Cumulative Incidence

Cumulative Incidence as a Measure of Risk in Health Economic Models

Cumulative incidence is the epidemiological name for risk, the quantity behind a statement such as a 20% five-year risk of fracture, and the CDC's teaching text lists attack rate, risk and probability of getting disease as its synonyms. Health economic models depend on it because most event probabilities in a decision tree or a Markov model start life as a cumulative incidence reported by a trial or cohort study over a fixed period. A reported risk becomes a usable model input only once its period, its handling of incomplete follow-up and its treatment of competing events are clear. It must then be converted to the model's cycle length on the right scale, and an illustrative example shows how dividing a five-year risk by five understates it, and moving a relative risk to the wrong time scale shrinks the events a treatment avoids.

Risk, rate and the denominator that separates them

An incidence proportion and an incidence rate share a numerator, the new cases arising during follow-up, and differ in the denominator. The CDC's Principles of Epidemiology in Public Health Practice describes the incidence proportion as the share of a population free of disease at the outset who develop disease, are injured or die within a specified period, and gives cumulative incidence as a synonym. The denominator is the number of people at the start of the period, restricted to those who could have the event, so every case in the numerator also sits in the denominator.

$$\text{CI}(0,t] = \frac{D(0,t]}{N_0}$$

where $\text{CI}(0,t]$ is the cumulative incidence from the start of follow-up to time $t$, $N_0$ is the number of people at risk at the start and $D(0,t]$ is the number of them with a new event by time $t$. In one of the CDC's examples, 100 of 189 diabetic men died during 13 years of follow-up, a risk of 52.9% over that period.

An incidence rate divides the same cases by the total person-time at risk, so it has units of inverse time and describes how fast events occur. The CDC lesson notes that person-time assumes the probability of disease is constant during the study period, so that 10 people followed for one year count the same as one person followed for 10 years, an assumption it calls often invalid because the risk of many chronic diseases rises with age. Cumulative incidence carries no unit but always carries a period.

Estimating cumulative incidence when follow-up is incomplete

The simple proportion works only when everyone is followed for the whole period. In the CDC lesson's heart disease example, 16 cases among 2,100 women, of whom 1,384 were lost to follow-up before four years, give an incidence proportion of 7.6 per 1,000 over four years. The lesson states that this figure ignores the people lost to follow-up and assumes they remained disease free for all four years.

Trials and cohort studies with staggered entry and dropout therefore estimate cumulative incidence with survival methods. Austin, Lee and Fine explain that, in the absence of competing risks, one minus the Kaplan-Meier estimator of the survival function estimates the cumulative incidence of events over time, and that estimating incidence as a function of follow-up time gives important information on the absolute risk of an event. Andersen and colleagues point out that the Kaplan-Meier and Nelson-Aalen estimators rest on the same counts, the number of failures and the number at risk at each failure time, and they give the Nelson-Aalen estimator under independent censoring. People who are censored are assumed to have the same future risk as those still followed, and independent censoring has its own page.

Competing events and the cumulative incidence function

When another event, most often death from other causes, can happen first and rule out the event of interest, the risk of the event of interest is described by the cause-specific cumulative incidence function. Andersen and colleagues show that the one-to-one link between rate and risk is then lost, because the cumulative incidence of one cause depends on the hazards of every cause. For two causes the function takes the following form.

$$F_1(t) = \int_0^t S(u)\, h_1(u)\, du$$

where $F_1(t)$ is the cumulative incidence of the event of interest by time $t$, $S(u)$ is the probability of being free of both events at time $u$ and $h_1(u)$ is the cause-specific hazard of the event of interest at time $u$.

Austin and colleagues recommend the cumulative incidence function in place of one minus Kaplan-Meier, which biases incidence upward whether or not the competing events are independent. In their heart failure cohort the five-year incidence of cardiovascular death was 43.0% from one minus Kaplan-Meier and 36.8% from the cumulative incidence function. They add that the cumulative incidence of one event type need not approach 1 as time grows, and that the cumulative incidences of all the competing events sum to the incidence of the composite outcome. Regression models and the conversion of several competing exits into cycle probabilities are covered on the competing risks page.

