosen threshold at each visit. The observed values are zero or one, and measurements from the same person are generally related. A logistic mixed model addresses both features: it models the probability of a positive measurement and represents subject-to-subject variation through random effects.

The interpretation is conditional on a subject’s random effects. This matters: a coefficient describes a change in log odds for subjects with the same random effects, not automatically a population-average change in odds. The earlier material on exponential-family longitudinal submodels provides the broader context for choosing a response distribution and link; here the focus is the repeated binary case.

Definition: Let $Y_{ij}$ be the binary biomarker for subject $i$ at visit $j$, where $i=1,\ldots,N$ and $j=1,\ldots,n_i$. A logistic mixed model specifies $Y_{ij}\mid b_i$ as Bernoulli with probability $\pi_{ij}$, and models its conditional log odds as $$ \operatorname{logit}(\pi_{ij}) =\log\left(\frac{\pi_{ij}}{1-\pi_{ij}}\right) =\mathbf{x}_{ij}^{\mathsf T}\boldsymbol{\beta} +\mathbf{z}_{ij}^{\mathsf T}b_i, \qquad b_i\sim N_q(\mathbf{0},D). $$ Here $\mathbf{x}_{ij}$ is the fixed-effects design vector, $\boldsymbol{\beta}$ is its coefficient vector, $\mathbf{z}_{ij}$ is the random-effects design vector, and $D$ is the random-effects covariance matrix.

Model Structure and Assumptions

The inverse-logit transformation converts the linear predictor to a probability:

\pi_{ij} =\frac{\exp(\mathbf{x}_{ij}^{\mathsf T}\boldsymbol{\beta} +\mathbf{z}_{ij}^{\mathsf T}b_i)} {1+\exp(\mathbf{x}_{ij}^{\mathsf T}\boldsymbol{\beta} +\mathbf{z}_{ij}^{\mathsf T}b_i)}.

The model is built on several assumptions. Conditional on $b_i$ and the included covariates, each response is Bernoulli with the stated probability, and repeated responses within a subject are conditionally independent. Random effects are independent across subjects and follow the specified mean-zero multivariate normal distribution. The fixed-effects and random-effects design terms must represent the systematic structure adequately. These assumptions are substantive choices, not consequences of using a logistic link.

Conditional independence does not mean that the observed repeated outcomes are independent without qualification. Integrating over a shared random effect induces association between responses from the same subject. For example, a subject with a high random intercept tends to have higher probabilities of a positive biomarker at multiple visits.

A common starting model has a random intercept:

\operatorname{logit}(\pi_{ij}) =\beta_0+\beta_1 t_{ij}+\beta_2 x_i+b_{0i}, \qquad b_{0i}\sim N(0,\sigma_b^2).

Here $t_{ij}$ is time and $x_i$ is a subject-level covariate. The intercept $b_{0i}$ shifts that subject’s log odds at every visit. A random time slope can allow subjects’ time trends to differ:

\operatorname{logit}(\pi_{ij}) =\beta_0+\beta_1 t_{ij}+b_{0i}+b_{1i}t_{ij}, \qquad \begin{pmatrix}b_{0i}\\b_{1i}\end{pmatrix} \sim N_2(\mathbf{0},D).

The off-diagonal entry of $D$ allows a subject’s intercept deviation and slope deviation to be associated. A random slope is not required merely because the outcome is repeated; it should be supported by the design and data, and it can be difficult to estimate when visits are sparse.

Fixed Effects on the Conditional Log-Odds Scale

Suppose a fixed covariate $x$ enters the linear predictor with coefficient $\beta_x$, without an interaction. Holding the other covariates and the random effects fixed, increasing $x$ by one unit changes the linear predictor by $\beta_x$. Since the linear predictor is the log odds, the conditional odds are multiplied by $\exp(\beta_x)$. For a change of $c$ units, the multiplier is $\exp(c\beta_x)$.

Worked Example: Time, Treatment, and a Conditional Probability

Consider the model $$ \operatorname{logit}(\pi_{ij}) =-2+0.4t_{ij}+0.3z_i-0.1t_{ij}z_i+b_{0i}, $$ where time $t_{ij}$ is measured in years and $z_i=1$ denotes the treatment group while $z_i=0$ denotes the reference group. For the calculation, take a subject with $b_{0i}=0$ at year $t_{ij}=2$ in the treatment group.

