Why Rescale Visit Times?
A visit-intensity model describes how the instantaneous rate of a visit changes with follow-up time and the history available just before that time. A raw gap of two months can be unusually long when visits are expected weekly, but quite ordinary when they are expected every six months. Time-rescaling puts gaps on a common scale: under a correctly specified conditional intensity, each completed rescaled gap should behave like an exponential random variable with mean one. The ordered rescaled event times should behave like those of a unit-rate Poisson process.
This is a model diagnostic, not a way to prove that the visit process is noninformative or that a joint model is correct. It checks the fitted intensity against the observed timing pattern, conditional on the information used to define that intensity. The earlier material on diagnosing dependence between visits and outcomes motivates why a plausible visit model matters; here the focus is on assessing its timing implications.
The Time-Rescaling Result and Its Conditions
The time-rescaling theorem says that, when a simple point process has a correctly specified conditional intensity and a continuous compensator, changing the time scale from calendar time to integrated-intensity time turns the process into a unit-rate Poisson process. In particular, its successive complete rescaled gaps are independent and each has an exponential distribution with rate one. Equivalently, each gap has mean one and variance one. Transforming each gap by \(u_{ij}=1-\exp(-r_{ij})\) gives independent Uniform(0,1) values under the same ideal conditions.
These claims require care about what is conditioned on. The intensity must be the conditional rate relative to the filtration used for the model and diagnostic. If the intensity includes time-varying covariates, they must be available predictably, and their observed paths must be used in computing the integral. A random future intensity path cannot simply be treated as fixed when conditioning only on the history at the start of a gap. In general, the probability of no visit conditional only on that starting history is not the exponential of a random future integrated intensity. The result used here is the time-change theorem for the full predictable intensity, not that incorrect shortcut.
For the standard application, assume visits form a non-explosive, simple process (no simultaneous visits), the intensity is nonnegative and predictable, and the compensator is continuous and increases through the observed exposure period. The model must include the relevant observed history and covariates in its conditional intensity. If the process has an atom, such as a separately modeled scheduled visit at an exact time, the continuous-intensity result does not directly describe that atom. It must be handled as a discrete or scheduled component rather than hidden inside a continuous hazard.
Specify the rate using only information available just before each time, including any modeled dependence on past visits, treatment, or biomarker history.
For every complete interval between consecutive visits, calculate the fitted intensity’s integral along that interval, updating covariates or history as specified by the model.
Compare the rescaled gaps with Exp(1), and examine their serial pattern and the cumulative rescaled event process. Distributional agreement alone does not establish independence.
A short justification clarifies why the integrated rate, rather than raw elapsed time, is the relevant scale. Write \(u=\Lambda_i(0,t)\) and let \(t(u)\) be its inverse where the compensator increases. Over an increment \(du\), the corresponding calendar increment satisfies \(du=\lambda_i(t\mid\mathcal{F}_i(t-))dt\). The conditional expected event increment is therefore approximately $$ \lambda_i(t\mid\mathcal{F}_i(t-))dt=du. $$ Thus the time-changed process has conditional intensity one. Under the simple-process and regularity conditions above, the unit-intensity counting-process characterization gives a unit-rate Poisson process. Its gaps are consequently independent Exp(1) variables. This argument uses the conditional intensity of the full process; it does not assert a no-event formula conditional only on the history at a gap’s start.
Worked Examples
Worked Example: Integrating a Piecewise Visit Rate
Suppose, hypothetically, that a fitted model gives the following conditional visit rates for one subject during three consecutive intervals: \(0.20\) visits per month from month 0 to 2, \(0.10\) visits per month from month 2 to 5, and \(0.30\) visits per month from month 5 to 8. Assume the observed visits occur at months 2, 5, and 8. For this calculation, the fitted rate is constant within each stated interval and is the appropriate predictable rate there.
The first rescaled gap is the rate times its two-month duration: $$ r_{i1}=\int_0^2 0.20\,ds=0.20(2)=0.40. $$ The second and third are $$ r_{i2}=\int_2^5 0.10\,ds=0.10(3)=0.30,\qquad r_{i3}=\int_5^8 0.30\,ds=0.30(3)=0.90. $$ The rescaled gaps are therefore \(0.40, 0.30, 0.90\), and their mean is \((0.40+0.30+0.90)/3=0.533\).
All three gaps are below one, which could be a sign that the fitted rate is too high for this subject’s observed timing. But three gaps provide very little evidence: random Exp(1) gaps can all be below one, and the mean is highly variable at this sample size. This calculation illustrates the scale change; it does not justify a formal rejection or a strong conclusion about fit.
Worked Example: Exponential and Uniform Residual Views
Consider four hypothetical complete rescaled gaps from a fitted visit model: \(0.20, 0.55, 1.10,\) and \(1.45\). Under the model’s ideal reference, each has an Exp(1) distribution. Transform each value using \(u=1-\exp(-r)\):
For example, the first transformed value is \(1-\exp(-0.20)=0.1813\). The four values are approximately \(0.1813, 0.4231, 0.6671,\) and \(0.7654\). They are not sorted, but a uniform Q–Q plot would sort them and compare them with uniform quantiles. With only four observations, apparent deviations from the diagonal are unsurprising and should not be overinterpreted.
