Common Errors In Transition-Probability Estimation

Why Transition-Probability Estimates Need Careful Checks

A transition-probability matrix can look plausible while encoding the wrong calculation. A denominator drawn from the wrong state changes the estimated transition intensity; reversing matrix order changes which paths are possible; and treating an intensity as a probability ignores the time interval over which the transition can occur. Even a correctly computed estimate can be reported too far beyond the follow-up that supports it.

The checks in this tutorial use the product-integral setting for transition probabilities. They assume states and transition times are recorded correctly, the transition structure is specified, and the censoring assumptions required for the estimator are reasonable. At each event time, the risk set for a transition is determined by the state occupied immediately before that time. The earlier material on independent censoring and product-integral estimation gives the broader estimation context.

Definition: Let \(Y_i(u)\) be the number of individuals observed and in state \(i\) immediately before time \(u\), and let \(dN_{ij}(u)\) be the number of observed transitions from state \(i\) to state \(j\) at \(u\). The estimated intensity increment for \(i\to j\) is \(d\widehat A_{ij}(u)=dN_{ij}(u)/Y_i(u)\), when \(Y_i(u)>0\). For \(i\ne j\), the diagonal increment is \(d\widehat A_{ii}(u)=-\sum_{j\ne i}d\widehat A_{ij}(u)\). Each factor \(I+d\widehat A(u)\) has rows indexed by origin states and columns by destination states.

The transition-probability matrix from time \(s\) to time \(t\), written \(\widehat P(s,t)\), uses the same indexing: its \((i,j)\) entry estimates the probability of occupying state \(j\) at \(t\), conditional on occupying state \(i\) at \(s\). Under this row-origin convention, successive factors are multiplied from left to right in chronological order. A valid probability matrix has nonnegative entries and rows summing to one, but those checks alone cannot establish that its denominators, order, or time range are correct.

Use the Risk Set for the Transition's Origin

The denominator \(Y_i(u)\) counts those at risk of making a transition out of state \(i\) at time \(u\), not everyone still under observation and not everyone who began in the study. Different origin states can have different risk sets at the same time. Using the wrong denominator changes the transition increment and can distort every later product-integral factor that depends on it.

Worked Example: A State-Specific Denominator

Consider a hypothetical study with states A, B, and C. By day 3, four individuals are observed in B immediately before the event time; one makes a transition from B to C at day 3. Seven others are still observed in A, and one additional individual was censored earlier. Assume no other transitions out of B occur at that time.

For the B-to-C increment, the origin state is B, so \(Y_B(3)=4\) and \(dN_{BC}(3)=1\). Thus

d\widehat A_{BC}(3)=\frac{1}{4}=0.25.

The corresponding product-integral factor has a 0.25 entry in row B, column C, and a 0.75 diagonal entry in row B. Its row for B is therefore \((0,0.75,0.25)\), in the state order A, B, C. The seven individuals in A do not belong in this denominator: they were not at risk of a B-to-C transition at day 3.

If someone instead divides by all 11 individuals still observed just before day 3, the result is \(1/11=0.09091\), which understates the B-to-C increment. The issue is not a small-sample adjustment; it is that the denominator describes the wrong risk set. If \(Y_B(3)=0\), the increment is undefined, not zero.

Useful diagnostics include tabulating each event by origin state and checking that its denominator equals the number in that state immediately before the event. A common warning sign is a denominator that includes individuals who are in another state, have already experienced an absorbing event, or have been censored. Sparse risk sets are another concern: a single transition among two people gives an increment of 0.5, so estimates may be highly variable even when the calculation is correctly defined.

Multiply Transition Factors in Chronological Order

With row vectors, a distribution is updated by multiplying on the right. If \(B_1\) is the factor at the earlier time and \(B_2\) the factor at the later time, the update is \(pB_1B_2\). Reversing the factors describes a different sequence of transitions. Matrix multiplication is generally not commutative: \(B_1B_2\ne B_2B_1\).

Worked Example: Correcting a Reversed Product

Suppose three states are A, B, and C. At an earlier time, half the individuals in A move to B, while B remains in B and C remains in C. At a later time, 40% of those in B move to C and 60% remain in B. The hypothetical transition factors are

B_1= \begin{pmatrix} 0.5&0.5&0\\ 0&1&0\\ 0&0&1 \end{pmatrix}, \qquad B_2= \begin{pmatrix} 1&0&0\\ 0&0.6&0.4\\ 0&0&1 \end{pmatrix}.

For an individual starting in A, the initial row vector is \(e_A=(1,0,0)\). Applying the earlier factor and then the later factor gives

e_AB_1B_2=(0.5,0.5,0) \begin{pmatrix} 1&0&0\\ 0&0.6&0.4\\ 0&0&1 \end{pmatrix} =(0.5,0.3,0.2).

Thus the estimated probabilities after both times are 0.5 in A, 0.3 in B, and 0.2 in C. The C probability is \(0.5(0)+0.5(0.4)=0.2\): only the half that first reached B can then move to C.

The reversed product gives a different answer. In particular,

e_AB_2B_1=(1,0,0) \begin{pmatrix} 0.5&0.5&0\\ 0&0.6&0.4\\ 0&0&1 \end{pmatrix} =(0.5,0.5,0).

The reversed product puts the later B-to-C factor before the earlier A-to-B factor, so it does not represent the stated sequence. The corrected matrix product \(B_2B_1\) has row B equal to \((0,0.6,0.4)\): its C entry is \(0.6(0)+0.4(1)=0.4\), not 0.7. The correct chronological product for the example remains \(B_1B_2\).

A practical check is to write the time attached to every factor before multiplying, then verify that the first update acts on the starting distribution. Also check that each factor is row-stochastic: entries are nonnegative and each row sums to one. These checks can catch arithmetic and orientation errors, but a row-stochastic reversed product is still in the wrong order.

