kmms function

Kaplan-Meier (KM) Mean and Standard Error

Kaplan- Meier Estimate of Mean and Standard Error of the Mean for Left Censored Data

kmms(dd, gam = 0.95)

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

• gam: one-sided confidence level $\gamma$. Default is 0.95

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 by Schmoyer et al. (1996)--see plend.

The mean of the PLE is a censoring-adjusted point estimate of E(X) the mean of X. An approximate standard error of the PLE mean can be obtained using the method of Kaplan and Meier (1958), and the $100\gamma\%$ UCL is c("$KM.mean + t(\\gamma -1,\n$", "$m-1) sp$"), where sp is the Kaplan-Meier standard error of the mean adjusted by the factor $m/(m-1)$, where m is the number of detects in the sample. When there is no censoring this reduces to the second approximate method described by Land (1972).

Returns

A LIST with components: - KM.mean: Kaplan- Meier(KM) estimate of mean E(X)

• KM.LCL: KM estimate of lower confidence limit

• KM.UCL: KM estimate of upper confidence limit

• KM.se: estimate of standard error of KM-mean

• gamma: one-sided confidence level $\gamma$. Default 0.95

References

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.

Author(s)

E. L. Frome

Note

Error in KM.se corrected on 12 June 2007. KM standard error is adjusted by multiplying by sqrt(m/(m-1)) where m is number of detected values. Error occurred if there were ties in detected values by calculating the number of unique detected values. For example, for beTWA sqrt(m/(m-1)) is 1.004796 . Due to error 1.008032 was used. The sqrt(m/(m-1)) will always be smaller after correction, depending on value of m and the number of ties. See the example.

See Also

plend, plekm

Examples

# results for beTWA data using kmms in stand Ver 1.1 with error
#    KM.mean      KM.LCL      KM.UCL       KM.se       gamma 
# 0.018626709 0.014085780 0.023167637 0.002720092 0.950000000
#
data(beTWA) # Use data from Example 2 in ORNLTM2002-51
unlist(kmms(beTWA))