Moving between cumulative incidence, rates and cumulative hazards

Without competing events, cumulative incidence and the cumulative hazard determine each other. Andersen and colleagues describe this one-to-one correspondence in studies of all-cause mortality, and it takes the following form.

$$F(t) = 1 - S(t) = 1 - \exp\{-H(t)\}$$

where $F(t)$ is the cumulative incidence by time $t$, $S(t)$ is the probability of remaining event free beyond $t$ and $H(t)$ is the cumulative hazard, the hazard rate summed from the start of follow-up to $t$.

With a constant rate the cumulative hazard is the rate multiplied by time. Gidwani and Russell, in their tutorial on estimating transition probabilities from published evidence, give the conversions in both directions.

$$r = \frac{-\ln(1 - p)}{t}, \qquad p = 1 - \exp(-rt)$$

where $p$ is the probability of the event over a period of length $t$ and $r$ is the constant rate per unit of time. A risk is therefore always smaller than the rate multiplied by time; with an illustrative rate of 0.1 a year, the five-year cumulative incidence is $F(5) = 1 - e^{-0.5} = 0.393$, not 0.5. The ISPOR-SMDM report on state-transition modelling recommends that probabilities be converted from one time unit to another through rates, and that probabilities never be called rates.

Cumulative incidence as an input to decision models

A decision tree with a fixed horizon can use a cumulative incidence directly as a branch probability when the reported period matches the horizon. A Markov model needs a transition probability for each cycle, which is conditional on being event free at the start of that cycle, whereas a cumulative incidence runs from the original baseline. Gidwani and Russell give the typical case of a trial reporting outcomes at two years when the model has an annual cycle, and for a state with only two possible transitions they convert the period probability to the cycle length in one step.

$$p_{\text{cycle}} = 1 - (1 - p_t)^{1/n}$$

where $p_t$ is the cumulative incidence over the reported period, $n$ is the number of model cycles in that period and $p_{\text{cycle}}$ is the probability per cycle. In their example a four-year probability of death of 0.40 gives an annual probability of 0.119888, which leaves exactly 60 of 100 people alive after four annual cycles. The formula assumes a constant rate across the period. Gidwani and Russell also show that it gives incorrect probabilities once three or more transitions can occur from a state, which is the competing risks case.

When a study reports a full curve and there are no competing events, the cycle probability for the interval from $t_{k-1}$ to $t_k$ is $[F(t_k) - F(t_{k-1})] / [1 - F(t_{k-1})]$, the share of the cohort with the event in the interval divided by the share still event free at its start. Using the unconditional difference $F(t_k) - F(t_{k-1})$ on its own understates the transition probability, more so as the cumulative incidence grows.

Treatment effects are often reported on the same scale. The cumulative incidences in each trial arm give the absolute risk reduction and the number needed to treat over the trial period, and a cost-offset or budget calculation multiplies the events avoided by the cost per event. Gidwani and Russell define the relative risk as the probability of the event in the exposed group divided by the probability in the unexposed group, so a relative risk built from five-year cumulative incidences describes five-year risks.

Worked example of a five-year risk in an annual-cycle model

All numbers in this example are illustrative. A cohort study reports a five-year cumulative incidence of 0.20 for a first complication in a younger population in which deaths from other causes over five years are negligible, so the two-transition formula applies. A trial reports a five-year cumulative incidence of 0.15 with a new treatment, a relative risk of 0.75, and each complication costs GBP 6,000. The model uses annual cycles.

1. Convert the five-year risk to an annual probability. With $n = 5$ the conversion is as follows.

$$p_{\text{cycle}} = 1 - (1 - 0.20)^{1/5} = 1 - 0.956352 = 0.043648$$

where $p_{\text{cycle}}$ is the annual probability without treatment. Five cycles at this probability leave $(0.956352)^5 = 0.80$ event free, reproducing the source. For the treated arm, $p_{\text{cycle}} = 1 - 0.85^{1/5} = 0.031981$.

2. Check the shortcut of dividing by five. Dividing gives 0.20 / 5 = 0.04 a year, and five cycles give a five-year risk of $F(5) = 1 - 0.96^5 = 0.184627$. The model then has 15.4 fewer complications per 1,000 people than the source reports, an understatement of 7.7% (0.015373 / 0.20).

