Why Model Visits Using the Current State?
A patient’s biomarker may change between clinic visits. If an unusually high value prompts a clinician or patient to arrange an assessment, then the instantaneous chance of a visit may depend on the biomarker value at that time. This is a different assertion from saying that visit timing depends on recorded history: the value that might prompt a visit is generally not observed until a visit takes place.
A current-state-triggered visit intensity makes that assertion precise. Its key interpretive challenge is also immediate: a rate conditional on the biomarker’s current value is not necessarily a rate that can be calculated from the observed data at every time. Between visits, the current value may be unknown.
Definition and Model
For patient \(i\), let \(N_i^V(t)\) count visits through time \(t\). Let \(Y_i(t)\) indicate that the patient is under observation and at risk of a visit immediately before time \(t\), and let \(X_i(t-)\) be the biomarker value immediately before \(t\). The minus sign emphasizes that the value used to predict a visit at \(t\) must be available just before that visit. Let \(Z_i(t-)\) denote other predictable covariates, and let \(x_0\) be a reference biomarker value.
\(\lambda_i^V(t\mid\mathcal F_{i,t-})=Y_i(t)\lambda_0^V(t)\exp\{\gamma^\mathsf{T}Z_i(t-)+\alpha[X_i(t-)-x_0]\}\).
Here \(\mathcal F_{i,t-}\) is the full information available just before \(t\), \(\lambda_0^V(t)\) is a nonnegative baseline visit rate, \(\gamma\) is a vector of coefficients for \(Z_i(t-)\), and \(\alpha\) is the coefficient for the current biomarker. The intensity is zero when the patient is not at risk.
Formally, for a small interval of length \(dt\), the model says
The conditional probability of two or more visits in that interval is \(o(dt)\). An intensity has units of visits per unit time; since visits are counted as events, its rate is usually reported simply in inverse-time units. The exponent must be dimensionless. If \(X_i\) is measured in milligrams per litre, then \(\alpha\) has units of inverse milligrams per litre.
Conditional on the other terms in the model, a one-unit increase in the current biomarker multiplies the instantaneous visit rate by \(\exp(\alpha)\). This is a conditional rate comparison, not a causal effect. The baseline rate may vary over time, and \(Z_i(t-)\) and the at-risk indicator are held fixed in this comparison. Using \(x_0\) centers the biomarker; it makes the biomarker contribution to the rate equal to one when \(X_i(t-)=x_0\).
The model is a specification of dependence, not a complete description of how the biomarker evolves. It assumes that, conditional on the stated terms, the current biomarker and covariates are sufficient for this intensity. If earlier visit history or other predictors still matter, the model may need additional terms. The current-state claim is specifically about \(X_i(t-)\), not about the most recent recorded value.
Worked Example: Interpreting a Biomarker Rate Ratio
Suppose a hypothetical model has \(\alpha=0.35\) per milligram per litre. Compare two patients at the same time, with the same \(Z_i(t-)\) and at-risk status, whose current biomarker values are 12 and 8 milligrams per litre.
The conditional rate ratio for the 12-unit state versus the 8-unit state is
Under this model, the instantaneous visit rate at a current value of 12 is about 4.055 times the rate at a current value of 8, conditional on the stated covariates and time. This comparison does not say that the probability of a visit over a long interval is 4.055 times larger: probabilities over intervals depend on the rates throughout those intervals.
From an Instantaneous Rate to an Interval Probability
A rate at one instant is not itself a probability over a period. If the conditional intensity is specified along a biomarker path and the rate depends on no additional evolving visit-history terms, the cumulative intensity over an interval is the integral of the rate. The corresponding probability of no visit over that interval, conditional on the specified path and covariates, is the exponential of minus that integral.
\(1-\exp\{-\int_a^b \lambda_i^V(u)\,du\}\).
This calculation requires the rate over the interval, not just the value at its beginning or end.
Worked Example: A Changing Biomarker Path
Consider a hypothetical patient who remains at risk for two days. Let \(\lambda_0^V(t)=0.05\) visits per day, let \(Z_i(t-)=0\), and let \(\alpha=0.20\) per biomarker unit with \(x_0=5\). Suppose the current biomarker is 5 for the first day and 8 for the second day. Assume this path is known for the calculation and the stated intensity applies throughout.
The rate is \(0.05\) visits per day on day one. On day two it is \(0.05\exp\{0.20(8-5)\}\) visits per day. Therefore, the cumulative intensity is
The probability of at least one visit over the two days is
Thus the model gives about a 13.16% probability of at least one visit over these two days, conditional on the assumed path and model. This example illustrates why a changing state requires integrating the changing rate; substituting only one biomarker value for the whole period would not produce this calculation.
Why an Unobserved Current Value Changes the Interpretation
The intensity above is conditional on the full information \(\mathcal F_{i,t-}\), including \(X_i(t-)\). In practice, the analyst may instead have only the observed history: previous visit times and biomarker measurements, known covariates, and the fact that no new visit has occurred yet. Call this observed information \(\mathcal H_{i,t-}\). If the current biomarker is not measured between visits, then \(\mathcal H_{i,t-}\) does not reveal \(X_i(t-)\).
The rate conditional on observed information is therefore an average over uncertainty about the current state. In general, the observed-history intensity is the conditional expectation of the full-information intensity given the observed history. This projection is not usually obtained by putting the last recorded biomarker into the current-state formula.
For the model above, when \(Y_i(t)\), \(Z_i(t-)\), and the baseline rate are known from the observed history, this becomes
The conditional distribution in this expectation is based on the information available just before \(t\), including the absence of a visit up to that point. It is not generally enough to know the last measurement alone. Also, because the exponential is nonlinear, averaging the rate multiplier is not generally the same as applying the model to the average biomarker value.
