# plend: Compute Product Limit Estimate for Non-detects In STAND: Statistical Analysis of Non-Detects

## Description

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

## Usage

 `1` ```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]< … < 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 ≤ x ≤ 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 ≤ x < a[m] the PLE is F[j]= ∏ (n[j] -- r[j])/n[j], where the product is over all a[j] > x, and the PLE is 1 for x ≥ 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 ≤ 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].

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.

`plekm`, `pleicf`, `qq.lnorm`

## Examples

 ```1 2 3 4 5``` ```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 ```

### Example output

```Loading required package: survival
a        ple  n r       surv
1  0.015 0.09677419  3 0 0.90322581
2  0.025 0.16129032  5 2 0.83870968
3  0.040 0.25806452  8 3 0.74193548
4  0.045 0.29032258  9 1 0.70967742
5  0.050 0.35483871 11 2 0.64516129
6  0.070 0.38709677 12 1 0.61290323
7  0.075 0.41935484 13 1 0.58064516
8  0.095 0.45161290 14 1 0.54838710
9  0.100 0.48387097 15 1 0.51612903
10 0.125 0.54838710 17 2 0.45161290
11 0.145 0.61290323 19 2 0.38709677
12 0.150 0.67741935 21 2 0.32258065
13 0.165 0.70967742 22 1 0.29032258
14 0.270 0.74193548 23 1 0.25806452
15 0.290 0.77419355 24 1 0.22580645
16 0.345 0.80645161 25 1 0.19354839
17 0.395 0.87096774 27 2 0.12903226
18 0.420 0.90322581 28 1 0.09677419
19 0.495 0.93548387 29 1 0.06451613
20 0.840 0.96774194 30 1 0.03225806
21 1.140 1.00000000 31 1 0.00000000
```

STAND documentation built on May 2, 2019, 3:39 p.m.