Composite Partial Likelihood Estimation Under Length-Biased Sampling, With Application to a Prevalent Cohort Study of Dementia

Abstract

The Canadian Study of Health and Aging (CSHA) employed a prevalent cohort design to study survival after onset of dementia, where patients with dementia were sampled and the onset time of dementia was determined retrospectively. The prevalent cohort sampling scheme favors individuals who survive longer. Thus, the observed survival times are subject to length bias. In recent years, there has been a rising interest in developing estimation procedures for prevalent cohort survival data that not only account for length bias but also actually exploit the incidence distribution of the disease to improve efficiency. This article considers semiparametric estimation of the Cox model for the time from dementia onset to death under a stationarity assumption with respect to the disease incidence. Under the stationarity condition, the semiparametric maximum likelihood estimation is expected to be fully efficient yet difficult to perform for statistical practitioners, as the likelihood depends on the baseline hazard function in a complicated way. Moreover, the asymptotic properties of the semiparametric maximum likelihood estimator are not well-studied. Motivated by the composite likelihood method (Besag 1974), we develop a composite partial likelihood method that retains the simplicity of the popular partial likelihood estimator and can be easily performed using standard statistical software. When applied to the CSHA data, the proposed method estimates a significant difference in survival between the vascular dementia group and the possible Alzheimer's disease group, while the partial likelihood method for left-truncated and right-censored data yields a greater standard error and a 95% confidence interval covering 0, thus highlighting the practical value of employing a more efficient methodology. To check the assumption of stable disease for the CSHA data, we also present new graphical and numerical tests in the article. The R code used to obtain the maximum composite partial likelihood estimator for the CSHA data is available in the online Supplementary Material, posted on the journal web site.

KEY WORDS:

• Backward and forward recurrence time
• Cross-sectional sampling
• Random truncation
• Renewal processes

Acknowledgments

The authors thank Professors Ian McDowell, Masoud Asgharian, and Christina Wolfson for kindly sharing the Canadian Study of Health and Aging data. The core study was funded by the National Health Research and Development Program (NHRDP) of Health Canada Project 6606-3954-MC(S). Additional funding was provided by Pfizer Canada Incorporated through the Medical Research Council/Pharmaceutical Manufacturers Association of Canada Health Activity Program, NHRDP Project 6603-1417-302(R), Bayer Incorporated, and the British Columbia Health Research Foundation Projects 38 (93-2) and 34 (96-1). The authors also thank the Associate Editor, the referee, Dr Dean Follmann, and Dr Michael Proschan for their comments that improved the presentation of this article.

Notes

NOTE: and are the estimated regression coefficients, where the true parameter values are (1, 1). PL, the maximum partial likelihood estimator; QS, the estimator studied by Qin and Shen (2010); PPL, the maximum pseudo-partial likelihood estimator studied by Tsai (2009); CPL, the proposed maximum composite partial estimator; Bias and ES are the empirical bias (× 1000) and empirical standard deviation (× 1000) of 2000 regression parameter estimates; ASE is the averaged robust standard error estimate; RE is the empirical variance of the maximum partial likelihood estimator divided by that of the maximum composite partial likelihood estimator.

NOTE: PL, the maximum partial likelihood estimator; QS, the estimator studied by Qin and Shen (2010); PPL, the maximum pseudo-partial likelihood estimator studied by Tsai (2009); CPL, the proposed maximum composite partial estimator; Coef and SE are the estimated coefficient and the empirical standard deviation of 2000 regression parameter estimates; 95% CI is the 95% bootstrap confidence interval given by the 2.5th and 97.5th percentiles of the 2000 estimates.