Statistical issues in survival analysis (Pseudo observations length-bias Cox model)
November 6, 2025 For these authors, the goal of this paper was to apply pseudo-observations to estimate the regression coefficients in the Cox proportional hazards model under length-biased right-censored (LBRC) data. On some occasions as they found, observations may not be representative of the original data, and the selection of an observation is proportional to its length or duration. This is known as length-biased sampling. Length-biased data can be thought of as a special case of lefttruncated data, where the truncation variable follows a uniform distribution, known as the stationarity assumption. Sampling bias may be inherent in the data as well. In addition to sampling bias, some individuals may be lost to follow-up or drop out of the study before the terminating event occurs. Therefore, in this case, they were dealing with length-biased rightcensored (LBRC) data. In Equation 6 they define a G(u) but they did not describe what it represents. In 2003, Andersen et al. had proposed a general approach to censored data regression based on pseudo-observations. The pseudo-observations were derived from jackknife theory and they also could be through standard regression methods such as the generalized estimating equations (GEE) approach (Liang and Zeger, 1986) to estimate regression coefficients. Also, this approach has been applied to various survival analysis models, including regression models for the cumulative incidence functions in competing risks. One can define the pseudo observations to be computed from the estimate of the survival function and that would be computed at the available time points. In their study, they compared the pseudo-observation methods with two prominent standard approaches proposed by Qin and Shen (2010) and Huang and Qin (2012) for estimating the coefficients of a Cox proportional hazards model under LBRC data. Vardi (1982a) derived the nonparametric maximum likelihood estimator (NPMLE) of the survival function under LBRC data. This NPMLE does not have a closed form and would have to be obtained through implementing the expectation-maximization (EM) algorithm (Vardi 1989). There focus was on the Cox model. They then proposed a generalized linear model for the pseudo responses as survival at a fixed time point and then they estimated this regression model by GEE with a loglink function; they defined the estimating equation for this. Since the conditional survival