Competing Destinations From a Shared State

Why Distinguish Destinations?

A person in an illness state may next recover, progress to a more severe state, or die. These are distinct next transitions from the same current state. At the instant the person leaves the illness state, only one can be the next destination. Describing the destinations separately preserves information about how the state can change; combining them into “any exit” does not.

The destination of a transition is not necessarily a final outcome. Recovery or progression may lead to further transitions, and death may be an absorbing state in a particular model. Here the focus is narrower: the possible next destinations from one specified state, and the rates and probabilities associated with that first departure. No assumption that the current state alone summarizes all relevant past history is needed to define these outgoing transitions.

Definition: For a current state $i$, a set of distinct states $j\ne i$ that can be reached directly from $i$ are its outgoing destinations. A transition intensity $q_{ij}(t)$ is the instantaneous rate of transition from $i$ to $j$ at time $t$, conditional on being in $i$ just before $t$ and on the specified history or covariates. Formally, when the limit exists, $$ q_{ij}(t)=\lim_{h\downarrow 0}\frac{P\{X(t+h)=j\mid X(t-)=i,\ \text{specified history and covariates}\}}{h}. $$ The total outgoing intensity is $q_i^+(t)=\sum_{j\ne i}q_{ij}(t)$, where the sum includes only allowed destinations. Each $q_{ij}(t)$ is nonnegative, and a structurally disallowed transition has intensity zero.

The intensity has units of inverse time: for example, $0.04$ per month. For a sufficiently short interval $h$, the probability of a transition to $j$ is approximately $q_{ij}(t)h$, conditional on still being in $i$ at $t$. This is a local approximation, not a probability for a long interval. Over a longer interval, some people leave the state, so the set still at risk changes.

From Outgoing Rates to First-Destination Probabilities

Suppose a person is in state $i$ at the start of an interval. Let $T_i$ be the time to the first departure from $i$, and let $J$ denote the destination of that departure. If the outgoing intensities $q_{ij}(v)$ are specified over time while the person remains in $i$, the total intensity of departure at time $v$ is $q_i^+(v)$. The probability of no departure up to elapsed time $u$ is

S_i(u)=P(T_i>u)=\exp\left\{-\int_0^u q_i^+(v)\,dv\right\}.

The probability that the first departure occurs by $u$ and goes to destination $j$ is

F_{ij}(u)=P(T_i\le u,\ J=j) =\int_0^u S_i(v)q_{ij}(v)\,dv.

These expressions use the conditional departure intensities for people still in state $i$; they do not require that the rates be constant over time. They do require the outgoing destinations to be specified clearly and the rates to describe the relevant risk set. In particular, this calculation concerns the first transition out of $i$, not all transitions that may happen afterward.

Key result: The destination-specific probabilities partition the probability of leaving the state by time $u$: $$ \sum_{j\ne i}F_{ij}(u)=1-S_i(u). $$ Thus the total exit intensity determines how quickly people leave, while the separate $q_{ij}$ values determine how that departure probability is allocated among destinations.

The partition follows because $S_i(v)$ is the probability of still being in $i$ just before $v$. Multiplying it by $q_{ij}(v)$ gives the probability density for a first departure to $j$ at $v$. Summing over all destinations gives the density of any first departure. Since $S_i'(v)=-S_i(v)q_i^+(v)$, integrating that total density gives

\sum_{j\ne i}F_{ij}(u) =\int_0^u S_i(v)q_i^+(v)\,dv =-\int_0^u S_i'(v)\,dv =1-S_i(u).

This identity is also a useful check on calculations: the probabilities of all first destinations plus the probability of remaining in the state must equal one.

Worked Example: Three Destinations With Constant Rates

Consider a hypothetical illness state with constant transition intensities of $0.040$ per month to recovery, $0.030$ per month to progression, and $0.010$ per month to death. Assume these rates apply to people while they remain in the illness state, all three destinations are observed and classified, and the rates are constant over the six-month interval. Calculate the probability of each first destination within six months.

The total outgoing intensity is $$ q_i^+=0.040+0.030+0.010=0.080\text{ per month}. $$ Therefore, the probability of remaining in the illness state for more than six months is $$ S_i(6)=e^{-(0.080)(6)}=e^{-0.48}=0.6188. $$ So the probability of any departure by six months is $1-0.6188=0.3812$.

