The ICLIFETEST Procedure

Example 50.1 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 50.1.1. Note that because the first Turnbull interval is $(0,3)$, the failure probability function is undefined within that interval.

Output 50.1.1: Failure Probability Plot for Fictitious Nondetection Data

Failure Probability Plot for Fictitious Nondetection Data

Output 50.1.2 presents the estimated failure probability, with standard errors that are estimated by the method of multiple imputations.

Output 50.1.2: 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