A Worked Logistic Random-Intercept Interpretation

From a Cluster’s Linear Predictor to Its Probability

A logistic random-intercept model gives each cluster its own baseline on the log-odds scale. To interpret a prediction, specify both the covariate value and the cluster’s random-intercept value, then transform the resulting linear predictor into odds and a probability. The calculation is simple, but the conditioning matters: the result describes a response probability for a specified cluster deviation, not an average over all clusters.

Definition: For binary response $Y_{ij}$, covariate $x_{ij}$, and cluster $j$, a logistic random-intercept model is $$ Y_{ij}\mid b_j,x_{ij}\sim\operatorname{Bernoulli}(p_{ij}),\qquad \operatorname{logit}(p_{ij})=\log\left(\frac{p_{ij}}{1-p_{ij}}\right)=\beta_0+\beta_1x_{ij}+b_j. $$ Here $p_{ij}=P(Y_{ij}=1\mid x_{ij},b_j)$, $\beta_0$ is the fixed intercept, $\beta_1$ is the fixed covariate coefficient, and $b_j$ is cluster $j$'s random-intercept deviation. The conventional random-effects assumption is $b_j\sim N(0,\tau_{00})$, with independent random effects across clusters. Responses are assumed independent conditional on the random effects and covariates.

The conditional linear predictor is $\eta_{ij}=\beta_0+\beta_1x_{ij}+b_j$. For any specified $\eta_{ij}$, the conditional odds are $\exp(\eta_{ij})$, and the conditional probability is

p_{ij}=\frac{\exp(\eta_{ij})}{1+\exp(\eta_{ij})} =\frac{1}{1+\exp(-\eta_{ij})}.

These are two equivalent forms of the inverse-logit transformation. A positive $b_j$ raises the linear predictor and hence the cluster’s odds at every fixed $x_{ij}$; a negative $b_j$ lowers them. The value $b_j=0$ represents the model’s reference cluster on the link scale. It does not necessarily represent the average cluster’s probability.

Why the Transformation Works

Starting with the definition of the logit, $\log(p/(1-p))=\eta$, exponentiate both sides to obtain $\exp(\eta)=p/(1-p)$. Multiplying by $1-p$ and collecting the terms involving $p$ gives

\exp(\eta)(1-p)=p \quad\Longrightarrow\quad \exp(\eta)=p\{1+\exp(\eta)\} \quad\Longrightarrow\quad p=\frac{\exp(\eta)}{1+\exp(\eta)}.

The odds-to-probability conversion follows from the same algebra. If the odds are $o=p/(1-p)$, then $o(1-p)=p$, so $p=o/(1+o)$. Odds and probabilities therefore contain the same information, but they are not numerically interchangeable: odds can exceed 1, whereas probabilities lie between 0 and 1.

For a numerical prediction, the model is conditional on a specified $b_j$. The normal distribution for $b_j$ describes variation across clusters; it is not needed to perform the inverse-logit calculation once a particular value of $b_j$ has been selected. When fitted coefficients are used, the resulting probabilities are model-based plug-in predictions, with parameter-estimation uncertainty not included unless it is separately calculated.

Worked Example: Two Cluster Baselines at the Same Covariate Value

Suppose a hypothetical fitted model for a binary outcome is $\operatorname{logit}(p_{ij})=-1.5+0.8x_{ij}+b_j$. Calculate the conditional odds and probability for $x_{ij}=1$ in two clusters, one with $b_j=-0.7$ and one with $b_j=0.5$.

Cluster A. Its conditional linear predictor is $\eta=-1.5+0.8(1)-0.7=-1.4$. Thus its odds are $\exp(-1.4)\approx0.2466$, and its conditional probability is \[ p=\frac{\exp(-1.4)}{1+\exp(-1.4)}\approx0.1978. \] The probability is about 19.78% for an observation with $x=1$ in a cluster with this specified intercept deviation.

