Effects of informative censoring on the hazard function: Application to the proportional hazards regression model

The standard analyses of survival data involve the assumption that survival and censoring are independent. When censoring and survival are related, the phenomenon is known as informative censoring. This paper examines the effects of an informative censoring assumption on the hazard function and the estimated hazard ratio provided by the Cox model.^ The limiting factor in all analyses of informative censoring is the problem of non-identifiability. Non-identifiability implies that it is impossible to distinguish a situation in which censoring and death are independent from one in which there is dependence. However, it is possible that informative censoring occurs. Examination of the literature indicates how others have approached the problem and covers the relevant theoretical background.^ Three models are examined in detail. The first model uses conditionally independent marginal hazards to obtain the unconditional survival function and hazards. The second model is based on the Gumbel Type A method for combining independent marginal distributions into bivariate distributions using a dependency parameter. Finally, a formulation based on a compartmental model is presented and its results described. For the latter two approaches, the resulting hazard is used in the Cox model in a simulation study.^ The unconditional survival distribution formed from the first model involves dependency, but the crude hazard resulting from this unconditional distribution is identical to the marginal hazard, and inferences based on the hazard are valid. The hazard ratios formed from two distributions following the Gumbel Type A model are biased by a factor dependent on the amount of censoring in the two populations and the strength of the dependency of death and censoring in the two populations. The Cox model estimates this biased hazard ratio. In general, the hazard resulting from the compartmental model is not constant, even if the individual marginal hazards are constant, unless censoring is non-informative. The hazard ratio tends to a specific limit.^ Methods of evaluating situations in which informative censoring is present are described, and the relative utility of the three models examined is discussed. ^

Measurement of isotopic uranium in water for compliance monitoring by liquid scintillation counting with alpha/beta discrimination

A simple and inexpensive method is described for analysis of uranium (U) activity and mass in water by liquid scintillation counting using $\alpha$/$\beta$ discrimination. This method appears to offer a solution to the need for an inexpensive protocol for monitoring U activity and mass simultaneously and an alternative to the potential inaccuracy involved when depending on the mass-to-activity conversion factor or activity screen.^ U is extracted virtually quantitatively into 20 ml extractive scintillator from a 1-$\ell$ aliquot of water acidified to less than pH 2. After phase separation, the sample is counted for a 20-minute screening count with a minimum detection level of 0.27 pCi $\ell\sp{-1}$. $\alpha$-particle emissions from the extracted U are counted with close to 100% efficiency with a Beckman LS6000 LL liquid scintillation counter equipped with pulse-shape discrimination electronics. Samples with activities higher than 10 pCi $\ell\sp-1$ are recounted for 500-1000 minutes for isotopic analysis. Isotopic analysis uses events that are automatically stored in spectral files and transferred to a computer during assay. The data can be transferred to a commercially available spreadsheet and retrieved for examination or data manipulation. Values for three readily observable spectral features can be rapidly identified by data examination and substituted into a simple formula to obtain $\sp{234}$U/$\sp{238}$U ratio for most samples. U mass is calculated by substituting the isotopic ratio value into a simple equation.^ The utility of this method for the proposed compliance monitoring of U in public drinking water supplies was field tested with a survey of drinking water from Texas supplies that had previously been known to contain elevated levels of gross $\alpha$ activity. U concentrations in 32 samples from 27 drinking water supplies ranged from 0.26 to 65.5 pCi $\ell\sp{-1}$, with seven samples exceeding the proposed Maximum Contaminant Level of 20 $\mu$g $\ell\sp{-1}$. Four exceeded the proposed activity screening level of 30 pCi $\ell\sp{-1}$. Isotopic ratios ranged from 0.87 to 41.8, while one sample contained $\sp{234}$U activity of 34.6 pCi $\ell\sp{-1}$ in the complete absence of its parent, $\sp{238}$U. U mass in the samples with elevated activity ranged from 0.0 to 103 $\mu$g $\ell\sp{-1}$. A limited test of screening surface and groundwaters for contamination by U from waste sites and natural processes was also successful. ^