Do Not Treat an Intensity as a Probability

An intensity is a rate, not a probability over an unspecified interval. In continuous time, an intensity has units of inverse time, whereas a transition probability is dimensionless and refers to a stated time interval. For a sufficiently short interval of length \(\Delta t\), an off-diagonal transition probability is approximately the corresponding intensity times \(\Delta t\); it is not generally equal to the intensity.

Worked Example: Converting a Constant Rate to a Probability

Suppose a hypothetical two-state process has a constant transition intensity of \(0.08\) per day from state A to absorbing state B, with no other possible transition. Under this constant-rate model, the probability of moving from A to B within one day is

P_{AB}(0,1)=1-\exp(-0.08\times 1) =1-\exp(-0.08) \approx 0.07688.

The one-day transition probability is about 0.0769, or 7.69%, not 0.08 by definition. The rate is \(0.08\) per day; the probability is for the particular one-day interval. For a short interval of length \(\Delta t\), \(1-\exp(-0.08\Delta t)\approx0.08\Delta t\), which explains the approximation and its dependence on interval length.

Under the stated two-state, constant-intensity model, the probability matrix for one day is

P(0,1)= \begin{pmatrix} \exp(-0.08)&1-\exp(-0.08)\\ 0&1 \end{pmatrix} \approx \begin{pmatrix} 0.9231&0.07688\\ 0&1 \end{pmatrix}.

Its rows sum to one. This calculation relies on the constant-intensity model over the full day; it is not a license to use an intensity estimate as a finite-interval probability in a different model or over a different duration.

For time-varying intensities, the transition probabilities depend on the accumulation and ordering of changes over the interval, which is why the product-integral construction uses successive factors. A generator or intensity matrix has off-diagonal rate entries and diagonal entries chosen so rows sum to zero. A transition-probability matrix instead has entries between zero and one and rows summing to one. Confusing these matrices can produce impossible entries, incorrect units, or an invalid interpretation even when some numbers appear close.

Respect the Follow-Up Support

A nonparametric transition estimate is supported only where the relevant origin-state risk sets provide information. If no one is observed in state \(i\) after a time \(\tau\), then transitions out of \(i\) after \(\tau\) cannot be estimated from those data using the risk-set increment. An absent event is not evidence that the transition probability is zero when nobody remained at risk to experience it.

Worked Example: A Transition Beyond Observed Support

In a hypothetical follow-up, some individuals occupy state B through day 4, but after day 4 none remain under observation in B. A report requests the probability of a B-to-C transition between days 4 and 6, estimated nonparametrically from these data.

At times after day 4, \(Y_B(u)=0\). Therefore \(dN_{BC}(u)/Y_B(u)\) is not defined, and the data supply no B-origin increments for that interval. The requested probability is not estimable from this risk-set calculation; reporting it as zero, or carrying the day-4 estimate forward as if it were observed, would be unjustified.

A parametric model might produce a prediction beyond day 4, but that result would depend on assumptions about the unobserved period and should be identified as model-based extrapolation. It is not the same as an estimate supported by observed transitions and risk sets.

Choosing Checks and Interpreting Their Limits

No single diagnostic establishes that a transition-probability estimate is correct. The following checks target distinct failure modes:

Risk-set check: confirm that each \(dN_{ij}(u)\) is divided by \(Y_i(u)\), with membership assessed immediately before \(u\).
Order check: list factors in time order and confirm that the earliest factor acts first on the row distribution.
Scale check: verify that intensities have inverse-time units and that probabilities refer to a specific interval.
Support check: report the last times with adequate origin-state risk sets; do not silently extend a nonparametric estimate past them.
Matrix check: verify nonnegative probability entries and rows summing to one, while remembering that these properties do not detect every conceptual error.

The product-integral estimate is a nonparametric or empirical construction based on observed transition increments and the risk sets. A parametric intensity model can smooth sparse data and provide predictions beyond observed support, but its conclusions require the assumed model to be credible. A fixed time-grid approximation can simplify reporting, but a coarse grid can conceal changes in transition order or timing. These approaches address different needs; none makes a wrong origin-state denominator or reversed multiplication valid.

The principal limitation is that transition-probability estimates may be unstable when origin-state risk sets are small and unavailable when those risk sets disappear. The earlier material on uncertainty quantification and numerical stability is relevant when assessing variability and computation. A mathematically valid matrix is not, by itself, evidence that an estimate is precise, supported, or correctly interpreted.

Key takeaway: For every reported transition probability, check the origin-state risk set, the chronological order of factors, the distinction between rates and probabilities, and the time range supported by observed follow-up. Each check answers a different question.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

At a particular time, 3 people are in state 1, 5 are in state 2, and one observed transition occurs from state 1 to state 3. What denominator belongs in the estimated intensity increment?
A row distribution is updated first by \(C_1\) at an earlier time and then by \(C_2\) at a later time. Which product represents the update under the row-vector convention?
A constant transition intensity is \(0.12\) per month. Is the probability of transition over one month exactly 0.12? Give the probability under a constant-rate, single-transition model.
No one remains at risk in origin state 2 after month 8. Can the risk-set product-integral estimate identify a transition probability from state 2 over months 8 to 10?

Solutions. (1) The denominator is \(Y_1=3\), so the increment is \(1/3\); the five people in state 2 are not in the origin-state risk set. (2) The product is \(C_1C_2\), because the earlier factor acts first on the row distribution. (3) No. Under the stated model it is \(1-\exp(-0.12)\approx0.1131\), a dimensionless one-month probability; \(0.12\) is the rate per month. (4) No. With no state-2 risk set after month 8, the required increments are undefined, so the nonparametric estimate is unsupported for that interval.