Why Model Duration in a State?
In a multi-state process, the risk of leaving a state can depend on how long the subject has occupied it. A constant transition rate is a simple starting point, but it cannot represent risk that rises or falls with elapsed duration. Parametric sojourn-time models provide a compact way to describe such patterns and, when their assumptions are reasonable, to make predictions beyond observed follow-up.
Let \(i\) be the current state, \(j\) a state reachable directly from \(i\), and \(u\) the time elapsed since entry into the current visit to state \(i\). The transition-specific hazard describes the instantaneous rate of moving from \(i\) to \(j\), conditional on still occupying \(i\) at duration \(u\). The exponential, Weibull, and log-normal families differ primarily in the shapes they allow this hazard to have.
For continuous durations with no instantaneous jumps at entry, the probability of still occupying state \(i\) after duration \(u\) is determined by the total exit hazard:
The density of leaving state \(i\) at duration \(u\) specifically by a transition to \(j\) is \(f_{ij}(u)=S_i(u)\lambda_{ij}(u)\). Thus a set of transition-specific hazards determines both the overall sojourn survival and the probabilities of the different exit routes.
Three Parametric Hazard Families
The formulas below can describe a single possible exit directly. With multiple possible destinations, they can instead be used as forms for the separate cause-specific hazards \(\lambda_{ij}(u)\); the resulting total sojourn survival is then computed from their sum. This distinction matters: in a competing-exit model, a chosen cause-specific hazard does not by itself imply that the observed sojourn time, conditional on that exit destination, follows the corresponding one-event survival distribution.
| Family | Hazard for a single exit | Duration-risk pattern |
|---|---|---|
| Exponential | \(\lambda(u)=r\), with \(r>0\) | Constant hazard; no duration dependence. |
| Weibull | \(\lambda(u)=\frac{\kappa}{\eta}(u/\eta)^{\kappa-1}\) | Decreases if \(\kappa<1\), is constant if \(\kappa=1\), and increases if \(\kappa>1\). |
| Log-normal | \(\lambda(u)=\frac{\phi(z)}{u\sigma[1-\Phi(z)]}\), where \(z=(\log u-\mu)/\sigma\) | Can rise and later fall; it allows a non-monotone hazard. |
For the exponential model, \(r\) is a rate in inverse time units. Its single-exit survival is \(S(u)=\exp(-ru)\). For the Weibull model, \(\eta>0\) is a scale in time units and \(\kappa>0\) is a dimensionless shape parameter; its single-exit survival is \(S(u)=\exp\{-(u/\eta)^\kappa\}\). The scale is not generally the mean: when \(\kappa=1\), \(\eta\) is the exponential mean, while for other shapes the mean is \(\eta\Gamma(1+1/\kappa)\).
For a log-normal duration, \(\log T\sim N(\mu,\sigma^2)\), where \(T\) is a positive duration and \(\sigma>0\). Its density is \(f(u)=\phi(z)/(u\sigma)\) and its survival is \(S(u)=1-\Phi(z)\), giving the hazard in the table as \(f(u)/S(u)\). Here \(\phi\) and \(\Phi\) are the standard normal density and distribution functions. The parameter \(\mu\) depends on the time unit used inside the logarithm: expressing the same duration in days rather than months shifts \(\mu\) by the logarithm of the unit conversion. The parameter \(\sigma\), the standard deviation of log duration, is unchanged by that conversion.
Worked Example: Constant Exponential Exit Risk
Suppose a hypothetical state has one possible exit and a fitted exponential hazard of \(r=0.08\) per month. The exponential model assumes the same instantaneous exit rate at every duration in the state. Its survival after \(u\) months is \(S(u)=\exp(-0.08u)\).
After 6 months, the estimated probability of remaining in the state is \(S(6)=\exp[-(0.08)(6)]=\exp(-0.48)=0.6188\). Thus, under this model, the probability of exiting by 6 months is \(1-0.6188=0.3812\). The hazard remains \(0.08\) per month at 6 months, just as it was immediately after entry. The corresponding mean sojourn time for this single-exit model is \(1/r=12.5\) months.
This form is suitable when a constant duration hazard is scientifically plausible, not merely because it is mathematically convenient. If exit risk changes systematically with time already spent in the state, the exponential assumption is restrictive.
