What a GLMM Results Table Must Make Clear
A GLMM results table is useful only if readers can tell what each number estimates. A coefficient without its predictor scale or reference category may be impossible to interpret; a variance placed among fixed effects may be mistaken for a treatment effect. A clear table therefore separates different kinds of model quantities and supplies the labels needed to read them.
The table should identify the outcome and its units, the model family and link, and the random-effects structure. Its fixed-effect section should state the predictor contrast or increment, the reference category where relevant, and whether reported effects are on the link scale or a transformed scale. Its random-effect section should give the grouping factor and the variance or standard deviation being summarized. A concise note can specify interval and test methods and define abbreviations.
For a binomial model with a logit link, a fixed-effect coefficient is a conditional log odds ratio. Exponentiating it gives a conditional odds ratio. For a Poisson model with a log link, a coefficient is a log rate ratio (when the model includes an appropriate exposure offset), and exponentiating it gives a rate ratio. These transformations do not turn a coefficient into a marginal population-average effect: they retain the model's conditional interpretation, given the random effects.
A standard table layout has a short model-description block, a fixed-effects panel, and a random-effects panel. The table note should identify the interval method and the meaning of any effect-scale column. Do not assume that labels such as “estimate” or “95% CI” are self-explanatory.
Model Information and Table Design
Before filling in estimates, write down what a reader must know to interpret them. State the response and its measurement units, the family and link, and the random-effects structure. For example, “30-day readmission (yes/no); binomial family, logit link; random intercept for hospital” is more informative than “logistic mixed model.” If a count model uses an exposure offset, mention it; the offset affects the interpretation of the rate.
In the fixed-effects panel, use a row label that specifies the unit or contrast. “Age, per 10 years” is preferable to “Age” if the model used age divided by 10. For a categorical predictor, include the comparison direction and the reference category, such as “Treatment: intervention vs usual care (reference: usual care).” If the outcome is binary, name the modeled event so that the direction of an odds ratio is clear.
A useful set of fixed-effect columns is estimate, standard error, confidence interval, and p-value. If the model uses a log link, an additional column can report the exponentiated estimate and its interval. Either report the link-scale coefficient or make the effect-scale transformation explicit; including both is often helpful because it preserves the fitted coefficient while giving a more directly interpretable effect measure.
For random effects, report the grouping factor and the fitted random-effect term, then its variance and, when useful, standard deviation. A random-intercept variance is measured on the linear-predictor scale squared; its standard deviation is on the linear-predictor scale. Do not call either quantity an odds ratio or a rate ratio. In a logistic model the random-intercept variance is on the latent logit scale, not a directly observed probability scale.
Worked Examples
Worked Example: A Binary Outcome and a Hospital Random Intercept
Suppose a hypothetical study models whether a patient is readmitted within 30 days, with a binomial family and logit link and a random intercept for hospital. The fixed-effect estimate for intervention versus usual care is \(\hat{\beta}=-0.510\), with \(SE=0.200\). The usual-care group is the reference category. The fitted hospital-intercept variance is \(0.160\).
Assume the reported interval and p-value use a large-sample Wald calculation. The 95% interval on the logit scale is
The Wald statistic is \(z=-0.510/0.200=-2.55\), giving a two-sided p-value of approximately \(0.0108\). Since the logit coefficient is a log odds ratio, exponentiating gives an odds ratio of \(\exp(-0.510)=0.600\). Exponentiating both interval endpoints gives \([\exp(-0.902),\exp(-0.118)]=[0.406,0.889]\).
| Fixed effects | Estimate (log odds) | SE | 95% CI (log odds) | Conditional OR (95% CI) | p-value |
|---|---|---|---|---|---|
| Intervention vs usual care (reference: usual care) | -0.510 | 0.200 | -0.902 to -0.118 | 0.600 (0.406 to 0.889) | 0.0108 |
| Random effects | Variance | SD |
|---|---|---|
| Hospital random intercept | 0.160 | 0.400 |
The table note should state that the outcome is 30-day readmission (yes/no), that the model uses a binomial family with logit link, and that intervals and p-values are Wald-based. The odds ratio compares conditional odds for intervention with usual care, at the same value of the hospital random intercept. It is not a statement that intervention causes a change in readmission.
Worked Example: A Count Outcome With an Exposure Offset
In a hypothetical ecological analysis, the outcome is the number of events recorded in each region, with person-time included as a log offset. A Poisson model with log link includes a random intercept for region. Suppose a predictor is measured in 5-unit increments and has estimate \(\hat{\beta}=0.262\), \(SE=0.110\). Its row compares a 5-unit increase with no increase. The region random-intercept variance is \(0.090\).
The Wald interval is \(0.262\pm1.96(0.110)=0.262\pm0.2156=[0.0464,0.4776]\). The Wald statistic is \(z=0.262/0.110=2.38\), with two-sided \(p\approx0.0172\). Exponentiating the coefficient and interval gives a rate ratio of \(\exp(0.262)=1.300\), with interval \([\exp(0.0464),\exp(0.4776)]=[1.048,1.612]\). Thus, for a 5-unit increase in the predictor, the conditional event rate is estimated to be 1.30 times as large, given the same region random intercept and exposure basis.
| Fixed effects | Estimate (log rate ratio) | SE | 95% CI (log rate ratio) | Conditional rate ratio (95% CI) | p-value |
|---|---|---|---|---|---|
| Predictor, per 5-unit increase | 0.262 | 0.110 | 0.0464 to 0.4776 | 1.300 (1.048 to 1.612) | 0.0172 |
| Random effects | Variance | SD |
|---|---|---|
| Region random intercept | 0.090 | 0.300 |
The model description should also say that person-time is included as an offset. Without that information, a reader might interpret the result as a comparison of event counts rather than rates. The standard deviation is \(\sqrt{0.090}=0.300\) on the log-rate scale.
