State-Occupation Probabilities From Panel Models

From Panel Transitions to Future State Occupation

A transition model describes how probabilities of moving between states change from one observation time to the next. A useful next question is: given a person's state now, what is the probability of occupying each state at a specified future time? Answering it requires combining transitions across observation intervals, not simply reading a single transition probability.

The resulting probabilities describe the person's state at the future observation time. They do not, by themselves, describe the route taken between visits. This distinction matters when the panel model allows transitions to occur between observations without recording when they occurred.

Definition: Let the finite state space be \(S=\{1,\ldots,K\}\). For observation times \(t_0<t_1<\cdots<t_m\), let \(P_r\) be the \(K\times K\) transition matrix from \(t_{r-1}\) to \(t_r\). Its entry \(p_{ij}^{(r)}\) is the probability that the state at \(t_r\) is \(j\), conditional on state \(i\) at \(t_{r-1}\), under the specified model. The state-occupation probability at \(t_m\), conditional on starting in state \(i\) at \(t_0\), is the probability of occupying each possible state at \(t_m\) given that starting state.

We use a row-vector convention: each row of a transition matrix corresponds to the state at the beginning of an interval, and each column to the state at the end. Each entry must be nonnegative and each row must sum to one. If the starting distribution is the row vector \(\boldsymbol{\pi}(t_0)\), then the distribution at the final observation time is

\boldsymbol{\pi}(t_m)=\boldsymbol{\pi}(t_0)P_1P_2\cdots P_m.

For a known starting state \(i\), \(\boldsymbol{\pi}(t_0)\) is the row vector with a 1 in position \(i\) and 0 elsewhere. Thus the required state-occupation probabilities are row \(i\) of the product \(P_1P_2\cdots P_m\). If the transition matrix is constant across intervals, the product is \(P^m\). That power is appropriate only when the same transition probabilities apply at each step.

Assumptions and the Composition Rule

The calculations rely on a model that defines valid transition probabilities for the observation intervals of interest. In particular, the state space must be consistent across intervals, and the rows of each matrix must represent probabilities conditional on the state at the start of that interval. Most importantly, the model must support the Markov property at the observation times: given the current state (and any covariates included in the model), the next state does not additionally depend on earlier states or history.

For a time-homogeneous model, the same matrix \(P\) applies at each step. For a model with interval-specific probabilities, the matrices may differ. They must be applied in chronological order. If covariates determine the transition probabilities, the calculation is conditional on the covariate values used to select or calculate those matrices; averaging predictions across covariate groups requires an additional averaging step.

Composition rule: Under the Markov property, the transition probabilities from \(t_0\) to \(t_m\) are given by the matrix product \(P_1P_2\cdots P_m\). The \((i,j)\) entry is the probability of being in state \(j\) at \(t_m\), conditional on being in state \(i\) at \(t_0\).

To justify the rule for two intervals, consider an intermediate state \(k\) at \(t_1\). The probability of being in state \(j\) at \(t_2\) given state \(i\) at \(t_0\) is found by summing over all possible states at \(t_1\). By the Markov property, the contribution through state \(k\) is the probability of moving from \(i\) to \(k\), multiplied by the probability of moving from \(k\) to \(j\):

P(X(t_2)=j\mid X(t_0)=i) =\sum_{k=1}^{K}p_{ik}^{(1)}p_{kj}^{(2)} =(P_1P_2)_{ij}.

Applying the same argument repeatedly gives the product across any number of intervals. This is the law of total probability applied at each intermediate visit. It also explains why the multiplication order matters: in general, \(P_1P_2\) and \(P_2P_1\) are not equal.

Worked Example: Two-Step Occupation Probabilities

Suppose a hypothetical panel model has three states: 1 is a disease-free state, 2 is a disease state, and 3 is an absorbing state. For each visit interval, the fitted transition matrix is

P= \begin{pmatrix} 0.70&0.20&0.10\\ 0.05&0.75&0.20\\ 0&0&1 \end{pmatrix}.

Assume the Markov property and that this same matrix applies over both intervals. Each row sums to 1, and the absorbing-state row leaves state 3 unchanged. Starting in state 1, the probabilities after two intervals are row 1 of \(P^2\). For each destination state, sum over the possible state at the intermediate visit:

\begin{aligned} P(X(t_2)=1\mid X(t_0)=1)&=(0.70)(0.70)+(0.20)(0.05)+(0.10)(0)=0.50,\\ P(X(t_2)=2\mid X(t_0)=1)&=(0.70)(0.20)+(0.20)(0.75)+(0.10)(0)=0.29,\\ P(X(t_2)=3\mid X(t_0)=1)&=(0.70)(0.10)+(0.20)(0.20)+(0.10)(1)=0.21. \end{aligned}

The two-interval occupation distribution is therefore \((0.50,0.29,0.21)\). Its entries are nonnegative and sum to 1. In this model, a person starting in state 1 has a 0.21 probability of occupying state 3 at the second future visit. The calculation does not say when, or by which sequence of intermediate events, that person entered state 3.

Changing Transition Probabilities Across Visits

A single matrix power is not appropriate when transition probabilities depend on the interval, for example because observation intervals differ or because the model allows probabilities to vary over time. In that case, use the matrix estimated for each interval in chronological order. The product describes endpoint state occupation across those intervals, provided the model's conditional transition probabilities and Markov assumption apply.

