Random Slopes With the Logit Link

Why a Logit-Scale Slope Is Not a Constant Probability Change

In a logistic mixed model, a covariate’s slope describes change on the log-odds scale. With a random slope, that slope can differ from cluster to cluster. Exponentiating a cluster’s slope gives its conditional odds multiplier for a one-unit covariate increase, but it does not give a fixed change in probability. The probability difference also depends on the cluster’s starting probability.

This distinction matters when translating model results into statements about people or events. An odds ratio can be constant over a covariate change within a cluster even while the corresponding probability change varies with the covariate value. The interpretation must therefore specify both the cluster-specific slope and the starting point for any probability comparison.

Model and Assumptions

Definition: For a binary outcome $Y_{ij}$ in cluster $j$, let $p_{ij}=P(Y_{ij}=1\mid b_{0j},b_{1j},x_{ij})$. A logistic random-intercept and random-slope model for one covariate $x_{ij}$ is $$ Y_{ij}\mid b_{0j},b_{1j},x_{ij}\sim \operatorname{Bernoulli}(p_{ij}),\qquad \operatorname{logit}(p_{ij})=\log\!\left(\frac{p_{ij}}{1-p_{ij}}\right) =\beta_0+b_{0j}+(\beta_1+b_{1j})x_{ij}. $$ Here $\beta_0$ and $\beta_1$ are fixed effects, while $b_{0j}$ and $b_{1j}$ are cluster-specific deviations in the intercept and slope. Thus the slope for cluster $j$ is $\beta_1+b_{1j}$.

The interpretation below is conditional on a cluster’s random effects: comparisons hold $b_{0j}$ and $b_{1j}$ fixed. The model assumes that, conditional on the covariates and random effects, the binary outcomes follow the stated Bernoulli model and are independent across observations. It also assumes a linear relationship between the covariate and the conditional log odds, with a slope that is constant within each cluster but may differ between clusters. The random effects are treated as draws from a specified distribution across clusters; commonly that distribution has mean zero. A cluster-specific interpretation uses the cluster’s random-effect values, not a population-average effect.

For a change in $x$ from $x$ to $x+\Delta x$, the change in the cluster-specific linear predictor is $(\beta_1+b_{1j})\Delta x$. Because the link is the logit, exponentiating this change yields the conditional odds ratio. For a one-unit increase, the conditional odds ratio is $\exp(\beta_1+b_{1j})$. A positive slope means higher conditional odds as $x$ increases; a negative slope means lower conditional odds. Neither direction alone specifies the size of the probability change.

From Slope to Odds Ratio and Probability

\operatorname{OR}_j(\Delta x) =\frac{p_j(x+\Delta x)/(1-p_j(x+\Delta x))} {p_j(x)/(1-p_j(x))} =\exp\!\left((\beta_1+b_{1j})\Delta x\right).

This is a conditional odds ratio: it compares the odds for two covariate values in the same cluster, with the same random effects. For a binary predictor coded 0 and 1, $\Delta x=1$ is the contrast from 0 to 1. For a continuous predictor, the units must be stated; a one-unit increase could mean one year, one centimetre, or another chosen unit.

To see how the odds ratio translates into probability, let $p_0$ be the conditional probability at the starting value and let $R$ be the odds ratio for the specified change. The starting odds are $p_0/(1-p_0)$. Multiplying those odds by $R$ and converting back to a probability gives

p_1=\frac{R\,p_0}{1-p_0+R\,p_0}, \qquad p_1-p_0=\frac{p_0(1-p_0)(R-1)}{1-p_0+R\,p_0}.

The probability difference depends on $p_0$, even when $R$ is unchanged. The formula also shows that if $R>1$, the probability increases, and if $R<1$, it decreases, provided $0<p_0<1$. It does not follow that the probability changes by $R-1$: that quantity is a relative change in odds, not a change in probability.

Worked Example: A Cluster-Specific Odds Ratio

Suppose a hypothetical model predicts whether a patient experiences a binary event, using a treatment score $x$ measured in points. For one cluster, let $\beta_1=0.40$ and $b_{1j}=-0.15$. Find the cluster-specific odds ratio for a one-point increase and interpret it.

The cluster-specific slope is $$ \beta_1+b_{1j}=0.40-0.15=0.25. $$

For a one-point increase, $\Delta x=1$, so $$ \operatorname{OR}_j(1)=\exp(0.25)\approx 1.284. $$

Within this cluster, and conditional on its random effects, a one-point increase in treatment score multiplies the event odds by about 1.284, or increases the odds by about 28.4%. This statement concerns odds, not a 28.4 percentage-point increase in event probability. To calculate a probability difference, the starting conditional probability is also needed.

Worked Example: The Same Odds Ratio at Two Starting Probabilities

For a hypothetical binary outcome, suppose a one-unit covariate increase has conditional odds ratio $R=2$ in a cluster. Compare the probability change when the starting probability is 0.10 with the change when it is 0.50.

When $p_0=0.10$, the updated probability is $$ p_1=\frac{2(0.10)}{1-0.10+2(0.10)} =\frac{0.20}{1.10}\approx 0.1818. $$ The probability difference is $0.1818-0.10=0.0818$, or about 8.18 percentage points.

When $p_0=0.50$, the updated probability is $$ p_1=\frac{2(0.50)}{1-0.50+2(0.50)} =\frac{1.00}{1.50}\approx 0.6667. $$ The probability difference is $0.6667-0.50=0.1667$, or about 16.67 percentage points.

The odds double in both comparisons, but the probability increases by different amounts. The starting probability is part of the probability-scale interpretation; the odds ratio alone cannot determine a single probability difference.

