Reading a Trajectory as a Cluster-Specific Model
A random-slope model does not give every cluster the same trend. It describes a shared pattern through fixed effects, then allows a cluster’s slope to differ through its random slope. To read a fitted trajectory, combine those terms before describing its direction or size. When the model includes a cross-level interaction, the fixed slope itself depends on the cluster-level covariate, so that term must also be included.
A trajectory is conditional on the cluster’s random effects: it describes the model’s expected response for a cluster when its random effects are held fixed. It is not the trajectory obtained by averaging over the random-effects distribution. This distinction matters especially with nonlinear links, where transforming an average link value does not generally give the average response.
Here \(g\) is the link function, \(\beta_0,\beta_w,\beta_x,\beta_{xw}\) are fixed effects, and \(b_{0j},b_{1j}\) are cluster \(j\)'s random intercept and random slope. In the usual specification, the random-effects vector has mean zero and a covariance matrix that allows the intercept and slope to be correlated. Conditional on the random effects, observations follow the chosen response distribution and have the conditional mean specified by the model. The interpretation below treats the covariates and the cluster’s random effects as fixed.
For a given cluster, the linear predictor is a line in \(x\). Its intercept at \(x=0\) and its slope are
Thus the fixed slope \(\beta_x+\beta_{xw}w_j\) is the model’s slope for a cluster with covariate value \(w_j\) and random slope zero. The cluster-specific slope adds \(b_{1j}\). With no cross-level interaction, set \(\beta_{xw}=0\); then clusters share the same fixed slope, and their slope differences arise from their random slopes.
What Trajectories Say on Each Scale
The linear predictor is linear in \(x\), but the conditional response mean is \(g^{-1}\) of that predictor. For the identity link, the response trajectory is a straight line and its slope is \(s_j\) response units per unit of \(x\). For a log link, a one-unit increase in \(x\) multiplies the conditional mean by \(\exp(s_j)\). For a logit link, it multiplies the conditional odds by \(\exp(s_j)\); the probability change depends on the starting probability.
A short derivation makes the comparison precise. For clusters \(j\) and \(k\), at the same \(x\), subtract their linear predictors. If their cluster-level covariates differ, this includes differences in fixed intercept and slope terms as well as random effects. As \(x\) changes from \(x_L\) to \(x_U\), the change in their link-scale difference is
where \(\eta_j(x)\) denotes cluster \(j\)'s linear predictor. This follows because each predictor is an intercept plus \(s_jx\), so subtracting the differences at the two values cancels both intercepts. Therefore, if \(s_j>s_k\), cluster \(j\)'s link-scale trajectory gains relative to cluster \(k\) as \(x\) increases. Their levels at any one \(x\) still depend on the intercepts. A greater slope does not imply that cluster \(j\) has the greater response throughout the range.
For standard GLMM links such as identity, log, and logit, \(g^{-1}\) is increasing. Consequently, a positive \(s_j\) means the conditional response mean increases with \(x\), and a negative \(s_j\) means it decreases. The rate of change on the response scale is not generally constant. For the log link it is \(s_j\mu_j(x)\); for the logit link it is \(s_j\mu_j(x)\{1-\mu_j(x)\}\), where \(\mu_j(x)\) is the conditional mean. These derivatives follow from the chain rule. The same link-scale slope can therefore produce different response-scale changes at different starting means.
Worked Example: Comparing Two Identity-Link Trajectories
Suppose \(x\) is study time in hours, observed from 0 to 4 hours, and the fitted model for a hypothetical continuous outcome measured in points is
Consider cluster A with \(w_A=0,\ b_{0A}=-1,\ b_{1A}=0.3\), and cluster B with \(w_B=1,\ b_{0B}=1,\ b_{1B}=-0.2\). Because the link is identity, each linear predictor is also the conditional response mean, in points.
For A, the intercept is \(10+2(0)-1=9\), and the slope is \(1.2-0.5(0)+0.3=1.5\) points per hour. Its trajectory is \(9+1.5x\). It predicts 9 points at \(x=0\) and \(9+1.5(4)=15\) points at \(x=4\).
For B, the intercept is \(10+2(1)+1=13\), and the slope is \(1.2-0.5(1)-0.2=0.5\) points per hour. Its trajectory is \(13+0.5x\), predicting 13 points at \(x=0\) and \(13+0.5(4)=15\) points at \(x=4\).
At the start of the observed range B is 4 points higher. The slope difference is \(1.5-0.5=1.0\) point per hour, so over four hours the difference changes by \(1.0(4)=4\) points. The fitted means meet at the upper endpoint; neither trajectory is higher there. This example shows why an intercept comparison alone, or a slope comparison alone, cannot describe the whole set of trajectories.
Nonlinear Links and Response-Scale Changes
For a nonlinear link, report both the scale on which the model is linear and the response-scale implication relevant to the question. A logit slope is a change in log odds per unit of \(x\); it is not a fixed change in probability. A log-link slope is a change in log mean; it is not a fixed increase in the mean. The following comparisons use hypothetical fitted values and hold the cluster random effects fixed.