Worked Example: Averaging Over an Unknown Current State
Suppose, hypothetically, that the current biomarker given the observed history has a normal distribution with mean 5 and variance 4, in biomarker units squared. Let \(x_0=5\), \(\alpha=0.20\) per unit, and let the baseline rate be \(0.10\) visits per day, with no other covariate contribution.
For a normal variable \(X\sim N(\mu,\sigma^2)\), \(E\{\exp(aX)\}=\exp(a\mu+a^2\sigma^2/2)\). Applying this identity to the centered biomarker gives
The observed-history visit intensity is \(0.10(1.083)=0.1083\) visits per day. Simply substituting the conditional mean, 5, into the current-state formula would give \(0.10\) visits per day. The difference arises because the conditional distribution has uncertainty and the rate depends exponentially on the state. The calculation is not a general recipe for learning that distribution; it shows why the unobserved current value matters.
Justification of the Observed-History Rate
The averaging rule follows directly from conditional expectation. Given the full information just before \(t\), the probability of one visit in \(dt\) is \(\lambda_i^V(t\mid\mathcal F_{i,t-})dt+o(dt)\). Given only the observed information, average that conditional probability over the possible full histories compatible with what has been observed:
The coefficient of \(dt\) is the observed-history intensity. This argument depends on defining the full intensity with respect to information available immediately before the visit and on the usual conditional-intensity approximation for a short interval. It also shows why the current-state model alone does not supply the observed-history rate: the conditional distribution of the unobserved state is needed.
Assumptions, Diagnostics, and Alternatives
The model is most useful when a current biomarker could plausibly affect the decision or need to attend a visit. Its interpretation relies on several conditions: the visit process is defined consistently; \(Y_i(t)\) correctly indicates when a visit is possible; covariates used at \(t\) are predictable; and the specified covariates and current state adequately describe the conditional intensity. If a biomarker is only recorded at visits, the value at a visit should not automatically be treated as known immediately before that visit without a timing convention that makes this valid.
A practical warning is that higher recorded biomarker values among visits do not, by themselves, establish a current-state trigger. The value may have changed since the previous measurement, and the set of observed values is shaped by when measurements were obtained. A model using the last recorded value instead describes dependence on an observed, lagged history value; it is not the same model with a convenient substitute for the current state.
Useful checks start with the mechanism and the data: are visits plausibly prompted by the current state, or by symptoms and earlier recorded results? Are there measurements close enough to visits to make the proposed trigger clinically interpretable? Do visit patterns differ across the range of recorded values in a way consistent with the proposed direction, while allowing for time and other covariates? These checks can reveal implausibility, but sparse measurements cannot directly verify the biomarker value at unobserved times. A misspecified intensity may therefore remain difficult to diagnose from visit records alone.
| Approach | What the visit rate depends on | Key distinction |
|---|---|---|
| History-conditional intensity | Observed patient history available before the visit | Predictors are observed history, not necessarily the current biomarker state. |
| Current-state-triggered intensity | The biomarker value immediately before the visit, possibly with covariates | Directly represents a current-state trigger, but that state may be unobserved between visits. |
| Latent-trajectory-driven intensity | A modeled underlying biomarker trajectory | Provides a way to represent an evolving unobserved state, but requires additional trajectory assumptions. |
| Externally scheduled visits | A prespecified schedule, rather than a patient’s biomarker state | Appropriate when timing is set independently of the current biomarker, subject to the study’s assumptions. |
A current-state intensity is not automatically identifiable from visit times and occasional measurements: different unobserved paths can be compatible with the same recorded data. Nor does a positive \(\alpha\) prove that changing the biomarker would cause more visits. It describes a conditional association in the specified intensity. A separate model for the unobserved biomarker process and additional assumptions may be needed for estimation and interpretation; the form of that model is beyond the current-state specification itself.
Common Mistakes
- Replacing the current state with the last measurement. This changes the predictor from \(X_i(t-)\) to an observed past value. They coincide only under additional assumptions, such as no change since measurement.
- Reading \(\exp(\alpha)\) as a probability ratio. It is a conditional instantaneous rate ratio for a one-unit difference, not a ratio of interval probabilities.
- Treating a positive coefficient as proof of causation. It indicates conditional rate association under the model, not that intervening on the biomarker would alter visit timing.
- Ignoring uncertainty between visits. When the current state is unknown, the observed-history rate averages the state-dependent rate; it is generally not the rate evaluated at the last observed value or at the conditional mean.
Check Your Understanding
Answer each question, then compare with the solutions below.
- If \(\alpha=0.10\) per biomarker unit, what is the conditional instantaneous rate ratio for a two-unit increase in current biomarker, with other model terms fixed?
- Why is a rate based on the last recorded biomarker not necessarily a current-state-triggered rate?
- If the conditional distribution of the current biomarker is uncertain, what operation gives the visit intensity conditional on observed history?
- Suppose the current-state model has a positive \(\alpha\). State one valid interpretation and one conclusion that does not follow.
Solutions. (1) The rate ratio is \(\exp(0.10\times2)=\exp(0.20)\approx1.221\), so the rate is about 1.221 times as high, conditional on the other terms. (2) The biomarker may have changed since its last measurement; the recorded value is historical, whereas the model uses the value immediately before \(t\). (3) Take the conditional expectation of the full-information intensity given the observed history. In general, this is not the intensity evaluated at the last measurement or at the conditional mean. (4) A valid interpretation is that a one-unit higher current value is associated with an instantaneous rate multiplied by \(\exp(\alpha)\), holding the other terms fixed. It does not establish that changing the biomarker would cause the visit rate to change.