The ICLIFETEST Procedure

Example 49.1 Analyzing Data With Observations below a Limit of Detection

Data that have certain values below a limit of detection (LOD) are frequently encountered by toxicologists and environmental scientists. Such data are usually analyzed by imputing the unobserved values by LOD/2 or LOD/$\sqrt {2}$. This type of practice often raises the question of whether the population distributions can be estimated without bias. Gillespie et al. (2010) propose using a reverse Kaplan-Meier estimator, or equivalently, Turnbull’s method (1976) by treating the unobserved data as left-censored. When the assumption of independent censoring holds, these estimators can unbiasedly estimate the population distribution functions.

The following hypothetical data have two values, 3 and 10, that are below the limit of detection:

data temp;  
   input C1 C2;
  .     3
  4     4
  6     6
  8     8
  .     10
  12    12

The following statements invoke PROC ICLIFETEST to estimate the population distribution function by using Turnbull’s method:

proc iclifetest data=temp method=turnbull plots=survival(failure)
   time (c1,c2);

Specifying the PLOTS=SURVIVAL(FAILURE) option requests a failure probability plot. Results are shown in Output 49.1.1. Note that because the first Turnbull interval is $(0,3)$, the failure probability function is undefined within that interval.

Output 49.1.1: Failure Probability Plot for Fictitious Nondetection Data

Output 49.1.2 presents the estimated failure probability, with standard errors that are estimated by the method of multiple imputations.

Output 49.1.2: Cumulative Probability Estimates

The ICLIFETEST Procedure

Nonparametric Survival Estimates
  Probability Estimate Imputation
Time Interval Failure Survival
3 4 0.2083 0.7917 0.1811 0.0000
4 6 0.4167 0.5833 0.2179 0.0000
6 8 0.6250 0.3750 0.2099 0.0000
8 12 0.8333 0.1667 0.1521 0.0000
12 Inf 1.0000 0.0000 0.0000 0.0000