1: Event status (D=dead, A=alive) and survival age
2: Event status (D=dead, A=alive) and age in 1975 (at start of observation period)
Analytic approaches that rely on person-years at risk can underestimate hazards and incidence rates. For example, the hazard at age 70 years is the conditional probability of event onset at age 70, or the number of events at age 70 divided by the number of individuals at risk at age 70. Individuals who contribute time that occurred before the observation period (in other words, individuals who were older than 70 when the study began) appear in the denominator, but not in the numerator.
The bias in estimating hazards and incidences can be removed by excluding follow-up time that occurs before the observation period, by left-truncation. The risk set, at any age, must include only those individuals who were at risk at that age during the observation period.
Consider, for example, a hypothetical population of 12 individuals whose lifetimes are displayed in Figure 1. Suppose the event of interest is death. The population hazard at age 70 is defined as the ratio of the number of deaths at age 70 to the number of subjects in the risk set (defined as the number of subjects at risk entering age 70) and is equal to 4/12 = 0.33. If an investigation is initiated in 1975 (see Figure 1), then 8 subjects are included in the investigation (subjects 5-12 who are alive at the beginning of the observation period in 1975). In a standard survival analysis of this 1975 cohort using survival age as the dependent variable, each of the 8 subjects contributes the number of years from his/her birth to survival age. The risk set entering age 70 includes 8 subjects, of whom 2 died at age 70 (Subjects 7 and 10). The population hazard is underestimated as 2/8 = 0.25. Subjects 5 and 6 are older than 70 years of age at the beginning of the observation period (1975); they were both at risk at age 70, however, they were not at risk at age 70 during the observation period. These subjects should be excluded from the risk set at age 70 (in 1975). If we restrict the risk set at age 70 to only those who are at risk at age 70 during the observation period, the risk set includes 6 subjects, and we correctly estimate the population hazard as 2/6 = 0.33. This approach reflects a selected risk set strategy and we use this strategy in
The estimation of cumulative incidence of AD is complicated by a fairly common situation: the development of AD is subject to the competing risk of death. Subjects who die during the observation period are treated as censored observations in traditional survival analytic techniques such as the Kaplan-Meier method , This method is inappropriate as it assumes that failure from the event of interest is still possible beyond the time at which the censoring occurred. For example, a person who dies of cardiovascular disease cannot develop AD and should not contribute to the estimate of development of AD. Gooley  shows that the potential contribution of censored observations to the probability of failure from the event of interest is distributed among those subjects remaining at risk. However, the potential contribution of a subject who has died should be zero. Treating such subjects as censored inflates the estimate of cumulative incidence. Various analytic solutions to the problem of competing risks have been proposed and implemented [12-17], but there is still no software available that addresses this issue.
We provide estimates of both the unadjusted cumulative incidence (UCI) and the cumulative incidence adjusted for the competing risk of death (ACI). (Note that these are generally referred to in the literature as 1-KM - the complement of the Kaplan-Meier estimate of survival - versus CI [12,13]). The ACI is useful as an estimate of the probability of actually developing AD, while the UCI estimates the probability of developing AD assuming no competing risk (that is, all subjects living for the entire lifespan). The former estimator is particularly useful from a public health standpoint as it allows the estimation of the numbers of cases of AD one can expect in a given population. Further, by adjusting the observed cumulative incidence using the mortality experience of a 'standard population' one can estimate a standardized lifetime risk. In some diseases which appear to be associated with aging per se, the exponential rise in annual incidence with increasing age is balanced by the exponential decrease in life expectancy seen with age, resulting in a relatively invariant estimate of the lifetime risk in elderly individuals. Thus for instance, in the Framingham Study, the lifetime risks of Alzheimer's disease  and congestive heart failure  were found to remain relatively constant with increasing age beyond 65 years. The ability to generate a single sex-specific estimate of lifetime risk is useful in educating the public regarding the true risk of the disease. The unadjusted cumulative incidence may be useful in the pathophysiological investigation of potential risk factors for AD. As an example, cigarette smoking may appear to provide protection from AD when the ACI is used to estimate cumulative incidence. This could be a simple consequence of the fact that cigarette smoking is associated with increased mortality, thus decreasing the observed incidence of AD. Smoking may, however, increase the physiological risk of AD; this would be seen only if the UCI is used to estimate cumulative incidence (Seshadri et al., submitted to the American Academy of Neurology, 2000).
Many standard statistical computing packages do not handle these issues in a straightforward manner, if at all. Even the calculation of one-year incidence rates and age-adjusted rates using direct standardization requires a certain amount of programming. Many standard statistical software packages do not exclude follow-up that occurred before the observation period and thus underestimate hazards and cumulative incidence. For example, SAS Proc Lifetest provides estimates of cumulative incidence using the Kaplan-Meier method, but has no mechanism for left-truncation. SAS Proc Phreg performs proportional hazards modelling and allows left-truncation but does not consider the adjustment for competing risk that is often necessary. We developed an SAS macro that produces: one-year incidence rates by age group; age-adjusted rates to compare rates among the levels of a grouping variable; estimates of traditional Kaplan-Meier cumulative incidence; and estimates of cumulative incidence adjusted for the competing risk of another event. Confidence intervals for each of the cumulative incidence estimates can also be provided.
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