Cluster B. Its predictor is $\eta=-1.5+0.8(1)+0.5=-0.2$. Its odds are $\exp(-0.2)\approx0.8187$, and its conditional probability is \[ p=\frac{\exp(-0.2)}{1+\exp(-0.2)}\approx0.4502. \] The displayed expressions retain the unrounded exponentials in the probability calculations; the odds are shown rounded for reporting.

At the same covariate value, Cluster B has a higher conditional probability than Cluster A. Their odds ratio is $\exp\{0.5-(-0.7)\}=\exp(1.2)\approx3.320$, because the difference in their random intercepts is 1.2. The conditional probability difference, calculated from the unrounded probabilities, is approximately $0.450166-0.197816=0.2524$, or 25.24 percentage points. This probability difference is specific to this covariate value and these cluster deviations.

Changing the Covariate and the Cluster Deviation

A fixed covariate coefficient changes the conditional log odds by the same amount for each one-unit covariate increase, regardless of the cluster. The corresponding odds multiplier is $\exp(\beta_1)$. However, the probability change is not generally constant across covariate values or clusters: the inverse-logit transformation is nonlinear. Thus a single odds ratio does not imply a single probability difference.

Worked Example: A Covariate Contrast Within One Cluster

Consider the hypothetical model $\operatorname{logit}(p_{ij})=-2+0.45x_{ij}+b_j$. For a cluster with $b_j=0.4$, compare conditional predictions at $x_{ij}=0$ and $x_{ij}=2$.

At $x=0$, $\eta=-2+0.45(0)+0.4=-1.6$. The odds are $\exp(-1.6)\approx0.2019$, and the probability is \[ p=\frac{\exp(-1.6)}{1+\exp(-1.6)}\approx0.1680. \] At $x=2$, $\eta=-2+0.45(2)+0.4=-0.7$. The odds are $\exp(-0.7)\approx0.4966$, and the probability is \[ p=\frac{\exp(-0.7)}{1+\exp(-0.7)}\approx0.3318. \]

Increasing $x$ by two units multiplies the conditional odds by $\exp\{0.45(2)\}=\exp(0.9)\approx2.460$. This agrees with the odds computed before rounding: $\exp(-0.7)/\exp(-1.6)=\exp(0.9)$. The probability increases by about $0.331812-0.167982=0.1638$, or 16.38 percentage points. That probability change is the result for this cluster and this covariate contrast; it is not a universal effect of a two-unit increase.

Worked Example: Comparing Two Clusters at a Fixed Covariate Value

Suppose a second hypothetical model is $\operatorname{logit}(p_{ij})=-0.8-0.6x_{ij}+b_j$. At $x=1$, compare a cluster with $b_j=0.3$ with a cluster having $b_j=-0.4$.

For the first cluster, $\eta=-0.8-0.6(1)+0.3=-1.1$. Its odds are $\exp(-1.1)\approx0.3329$, and its probability is \[ p=\frac{\exp(-1.1)}{1+\exp(-1.1)}\approx0.2497. \] For the second cluster, $\eta=-0.8-0.6(1)-0.4=-1.8$. Its odds are $\exp(-1.8)\approx0.1653$, and its probability is \[ p=\frac{\exp(-1.8)}{1+\exp(-1.8)}\approx0.1419. \]

At this fixed $x$, the first cluster’s odds are $\exp\{0.3-(-0.4)\}=\exp(0.7)\approx2.014$ times the second cluster’s odds. The probability difference, using the unrounded inverse-logit values, is approximately $0.249740-0.141851=0.1079$, or 10.79 percentage points. The negative coefficient $-0.6$ describes the covariate’s conditional log-odds change per unit; it does not mean every one-unit increase reduces probability by the same number of percentage points.

Assumptions, Diagnostics, and Interpretation Checks

These calculations rely on a binary response and the specified logit conditional mean model. They also rely on the random-intercept structure being an adequate description: after conditioning on $b_j$ and the covariates, the model assigns the stated Bernoulli probability to each response, and the conditional-independence assumption is appropriate for the observations. The normal random-effects distribution is part of the usual model specification, though it does not change the arithmetic for a selected $b_j$.