Derivation of the Conditional Contrast

The odds-ratio result follows directly from the model. Write the conditional linear predictor in cluster $j$ at covariate value $x$ as $\eta_j(x)=\beta_0+b_{0j}+(\beta_1+b_{1j})x$. Since the inverse-logit model implies that the odds equal $\exp(\eta_j(x))$, the odds at $x+\Delta x$ divided by the odds at $x$ are

Write the two linear predictors.
$\eta_j(x+\Delta x)-\eta_j(x)=(\beta_1+b_{1j})\Delta x$. The intercept terms cancel because the comparison is within the same cluster.

Convert the linear-predictor contrast to an odds ratio.
The odds are $\exp(\eta)$, so their ratio is $\exp\{\eta_j(x+\Delta x)-\eta_j(x)\}=\exp\{(\beta_1+b_{1j})\Delta x\}$.

Convert the odds to a probability.
If the starting odds are $o_0=p_0/(1-p_0)$, the ending odds are $R o_0$. Therefore $p_1=R o_0/(1+R o_0)=R p_0/(1-p_0+R p_0)$.

The derivation depends on comparing covariate values while keeping the cluster’s random effects fixed. If different clusters are compared, their intercept and slope deviations may both differ, so the contrast is not described by the slope alone. Likewise, a marginal comparison that averages over the random-effects distribution is not the same quantity as this conditional odds ratio.

Worked Example: Comparing Conditional Slopes Across Clusters

Suppose a hypothetical model uses hours of an exposure as $x$, and has fixed slope $\beta_1=0.30$. For cluster A, let $b_{1A}=-0.10$; for cluster B, let $b_{1B}=0.20$. Calculate each cluster’s odds ratio for a two-hour increase.

For cluster A, the slope is $0.30-0.10=0.20$ per hour. Thus $$ \operatorname{OR}_A(2)=\exp(0.20\times 2)=\exp(0.40)\approx 1.492. $$ Conditional on cluster A’s random effects, a two-hour increase multiplies the event odds by about 1.492.

For cluster B, the slope is $0.30+0.20=0.50$ per hour. Thus $$ \operatorname{OR}_B(2)=\exp(0.50\times 2)=\exp(1.00)\approx 2.718. $$ The same two-hour increase multiplies the event odds by about 2.718 in cluster B. These are odds comparisons, not probability differences; computing either probability difference would require the starting probability in that cluster at its starting exposure value.

Assumptions, Diagnostics, and Alternatives

A numerical slope interpretation is only as useful as the model and the information supporting it. First, check the coding and units of $x$: a slope per one unit is not interpretable until that unit is known. Second, assess whether a linear trend in conditional log odds is plausible over the covariate range being interpreted. A single linear slope can be misleading if the conditional relationship bends or changes direction. Third, check whether the data contain enough outcome variation and covariate variation within clusters to support cluster-specific slope estimation. A cluster with few observations or nearly all outcomes equal to 0 or all equal to 1 may contribute little information about its slope.

Sparse or separated data can lead to unstable estimates, very large coefficients, or numerical warnings. Convergence problems are not merely cosmetic: they can signal that the model is poorly supported by the data or difficult to estimate. Inspect fitted values and residual-based diagnostics appropriate for binary mixed models, investigate influential clusters, and compare observed patterns with model-implied patterns. A model that fits poorly should not be rescued by presenting a precise-sounding odds ratio. Uncertainty in estimated random effects also matters: a cluster-specific estimate is not known without error, particularly when that cluster has limited data.

A random-intercept-only logistic model is a simpler alternative, but it imposes the same conditional log-odds slope in every cluster. It can be reasonable when there is no substantive or empirical reason to allow the covariate effect to vary. The random-slope model relaxes that restriction, at the cost of estimating additional cluster-level structure and requiring adequate information. A population-averaged analysis targets a different summary from a cluster-conditional model; those summaries should not be substituted for one another without stating the change in estimand. Ordinary logistic regression also lacks the cluster-specific random-effects structure of this model.

Common pitfall: “The odds ratio is 2, so the probability rises by 100 percentage points.” This is incorrect: an odds ratio of 2 doubles odds. The probability increase must be calculated from the starting probability using $p_1=2p_0/(1-p_0+2p_0)$, and it will generally be less than 100 percentage points.

The main practical limitation is that a cluster-specific odds ratio does not by itself give a reader an intuitive probability-scale effect. When probabilities are central to the question, report conditional probabilities at clearly stated covariate values and for specified random effects, alongside the conditional odds ratio. Do not describe those conditional quantities as population averages. The logit-scale linearity assumption, available data within clusters, and uncertainty in the fitted parameters all limit how broadly the result can be generalized.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A cluster-specific slope is $-0.20$ per unit. What is the conditional odds ratio for a three-unit increase?
If an odds ratio is 3 and the starting probability is 0.20, calculate the ending probability and probability difference.
Why does a random-intercept-only model give the same conditional odds ratio for a given covariate change in every cluster?
True or false: if two clusters have the same conditional odds ratio, they must have the same probability difference. Explain.

Solutions. (1) The log-odds contrast is $(-0.20)(3)=-0.60$, so the odds ratio is $\exp(-0.60)\approx0.549$. (2) $p_1=3(0.20)/(1-0.20+3(0.20))=0.60/1.40\approx0.4286$; the increase is about $0.4286-0.20=0.2286$, or 22.86 percentage points. (3) With no random slope, the cluster-specific slope is the common fixed slope $\beta_1$; random intercepts cancel in a within-cluster covariate contrast. (4) False. The same odds ratio can produce different probability differences at different starting probabilities, as shown by $p_1=R p_0/(1-p_0+R p_0)$.