An exponential Q–Q check gives another view. The sorted gaps are \(0.20, 0.55, 1.10, 1.45\). Using plotting positions \(p_j=(j-0.5)/4\), the corresponding Exp(1) quantiles are \(-\log(1-p_j)\), approximately \(0.1335, 0.4700, 0.9808,\) and \(2.079\). These values are not far from the observed gaps except that the largest observed gap is much smaller than the last reference quantile. With \(n=4\), this is descriptive rather than decisive evidence. A proper assessment uses many subjects or gaps and considers uncertainty from estimating the intensity.
Worked Example: A Good Mean but a Suspicious Sequence
Suppose, hypothetically, a model assigns a constant visit intensity of \(0.25\) visits per month over a subject’s observed follow-up. Five observed calendar-time gaps are \(0.8, 1.0, 8.0, 0.6,\) and \(7.6\) months. Since the rate is constant, each transformed gap is \(r=0.25\times\text{gap}\), giving \(0.20, 0.25, 2.00, 0.15,\) and \(1.90\).
Their mean is \((0.20+0.25+2.00+0.15+1.90)/5=0.90\), near the Exp(1) mean of one. Yet the sequence alternates between short and long gaps. The corresponding uniform values are approximately \(0.1813, 0.2212, 0.8647, 0.1393,\) and \(0.8504\). This alternating pattern suggests that the gaps may not be independent as the fitted process requires. It could reflect a missing predictable time-varying driver, an omitted history effect, chance variation, or another feature of the observation process. Five gaps cannot distinguish these explanations, but the example shows why checking only the average rescaled gap is insufficient.
Reading Diagnostics and Recognizing Failure Modes
No single plot or summary diagnoses every departure. Examine both the distribution of rescaled gaps and their order. An exponential Q–Q plot can reveal systematic excesses of very small or very large gaps. A uniform Q–Q plot of \(1-\exp(-r_{ij})\) expresses the same marginal check on a bounded scale. The cumulative rescaled event count against cumulative rescaled time should be broadly compatible with a unit-rate Poisson process: its expected count over an interval of rescaled length \(a\) is \(a\). These checks are complementary, not independent proofs of model adequacy.
Inspect serial patterns within subjects as well as pooled residuals. A pooled histogram can look approximately exponential even when a sequence contains clustering or alternating gaps. Conversely, pooling subjects with different unmodeled rate levels can create apparent overdispersion. Stratifying plots by treatment, follow-up phase, or clinically relevant history can help identify where the fitted intensity misses systematic structure. Diagnostics should respect the repeated observations within a subject; rescaled gaps from one person are not automatically a substitute for independent subjects.
Several practical details matter. A gap cut short by administrative end of follow-up is right-censored, not a completed inter-visit gap. Its integrated intensity up to censoring is a partial exposure and should not be entered as though a visit occurred at its end. Delayed entry requires starting the process at the subject’s actual entry time. If covariates change during a gap, integrate the fitted rate over each segment using the covariate values and history applicable to that segment; multiplying the final rate by the entire gap is generally wrong.
Finally, these are fitted residuals. When the same data estimate intensity parameters and assess the rescaled gaps, the residuals are not exactly independent Exp(1) draws even if the model family is correct. Parameter fitting can make observed diagnostics look better than they would on new data. Formal calibration should account for parameter estimation and the analysis design; an unadjusted reference line or test is best treated as exploratory. Unobserved triggers and misspecified latent dependence can also undermine the conditional intensity, even when observed covariate patterns look acceptable.
How This Diagnostic Differs from Other Checks
| Approach | What it emphasizes | Key limitation |
|---|---|---|
| Raw-gap summaries | Elapsed time between visits in calendar units | Gaps are not comparable when fitted rates change over time or by history |
| Time-rescaled gaps | Whether integrated fitted exposure between visits resembles Exp(1), including the sequence’s Poisson implications | Depends on the fitted conditional intensity and its information set; estimation affects the reference |
| Visit-count comparisons | Observed versus expected numbers of visits over selected windows or groups | Can miss within-window timing patterns and serial structure |
Time rescaling is especially useful when the visit rate is nonconstant, because it tests timing relative to the model’s changing expected rate rather than treating all calendar gaps as exchangeable. Count checks can be easier to communicate and can highlight group-level discrepancies, while raw gaps are useful descriptively. Neither replaces checking whether the conditional intensity represents the scientific observation mechanism. A diagnostic cannot establish that visits are independent of the biomarker after conditioning, nor can it repair unmeasured visit triggers.
Common Pitfall
Check Your Understanding
Answer each question, then compare with the brief solutions.
- A fitted rate is constant at \(0.4\) visits per week during a completed gap of 3 weeks. What is the rescaled gap?
- What reference distribution should \(1-\exp(-r_{ij})\) follow under the ideal time-rescaling conditions?
- Why is a final partial gap ending at administrative censoring not treated as a completed exponential residual?
- Rescaled gaps alternate between small and large values, while their mean is near one. What aspect of the model does this pattern raise concern about?
- Why is the no-visit probability from a gap’s starting history not generally obtained by exponentiating minus a random future integrated intensity?
Solutions. (1) \(r=0.4(3)=1.2\), dimensionless integrated intensity. (2) Uniform(0,1), provided the conditional intensity and regularity conditions hold. (3) No event was observed at the censoring time; the interval is right-censored and supplies only partial exposure, not a complete inter-visit gap. (4) Independence or serial structure may be wrong even if the marginal mean is plausible. (5) Future intensity paths can be random given the starting history, so the conditional probability requires accounting for those paths; the theorem applies through the full predictable intensity and its compensator transformation.