Observation Intensities Conditional On Patient History

Why Condition Visit Rates on Observed History?

A patient may have more frequent clinical visits after a medication change, a recent symptom report, or a sequence of elevated biomarker measurements. A single average visit rate hides these differences. A conditional visit intensity describes how the instantaneous rate of another visit varies with information already observed for that patient. It makes the time ordering explicit: the predictors are known before the visit whose rate is being described.

The focus here is a model based on observed covariates and observed measurement history. It does not assume that the measurements reveal an unobserved current health state, and it does not make a causal claim about why visits occur. Rather, it specifies a conditional rate given a defined history. As in the earlier material on counting processes and separating visits from terminal events, the visit process is considered only while the patient remains under observation and at risk of another visit.

The term \(o(\Delta t)\) represents a remainder that becomes negligible relative to \(\Delta t\) as the interval shrinks. Under the usual simple-process condition that the probability of multiple visits in an interval is of smaller order than its length, the intensity captures the first-order chance of one visit. For a sufficiently short interval, the probability is approximately the intensity multiplied by the interval length. For a longer interval, that multiplication is generally not an exact visit probability.

A Conditional Model Using Observed History

A common specification uses a log link so that the modeled intensity is positive. Write \(X_i(t)\) for observed covariates available just before time \(t\), and \(g\{H_i^{\mathrm{obs}}(t-)\}\) for one or more summaries of the patient's past observed measurements or events. A basic model is

Here \(\lambda_0^V(t)\) is a baseline intensity, \(\beta\) and \(\gamma\) are regression coefficients, and the history summary is specified by the analyst. For example, it could indicate whether a medication was changed before \(t\), or record the most recently observed biomarker category. A coefficient describes the conditional association of its predictor with visit intensity, holding the other included predictors and the baseline intensity fixed.

The predictors must be available before the time they are used. If a biomarker is measured at a visit, that measurement cannot be treated as a predictor of the intensity of that same visit: it is observed because the visit occurred. It can contribute to the history used to model visits after the measurement. This distinction prevents information from the event being used to predict itself.

The model requires a correctly defined risk period and a well-specified observed history. Its interpretation also relies on the chosen functional form being adequate: a linear term on the log-intensity scale assumes a constant multiplicative change per unit of that predictor. A baseline intensity may vary over calendar time even when the predictors stay fixed. If the patient is no longer under observation or is no longer eligible for a visit, \(Y_i(t)=0\) and the visit intensity is zero for the modeled risk process.

Why the Coefficient Gives a Rate Ratio

The rate-ratio interpretation follows directly from comparing two histories at the same time. Suppose the histories differ only in a predictor \(x\), which changes from \(x\) to \(x+1\), and its model coefficient is \(\beta\). The common baseline and other terms cancel:

where \(C\) collects the unchanged terms. Thus, if \(\beta=\log(1.5)\), a one-unit increase in \(x\) corresponds to 1.5 times the conditional intensity, or a 50% higher intensity, at that time and for otherwise equal modeled histories. This ratio concerns instantaneous rates. It does not say that the patient is 50% more likely to have a visit over an entire month, because the rate and the patient’s history may change during that month.

Time-Varying Predictors and History Summaries

A time-varying predictor can change the intensity during follow-up, even if the model coefficients remain constant. The update occurs when the predictor’s value changes in the observed history. A past measurement can likewise be represented by a defined summary, such as whether the most recent measured value exceeded a threshold. Such a summary is not the same as a continuously observed current measurement: between measurement visits, the recorded summary remains based on what was last observed.

If an intensity is approximately constant over a short interval of length \(\Delta t\), then the probability of at least one visit in that interval is approximately \(\lambda^V\Delta t\). The approximation is useful only when the interval is sufficiently short for the neglected higher-order terms to be small. The following calculation illustrates both the conversion and its limitation.

Assumptions, Diagnostics, and Failure Modes

Before interpreting a fitted conditional intensity, check the structure that gives it meaning. The observation window and terminal-event rules must define when a patient is at risk. The history used at time \(t\) must contain only information available before \(t\). The assumed log-linear relationship and any baseline-time structure must be plausible over the range of observed histories. Interpretation also presumes that visit times and predictor updates have been recorded with enough accuracy to establish their order.

Useful diagnostics focus on whether the model describes observed visit patterns across time and history groups. Compare observed and model-predicted visit counts or rates over follow-up intervals and across predictor categories. Examine whether discrepancies persist for particular patients, time periods, or ranges of the history summaries. Large residual patterns, systematic underprediction after a particular type of measurement, or strong variation in observed rates within a model-defined group can indicate omitted predictors, an inadequate time pattern, or a poor functional form. Sparse data may make these checks imprecise, so an apparently small discrepancy is not proof that the model is adequate.

The following assumptions and limitations are especially important:

• Correct ordering: predictors must be recorded before the visit whose intensity they predict. Using a visit’s own measurement to explain the occurrence of that visit creates outcome-dependent information leakage.
• Appropriate risk status: a patient who is no longer followed or cannot have another observed visit must not be treated as continuing to contribute at-risk time. Visits and terminal events must be distinguished.
• Adequate model form: the log-linear specification assumes multiplicative effects unless interactions, nonlinear terms, or time-varying coefficients are included. A single coefficient may conceal changing associations over follow-up.
• Observed-history limitation: conditioning on recorded history does not guarantee that all determinants of visit timing have been measured. The model describes rates given the included history, not a complete account of the clinical decision process.
• Association, not intervention: a higher conditional rate after an observed event or measurement does not show that changing that predictor would change the rate. Clinical severity, access, or other factors may be related to both.

How This Model Differs From Alternatives

A fixed appointment schedule describes intended observation times, not necessarily the rate of all visits that actually occur. A marginal visit-rate summary averages across patients and histories; it can be useful descriptively but does not show how rates vary with a particular patient’s observed history. A conditional intensity model instead compares instantaneous rates at the same time for histories that differ in specified predictors. These approaches answer different questions and need not agree numerically.

The choice to condition on observed history also sets a boundary on interpretation. This tutorial does not replace observed measurements with a latent underlying trajectory, nor does it claim that the most recent measurement equals the patient’s exact current state. Nor does a rate model alone describe the full joint behavior of visits, measurements, and terminal events. Its value is narrower and concrete: it gives a precise, time-ordered way to express how the observed information available before \(t\) is associated with the instantaneous rate of a visit at \(t\).

Common Mistake

A common error is to read a coefficient of \(0.70\) as an increase of 0.70 visits per month, or as a 70% increase in the probability of a visit. In a log-intensity model, the rate ratio is \(\exp(0.70)=2.014\), not \(1+0.70\); the corresponding conditional intensity is about twice as large. It is still not a probability ratio. First interpret the coefficient on the log-rate scale, exponentiate it to obtain a rate ratio, and then distinguish that rate ratio from any probability over a specified interval.