Cause Misclassification And Hazard Interpretation

Why Cause Misclassification Changes the Quantity

A first event can occur at the right time and still be assigned the wrong cause. For example, a hypothetical clinical registry might record the time of a first event accurately but confuse two event types. The event-free survival pattern may then be unchanged, while the recorded counts and rates for each cause differ from the quantities for the true causes. Interpreting a hazard therefore requires knowing whether “cause” means the underlying event type or the label recorded in the data.

This tutorial considers first events with mutually exclusive true causes. The event time is observed correctly, and every event receives exactly one recorded cause label. The question is how the hazards and cumulative incidence for recorded labels relate to those for true causes. The calculations describe population quantities; estimating them from incomplete follow-up also requires appropriate handling of censoring.

Definition: Let \(k\) index the true cause and \(j\) the recorded cause label. The cause-classification probability \(q_{kj}(t)\) is the probability that an event of true cause \(k\) occurring at time \(t\) is recorded as label \(j\). For each true cause \(k\), the probabilities satisfy \(q_{kj}(t)\geq 0\) and \(\sum_j q_{kj}(t)=1\). When these probabilities do not vary with time, write \(q_{kj}\).

Let \(\lambda_k(t)\) be the true cause-specific hazard for cause \(k\): the instantaneous rate of a true cause-\(k\) event at time \(t\), conditional on having had no event by then. Assume event times have continuous distributions, the classification mechanism does not alter whether or when an event occurs, and the recorded label is assigned according to the specified probabilities conditional on the true cause and event time. Under these assumptions, the event-free survival function \(S(t)\) is the same whether events are described by their true or recorded causes. In particular, its total hazard is \(\sum_k\lambda_k(t)\).

The classification probabilities may also depend on measured group or covariate values. A single pooled classification matrix is appropriate only when that matrix describes the relevant population and remains stable over the times and groups being compared.

From True Hazards to Recorded-Cause Quantities

In a short interval of length \(dt\), the probability of a true cause-\(k\) event among people still event-free is approximately \(\lambda_k(t)dt\). Of those events, the fraction recorded as label \(j\) is \(q_{kj}(t)\). Adding the contributions from all true causes gives the cause-specific hazard for recorded label \(j\):

\lambda^{\mathrm{obs}}_j(t)=\sum_k q_{kj}(t)\lambda_k(t).

This pointwise mixture is a statement about rates at time \(t\). The observed cumulative incidence \(F^{\mathrm{obs}}_j(t)\), in contrast, accumulates events over the entire interval from 0 to \(t\). At each earlier event time \(s\), the appropriate classification probabilities are those applying at \(s\). Thus:

F^{\mathrm{obs}}_j(t) = \sum_k\int_0^t q_{kj}(s)S(s)\lambda_k(s)\,ds.

For constant classification probabilities, this reduces to \(F^{\mathrm{obs}}_j(t)=\sum_k q_{kj}F_k(t)\), where \(F_k(t)\) is the true cumulative incidence of cause \(k\). If classification probabilities vary with time, it is generally incorrect to multiply each true cumulative incidence by the classification probability at the final time. Earlier events may have been classified under different probabilities.

For continuous event times, the subdistribution hazard for recorded label \(j\) is defined by the rate of increase in its cumulative incidence relative to the probability of not yet having had that recorded cause:

\lambda^{\mathrm{obs}}_{j,\mathrm{sub}}(t) = \frac{dF^{\mathrm{obs}}_j(t)/dt}{1-F^{\mathrm{obs}}_j(t)} = \frac{S(t)\sum_k q_{kj}(t)\lambda_k(t)} {1-F^{\mathrm{obs}}_j(t)}.

The denominator is determined by the cumulative incidence for the recorded label, so misclassification can change it as well as the numerator. This subdistribution hazard has units of inverse time; it is not a probability. The introductory material on subdistribution hazards establishes why its denominator differs from the event-free risk set used for a cause-specific hazard.

Worked Calculations

Worked Example: Constant Misclassification Between Two Causes

Suppose a hypothetical population has two true causes, A and B, with constant cause-specific hazards \(\lambda_A=0.10\) per year and \(\lambda_B=0.20\) per year. The classification probabilities are constant: 90% of true A events are recorded as A, and 10% as B; 10% of true B events are recorded as A, and 90% as B. Assume all events are classified and their times are recorded accurately. Find the observed cause-specific hazards and the two-year observed cumulative incidence and subdistribution hazard for recorded label A.

