Pseudo-Observations For Transition-Probability Calibration

Why Use Pseudo-Observations for Calibration?

A transition model produces predicted probabilities, but validation requires an observed-data quantity to compare with them. At a fixed horizon, a subject’s eventual state may be unknown because follow-up ended earlier. The Aalen–Johansen estimator uses the observed transition history and censoring information to estimate transition probabilities for the validation cohort. Pseudo-observations turn this cohort-level estimate into one value per subject, allowing calibration to be examined against subject-specific predictions.

A pseudo-observation is not necessarily an observed outcome. It is a jackknife quantity constructed from the change in an estimator when one subject is removed. It may be below zero or above one, and its sample mean is not generally equal to the original Aalen–Johansen estimate. Those features matter when choosing a calibration analysis and interpreting its results.

Definition: Consider subjects who begin in state $r$ at a common time origin, and let $\tau$ be a fixed prediction horizon. The transition probability $P_{rs}(0,\tau)$ is the probability of occupying state $s$ at time $\tau$, given the specified origin in state $r$. It is an occupancy probability, not necessarily the probability of ever entering state $s$. For an estimator $\hat\theta$ of this quantity based on $n$ subjects, let $\hat\theta_{(-i)}$ be the estimate after deleting subject $i$. The jackknife pseudo-observation for subject $i$ is $$ \hat\theta_i^{\mathrm{pseudo}}=n\hat\theta-(n-1)\hat\theta_{(-i)}. $$

The Aalen–Johansen Estimate at a Fixed Horizon

For a finite-state process, write $Y_j(u)$ for the number at risk in state $j$ immediately before event time $u$, and $dN_{jk}(u)$ for the number of observed transitions from state $j$ to state $k$ at that time. The estimated transition-hazard increment from $j$ to $k$ is $dN_{jk}(u)/Y_j(u)$. The diagonal increment is the negative sum of the off-diagonal increments in that row, so each row of the increment matrix sums to zero.

The Aalen–Johansen transition matrix estimate is the ordered product of the corresponding increment matrices. Under the row-current-state, column-destination convention, its entry in row $r$, column $s$ estimates $P_{rs}(0,\tau)$. At a fixed horizon, the pseudo-observations are obtained by recalculating this same estimator after each subject is deleted. The full and deleted estimates must use the same time origin, horizon, state definitions, and analysis rules.

Worked Example: Reading a Fixed-Horizon Transition Probability

Suppose a hypothetical cohort starts in state 0, and the only possible states are 0, 1, and 2. At the first event time, four subjects are at risk in state 0; one moves to state 1 and one to state 2. At a later time, two subjects remain in state 0 and one is in state 1; one moves from state 0 to state 1, while the subject in state 1 moves to state 2. There is no censoring before the horizon.

At the first time, the estimated transition matrix increment gives the following product factor. States 1 and 2 have no departures at that time, so their rows are unchanged:

\hat P_1= \begin{pmatrix} 0.50&0.25&0.25\\ 0&1&0\\ 0&0&1 \end{pmatrix}.

At the later time, half of those at risk in state 0 move to state 1, and the subject at risk in state 1 moves to state 2:

\hat P_2= \begin{pmatrix} 0.50&0.50&0\\ 0&0&1\\ 0&0&1 \end{pmatrix}.

Multiplying in time order gives the estimated transition matrix at the horizon:

\hat P_1\hat P_2= \begin{pmatrix} 0.25&0.25&0.50\\ 0&0&1\\ 0&0&1 \end{pmatrix}.

Thus the estimated probability of occupying state 1 at the horizon, starting in state 0, is $0.25$. The estimated probability of occupying state 2 is $0.50$. The remaining $0.25$ is the probability of still occupying state 0. The entries in the starting-state row sum to one, as they should.

Constructing and Interpreting Pseudo-Observations

For each subject, delete that subject from the validation data, recalculate the Aalen–Johansen estimate at $\tau$, and substitute the full and deleted estimates into the jackknife formula. This calculation is repeated for all $n$ subjects. The subject index labels which deletion was made; it does not mean the resulting quantity is an independently observed transition outcome.

The motivation is that, for a sufficiently regular estimator, the effect of deleting one subject reflects that subject’s contribution to the estimated quantity. Under standard large-sample conditions, pseudo-observations can therefore act as response values in regression analyses. This is an approximation for inference and calibration, not a guarantee that every pseudo-value behaves like a binary outcome.

