Nonparametric incidence estimation from prevalent cohort survival data

Abstract

SUMMARYIncidence is an important epidemiological concept most suitably studied using an incident cohort study. However, data are often collected from the more feasible prevalent cohort study, whereby diseased individuals are recruited through a cross-sectional survey and followed in time. In the absence of temporal trends in survival, we derive an efficient nonparametric estimator of the cumulative incidence based on such data and study its asymptotic properties. Arbitrary calendar time variations in disease incidence are allowed. Age-specific incidence and adjustments for both stratified sampling and temporal variations in survival are also discussed. Simulation results are presented and data from the Canadian Study of Health and Aging are analysed to infer the incidence of dementia in the Canadian elderly population.

Estimating the Lifetime Risk of Dementia in the Canadian Elderly Population Using Cross-Sectional Cohort Survival Data

“…We can show, as in Carone et al (2012), that d Λ 0 ( u ) ∝ dG ( t 0 − u ), where G is the marginal distribution function of age at recruitment W *, and W * has support contained in ( t 0 − τ 1 , t 0 − τ 0 ). It follows then that $$\begin{array}{ccc}left\frac{{\mathrm{normal\Lambda }}_1(t_0)}{{\mathrm{normal\Lambda }}_2(t_0)}& =& \frac{{italic\int }_{{\tau }_0}^{{\tau }_1}\mathrm{pr}false(Z^{\ast }\ge t_0-u,D^{\ast }false(t_0-ufalse)=1\mid W^{\ast }=t_0-ufalse)\mathrm{italicdG}false(t_0-ufalse)}{{italic\int }_{{\tau }_0}^{{\tau }_1}\mathrm{pr}false(Z^{\ast }\ge t_0-u\mid W^{\ast }=t_0-ufalse)\mathrm{italicdG}false(t_0-ufalse)}\\ & =& \frac{{italic\int }_{{\tau }_0}^{{\tau }_1}\mathrm{pr}false(X^{\ast }\le t_0-u\le Z^{\ast },D^{\ast }=1\mid W^{\ast }=t_0-ufalse)\mathrm{italicdG}false(t_0-ufalse)}{{italic\int }_{{\tau }_0}^{{\tau }_1}\mathrm{pr}false(Z^{\ast }\ge t_0-u\mid W^{\ast }=t_0-ufalse)\mathrm{italicdG}false(t_0-ufalse)}\\ & =& \frac{{\mathit{\int }}_{t_0-{\tau }_1}^{t_0-{\tau }_0}\mathrm{pr}(X^{\ast }\le w\le Z^{\ast }\mid W^{\ast }=w,D^{\ast }=1)\mathrm{pr}(D^{\ast }=1\mid W^{\ast }=w)}{}\end{array}$$…”

“…Uniform consistency of the proposed estimators at n 1/2 -rate can be proved using the univariate integration-by-parts formula, its bivariate version derived above, the Continuous Mapping Theorem and the uniform consistency of G n (Wang, 1991; Woodroofe, 1985; Carone et al, 2012), $H_n^{\prime }$ and Q n (Wang, 1987). Weak convergence to a Gaussian limit can then be established using the functional delta method and the techniques in Chapter 12 of Kosorok (2008).…”

“…Setting $$\mathrm{normal\Gamma }false(u,\upsilon false)={italic\int }_{u\le w\le \upsilon }Qfalse(\upsilon -wfalse)Gfalse(\mathrm{italicdw}false),$$conditions 3 and 4 ensure that Γ is uniformly bounded away from zero on the support of H and thus negates the need for unnecessarily technical arguments to handle the singularities of Γ( u, υ ) on u = υ . We conjecture that condition 4 may be weakened to requiring that some integral be finite, in the same spirit as the integral condition in Theorem 1 of Carone et al (2012). A bivariate integration by parts formula, shared by James A. Fill in a personal communication, is useful for studying these estimators.…”

“…To assist both, epidemiologists have investigated epidemiological parameters of dementia in populations. Measures including the prevalence (McDowell et al, 1994) and incidence of dementia (Addona et al, 2009; Carone et al, 2012) have been studied in the Canadian population and reported in the literature. However, despite its importance, the lifetime risk of dementia in Canada, defined as the proportion of Canadians developing dementia during their lifetime, does not seem to have been previously described.…”

Analyzing multiple cross‐sectional samples with application to hospitalization time after surgeries

“…These assumptions are not essential and can be easily relaxed. We further assume that {N sk } s=1,…,S ; k=1,…,K are independent and that N sk ∼ Pois(c sk ), where sk is known, that is, the entrance rates are known up to a multiplicative constant c. The discrete Poisson model may be obtained by grouping a continuous Poisson entrance process, a model that has been considered by many authors in similar contexts (see, for example, [9][10][11][12][13][14][15][16]). Besides its biological plausibility, the Poisson assumption ensures independence of observed hospitalization times [8].…”