plend function

Compute Product Limit Estimate for Non-detects

Compute Product Limit Estimate(PLE) of F(x) for positive data with non-detects (left censored data)

plend(dd)

Arguments

• dd: An n by 2 matrix or data frame with

  x (exposure) variable in column 1, and

  det = 0 for non-detect or 1 for detect in column 2

Details

The product limit estimate (PLE) of the cumulative distribution function was first proposed by Kaplan and Meier (1958) for right censored data. Turnbull (1976) provides a more general treatment of nonparametric estimation of the distribution function for arbitrary censoring. For randomly left censored data, the PLE is defined as follows [Schmoyer et al. (1996)]. Let $a[1]< \ldots < a[m]$ be the m distinct values at which detects occur, r[j] is the number of detects at a[j], and n[j] is the sum of non-detects and detects that are less than or equal to a[j]. Then the PLE is defined to be 0 for $0 \le x \le a0$, where a0 is a[1] or the value of the detection limit for the smallest non-detect if it is less than a[1]. For $a0 \le x < a[m]$ the PLE is c("$F[j]= \\prod (n[j] --\n$", "$r[j])/n[j]$"), where the product is over all $a[j] > x$, and the PLE is 1 for $x \ge a[m]$. When there are only detects this reduces to the usual definition of the empirical cumulative distribution function.

Returns

Data frame with columns - a(j): value of jth detect (ordered)

• ple(j): PLE of F(x) at a(j)

• n(j): number of detects or non-detects $\le$ a(j)

• r(j): number of detects equal to a(j)

• surv(j): 1 - ple(j) is PLE of S(x)

References

Frome, E. L. and Wambach, P. F. (2005), "Statistical Methods and Software for the Analysis of Occupational Exposure Data with Non-Detectable Values," ORNL/TM-2005/52,Oak Ridge National Laboratory, Oak Ridge, TN 37830. Available at: http://www.csm.ornl.gov/esh/aoed/ORNLTM2005-52.pdf

Kaplan, E. L. and Meier, P. (1958), "Nonparametric Estimation from Incomplete Observations," Journal of the American Statistical Association, 457-481.

Schmoyer, R. L., J. J. Beauchamp, C. C. Brandt and F. O. Hoffman, Jr. (1996), "Difficulties with the Lognormal Model in Mean Estimation and Testing," Environmental and Ecological Statistics, 3, 81-97.

Turnbull, B. W. (1976), "The Empirical Distribution Function with Arbitrarily Grouped, Censored and Truncated Data," Journal of the Royal Statistical Society, Series B (Methodological), 38(3), 290-295.

Author(s)

E. L. Frome

Note

In survival analysis $S(x) = 1 - F(x)$ is the survival function i.e., $S(x) = P[X > x]$. In environmental and occupational situations $1 - F(x)$ is the "exceedance" function, i.e., $C(x) = 1 - F(x) = P [X > x]$.

See Also

plekm, pleicf, qq.lnorm

Examples

data(SESdata) #  use SESdata data set Example 1 from ORNLTM-2005/52
pnd<- plend(SESdata)
Ia<-"Q-Q plot For SESdata "
qq.lnorm(pnd,main=Ia) #  lognormal q-q plot based on PLE 
pnd