plend {STAND}

Compute Product Limit Estimate for Non-detects

Description

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

Usage

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 F[j]= \prod (n[j] -- 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.

Value

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)

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].

Author(s)

E. L. Frome

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.

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