Description Usage Arguments Details Value Note Author(s) References See Also Examples

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

1 | ```
plend(dd)
``` |

`dd` |
An n by 2 matrix or data frame with |

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.

Data frame with columns

`a(j)` |
value of j |

`ple(j)` |
PLE of F(x) at a(j) |

`n(j)` |
number of detects or non-detects |

`r(j)` |
number of detects equal to a(j) |

`surv(j)` |
1 - ple(j) is PLE of S(x) |

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

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.

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STAND documentation built on May 30, 2017, 7:22 a.m.

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