What an Andersen-Gill Coefficient Describes
A recurrent-event coefficient is easy to misstate as an effect on “the number of events.” The Andersen-Gill (AG) coefficient has a more specific meaning: it describes a covariate association with the instantaneous recurrence intensity, conditional on a subject’s observed history and covariates. It is not, by itself, a contrast in expected event counts or in the probability of experiencing at least one recurrence.
Let \(N_i(t)\) be the number of recurrent events experienced by subject \(i\) by time \(t\), and let \(Y_i(t)\) indicate whether that subject is under observation and at risk immediately before \(t\). Write \(Z_i(t)\) for the covariate vector, with values known just before \(t\), and let \(\mathcal{H}_i(t-)\) denote the observed event and covariate history up to just before \(t\). The AG model specifies the conditional intensity
Here \(\lambda_0(t)\) is an unspecified baseline intensity, and \(\beta\) is the vector of regression coefficients. In a short interval of length \(dt\), while subject \(i\) is at risk, the conditional expected event increment is approximately \(\lambda_i(t)\,dt\). Thus intensity has units of events per unit time; a ratio of intensities has no units. In the standard AG specification, the same coefficient applies across recurrences. Event history defines the information being conditioned on and who remains at risk; it does not automatically give each event order its own coefficient.
From Coefficient to Intensity Ratio
The ratio interpretation follows by comparing two covariate values at the same time. Suppose two at-risk subjects have the same other covariates, and their values of \(Z_j\) differ by \(\Delta\). Under the model, the baseline intensity and the contributions from the other covariates cancel:
For a one-unit contrast, \(\Delta=1\), so the ratio is \(\exp(\beta_j)\). For a contrast of \(k\) units, it is \(\exp(k\beta_j)\). This calculation depends on the log-linear covariate effect and proportional-intensity form: the ratio does not vary with time unless the model includes a time interaction or otherwise allows a time-varying coefficient. It also applies only where both covariate values are meaningful and subjects are at risk.
For a binary covariate, \(\exp(\beta_j)=1.4\) means the modeled conditional intensity is 1.4 times as high in the category coded 1 as in the reference category, for the same values of the other modeled covariates. Equivalently, the intensity is 40% higher. A ratio of 0.7 means it is 30% lower, not 70% lower. These percentage descriptions concern intensity, not the probability of an event in a specified period.
Worked Example: Binary Treatment
Suppose a hypothetical AG analysis compares a treatment group coded 1 with a control group coded 0. The fitted treatment coefficient is \(\hat{\beta}=-0.357\), with standard error \(0.150\). Assume the model includes the relevant baseline covariates and that the stated standard error is appropriate for the analysis.
The estimated conditional intensity ratio is
In this fitted model, at a given time when subjects are at risk, the treatment group’s conditional recurrence intensity is estimated to be 0.700 times the control group’s, for the same modeled covariates and observed-history conditioning. This corresponds to an estimated 30.0% lower intensity, since \(100(1-0.700)=30.0\%\). It does not establish that treatment reduces each subject’s total events by 30.0%, or that a subject’s probability of recurrence is 30.0% lower.
A Wald 95% confidence interval for the log coefficient is \(-0.357\pm1.96(0.150)\), or \((-0.651, -0.063)\). Exponentiating the endpoints gives \((\exp(-0.651),\exp(-0.063))=(0.522,0.939)\). Under the model and inference procedure, this interval gives a range of values compatible with the data for the conditional intensity ratio. It excludes 1, but its interpretation still depends on the model and the validity of the standard error.
Units, Covariate Coding, and Time-Varying Predictors
The meaning of a coefficient depends on how its covariate is coded and measured. If age is entered in years, \(\exp(\beta)\) is the ratio for a one-year difference. If age is entered in decades, it is the ratio for a ten-year difference. Reporting the measurement unit is essential; a large-looking coefficient per one unit may describe a very small real-world contrast.
Worked Example: Continuous Exposure Measured in Five-Unit Increments
In a hypothetical study, a continuous exposure is recorded in milligrams per day. To make the coefficient easier to interpret, the analyst defines \(Z\) as exposure divided by 5, so that a one-unit increase in \(Z\) means an increase of 5 mg/day. The fitted coefficient is \(0.080\), with standard error \(0.030\).