The conditional linear predictor is $$ \eta=-2+(0.4)(2)+(0.3)(1)+(-0.1)(2)(1)=-1.1. $$ Thus the conditional odds of a positive biomarker measurement are $$ \exp(-1.1)=0.3329, $$ and the conditional probability is $$ \pi_{ij}=\frac{0.3329}{1+0.3329}\approx 0.2497. $$

At the same time, the reference-group subject with $b_{0i}=0$ has $\eta=-2+(0.4)(2)=-1.2$, giving odds $\exp(-1.2)=0.3012$. The treatment-versus-reference conditional odds ratio at year 2 is therefore $\exp(0.3-0.1(2))=\exp(0.1)=1.105$, matching the ratio $0.3329/0.3012\approx 1.1052$ up to rounding. The treatment coefficient alone is not the treatment effect at every time because the model includes a time-by-treatment interaction.

Interpretation with an interaction: In a model with terms $\beta_t t+\beta_z z+\beta_{tz}tz$, the conditional treatment-versus-reference log odds ratio at time $t$ is $\beta_z+\beta_{tz}t$. The corresponding conditional odds ratio is $\exp(\beta_z+\beta_{tz}t)$. The time coefficient $\beta_t$ is the time slope in the reference group; in the treatment group it is $\beta_t+\beta_{tz}$.

What the Random Effects Mean

Random effects describe subject-specific departures from the fixed-effects pattern. In the random-intercept model, $b_{0i}$ is a subject’s deviation in log odds from the fixed-effects predictor. A one-unit increase in $b_{0i}$, holding the fixed predictor constant, multiplies that subject’s odds by $e$. This does not mean the subject’s probability increases by a constant amount: the probability scale is nonlinear.

Worked Example: Interpreting a Subject’s Random Intercept

At a particular visit, suppose a subject’s fixed-effects linear predictor is $-1.1$. A subject with random intercept $b_{0i}=0.5$ has conditional linear predictor $-1.1+0.5=-0.6$. Their conditional odds are $\exp(-0.6)=0.5488$, and their conditional probability is $0.5488/(1+0.5488)=0.3543$.

For $b_{0i}=0$, the odds are $\exp(-1.1)=0.3329$, and the probability is $0.2497$. The odds ratio comparing the $b_{0i}=0.5$ subject with the $b_{0i}=0$ subject, at the same fixed-effects covariates, is $$ \frac{\exp(-0.6)}{\exp(-1.1)}=\exp(0.5)=1.649. $$ The probabilities differ by about $0.1046$, not by a factor of $1.649$. The random intercept is measured on the log-odds scale, not the probability scale.

With a random slope, the subject-specific time slope is a sum of a fixed slope and a random deviation. It describes how that subject’s conditional log odds change with time, given the model; it does not assert that every individual follows a perfectly linear biological trajectory.

Worked Example: A Subject-Specific Time Slope

Suppose $$ \operatorname{logit}(\pi_{ij})=-1+0.25t_{ij}+b_{0i}+b_{1i}t_{ij}. $$ For a subject with $b_{0i}=0.2$ and $b_{1i}=-0.15$, the subject-specific intercept is $-1+0.2=-0.8$, and the subject-specific time slope is $0.25-0.15=0.10$ log-odds units per year.

At year 0, $\eta=-0.8$, so the conditional probability is $\exp(-0.8)/(1+\exp(-0.8))=0.3100$. At year 2, $\eta=-1+(0.25)(2)+0.2+(-0.15)(2)=-0.6$, so the conditional probability is $\exp(-0.6)/(1+\exp(-0.6))=0.3543$. Across those two years, the conditional odds ratio is $\exp(2(0.10))=\exp(0.2)=1.221$. These calculations are conditional on the stated random effects and other model terms.

Likelihood and a Short Justification

The subject-level likelihood contribution is obtained by multiplying the Bernoulli probabilities conditional on the random effects and then integrating over their distribution. This integration is what accounts for the unobserved subject-specific effects:

L_i(\boldsymbol{\beta},D) =\int \left\{ \prod_{j=1}^{n_i} \pi_{ij}(b_i)^{Y_{ij}} [1-\pi_{ij}(b_i)]^{1-Y_{ij}} \right\} \phi_q(b_i;\mathbf{0},D)\,db_i, \qquad L=\prod_{i=1}^{N}L_i.