A useful way to check a calculation is to examine its direction and scale. If $\eta=0$, odds equal 1 and probability equals 0.5. Positive $\eta$ must yield probability above 0.5; negative $\eta$ must yield probability below 0.5. Larger $\eta$ must produce larger odds and probability. Probabilities must remain strictly between 0 and 1 for finite $\eta$. For very large positive or negative predictors, direct computation of $\exp(\eta)$ can overflow or underflow; the equivalent form $1/\{1+\exp(-\eta)\}$, or a numerically stable software implementation, avoids some such issues.

A calculated value is only as relevant as its covariate reference point. If $x=0$ has no substantive meaning, the fixed intercept and the prediction at $x=0$ may be mathematically valid but scientifically unhelpful. Likewise, a chosen $b_j$ should be identified as a specified deviation, such as a hypothetical value or an estimated cluster effect, rather than described as a known cluster characteristic without qualification. The random-effects distribution can help contextualize whether a selected value is typical, but the conditional calculation itself does not establish that.

The main alternative interpretations answer different questions. A logistic regression without a random intercept gives a common intercept rather than cluster-specific baselines. A logistic random-intercept prediction conditions on a given $b_j$. A population-averaged probability instead averages the conditional probabilities over the random-effects distribution, for example $E_{b_j}\{\operatorname{logit}^{-1}(\beta_0+\beta_1x+b_j)\}$. Because the inverse-logit function is nonlinear, this average is generally not equal to $\operatorname{logit}^{-1}(\beta_0+\beta_1x)$. The latter is the prediction at $b_j=0$, not automatically the population average. Estimating or interpreting marginal effects requires an explicit averaging calculation and is a distinct target from the conditional predictions worked here.

Common Mistakes

Calling a conditional prediction a population average. A value computed at $b_j=0$ is conditional on that link-scale deviation. It is not generally the average probability across clusters.
Treating $b_j$ as a probability adjustment. The random intercept is added to the log odds. It must be included in $\eta$ before applying the inverse-logit transformation.
Confusing odds and probability. If the odds are $o$, the probability is $o/(1+o)$, not $o$. Odds of 0.5 correspond to probability $0.5/1.5=0.3333$, not 0.5.
Assuming a constant probability difference. The coefficient $\beta_1$ implies a fixed change in conditional log odds per unit of $x$; the corresponding probability change depends on the starting predictor.
Rounding intermediate values too early. Use the unrounded exponential or probability in subsequent arithmetic, then round the final reported result. A result calculated from rounded odds can differ in its last reported digit.

Key takeaway: To interpret a logistic random-intercept prediction, state the covariate value and random-intercept value, calculate $\eta=\beta_0+\beta_1x+b_j$, and report the conditional odds and probability from that same predictor. Label the result as cluster-specific; do not silently replace it with a population average.

Check Your Understanding

Try each calculation, then compare with the brief solutions below.

For $\operatorname{logit}(p)=-1.2+0.5x+b_j$, calculate the odds and probability at $x=2$ and $b_j=-0.3$.
If a cluster’s random intercept increases by 0.4 while $x$ is fixed, by what factor do its conditional odds change?
For $\operatorname{logit}(p)=-2+0.7x+b_j$, calculate the probability at $x=1$, $b_j=0.2$. Is it above or below 0.5?
Explain why the prediction at $b_j=0$ is not necessarily the population-averaged probability.

Solutions. (1) $\eta=-1.2+0.5(2)-0.3=-0.5$; the odds are $\exp(-0.5)\approx0.6065$, and $p=\exp(-0.5)/\{1+\exp(-0.5)\}\approx0.3775$. (2) The odds multiplier is $\exp(0.4)\approx1.492$. (3) $\eta=-2+0.7(1)+0.2=-1.1$; the odds are $\exp(-1.1)\approx0.3329$, and $p=\exp(-1.1)/\{1+\exp(-1.1)\}\approx0.2497$, below 0.5 because the predictor is negative. (4) The population average averages inverse-logit probabilities over the random-intercept distribution, whereas $b_j=0$ evaluates the inverse logit at one particular deviation. Nonlinearity means these operations are generally unequal.