In this paper, we have focused on the estimation of cumulative incidence function for an event of interest in the presence of competing risk events. We have outlined nonparametric estimation using a Kaplan–Meier approach, which assumes noninformative censoring, as well as an alternative approach that accounts for informative censoring. Calculating the standard error of the estimates and the corresponding confidence intervals are discussed by Collett (1994)
and Marubini and Valsecchi (1995)
. The Kaplan–Meier approach results in one curve that portrays the estimated cumulative probability of the event of interest (i.e. one minus the estimated survival probability) with a jump in the curve corresponding to the occurrence of an event of interest at a specific time. The competing risk approach generates two curves, one representing the event of interest and the other representing the competing risk event. The curve representing the cumulative incidence of the event of interest has jumps at times where an event of interest occurs. Likewise, the curve corresponding to the incidence of the competing risk event has jumps at times where competing risk events occur.
The estimated cumulative incidence of an event of interest derived using the Kaplan–Meier approach is, in general, larger than estimates obtained when accounting for competing risks. This is due to the following reason. In the Kaplan–Meier estimation approach, when an individual experiences a competing risk event, this individual is treated as censored and is eliminated from the risk set. On the other hand, in the competing risk approach, this individual is an event in the calculation of the overall survival probability of any event (Step 1). The estimated overall survival of any event is lowered when this individual experiences a competing risk event. Recall that the incidence of the event of interest in a specific time interval is the probability of surviving any event up to that time interval and experiencing the event of interest in that interval. Since the overall survival is reduced when any event occurs, the resulting incidence of the event of interest is reduced as well.
In the FA data set, a total of 120 out of the 755 patients were diagnosed with HM. The remaining 635 patients not diagnosed with HM were considered censored under the Kaplan–Meier approach. The estimated cumulative incidences using the Kaplan–Meier approach are 6.3, 22.6, 39.0 and 47.8% at 10, 20, 30 and 40 years, respectively, since birth (). A total of 199 out of the 635 ‘censored’ patients died prior to the onset of HM. The estimated cumulative incidences of HM using the competing risk approach are 5.9, 17.8, 27.8 and 31.8%, respectively, at 10, 20, 30 and 40 years since the diagnosis of FA. Thus, the estimates are lower when accounting for the competing risk event. summarises the cumulative incidence estimates for the breast cancer data set. A total of 43 out of the 305 patients died due to breast cancer. The remaining 262 individuals were considered censored when estimating the cumulative mortality without accounting for competing risk using the Kaplan–Meier approach. Of these 262 individuals, 25 women died due to causes unrelated to breast cancer. The estimated cumulative breast cancer-specific mortality without accounting for competing risk are 0.3, 4.4, 14.2 and 18.6% at 1, 5, 10 and 15 years, respectively. The corresponding estimates when accounting for competing risks are 0.3, 4.3, 13.6 and 17.6%, respectively. As before, it is evident that the estimates are lower when accounting for the competing risk event.
In certain situations, the cumulative incidence of an event of interest estimated using the Kaplan–Meier approach and the competing risk approach can be similar. The difference between the estimated breast cancer specific-mortality derived from the Kaplan–Meier and the competing risk approach is small for the breast cancer data set. However, the incidence of HM estimated using the Kaplan–Meier approach is substantially larger than that derived from the competing risk approach. Fanconi anaemia patients are at increased risk for HM as well death, relative to the general population (Kutler et al, 2003
), due to their underlying disease, that is, FA. It is, therefore, important in this setting to appropriately account for the competing causes of risk when estimating the cumulative incidence of HM. On the other hand, death due to other causes may not be related to having breast cancer unlike breast cancer-specific mortality. In this case, ignoring the informative censoring mechanism does not substantially influence the estimates of breast cancer-specific mortality.
When there are no competing risk events, that is, when there is only one type of failure, the estimate of the cumulative incidence of the event derived using the Kaplan–Meier approach and the competing risk approach will be identical. Similarly, in the setting when it is of interest to estimate the cumulative incidence of the first event in the presence of multiple types of failure, there are no competing risk events. This is because any event that occurs subsequent to the first event is not relevant to the analysis. Only those patients not experiencing any event will be censored. For example, suppose an FA patient dies after experiencing HM, and it is of interest to estimate the cumulative incidence of the first event (HM or death). In this setting, only the time to HM would be of interest for this patient in the cumulative incidence calculation. The estimated cumulative incidence of the first event obtained using the Kaplan–Meier approach will be the same as that obtained using the competing risk approach.
One minus the Kaplan–Meier survival probability can be interpreted as the cumulative probability of failure. The cumulative incidence of an event of interest estimated by accounting for competing risk events is the probability of experiencing the event of interest by a given time and not experiencing a competing risk event by this time. One minus the cumulative incidence is the probability of surviving the event of interest up to a specific time. This can occur if the patient did not experience both the event of interest and the competing risk event, or if the patient had the competing risk event before the onset of the event of interest. As a result, one minus the cumulative incidence adjusted for competing risk events cannot be interpreted as the probability of surviving any event.
The topics of competing risk events and the estimation of cumulative incidence of an event of interest have been discussed by several authors. Gail (1975)
reviews the theoretical concepts underlying the estimation of cumulative incidence of an event using a variety of models. Prentice et al (1978)
discuss likelihood inference to examine the effect of prognostic factors on the event of interest in the presence of competing risk events. Pepe and Mori (1993)
describe various probability models for summarising competing risk data. Tai et al (2001)
developed a method to estimate the cumulative incidence of a specific event based on an extension of the Cox proportional hazards regression model. They compare their estimates to the Kaplan–Meier estimate of cumulative incidence as well as the cumulative incidence accounting for competing risk as described above. Their findings show that the estimates obtained using the Kaplan–Meier approach are numerically larger than those accounting for competing risk events. Clark et al (2003a
) and Bradburn et al (2003a
) provide a detailed tutorial review of various survival analysis concepts, including a brief summary of competing risk analysis.
Several softwares are available to estimate the overall survival probabilities and cumulative incidence of an event of interest. The Kaplan–Meier estimate of survival probability can be easily obtained using standard statistical analysis softwares including R (http://www.r-project.org
), S-plus (Insightful Corp., 2003), SAS (SAS Institute, 2002) and SPSS (SPSS Inc., 2003). Software has been made available in R by Gray (1988
) for obtaining the estimate of cumulative incidence in the presence of competing risk events.
In summary, it is important to account for competing risk events when estimating disease incidence. Failure to account for such competing events results in an overestimate of the cumulative incidences. This could be substantial when the competing risk event is related to or is a result of the underlying disease.