Here $\phi_q$ is the $q$-dimensional normal density and $\pi_{ij}(b_i)$ is the inverse-logit probability for a given $b_i$. In general, the integral does not simplify to a logistic likelihood with the same fixed coefficients. This follows because the inverse-logit is nonlinear: averaging probabilities over random effects is not equivalent to applying the inverse-logit to the average linear predictor. Thus $\boldsymbol{\beta}$ gives conditional, subject-specific effects; a population-average probability or odds ratio requires averaging predictions over the random-effects distribution and then forming the desired contrast.

Specify the response model.
State that each $Y_{ij}\mid b_i$ is Bernoulli and identify the conditional probability and logit link.

Choose fixed and random terms.
Define time, covariates, interactions, and the random-effects design; give the distribution and covariance structure for $b_i$.

Interpret conditionally.
Translate a fixed log-odds contrast into a conditional odds ratio, specifying the covariates and time at which it applies.

Diagnostics, Alternatives, and Limitations

Model checking should examine both the binary response pattern and the repeated-subject structure. Compare observed and model-predicted positive proportions across time and important covariate groups; examine whether the model misses systematic time trends or treatment-by-time differences. Check for sparse combinations, extreme fitted probabilities, convergence warnings, and estimated random-effect variances near zero. Such findings may indicate weak information for the chosen random-effects structure or a mismatch between the model and the data. Because random effects are latent, their estimated distribution should not be treated as a direct set of measured subject traits.

A logistic mixed model is not the only way to analyze repeated binary outcomes. A marginal logistic approach, such as generalized estimating equations, targets population-average associations and uses a working within-subject correlation structure. The mixed model instead specifies subject-specific conditional effects through random effects. An ordinary logistic regression that treats every visit as independent ignores within-subject dependence and can produce misleading uncertainty estimates. Which approach is appropriate depends partly on whether the scientific target is conditional or population-average.

Important limitations include sensitivity to the assumed random-effects distribution and the difficulty of estimating complex random-effects covariance structures with few subjects or visits. Conditional odds ratios are also not generally interchangeable with population-average odds ratios. Finally, a well-fitting model describes conditional associations under its assumptions; by itself, it does not establish that a covariate causes changes in biomarker status.

Common Mistake

Pitfall: “The treatment coefficient is 0.3, so treatment multiplies the odds by $\exp(0.3)$ at every visit, and the probability rises by that same factor.” This reasoning ignores the interaction and confuses odds with probability. In the example model, the conditional treatment odds ratio at time $t$ is $\exp(0.3-0.1t)$; even where an odds ratio is constant, it does not multiply the probability. Always identify the full linear-predictor contrast, exponentiate it for the odds ratio, and compute probabilities separately if needed.

Key takeaway: A logistic mixed model represents a repeated binary biomarker through a conditional Bernoulli model and subject-level random effects. Fixed effects and random effects act on the conditional log-odds scale; their interpretations must retain that conditioning.

Check Your Understanding

Answer each question, then compare with the solutions below.

A fixed coefficient for a continuous biomarker covariate is $0.4$, with no interaction. What is the conditional odds ratio for a one-unit increase?
For a subject, the fixed-effects predictor is $-1.0$ and the random intercept is $0.2$. Calculate the conditional predictor and probability.
Why does setting $b_i=0$ not generally give the population-average probability?
What does conditional independence mean in a logistic mixed model, and how can repeated outcomes still be associated marginally?

Solutions. (1) The one-unit conditional odds ratio is $\exp(0.4)=1.492$, with other terms and random effects held fixed. (2) The conditional predictor is $-1.0+0.2=-0.8$; the probability is $\exp(-0.8)/(1+\exp(-0.8))=0.3100$. (3) The population-average probability integrates the inverse-logit probability over the random-effects distribution; because inverse-logit is nonlinear, this average is generally not the inverse-logit at $b_i=0$. (4) Given the random effects and covariates, the responses are independent Bernoulli variables. Without conditioning, responses share the same random effects, which induces within-subject association.