With constant rates, the probability of a first departure to destination $j$ by $u$ is $q_{ij}/q_i^+$ times the probability of any departure. Hence: $$ F_{i,\mathrm{recovery}}(6)=\frac{0.040}{0.080}(0.3812)=0.1906, $$ $$ F_{i,\mathrm{progression}}(6)=\frac{0.030}{0.080}(0.3812)=0.1429, \qquad F_{i,\mathrm{death}}(6)=\frac{0.010}{0.080}(0.3812)=0.04765. $$ The four probabilities sum to $0.6188+0.1906+0.1429+0.04765\approx1.000$, allowing for rounding. Among people who have left the illness state by six months, the destination proportions are $0.040/0.080=0.50$ for recovery, $0.375$ for progression, and $0.125$ for death. These are conditional proportions among departures, not the six-month probabilities in the full starting group.

Estimating Constant Destination-Specific Rates

A simple rate model assumes that, while a person remains in state $i$, each destination-specific intensity is constant over the observation period. For destination $j$, let $d_{ij}$ be the number of observed first transitions from $i$ to $j$, and let $Y_i$ be the total person-time spent in $i$ while under observation. Under constant intensities, correctly classified exits, and non-informative censoring conditional on the model, the likelihood contribution for the outgoing transitions is proportional to

L\ \propto\ \left(\prod_{j\ne i}q_{ij}^{\,d_{ij}}\right) \exp\left(-Y_i\sum_{j\ne i}q_{ij}\right).

Taking the log likelihood and maximizing separately with respect to each $q_{ij}$ gives the rate estimate $d_{ij}/Y_i$. This is an estimate of a rate, not a probability. Its units are inverse time, and it can exceed one per time unit without being an impossible probability.

Worked Example: Estimating Rates From State Occupancy Time

In a hypothetical cohort, individuals contribute a combined 400 person-months in an illness state before leaving it or being censored. There are 12 first transitions to recovery, 8 to progression, and 4 to death. Assume constant transition intensities over the observation period, accurate destination classification, and censoring that is non-informative given the model. Estimate the three rates and the rate of any exit.

The destination-specific estimates are event counts divided by time at risk in the illness state: $$ \widehat q_{i,\mathrm{recovery}}=\frac{12}{400}=0.0300\text{ per month}, \quad \widehat q_{i,\mathrm{progression}}=\frac{8}{400}=0.0200\text{ per month}, $$ $$ \widehat q_{i,\mathrm{death}}=\frac{4}{400}=0.0100\text{ per month}. $$ The estimated total outgoing rate is $0.0300+0.0200+0.0100=0.0600$ per month. The estimated rates allocate the observed exits in the proportions $12/24=0.500$, $8/24=0.3333$, and $4/24=0.1667$. These proportions summarize the observed destination mix under the stated model; they are not probabilities of each destination by a particular follow-up time.

A rate-based probability calculation also needs a time horizon. For example, if these estimated rates are treated as constant for six months, the estimated probability of any exit is $1-e^{-(0.0600)(6)}=0.3023$. The estimated six-month probability of progression as the first destination is $$ \frac{0.0200}{0.0600}(0.3023)=0.1008. $$ These are model-based estimates, not direct proportions from the 24 observed transitions.

Define and Record Destinations Carefully

A destination is defined by the state entered immediately after leaving the current state. The state definitions therefore determine which transitions count as distinct. For instance, if “illness” includes both mild and severe disease, a change from mild to severe illness is not a transition out of that broad state. If severity is represented as separate states, it can be. Neither representation is automatically correct: the distinction should match the scientific question and the information recorded.

Worked Example: Classifying the Next Transition

Suppose a hypothetical study defines three states: $I$, active illness; $R$, recovered; and $P$, progressed illness. Death is recorded as state $D$. A participant starts in $I$, later enters $P$, and dies afterward. Another participant starts in $I$ and enters $R$. What are the first destinations from $I$?

For the first participant, the first destination from $I$ is $P$, not $D$: death occurs after the person has already left $I$. For the second participant, the first destination is $R$. Thus the allowed outgoing destinations from $I$ in this definition are $R$, $P$, and $D$, but a particular observed history contributes only one first destination from $I$. The subsequent transition from $P$ to $D$ belongs to the outgoing transitions from $P$, not to the first-destination count from $I$.