For a 5 mg/day increase, the estimated ratio is \(\exp(0.080)=1.083\), or an estimated 8.33% higher conditional recurrence intensity. For a 10 mg/day increase, the contrast is two units, so the ratio is \(\exp(2(0.080))=\exp(0.160)=1.174\), an estimated 17.4% higher intensity. The model assumes a linear relationship between the log intensity and exposure over the contrast; the 10 mg/day interpretation is not justified if that functional form is badly wrong.
For the five-unit contrast, the Wald 95% interval on the coefficient scale is \(0.080\pm1.96(0.030)=(0.0212,0.139)\). Exponentiating gives an intensity-ratio interval of \((1.021,1.149)\). In this hypothetical analysis, the interval corresponds to approximately 2.14% to 14.9% higher conditional intensity per 5 mg/day. It is an interval for a model-based contrast, not a statement that the true ratio has a 95% probability of lying inside the realized interval.
A time-varying covariate uses the value available immediately before the event time. For example, if medication status changes during follow-up, its coefficient compares the conditional intensities associated with current medication statuses, given the modeled covariates and observed history. The interpretation is not automatically the effect of ever receiving medication, nor does fitting a time-varying covariate by itself resolve confounding caused by treatment changes over time. The covariate must be defined so it is available before the event increment being modeled.
Worked Example: Current Status and a Conditional Contrast
In a hypothetical recurrent-infection cohort, medication status is coded 1 while a participant is currently taking medication and 0 otherwise. The fitted AG coefficient is \(-0.223\). Then \(\exp(-0.223)=0.800\). At times when participants are at risk, the model estimates an 0.800 ratio of conditional recurrence intensities for currently taking versus not currently taking medication, holding other modeled covariates fixed. That is a 20.0% lower estimated intensity during current use under the model.
The comparison is conditional on the information represented by the model and the observed history; it does not imply that starting medication causes a 20.0% reduction. If medication status changes in response to worsening health or prior recurrences, those factors can influence both current status and subsequent intensity. The coefficient’s interpretation as a conditional association remains, but a causal interpretation requires additional design and identification assumptions.
Assumptions, Inference, and Checks
An intensity-ratio interpretation is only as credible as the model and the observation process supporting it. Before interpreting an estimated coefficient, check the following conditions:
- Correct intensity structure. The multiplicative form, covariate functional forms, and any assumed common effect across recurrences should adequately represent the conditional intensity. A single \(\beta\) does not allow recurrence-specific effects unless the model is extended.
- Proportionality. Without a time interaction, the covariate intensity ratio is assumed constant over analysis time. A changing association can make one fitted coefficient a misleading summary.
- Predictable covariates. Covariate values used at time \(t\) must be known immediately before \(t\); using future information to define a predictor can invalidate the intended conditional interpretation.
- Observation and censoring. Censoring and loss to follow-up must be handled in a way that supports the modeled intensity. Informative observation or censoring not addressed by the analysis can distort the estimated association.
- Suitable risk sets and variation. Covariate contrasts require adequate representation among subjects at risk. Sparse data or near-separation can produce unstable estimates and wide or unreliable intervals.
Useful diagnostics include checking whether the covariate effect appears to change over time, examining the functional form of continuous predictors, and looking for influential subjects or unusual event patterns. Plotting or summarizing observed recurrence patterns by covariate groups can reveal gross mismatches, although such summaries do not by themselves verify the conditional model. A coefficient that changes substantially under reasonable covariate specifications is a warning that the result depends on modeling choices.
For recurrent events, events from one subject are not independent observations. The coefficient’s interpretation and its standard error are separate questions: an estimated coefficient can describe the intended conditional contrast while an inappropriate variance estimate gives misleading uncertainty. The analysis must use inference suited to the dependence and sampling structure. In particular, treating every recurrence as an independent subject can make standard errors too small. Also inspect the number of subjects and the distribution of recurrences, not only the total event count.