Worked Example: Unequal Intervals With Different Matrices

Consider a hypothetical three-state model with state 1 disease-free, state 2 disease, and state 3 an absorbing state. The visit times are \(t_0=0\), \(t_1=1\), and \(t_2=3\) years. Suppose the fitted matrices for the one-year and two-year intervals are, respectively,

P_1= \begin{pmatrix} 0.80&0.20&0\\ 0&0.70&0.30\\ 0&0&1 \end{pmatrix}, \qquad P_2= \begin{pmatrix} 0.60&0.30&0.10\\ 0&0.50&0.50\\ 0&0&1 \end{pmatrix}.

Both matrices have nonnegative entries and rows summing to 1. Suppose a person starts in state 1. After the first interval, the state distribution is row 1 of \(P_1\), or \((0.80,0.20,0)\). Propagating that distribution through the second interval gives

\begin{aligned} (0.80,0.20,0)P_2 &=\bigl((0.80)(0.60),\ (0.80)(0.30)+(0.20)(0.50),\\ &\qquad (0.80)(0.10)+(0.20)(0.50)\bigr)\\ &=(0.48,0.34,0.18). \end{aligned}

Thus the predicted probabilities at year 3 are 0.48 in state 1, 0.34 in state 2, and 0.18 in state 3, conditional on starting in state 1 at year 0. The probabilities sum to 1. Using \(P_1P_2\), rather than \(P_2P_1\), respects the order in which the intervals occur. The matrices already describe transitions across intervals of different lengths; the calculation does not treat the two intervals as equal in duration.

Comparing Predictions Across Starting States

A comparison of state-occupation predictions should hold the future time and the transition model fixed, then change the starting state. This produces distributions conditional on different starting states. Differences can be summarized for a particular state or displayed as full probability vectors; they should not be reduced to a single probability unless the question identifies the outcome of interest.

Worked Example: Comparing Two Starting States

For a hypothetical model with three states, suppose the one-interval transition matrix is

Q= \begin{pmatrix} 0.75&0.20&0.05\\ 0.15&0.70&0.15\\ 0&0&1 \end{pmatrix}.

Assume the same matrix applies over two intervals and the Markov property holds. For someone starting in state 1, the probability of occupying state 2 after two intervals is

$$ (0.75)(0.20)+(0.20)(0.70)+(0.05)(0)=0.29. $$

For someone starting in state 2, the corresponding probability is

$$ (0.15)(0.20)+(0.70)(0.70)+(0.15)(0)=0.52. $$

Under this model, the predicted probability of occupying state 2 at the second future visit is 0.23 higher for starting state 2 than for starting state 1. This is a comparison of model-based conditional predictions, not evidence that changing a person's starting state would cause that difference.

If the starting distribution is instead 60% in state 1 and 40% in state 2, the predicted state-2 probability is a weighted average of the two conditional probabilities: \((0.60)(0.29)+(0.40)(0.52)=0.382\). The starting distribution must be stated because this marginal prediction is not the same quantity as either state-specific prediction.

Checks, Alternatives, and Limitations

Several basic checks can catch mistakes in calculation or interpretation:

Check that every transition matrix has nonnegative entries and rows summing to 1, allowing for rounding.
After multiplication, check that every predicted occupation probability is between 0 and 1 and that the full distribution sums to 1.
Confirm that the matrices match the intervals and covariate conditions being predicted, and that they are multiplied in chronological order.
Check whether the Markov assumption is credible. If future transition probabilities depend on time spent in a state, earlier history, or other omitted information, a model that conditions only on the current state may give misleading predictions.

This propagation approach uses the transition probabilities estimated by the panel model. An alternative is to estimate future state proportions directly for groups defined by their starting state. Direct estimates may be useful as a descriptive check, but they do not automatically provide the same model-based conditional predictions or handle sparse state-by-visit patterns as flexibly. Continuous-time models can instead represent transitions over elapsed time using transition intensities; those models require their own assumptions and are not interchangeable with a discrete-time matrix unless the model structure justifies the conversion.

The predictions inherit uncertainty and limitations from the fitted transition probabilities. A matrix product based on point estimates is a point prediction, not an interval that accounts for estimation uncertainty. Predictions may also be sensitive to the chosen state definitions, visit schedule, and model specification. In particular, panel data may leave intermediate transitions unobserved, so an endpoint transition probability should not be interpreted as a probability of exactly one transition or as a description of the unobserved path.

Common pitfall: Treating a two-interval prediction as the entry from the first-interval matrix, or multiplying matrices in reverse order, ignores possible transitions in later intervals. The correction is to propagate the entire state distribution through each interval, in chronological order, using the model's matrix for that interval.

Check Your Understanding

Answer each question, then compare with the solutions below.

A person starts in state 1. Explain how to obtain the state-occupation probabilities after three intervals if the transition matrices are \(P_1,P_2,P_3\).
Using the first matrix \(P\) in the worked two-step example, what is the probability of occupying state 2 after two intervals when starting in state 2?
Why is \(P_1P_2\), rather than \(P_2P_1\), used when \(P_1\) and \(P_2\) describe consecutive intervals in that order?
If two predicted occupation probabilities are 0.38 and 0.51 for different starting states, what is the difference, and what does it mean?

Solutions. (1) Use row 1 of \(P_1P_2P_3\), since row 1 represents the specified starting state and the matrices are applied in chronological order. (2) Sum over the possible intermediate states: \((0.05)(0.20)+(0.75)(0.75)+(0.20)(0)=0.5725\). (3) Matrix multiplication composes transitions in order: the first factor carries the initial state to the intermediate visit, and the next carries that intermediate distribution to the later visit. (4) The difference is \(0.51-0.38=0.13\), or 13 percentage points. It is the difference between the model's conditional probabilities of occupying the specified state at the stated future time, not a causal effect of starting state.