Why the Interval Transformation Works
When a coefficient has a log link-scale interpretation, exponentiating it gives the corresponding multiplicative effect. Because the exponential function is increasing, exponentiating the two endpoints of an interval preserves their order and gives an interval on the transformed scale. In symbols, if a coefficient interval is \([L,U]\), then its exponentiated interval is \([\exp(L),\exp(U)]\). It is not generally correct to exponentiate the interval's width or to add and subtract the original standard error from the exponentiated estimate.
The Wald interval used in the examples is justified as a large-sample approximation: if the estimator is approximately normally distributed with estimated standard error \(SE\), a 95% interval is \(\hat{\beta}\pm1.96SE\). This approximation may be poor for small samples, sparse outcomes, or estimates near a boundary. The table must not disguise an uncertain approximation as an exact result. State the interval method and use a method appropriate to the fitted model and data.
Worked Example: Correcting an Ambiguous Fixed-Effect Row
A hypothetical logistic GLMM has a binary outcome “symptom present,” a logit link, and a participant random intercept. The fitted treatment coefficient for active treatment versus control is \(-0.405\), with \(SE=0.180\); control is the reference. A draft table labels the estimate column “Odds ratio” and places \(-0.405\) in that row. This is incorrect: an odds ratio cannot be negative, because it is the exponentiated coefficient.
The corrected calculation is \(z=-0.405/0.180=-2.25\), with two-sided \(p\approx0.0244\). The Wald log-odds interval is \(-0.405\pm1.96(0.180)=[-0.758, -0.052]\). Thus the odds ratio is \(\exp(-0.405)=0.667\), and the transformed interval is \([\exp(-0.758),\exp(-0.052)]=[0.469,0.949]\).
| Fixed effects | Estimate (log odds) | SE | 95% CI (log odds) | Conditional OR (95% CI) | p-value |
|---|---|---|---|---|---|
| Active treatment vs control (reference: control) | -0.405 | 0.180 | -0.758 to -0.052 | 0.667 (0.469 to 0.949) | 0.0244 |
The row now identifies the modeled event, comparison direction, reference category, link-scale estimate, and conditional odds ratio. Its note should name the binomial family and logit link and define the random-intercept grouping factor. This correction matters because a reader should not have to infer which scale the displayed number uses.
Checks, Alternatives, and Limitations
Before finalizing a table, audit each row against the model output and the analysis specification. Check that transformed estimates and intervals are calculated from the same link-scale values; that a factor's reference category and contrast direction are correct; that continuous-predictor units match the fitted scaling; and that random-effect labels match the grouping variables. Also check that rounding has not changed a stated conclusion. The p-value should not have more apparent precision than the estimate and uncertainty justify.
- Do not mix scales silently. A coefficient of \(-0.510\) and an odds ratio of \(0.600\) are both valid, but only when their columns identify the different scales.
- Do not place a variance in the fixed-effects panel. A random-intercept variance describes between-group heterogeneity, not a predictor association.
- Do not report a random-effect standard deviation without its scale. It is measured on the linear-predictor scale; it is not a standard deviation in outcome units.
- Do not imply that a variance-component p-value is routine or definitive. Variance parameters are constrained to be nonnegative, so ordinary large-sample tests can behave poorly near zero. If inferential testing is important, state and justify the method rather than adding an unexplained p-value.
- Do not omit the family, link, or event definition. The same coefficient sign has different practical meaning under different outcome coding or link functions.
A compact table can show only transformed effects, such as odds ratios, and omit link-scale coefficients when space is limited. That approach can be readable, but the scale, interval method, and contrast still need explicit labels; a reader loses direct access to the fitted regression coefficient. Alternatively, a table can show only link-scale coefficients, which is concise and model-faithful but often less accessible to applied readers. Showing both is usually clearest when space allows. No table format replaces the model specification or establishes that the model assumptions are adequate; the table is a communication device, not a diagnostic.
Inference columns also require care. A Wald standard error, Wald interval, and Wald p-value are linked to a normal approximation. Likelihood-based or resampling intervals may be preferable in settings where that approximation is weak, but the chosen method should be named. For random-effect variances, a point estimate alone can hide substantial uncertainty; if an interval is shown, identify how it was obtained and remember that its distribution may be asymmetric.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- A logistic GLMM reports \(\hat{\beta}=0.300\) for a 1-unit increase in a predictor. What is the corresponding odds ratio, and what information is needed to interpret its direction?
- A table lists “clinic variance = 0.25” in a fixed-effect panel. Explain the error and give the corresponding standard deviation.
- A Poisson log-link model estimates \(\hat{\beta}=-0.200\) for a stated 2-unit increase in a predictor. What is the rate ratio? What predictor label should the table use?
- Why should a table report the outcome family and link, even when the coefficient estimates and standard errors are included?
Solutions. (1) \(\exp(0.300)=1.350\). The table must state the modeled event, the predictor increment, and the comparison direction; for a categorical predictor it must also name the reference category. (2) Clinic variance is a random-effect summary, so it belongs in the random-effects panel. Its standard deviation is \(\sqrt{0.25}=0.500\) on the linear-predictor scale. (3) \(\exp(-0.200)=0.819\). Label the row “predictor, per 2-unit increase,” and state the comparison direction; the negative coefficient corresponds to a conditional rate ratio below one. (4) The family and link determine the scale and interpretation of coefficients, so the numbers alone do not identify whether an effect is a log odds, log rate, or another quantity.