Important finite-sample fact: The mean of the pseudo-observations is $$ \frac{1}{n}\sum_{i=1}^n\hat\theta_i^{\mathrm{pseudo}} =n\hat\theta-\frac{n-1}{n}\sum_{i=1}^n\hat\theta_{(-i)}. $$ It equals $\hat\theta$ only if $\sum_i\hat\theta_{(-i)}=n\hat\theta$. That condition holds for some estimators, including the sample mean, but it is not guaranteed for the nonlinear Aalen–Johansen estimator.

A common calibration regression relates the pseudo-observation to the model’s predicted transition probability $p_i$. In a simple identity-link specification, the conditional mean is represented as $\alpha+\beta p_i$. Ideal calibration corresponds to $\alpha=0$ and $\beta=1$: predictions are neither systematically shifted nor compressed or exaggerated on that scale. An intercept-only assessment with slope fixed at one focuses on calibration-in-the-large. These are working regression summaries; a flexible calibration curve may be needed if the relationship is nonlinear.

Pseudo-observation regression requires appropriate uncertainty estimation. The pseudo-values share the full-sample estimate and are therefore dependent. Ordinary regression standard errors that treat them as independent observed outcomes can be misleading. Use an inference method justified for pseudo-observation regression, such as an appropriate sandwich approach, or resample subjects and recompute all pseudo-observations within each resample. The choice should reflect the sampling design and censoring structure.

Worked Example: Jackknife Values With Complete Follow-Up

Consider four hypothetical subjects, all beginning in state 0 and observed through the horizon. Subjects A and C occupy the target state 1 at the horizon; B and D occupy a competing state. With complete follow-up, the Aalen–Johansen estimate of the probability of state 1 is $2/4=0.50$.

Deleting A or C leaves one target-state subject among three, so the deleted estimate is $1/3$. Deleting B or D leaves two target-state subjects among three, so the deleted estimate is $2/3$. The pseudo-values are:

\hat\theta_A^{\mathrm{pseudo}} =\hat\theta_C^{\mathrm{pseudo}} =4(0.50)-3(1/3)=2-1=1,

\hat\theta_B^{\mathrm{pseudo}} =\hat\theta_D^{\mathrm{pseudo}} =4(0.50)-3(2/3)=2-2=0.

Here the pseudo-values happen to equal the binary state indicators, and their mean is $0.50$. This equality arises from this complete-follow-up sample-proportion setting; it should not be assumed for censored data or for every nonlinear estimator.

Suppose the four model predictions, in subject order A through D, are $0.20, 0.40, 0.60,$ and $0.80$. An ordinary least-squares line using these pseudo-values has $\bar p=0.50$ and $\bar y=0.50$. Its slope is

\hat\beta= \frac{\sum_i(p_i-\bar p)(\hat\theta_i^{\mathrm{pseudo}}-\bar y)} {\sum_i(p_i-\bar p)^2} =\frac{-0.20}{0.20}=-1, \qquad \hat\alpha=\bar y-\hat\beta\bar p=1.00.

This tiny hypothetical example has a sharply negative fitted slope, showing poor agreement between the ordering of predictions and outcomes in this sample. It does not establish a population calibration relationship: four subjects provide little precision, and valid inference must account for the data structure.

Assumptions and a Finite-Sample Pitfall

The Aalen–Johansen estimate and its pseudo-observations are useful only if the observed follow-up supports the target quantity. The cohort and time origin must define the prediction target correctly; transition states and event times must be recorded consistently; and censoring must be noninformative for the transition process under the analysis used. In particular, ordinary pooled Aalen–Johansen estimation generally requires censoring to be independent of the event history in the target population. If censoring is independent only conditional on covariates, an unadjusted estimator may not be valid; a suitably stratified or censoring-adjusted estimator may be needed before constructing pseudo-observations.

There must also be adequate numbers at risk in relevant states through $\tau$. If a state has no one at risk at an event time, its transition increment cannot be estimated there. Sparse transitions or heavy censoring can make leave-one-out estimates unstable. Inspect the risk sets, the full and deleted estimates, and the resulting pseudo-values. Extreme or out-of-range pseudo-values are warnings to examine; they are not automatically evidence of an arithmetic error.

Worked Example: A Censored Sample Whose Pseudo-Values Exceed One

Consider three hypothetical subjects starting in one state. A has the target event at time 1, B is censored at time 2, and C has the target event at time 3. Let the horizon be time 3. There is only one event type, so the Aalen–Johansen estimate equals the estimated probability of having had the event by the horizon.