This classification assumes the study can determine the order of events and distinguish the states. If recovery and progression are only assessed at widely spaced visits, their actual order may be unknown. The data then do not justify assigning an exact first destination without an explicit observation or interval-censoring model.

Diagnostics, Alternatives, and Limitations

Before interpreting estimated destinations, check that the state diagram allows exactly the transitions the data can represent. For each person, event records should identify the state immediately before and after a transition, the transition time or observation interval, and whether follow-up ended through censoring. Check that destination counts are mutually exclusive for each first departure and that the sum of destination-specific event counts agrees with the total number of departures. Large amounts of unknown destination or tied events can make the division among destinations unreliable.

The constant-rate model also needs scrutiny. Compare rates across clinically meaningful time periods, examine plots of transitions over time and person-time at risk, and check whether rates differ across important measured groups. Strong changes over time or substantial differences among groups warn that a single common constant rate may conceal important variation. Censoring related to an unobserved impending transition can also bias estimates; simple division of counts by observed person-time does not repair informative censoring.

Approach	What it summarizes	Main trade-off
Separate transition intensities	Instantaneous rates for each allowed destination from the state	Rates describe local event speed; time-specific probabilities require integrating them over a horizon.
One combined exit intensity	The rate of any departure from the state	Simpler, but discards which destination occurred.
Discrete-time destination probabilities	Probabilities of each destination over a specified interval	Directly interpretable for that interval, but can be sensitive to interval length and may not resolve multiple transitions within an interval.

Separate destination-specific rates are useful when both the timing and type of next transition matter. A combined rate may be adequate when the scientific question concerns only how quickly people leave the state. Discrete-time probabilities can be more natural when events are observed only at scheduled visits, but an interval containing multiple unobserved transitions cannot automatically be treated as a single known transition.

A further limitation is that an average rate for a group can mask variation between individuals. If groups with different risks are mixed, the observed rate among those still in the state can change over time as their composition changes, even when each subgroup has a constant rate. The equations remain useful when rates are defined for the actual risk set, but a constant-rate model for everyone may then fit poorly. The model should be refined or its target population made explicit rather than interpreting a pooled rate as universal.

Common Mistakes

Using one destination as though it were the total exit rate. The total is the sum across all allowed outgoing destinations; omitting an exit understates departure.
Calling a rate a probability. A rate such as $0.03$ per month is not a 3% monthly probability. A probability requires a time interval and, generally, the competing rates.
Counting a later event as the next destination. If someone progresses and then dies, death is not their first destination from the original illness state.
Assuming “mutually exclusive next destination” means destinations never co-occur in a history. A person can visit several states over time, but only one state is the immediate destination at a particular departure.
Ignoring observation limitations. If event order or destination is not observed, a precise first-destination classification cannot be recovered merely by assigning the most plausible label.

Key takeaway: For a shared current state, define each allowed next destination precisely, assign it a destination-specific intensity, and sum those intensities to obtain the total departure rate. Keep rates, interval-specific probabilities, and observed destination proportions distinct.

Check Your Understanding

Answer each question using the stated assumptions. Short solutions follow the questions.

A state has constant outgoing rates $0.02$ per week to recovery and $0.03$ per week to progression. What is the total exit rate, and what is the probability of remaining in the state for four weeks?
With the same rates, what is the four-week probability that progression is the first destination?
A study observes 15 recoveries and 5 progressions during 500 person-weeks in a state. Under a constant-rate model, calculate each estimated rate and the total rate.
A person moves from illness to progression and later dies. Which transition is counted as the first destination from illness?

Solutions. (1) The total rate is $0.02+0.03=0.05$ per week. The probability of remaining for four weeks is $e^{-(0.05)(4)}=e^{-0.20}=0.8187$. (2) The probability of any exit is $1-0.8187=0.1813$; progression receives the fraction $0.03/0.05=0.60$, so its first-destination probability is $0.60(0.1813)=0.1088$. (3) The estimated recovery rate is $15/500=0.0300$ per week, the progression rate is $5/500=0.0100$ per week, and the total is $0.0400$ per week. (4) Progression is the first destination from illness. Death is a later transition from progression.