Worked Example: Correctly Comparing Logistic Trajectories
Let \(x\) be an exposure measured in units, observed from 0 to 2, and suppose a hypothetical logistic random-slope model has fixed intercept \(-0.6\), fixed slope \(0.3\), and no cross-level interaction. Cluster A has \(b_{0A}=-0.4,\ b_{1A}=0.1\); cluster B has \(b_{0B}=0.4,\ b_{1B}=-0.2\). Thus
A’s log-odds slope is \(0.4\) per exposure unit, so its odds are multiplied by \(\exp(0.4)\approx1.492\) per unit. At \(x=0\), \(p_A(0)=1/\{1+\exp(1)\}=0.2689414214\). At \(x=2\), \(\eta_A(2)=-1+0.4(2)=-0.2\), so \(p_A(2)=1/\{1+\exp(0.2)\}=0.4501660027\). Using these unrounded probabilities, the increase is \(0.4501660027-0.2689414214=0.1812245813\), or about 0.1812, equivalent to 18.12 percentage points.
B’s log-odds slope is \(0.1\), corresponding to an odds multiplier of \(\exp(0.1)\approx1.105\) per unit. Its probability changes from \(p_B(0)=0.4501660027\) to \(p_B(2)=0.5\), an increase of \(0.0498339973\), or about 4.983 percentage points. A’s probability increase over this interval is larger, even though A starts with a lower probability. These are fitted conditional changes, not causal effects.
Worked Example: Reading Multiplicative Change Under a Log Link
Suppose \(x\) is follow-up time in years, observed from 0 to 3 years, and a hypothetical log-link model describes an expected event rate per fixed unit of observation time. For clusters A and B, the fitted linear predictors are
A’s conditional mean rate is \(\mu_A(x)=4\exp(0.3x)\). It is 4 at time zero and \(4\exp(0.9)\approx4(2.4596)=9.838\) at three years. Its rate is multiplied by \(\exp(0.3)\approx1.350\) per year. B’s rate is \(6\exp(0.1x)\): it is 6 at time zero and \(6\exp(0.3)\approx6(1.3499)=8.099\) at three years, a per-year multiplier of \(\exp(0.1)\approx1.105\).
B begins with the higher fitted rate, but A has the steeper log-mean slope and the larger rate at year 3. The response-scale increases are not constant: A’s increase is about \(9.838-4=5.838\) rate units over the interval, whereas B’s is about \(8.099-6=2.099\). The slope coefficients describe multiplicative change per year; the endpoint differences describe absolute rate change over this particular interval.
Use the Observed Range and Check the Fit
A slope describes a fitted trend, not a guarantee that the same pattern holds at every possible covariate value. State the range of \(x\) over which the cluster data support the comparison. Outside that range, the model may imply increasingly separated lines, probabilities approaching 0 or 1, or rates that grow rapidly, even though there are little or no data to assess those implications. When clusters have different observed ranges, comparisons at a shared \(x\) can be especially uncertain if one cluster contributes few observations near that value.
Interpretation also depends on the model’s linearity on the link scale. Residual patterns or systematic departures from the fitted trend may indicate that a single linear slope is inadequate over the observed range. A random-slope distribution estimated near zero variance, or a fit with a boundary or convergence problem, calls for caution in interpreting cluster-to-cluster slope differences. Cluster-specific random effects are estimated with uncertainty and are generally shrunk toward zero; they should not be treated as error-free descriptions of each cluster’s true trajectory.
A model with a fixed slope only imposes a common link-scale slope and cannot represent cluster-to-cluster slope variation. Separate regressions for each cluster allow unrelated slopes but do not, by themselves, model their shared distribution or partial pooling. A random-slope GLMM provides a structured middle ground: it estimates a population distribution of slopes while estimating cluster-specific deviations. A marginal, population-averaged response answers a different question because it averages over random effects rather than holding one cluster’s effects fixed.
A Practical Reading Sequence
For an interpretation that remains tied to the model and the data, work through the following sequence:
Substitute the cluster’s \(w_j\), \(b_{0j}\), and \(b_{1j}\) into the fixed and random terms before describing its trajectory.
Combine \(\beta_x\), \(\beta_{xw}w_j\), and \(b_{1j}\). State the units as link-scale units per unit of \(x\).
Use the inverse link. Describe an additive change for the identity link, a multiplicative mean change for the log link, or an odds multiplier for the logit link.
Compare fitted levels at relevant shared covariate values and avoid claiming that a slope alone ranks clusters at every value.
These steps keep the fixed slope, interaction, and random deviation distinct while still producing one cluster-specific trajectory. They also make clear which statements are on the link scale and which are on the response scale.
Check Your Understanding
Answer each question, then check the solutions below.
- In a model with link-scale slope \(\beta_x+\beta_{xw}w_j+b_{1j}\), what does \(b_{1j}\) contribute to cluster \(j\)'s trajectory?
- Two identity-link trajectories are \(5+2x\) and \(9+x\). At what \(x\) do they meet, and which has the larger fitted response at \(x=0\)?
- For a log-link slope of \(-0.2\), what is the conditional-mean multiplier for a one-unit increase in \(x\)?
- Why is a logit slope not a constant probability change per unit of \(x\)?
Solutions. (1) It adds \(b_{1j}\) to the fixed slope, changing the cluster’s link-scale change per unit of \(x\); it is not a separate observation-level slope. (2) Set \(5+2x=9+x\), giving \(x=4\). At \(x=0\), the second trajectory is 9, greater than 5. (3) The multiplier is \(\exp(-0.2)\approx0.8187\), so the conditional mean is multiplied by about 0.819 per unit increase. (4) The slope is a change in log odds. Converting odds to probabilities is nonlinear, and the probability change depends on the starting probability.