3. Compute the decision quantities over five years. The absolute risk reduction is $\text{ARR} = 0.20 - 0.15 = 0.05$, so the number needed to treat is 1 / 0.05 = 20 and 50 complications are avoided per 1,000 people treated. At GBP 6,000 each, the cost offset is GBP 300,000 per 1,000 people, or GBP 300 per person.

4. Check the effect of applying the relative risk to the annual probability. Multiplying the annual control probability by 0.75 gives 0.032736, and five cycles give $F(5) = 1 - (1 - 0.032736)^5 = 0.153307$, not 0.15. The absolute risk reduction falls to 0.046693, or 46.7 complications avoided per 1,000, which is 6.6% fewer than the trial shows, and the cost offset falls to GBP 280 per person. Converting each arm's own five-year risk, as in step 1, keeps the trial result.

5. Extend the horizon to ten years. Under the constant-rate assumption the ten-year risks are $F(10) = 1 - 0.80^2 = 0.36$ without treatment and $F(10) = 1 - 0.85^2 = 0.2775$ with it, an absolute risk reduction of 0.0825. These figures go beyond the observed five years and rest wholly on the assumption that the annual probabilities stay the same.

6. Interpretation. The shortcut in step 2 understates how often the complication occurs, and the one in step 4 overstates the treated risk and so understates the value of preventing it. The ten-year gain in step 5 is a projection, so it belongs in a scenario analysis that varies the rate after year five.

Cumulative incidence beside incidence rate, prevalence and absolute risk

Several neighbouring measures describe the same events in different ways. Choosing among them depends on whether the question concerns a probability over a period, a speed of occurrence or a count of existing cases.

Incidence is the umbrella term for new cases, and cumulative incidence is one of its two main measures, the other being the incidence rate with its person-time denominator. Prevalence differs in its numerator, which the CDC lesson notes includes all people ill during the interval regardless of when the illness began. Absolute risk applies the same probability to a person or group over a period, and in outbreak work the CDC treats attack rate as a synonym for risk. The cumulative hazard is not a probability and can exceed 1, whereas a cumulative incidence always lies between 0 and 1.

Common errors when reading a reported risk

Most errors with cumulative incidence come from losing track of its period or its denominator. Each of them carries through to costs and QALYs in a model.

A risk quoted without its period, or compared with a risk over a different period, cannot be placed in a model. Dividing a period risk by the number of cycles understates it, and the gap widens as the risk grows. A simple proportion from a cohort with heavy loss to follow-up understates risk, while one minus Kaplan-Meier overstates it when competing deaths are treated as censoring. A relative risk from period risks does not transfer unchanged to per-cycle probabilities. Extending a cumulative incidence beyond the follow-up that produced it is an assumption about future rates, not an observation, and it needs its own justification and testing.

Sources

• Andersen PK, Geskus RB, de Witte T, Putter H. Competing risks in epidemiology: possibilities and pitfalls. International Journal of Epidemiology. 2012;41(3):861-870. https://doi.org/10.1093/ije/dyr213
• Austin PC, Lee DS, Fine JP. Introduction to the analysis of survival data in the presence of competing risks. Circulation. 2016;133(6):601-609. https://doi.org/10.1161/CIRCULATIONAHA.115.017719
• Centers for Disease Control and Prevention. Principles of Epidemiology in Public Health Practice, 3rd ed. Lesson 3: Measures of Risk, Section 2 (Morbidity frequency measures). Atlanta: CDC; 2006, page last reviewed 2012. https://archive.cdc.gov/www_cdc_gov/csels/dsepd/ss1978/lesson3/section2.html
• Gidwani R, Russell LB. Estimating transition probabilities from published evidence: a tutorial for decision modelers. PharmacoEconomics. 2020;38(11):1153-1164. https://doi.org/10.1007/s40273-020-00937-z
• Siebert U, Alagoz O, Bayoumi AM, Jahn B, Owens DK, Cohen DJ, Kuntz KM. State-transition modeling: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-3. Value in Health. 2012;15(6):812-820. https://doi.org/10.1016/j.jval.2012.06.014