Worked Example: Increasing Weibull Risk
Consider a hypothetical single-exit Weibull model with shape \(\kappa=2\) and scale \(\eta=10\) months. These parameters imply increasing exit risk because \(\kappa>1\). At duration \(u=5\) months, the hazard is \(\lambda(5)=(2/10)(5/10)^{2-1}=0.1000\) per month. At 10 months it is \(\lambda(10)=(2/10)(10/10)^{1}=0.2000\) per month.
The probability of remaining in the state through 5 months is \(S(5)=\exp[-(5/10)^2]=\exp(-0.25)=0.7788\). At 10 months it is \(S(10)=\exp[-(10/10)^2]=\exp(-1)=0.3679\). The median sojourn duration \(m\) solves \(S(m)=0.5\), so \(\exp[-(m/10)^2]=0.5\), giving \(m=10\sqrt{\log 2}=8.326\) months.
The hazard values describe instantaneous exit rates among those still in the state at those durations; they are not probabilities of exiting during a whole month. The survival calculations give the cumulative probability of no exit up to each specified duration.
Worked Example: A Log-Normal Hazard That Can Turn Down
Suppose a hypothetical single-exit sojourn duration, measured in months, has \(\mu=\log(5)\) and \(\sigma=0.5\). At 5 months, \(z=\log(5/5)/0.5=0\), so \(\phi(z)=0.3989\) and \(1-\Phi(z)=0.5\). The hazard is \(\lambda(5)=0.3989/[(5)(0.5)(0.5)]=0.3192\) per month.
At 10 months, \(z=\log(10/5)/0.5=\log(2)/0.5=1.386\). Using \(\phi(z)\approx0.1526\) and \(1-\Phi(z)\approx0.08283\), \(\lambda(10)=0.1526/[(10)(0.5)(0.08283)]=0.3685\) per month.
The hazard has risen between 5 and 10 months in this example. A log-normal hazard is not constrained to increase forever: it tends toward zero at very long durations and can therefore rise and later decline. The log-normal median is \(\exp(\mu)=5\) months, but that median is not the duration at which the hazard reaches its maximum.
Using the Families With Competing Exits
Suppose state \(i\) has two possible exits. One may specify a parametric form for each hazard, then add them to obtain the total hazard. For example, two exponential cause-specific hazards \(\lambda_{i1}(u)=a_1\) and \(\lambda_{i2}(u)=a_2\) imply \(S_i(u)=\exp[-(a_1+a_2)u]\). The overall sojourn time is exponential in this special case, while the destination-specific exit densities are \(a_1S_i(u)\) and \(a_2S_i(u)\).
Worked Example: Two Competing Exponential Transitions
From a hypothetical state \(i\), suppose the hazards to states \(j=1\) and \(j=2\) are constant at \(0.06\) and \(0.04\) per month. The total exit hazard is \(0.06+0.04=0.10\) per month, so the probability of no exit by 8 months is \(S_i(8)=\exp[-(0.10)(8)]=0.4493\).
The probability of a transition to state 1 by 8 months is the integral of its cause-specific density: \(\int_0^8 \exp(-0.10u)(0.06)\,du =(0.06/0.10)[1-\exp(-0.8)]=0.3304\). The probability of a transition to state 2 by 8 months is \((0.04/0.10)[1-\exp(-0.8)]=0.2203\). These probabilities sum to \(0.5507\), which is \(1-0.4493\), the probability of any exit by 8 months. Among eventual exits under these constant hazards, the probability of destination 1 is \(0.06/0.10=0.6000\).
With non-constant cause-specific hazards, the same principle applies: use \(S_i(u)=\exp\{-\int_0^u\sum_j\lambda_{ij}(v)\,dv\}\), then integrate \(S_i(u)\lambda_{ij}(u)\) for the probability of exit to destination \(j\). In general, the total sojourn distribution will not be a member of the same simple family as each separate cause-specific hazard.
A related but different construction specifies a distribution for duration conditional on the next destination, together with a model for which destination occurs. That is not interchangeable with assigning separate cause-specific hazards. In particular, independently specifying log-normal durations for each possible exit and treating them as observed competing risks can produce an incoherent model unless the joint mechanism is defined. State clearly whether a parametric family describes a cause-specific hazard or a duration conditional on destination.