In the full sample, the first event occurs among three at risk, contributing $1/3$. After that event, two remain at risk; B is censored at time 2, and C’s event at time 3 contributes the remaining $2/3$. Thus $\hat\theta=1/3+2/3=1$.

Deleting A leaves B and C. B is censored before C’s event, so the event occurs among the sole subject still at risk; the deleted estimate is 1. Deleting B leaves two observed events, giving an estimate of 1. Deleting C leaves A’s event among two at risk (A and B), followed by B’s censoring, giving an estimate of $1/2$. The pseudo-values are therefore

3(1)-2(1)=1,\qquad 3(1)-2(1)=1,\qquad 3(1)-2(1/2)=2.

Their mean is $(1+1+2)/3=4/3$, not the full-sample estimate of 1. The value 2 is not a probability; it is a jackknife contribution associated with the nonlinear estimator and this small sample. Treating it as an observed binary outcome or assuming pseudo-values must lie in $[0,1]$ would be incorrect.

Calibration Checks and Alternatives

A useful analysis combines a numerical summary with a plot or grouped check. For a calibration plot, arrange subjects by predicted probability and compare predictions with estimated observed transition probabilities across prediction ranges. If pseudo-values are averaged within groups, describe those averages as pseudo-value summaries: they need not equal the Aalen–Johansen estimate calculated directly within each group. A direct grouped check instead estimates the transition probability with Aalen–Johansen separately within each prediction group, subject to adequate events and censoring support.

The main approaches answer related but not identical questions:

Approach	What it summarizes	Main caution
Pseudo-observation regression	Relationship between predicted probabilities and jackknife contributions to a fixed-horizon estimate	Requires suitable pseudo-value inference; values may be outside the probability range
Grouped Aalen–Johansen estimates	Observed transition experience in groups defined by predicted risk	Grouping loses information and estimates can be unstable in small groups
Direct comparison of cohort-level observed and mean predicted probability	Overall calibration-in-the-large	Agreement in the mean does not establish calibration across the prediction range

Calibration of one transition probability also does not ensure that a complete predicted transition matrix is coherent. For example, separately fitted calibration adjustments may produce entries that do not sum to one or otherwise fail to define valid probabilities. Check the full set of state probabilities and their constraints, not just one regression coefficient.

Common Mistakes

Assuming the pseudo-values average to the Aalen–Johansen estimate. The jackknife formula does not guarantee this for a nonlinear estimator. Verify the leave-one-out values rather than imposing an equality.
Calling a pseudo-value an individual outcome. It is a constructed quantity reflecting a deletion effect; it can be outside $[0,1]$.
Ignoring the censoring mechanism. Ordinary pooled Aalen–Johansen estimation may be biased if censoring is informative for the transition process.
Equating an overall mean match with good calibration. Predictions can have the correct average while being systematically too high in one risk range and too low in another.
Using ordinary independent-outcome standard errors without justification. Pseudo-values share information through the full-sample estimate; use inference appropriate to their construction.

Key takeaway: At a fixed horizon, calculate each pseudo-observation as $n\hat\theta-(n-1)\hat\theta_{(-i)}$, using the same Aalen–Johansen target in every deletion. Use these values as calibration surrogates only with explicit attention to censoring, dependence, and the fact that their finite-sample mean and range need not match those of probabilities.

Check Your Understanding

Try each problem, then compare your reasoning with the solutions below.

For $n=5$, the full-sample estimate is $0.40$, and the leave-one-out estimate after deleting subject 2 is $0.35$. Calculate that subject’s pseudo-observation.
What condition on the leave-one-out estimates makes the mean pseudo-observation equal the full-sample estimate?
Why can a pseudo-observation of $1.4$ occur without being a probability of 140%?
A pooled validation cohort has censoring that is independent of event history only after conditioning on a covariate. Is ordinary pooled Aalen–Johansen estimation automatically justified?
Predictions average to $0.30$, and a cohort-level Aalen–Johansen estimate is $0.30$. Does this establish calibration throughout the prediction range?

Solutions. (1) $5(0.40)-4(0.35)=2.00-1.40=0.60$. (2) The condition is $\sum_i\hat\theta_{(-i)}=n\hat\theta$, which is not generally guaranteed for Aalen–Johansen estimates. (3) A pseudo-observation is a jackknife quantity, not a literal event indicator or probability; subtraction of the leave-one-out estimate can produce values outside $[0,1]$. (4) No. Conditional independence alone does not automatically justify the unadjusted pooled estimator; use an appropriate stratified or censoring-adjusted method. (5) No. The two means assess overall agreement only; predictions may be miscalibrated in different parts of the risk range.