Monotonicity Checks For Cumulative Incidence

Why Monotonicity Is a Necessary Check

A cumulative-incidence curve reports the probability of having experienced a particular event by each time. For a fixed cause and a fixed prediction target, risk by a later time includes the possibility of experiencing that cause earlier. It therefore cannot be smaller than risk by an earlier time. This simple property is a useful diagnostic: a decreasing predicted curve is a signal to inspect how it was calculated, stored, or displayed.

Monotonicity is a structural property of cumulative incidence, not a claim that the hazard for the cause must increase over time. A cause-specific hazard can rise, fall, or vary irregularly while cumulative incidence remains nondecreasing. The relevant quantity is the probability added to the cause-specific risk over each time interval.

The definition concerns a fixed prediction target: the same cause, covariate profile or target population, and event definition at each time. When comparing curves for different profiles or populations, their ordering can change over time; that does not imply that either curve decreases. The earlier material on probability coherence concerns related checks across causes and event-free probability, while this tutorial focuses on the direction of change in each cause-specific curve.

The Mathematical Reason a Curve Cannot Decrease

Let \(S(t\mid x)\) denote the probability of remaining free of every event through time \(t\), and let \(\lambda_k(t\mid x)\) be the cause-specific hazard for cause \(k\). In continuous time, the cumulative incidence has the representation

where \(\Lambda_k\) is the cumulative cause-specific hazard and \(S(u-\mid x)\) is event-free survival immediately before \(u\). The survival weight is nonnegative, and a valid cause-specific hazard measure \(d\Lambda_k\) is nonnegative. Thus each contribution to the integral is nonnegative.

This is the monotonicity justification: the increase over an interval is the probability accumulated for cause \(k\) during that interval. It can be zero if there is no event contribution for that cause, but it cannot be negative. The same reasoning applies when hazards depend on time-varying covariates, provided the prediction follows a defined covariate path and uses nonnegative cause-specific hazard contributions.

On a discrete prediction grid, the same property is visible in the update. If \(q_{kj}\) is the conditional probability of cause \(k\) in interval \(j\), given that the individual is event-free at its start, then

Here \(S_{j-1}\geq 0\) and \(q_{kj}\geq 0\), so the increment cannot be negative. This expression assumes the interval probabilities are defined consistently with the model’s event-time convention; the calculation should not mix interval probabilities with hazard increments as though they were interchangeable.

Checking Predictions on a Time Grid

For a stored sequence of predicted values \(F_{k,0},F_{k,1},\ldots,F_{k,J}\), calculate adjacent increments:

Every increment should be nonnegative, apart from negligible floating-point discrepancies. Checking adjacent times is more informative than checking only the first and last values: a curve can have a net increase and still contain an erroneous local decrease. Keep the time points in their intended order, verify that the same cause and target are used throughout, and examine the increments as well as the plotted curve.

A tolerance can be useful when values are stored at finite precision. For example, an increment of \(-10^{-14}\) may be a rounding artifact if the values are otherwise consistent. A decrease of \(-0.01\) is not plausibly explained by ordinary floating-point rounding. The tolerance should be chosen in relation to the scale and precision of the calculation, and any values treated as negligible should be documented rather than silently altered.

Other Prediction Scales and Common Sources of Failure

When predictions are formed from hazards rather than directly from interval probabilities, the implementation must use a consistent update rule. For continuous hazards, the cumulative-incidence contribution over an interval is the integral of event-free survival against the cause-specific hazard measure. For a model with piecewise-constant cause-specific rates, that contribution can be calculated directly from the rates and the survival at the interval start.

Several implementation choices can produce apparent decreases. A negative estimated hazard increment, an incorrect sign in an update, or a survival value outside its intended range can make a computed contribution negative. Misordered time points, inconsistent units, and mixing outputs for different causes or covariate profiles can also create a decreasing sequence even if the underlying model is valid. Interpolation deserves attention: some interpolation methods can overshoot between grid points, so a curve that is monotone at stored prediction times may appear to dip when rendered at additional times.

If risks are standardized by averaging predictions over a fixed target population with nonnegative weights that sum to one, the average is also nondecreasing. For each interval, its change is a weighted sum of nonnegative individual changes. A decrease in the standardized curve therefore suggests a changing target or weights, a coding or alignment issue, or an error in the component predictions.

Confidence limits are different from point predictions. Monotonicity is required of the underlying cumulative-incidence function and its coherent point estimate; independently computed lower or upper confidence limits need not form monotone sequences. A small irregularity in a band is not, by itself, evidence that the estimated point curve decreases. Inspect the point estimates and the method used to construct uncertainty separately.

Diagnostics, Alternatives, and Limitations

A practical audit starts with the exact sequence of values used in the output, not just the graph. Check adjacent differences, then reconstruct one or more interval increments from the model components. For each suspected decrease, confirm the time ordering, cause label, target profile or population, survival weight, and cause-specific contribution. If the calculation is based on hazards, verify that the hazard contributions are nonnegative and that the survival and incidence updates follow the model’s specified convention. If the table is monotone but the graph is not, inspect interpolation and plotting transformations.

There are two useful levels of checking. The direct check—testing whether every adjacent difference is nonnegative—is quick and applies to any reported curve. The component check—recomputing each increase from survival and the cause-specific event contribution—helps locate the source when the direct check fails. Direct checking alone detects a problem but does not identify it; component checking is more diagnostic but requires access to intermediate model quantities.

Monotonicity is necessary, not sufficient. A nondecreasing curve can still be based on an incorrect model, an unsuitable target population, or a coding mistake that happens not to create a decrease. Conversely, a tiny negative increment may be a numerical artifact rather than a substantive modeling failure. The size, location, and reproducibility of a decrease should guide the investigation. Monotonicity checks do not validate model fit or resolve questions about uncertainty or follow-up support.