Nonparametric inference from censored prevalent cohort data with an application to survival with dementia

This thesis comprises three parts contributing to development of robust statistical methodologies for drawing simultaneous inference (uniform confidence bands) from survival data collected in prospective prevalent cohort studies. The analysis of prevalent cohort survival data poses challenges as they commonly include considerable loss to follow-up (censoring), and are not representatively drawn from the target population of interest owing to structural selection bias. The challenge of censored data is more complicated in prevalent cohort studies due to informative mechanism of censoring imposed by selection bias. The problem of constructing confidence bands under these complexities has not been addressed in the literature until recently (Shariati et al., 2023b).When the incidence (e.g. onset of disease) arises from a stationary point process, the so-called stationarity assumption, a more efficient approach can be adopted, the unconditional approach. The stationarity assumption has been verified to be supported by data and context of application in many studies. Using this assumption, all the methodologies developed in this thesis are based on the unconditional nonparametric maximum likelihood approach pioneered by Asgharian et al. (2002) and Asgharian and Wolfson (2005).The thesis devises a method for obtaining uniform confidence bands under selection bias and informative censoring for the cumulative hazard function and the survival function based on their unconditional nonparametric maximum likelihood estimators (NPMLEs). A detailed derivation of the asymptotic properties of the NPMLE of the hazard function, including uniform strong consistency, weak convergence and asymptotic efficiency, is presented. The intractable forms of the limiting processes of un- conditional NPMLEs preclude using the asymptotic behaviour in practice. One of the innovative ideas in this thesis is to numerically approximate the functionals of the asymptotic processes of the normalized NPMLEs.Age-specific life expectancy, known as mean residual life, is another key concept in survival analysis. The above methodology is further developed in two ways. The thesis first proposes a methodology for estimating the mean residual life function and its uniform confidence bands. For this objective, uniform strong consistency, weak convergence and asymptotic efficiency of the unconditional NPMLE of the mean residual life are derived. The estimation techniques are then extended to two-sample cases, providing the first method for constructing two-sample confidence bands, which can be applied to ascertain the effect of any categorical variable on life expectancy.The thesis also proposes an empirical likelihood approach for constructing confidence bands. This approach consists in establishing empirical likelihood functionals on the basis of the NPMLEs of the distribution function. This method is general: it derives an asymptotic Gaussian process for empirical likelihood functionals; it removes the need for variance estimation; it is applicable to a large class of functions, and small sample sizes; it determines the shape of bands solely based on data. This methodology is illustrated on examples embodying the survival function, the hazard function, the mean residual life function, and the quantile function which has not previously been studied under this setting.Comprehensive simulation studies are conducted validating all the procedures for small sample sizes. A large body of the thesis is dedicated to applying the proposed methodologies to analyse survival data collected on elderly population with dementia from the Canadian Study of Health and Aging. These analyses provide novel information on life expectancy, mortality rate, the quantile curve, and the pronounced effects of sex and type of dementia on life expectancy, all for elderly population with dementia in Canada. The extensions of the proposed methodologies under non-stationarity of the incidence process are discussed. All the developed methodologies are valid under the multiplicative censoring model which unifies several important estimating problems (Vardi, 1989).

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