The classification matrix, with rows for true causes and columns for recorded labels, is

Q= \begin{pmatrix} 0.90 & 0.10\\ 0.10 & 0.90 \end{pmatrix}.

Each row sums to 1, as required. The observed hazards are

\lambda^{\mathrm{obs}}_A =(0.90)(0.10)+(0.10)(0.20) =0.11\ \text{per year},

and \(\lambda^{\mathrm{obs}}_B=(0.10)(0.10)+(0.90)(0.20)=0.19\) per year. Their sum is \(0.30\) per year, the unchanged total event hazard. Therefore \(S(t)=\exp(-0.30t)\).

For label A, the observed cumulative incidence at two years is

F^{\mathrm{obs}}_A(2) =\int_0^2 0.11\exp(-0.30s)\,ds =\frac{0.11}{0.30}\left(1-\exp(-0.60)\right) \approx 0.1654.

The event density for label A at two years is \(0.11\exp(-0.60)\approx0.06037\) per year. Hence its subdistribution hazard then is

\lambda^{\mathrm{obs}}_{A,\mathrm{sub}}(2) = \frac{0.11\exp(-0.60)}{1-0.1654} \approx 0.07234\ \text{per year}.

For comparison, with true causes the two-year cumulative incidence of A is \((0.10/0.30)(1-\exp(-0.60))\approx0.1504\). The observed label-A incidence is higher because some true B events are recorded as A. The recorded-label subdistribution hazard likewise describes label A, not true cause A.

Worked Example: Classification Probabilities That Change Over Time

Keep the same hypothetical true hazards, \(0.10\) per year for A and \(0.20\) per year for B, so \(S(s)=\exp(-0.30s)\). Suppose 90% of true A events are recorded as A during the first year, but only 50% are recorded as A during the second year. At all times, 10% of true B events are recorded as A. Calculate the two-year cumulative incidence for recorded label A and its subdistribution hazard at year 2.

The observed cause-specific hazard for label A is \(0.90(0.10)+0.10(0.20)=0.11\) per year during the first year. During the second year it is \(0.50(0.10)+0.10(0.20)=0.07\) per year. The two-year cumulative incidence must use each rate over the time interval when it applies:

\begin{aligned} F^{\mathrm{obs}}_A(2) &=\int_0^1 0.11e^{-0.30s}\,ds+\int_1^2 0.07e^{-0.30s}\,ds\\ &=0.11\frac{1-e^{-0.30}}{0.30} +0.07\frac{e^{-0.30}-e^{-0.60}}{0.30}\\ &\approx (0.11)(0.8639)+(0.07)(0.6400)\\ &\approx 0.1398. \end{aligned}

The left-hand observed density for label A at year 2 is \(0.07e^{-0.60}\approx0.03842\) per year. Therefore its left-hand subdistribution hazard at year 2 is

\lambda^{\mathrm{obs}}_{A,\mathrm{sub}}(2) = \frac{0.07e^{-0.60}}{1-0.1398} \approx 0.04466\ \text{per year}.

Using the year-2 classification probability for the whole interval would instead give \(0.07(1-e^{-0.60})/0.30\approx0.1053\). That is not the cumulative incidence under the stated time-varying classification: it wrongly treats first-year events as if they were classified using the second-year probabilities.

Worked Example: Different Classification by Group

Consider two hypothetical groups with identical true hazards: \(0.10\) per year for cause A and \(0.20\) per year for cause B. In both groups, 90% of true A events are recorded as A. In group 1, 10% of true B events are recorded as A; in group 2, 40% are. All other events are recorded as B. Compare the observed cause-A hazards and one-year cumulative incidences.

For group 1, \(\lambda^{\mathrm{obs}}_A=0.90(0.10)+0.10(0.20)=0.11\) per year. For group 2, \(\lambda^{\mathrm{obs}}_A=0.90(0.10)+0.40(0.20)=0.17\) per year. In both groups the total hazard remains \(0.30\) per year, because every event is assigned one label. Thus the one-year event-free survival is \(e^{-0.30}\) for both.

For constant hazards and constant classification probabilities, \(F^{\mathrm{obs}}_A(1)=\lambda^{\mathrm{obs}}_A(1-e^{-0.30})/0.30\). Substitution gives

F^{\mathrm{obs},1}_A(1) =\frac{0.11}{0.30}(1-e^{-0.30}) \approx 0.09503, \qquad F^{\mathrm{obs},2}_A(1) =\frac{0.17}{0.30}(1-e^{-0.30}) \approx 0.1469.