Derivation and Likelihood Contributions
The survival formula follows directly from the definition of the total hazard. Over a short interval of length \(du\), the conditional probability of an exit is approximately \(\lambda_i(u)du\), so the conditional probability of staying is approximately \(1-\lambda_i(u)du\). The resulting differential equation and its solution are:
For a transition to \(j\) in the next interval, the process must first have remained in \(i\) through duration \(u\), with probability \(S_i(u)\), and then exit to \(j\) at rate \(\lambda_{ij}(u)\). This gives the event density \(f_{ij}(u)=S_i(u)\lambda_{ij}(u)\).
For independent right censoring conditional on the modeled information, an observed sojourn ending in a transition to \(j\) at duration \(t\) contributes \(S_i(t)\lambda_{ij}(t)\) to the likelihood. A sojourn censored while the subject remains in \(i\) at duration \(c\) contributes \(S_i(c)\). These terms show why fitting only durations among observed transitions, while ignoring censored visits, generally does not estimate the full sojourn model correctly. For recurrent visits, the relevant entry history and dependence between visits also require appropriate treatment; treating all visits as independent is not automatically justified.
Assumptions, Diagnostics, and Choice
A parametric duration model is meaningful only relative to its assumptions. At minimum, define the state-entry time and duration scale, specify which transitions are possible, and decide what history or covariate information is held fixed. The hazard is modeled as a function of time since entry to the current state visit. A chosen functional form is assumed to describe the duration-risk pattern over the period where it will be used. Standard likelihood-based inference also relies on a suitable censoring assumption, commonly independent right censoring conditional on modeled information.
Useful checks include comparing fitted and observed state-occupation probabilities and transition counts over duration, inspecting whether estimated risk patterns are compatible with the chosen shape, and examining residual or cumulative-hazard diagnostics. A systematic mismatch between fitted and observed duration patterns, or predictions that are highly sensitive to a few long visits, is a warning. A flexible empirical estimate can help reveal gross shape misspecification, but it may be unstable in the tail where few subjects remain at risk.
| Model | Strength | Main limitation |
|---|---|---|
| Exponential | Simple, interpretable, and constant hazard. | Cannot represent duration dependence. |
| Weibull | Allows a monotone increasing or decreasing hazard. | Does not represent a hazard that rises and then falls. |
| Log-normal | Allows a non-monotone hazard and skewed durations. | May be sensitive to tail behavior and has no constant-hazard interpretation. |
| Flexible duration model | Can accommodate more varied shapes with sufficient data. | Usually requires more parameters and careful smoothing or specification. |
The principal limitation of any fully parametric choice is extrapolation: a plausible fit over observed follow-up does not guarantee a correct tail beyond it. The exponential model is particularly restrictive. Weibull and log-normal models relax that restriction in different ways, but neither is universally preferable. The comparison should be guided by subject-matter knowledge, diagnostics, uncertainty, and the intended prediction horizon—not by goodness of fit alone. The introductory material on semi-Markov models provides the broader setting in which elapsed duration can affect transition risk.
Common Mistake
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A single-exit Weibull model has shape \(\kappa=0.7\). Does its hazard rise or fall with duration?
- A two-exit state has constant cause-specific hazards of \(0.03\) and \(0.07\) per month. What is the probability of remaining in the state for 5 months?
- Why is a log-normal cause-specific hazard not, by itself, proof that observed sojourn durations conditional on that destination are log-normal?
- What is the main duration-risk restriction imposed by an exponential sojourn model?
Solutions. (1) Since \(\kappa<1\), the Weibull hazard decreases with duration. (2) The total hazard is \(0.03+0.07=0.10\) per month, so \(S(5)=\exp[-(0.10)(5)]=\exp(-0.5)=0.6065\). (3) With competing exits, the observed sojourn survival is determined by the sum of all cause-specific hazards; the destination-specific event density is that survival times the hazard for the destination. A hazard shaped like the log-normal hazard does not make the conditional duration distribution given destination equal to a log-normal distribution. (4) The exponential model requires the hazard to be constant over elapsed duration.