Worked Example: Interpreting a Time Interaction
Suppose a hypothetical model includes treatment \(A\), coded 1 versus 0, and a treatment-by-time term:
For treated versus untreated subjects at time \(t\), the intensity ratio is \(\exp(\beta_A+\gamma t)\), not simply \(\exp(\beta_A)\) at every time. If \(\hat{\beta}_A=-0.20\) and \(\hat{\gamma}=0.05\) per year, then at \(t=2\) years the log ratio is \(-0.20+0.05(2)=-0.10\), giving \(\exp(-0.10)=0.905\). At \(t=6\) years it is \(-0.20+0.05(6)=0.10\), giving \(\exp(0.10)=1.105\). Thus this fitted model estimates a 9.52% lower intensity at year 2 and a 10.5% higher intensity at year 6, conditional on the model covariates. The coefficient \(\beta_A\) alone describes the contrast at \(t=0\), so reporting it as a constant treatment ratio would be incorrect.
What the Ratio Does Not Mean
An AG intensity ratio is not a marginal event-rate ratio. The earlier material on marginal event rates and the mean cumulative function concerns population-level summaries, such as average events per person-time or expected cumulative event counts. Those summaries average over people and histories in ways that a conditional AG coefficient does not. A conditional intensity ratio need not equal a marginal rate ratio, even when both are computed from the same cohort.
Nor is the AG coefficient a comparison of first-event hazards. Recurrences after earlier events contribute to the counting process while a subject remains under observation and at risk. The coefficient also does not directly compare expected numbers of events by a fixed horizon: that quantity depends on the entire intensity process, follow-up, and the distribution of subjects’ histories. If the scientific question concerns event-order-specific associations, a single common AG coefficient may be too restrictive; a model allowing effects to vary by event order may be needed.
| Quantity | What it compares | What an AG coefficient does not directly provide |
|---|---|---|
| Conditional intensity ratio | Instantaneous recurrence intensity for a covariate contrast, given modeled covariates and observed history | It does not, by itself, give a probability or cumulative event-count contrast |
| Marginal event-rate ratio | Population-average event rates, often events per person-time | It is not generally equal to a conditional AG intensity ratio |
| Mean cumulative function contrast | Difference in expected cumulative event counts over follow-up | It is not the exponentiated AG coefficient |
Common Misinterpretations
- Wrong: “A ratio of 0.70 means 30% fewer events per person.” Correction: It means 30% lower modeled conditional intensity for the stated covariate contrast. A person-level event-count contrast requires a separate estimand and analysis.
- Wrong: “The ratio is the same as a risk ratio.” Correction: Intensity is an instantaneous rate among those at risk, not a probability over a fixed interval.
- Wrong: “The coefficient is a treatment effect because treatment is in the model.” Correction: It is a conditional association unless the assumptions needed for a causal interpretation are also justified.
- Wrong: “A coefficient of 0.08 means an 8% increase.” Correction: The ratio is \(\exp(0.08)=1.083\), about an 8.33% increase, for a one-unit contrast on the covariate’s stated scale.
- Wrong: “A confidence interval that excludes 1 proves an effect.” Correction: It indicates incompatibility with a ratio of 1 under the specified model and inference procedure at the corresponding level; it does not establish the model assumptions or a causal claim.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- An AG model gives a binary covariate coefficient of \(\log(1.25)\). State the intensity-ratio interpretation and its percentage interpretation.
- A continuous predictor is measured in 2-unit increments and has coefficient \(-0.10\). What is the estimated ratio for a 6-unit increase?
- Why is it inaccurate to describe an AG ratio of 0.80 as a 20% lower probability of recurrence?
- A treatment-by-time interaction is included. Which quantity gives the treatment ratio at time \(t\)?
- Name two conditions or diagnostics that matter before interpreting the coefficient as a stable conditional association.
Solutions. (1) The estimated ratio is \(\exp\{\log(1.25)\}=1.25\): the conditional recurrence intensity is 25% higher in the category coded 1 than in the reference category, for the same modeled covariates. (2) Six units equal three increments of 2, so the ratio is \(\exp(3(-0.10))=\exp(-0.30)=0.741\), about 25.9% lower conditional intensity. (3) Intensity is an instantaneous rate among those at risk, not a probability over a specified period; a probability contrast requires additional information and a separate calculation. (4) With treatment coefficient \(\beta_A\) and interaction coefficient \(\gamma\), the ratio is \(\exp(\beta_A+\gamma t)\). (5) Suitable checks include proportionality over time, functional form for continuous predictors, adequate covariate variation in risk sets, predictable covariates, and appropriate handling of censoring and within-subject dependence.