The groups have the same true cause-specific hazards, but different observed label-A hazards and incidences because their classification differs. The recorded-label comparison alone cannot establish a difference in the true incidence of cause A.

Derivation, Diagnostics, and Alternatives

The hazard mixture follows directly from conditioning on the true event type. Conditional on being event-free at \(t\), the probability of a true cause-\(k\) event followed by recorded label \(j\) in the next \(dt\) is approximately \(\lambda_k(t)dt\,q_{kj}(t)\). Summing over mutually exclusive true causes and dividing by \(dt\) gives

\lim_{dt\to0} \frac{\sum_k \lambda_k(t)dt\,q_{kj}(t)}{dt} = \sum_k q_{kj}(t)\lambda_k(t).

The density of a recorded label-\(j\) event at time \(t\) is its cause-specific hazard times the probability of still being event-free, \(S(t)\lambda^{\mathrm{obs}}_j(t)\). Integrating this density from 0 to \(t\) gives the cumulative-incidence formula, including the time-specific classification probabilities inside the integral. Dividing that density by \(1-F^{\mathrm{obs}}_j(t)\) gives the stated subdistribution hazard.

Useful checks and diagnostics include:

Verify that each row of the classification matrix sums to 1. If events can be unclassified, include an explicit unclassified category rather than silently treating missing labels as a cause.
Compare the recorded labels with a validation sample in which the true cause is assessed. Estimate classification probabilities separately by relevant group and, when supported by data, by time.
Check whether a change in observed cause-specific hazards could reflect altered classification rather than altered true event rates. Stable overall event-free survival does not rule out substantial cause-label errors.
Assess uncertainty in the classification probabilities. A small validation sample or a classification matrix with nearly indistinguishable rows can make corrected true-cause quantities highly uncertain.

There are two distinct targets. An analysis of recorded labels estimates quantities for the labels as observed, which can be appropriate when the classification itself defines the operational outcome. An analysis of true causes requires information about the misclassification process, such as validation data, or explicit sensitivity analyses over plausible classification probabilities. Correcting labels is not automatic: even with a known matrix, recovering true cause-specific quantities may be unstable if different true causes are often assigned the same labels.

The formulas also have limits. They rely on correct event times, a complete and mutually exclusive set of recorded labels, and a classification mechanism described by \(q_{kj}(t)\) conditional on true cause and event time. If classification depends on group, use group-specific probabilities. If it depends on additional factors, those factors must be represented or the resulting mixture may not describe the target population. These formulas do not by themselves address bias from incomplete follow-up.

Common mistake: Replacing the true cause-specific hazard for A with the observed hazard for label A and interpreting it as the true A hazard. The observed rate is a mixture of all true-cause hazards weighted by their probabilities of receiving label A. Similarly, with time-varying classification, using only the final-time probabilities to calculate cumulative incidence ignores how earlier events were classified.

The central distinction is between a correctly measured event time and a correctly identified event cause. The first can preserve the overall event-free survival while the second is still needed for cause-specific interpretation. State whether a reported hazard or cumulative incidence refers to the recorded label or the true cause, and describe the classification assumptions used to connect them.

Check Your Understanding

Answer each question, then compare with the solutions below.

There are two true causes with hazards \(0.04\) and \(0.06\) per month. For the first true cause, 80% of events receive label 1; for the second, 25% receive label 1. What is the observed cause-specific hazard for label 1?
Why can the observed cumulative incidence at time \(t\) not generally be calculated by multiplying the true cumulative incidences by the classification probabilities at time \(t\) when those probabilities vary over time?
Suppose two groups have identical true hazards but different probabilities of assigning true cause B to label A. Can their observed label-A cause-specific hazards differ? Explain.
With continuous event times, what denominator is used for the subdistribution hazard of recorded label \(j\), and what does that denominator represent?

Solutions. (1) The mixture is \(0.80(0.04)+0.25(0.06)=0.047\) per month. (2) Cumulative incidence includes events occurring at every earlier time \(s\); each contribution must use the classification probabilities applying at \(s\), so the probabilities belong inside the time integral. (3) Yes. The observed label-A hazard weights each true cause-specific hazard by its probability of receiving label A; changing the weight for cause B changes the mixture even when the true hazards are unchanged. (4) The denominator is \(1-F^{\mathrm{obs}}_j(t)\), the probability that recorded label \(j\) has not occurred by time \(t\). The subdistribution hazard is a rate, not a probability.

Cause Misclassification And Hazard Interpretation

What You'll Learn