Description

Estimate the mean, standard deviation, and standard error of the mean nonparametrically given a sample of data from a positive-valued distribution that has been subjected to left- or right-censoring, and optionally construct a confidence interval for the mean.

Usage

Arguments

Details

Let \underline{x} = (x_1, x_2, …, x_N) denote a vector of N observations from some positive-valued distribution with mean μ and standard deviation σ. Assume n (0 < n < N) of these observations are known and c (c=N-n) of these observations are all censored below (left-censored) or all censored above (right-censored) at k censoring levels

T_1, T_2, …, T_k; \; k ≥ 1 \;\;\;\;\;\; (1)

Finally, let y_1, y_2, …, y_n denote the n ordered uncensored observations.

Estimation
It can be shown that the mean of a positive-valued distribution is equal to the area under the survival curve (Klein and Moeschberger, 2003, p.33):

μ = \int_0^∞ [1 - F(t)] dt = \int_0^∞ S(t) dt \;\;\;\;\;\; (2)

where F(t) denotes the cumulative distribution function evaluated at t and S(t) = 1 - F(t) denotes the survival function evaluated at t. When the Kaplan-Meier estimator is used to construct the survival function, you can use the area under this curve to estimate the mean of the distribution, and the estimator can be as efficient or more efficient than parametric estimators of the mean (Meier, 2004; Helsel, 2012; Lee and Wang, 2003). Let \hat{F}(t) denote the Kaplan-Meier estimator of the empirical cumulative distribution function (ecdf) evaluated at t, and let \hat{S}(t) = 1 - \hat{F}(t) denote the estimated survival function evaluated at t. (See the help files for ecdfPlotCensored and qqPlotCensored for an explanation of how the Kaplan-Meier estimator of the ecdf is computed.)

The formula for the estimated mean is given by (Lee and Wang, 2003, p. 74):

\hat{μ} = ∑_{i=1}^{n} \hat{S}(y_{i-1}) (y_{i} - y_{i-1}) \;\;\;\;\;\; (3)

where y_{0} = 0 and \hat{S}(y_{0}) = 1 by definition. It can be shown that this formula is eqivalent to:

\hat{μ} = ∑_{i=1}^n y_{i} [\hat{F}(y_{i}) - \hat{F}(y_{i-1})] \;\;\;\;\;\; (4)

where \hat{F}(y_{0}) = \hat{F}(0) = 0 by definition (USEPA, 2009, p. 15-10; Singh et al., 2010, pp. 109–111; Beal, 2010).

The formula for the estimated standard deviation is:

\hat{σ} = \{∑_{i=1}^n (y_{i} - \hat{μ})^2 [\hat{F}(y_{i}) - \hat{F}(y_{i-1})]\}^{1/2} \;\;\;\;\; (5)

(USEPA, 2009, p. 15-10), and the formula for the estimated standard error of the mean is:

\hat{σ}_{\hat{μ}} = [∑_{r=1}^{n-1} \frac{A_r^2}{(N-r)(N-r+1)}]^{1/2} \;\;\;\;\;\; (6)

where

A_r = ∑_{i=r}^{n-1} \hat{S}(y_{i}) (y_{i+1} - y_{i}) \;\;\;\;\;\; (7)

(Lee and Wang, 2003, p. 74). Kaplan and Meier suggest using a bias correction of n/(n-1) (Lee and Wang, 2003, p.75):

\hat{σ}_{\hat{μ}, BC} = \frac{n}{n-1} \hat{σ}_{\hat{μ}} \;\;\;\;\;\; (8)

When correct.se=TRUE, Equation (8) is used instead of Equation (6).

If the smallest value for left-censored data is censored and less than or equal to the smallest uncensored value then the estimated mean will be biased high, and if the largest value for right-censored data is censored and greater than or equal to the largest uncensored value, the the estimated mean will be biased low. In these cases, the above formulas can and should be modified (Barker, 2009; Lee and Wang, 2003, p. 74). For left-censored data, the smallest censored observation can be treated as observed and set to the smallest censoring level (left.censored.min="DL"), half of the smallest censoring level (left.censored.min="DL/2"), or some other value greater than 0 and the smallest censoring level. Setting left.censored.min="Ignore" uses the formulas given above (biased in this case) and is what is presented in Singh et al. (2010, pp. 109–111) and Beal (2010). USEPA (2009, pp. 15–7 to 15-10) on the other hand uses the method associated with left.censored.min="DL". For right-censored data, the largest censored observation can be treated as observed and set to the censoring level (right.censored.max="DL") or some value greater than the largest censoring level. In the survival analysis literature, this method of dealing with this situation is called estimating the restricted mean (Miller, 1981; Meier, 2004; Barker, 2009).

Confidence Intervals
This section explains how confidence intervals for the mean μ are computed.

Normal Approximation (ci.method="normal.approx")
This method constructs approximate (1-α)100\% confidence intervals for μ based on the assumption that the estimator of μ is approximately normally distributed. That is, a two-sided (1-α)100\% confidence interval for μ is constructed as:

[\hat{μ} - t_{1-α/2, m-1}\hat{σ}_{\hat{μ}}, \; \hat{μ} + t_{1-α/2, m-1}\hat{σ}_{\hat{μ}}] \;\;\;\; (9)

where \hat{μ} denotes the estimate of μ, \hat{σ}_{\hat{μ}} denotes the estimated asymptotic standard deviation of the estimator of μ, m denotes the assumed sample size for the confidence interval, and t_{p,ν} denotes the p'th quantile of Student's t-distribuiton with ν degrees of freedom. One-sided confidence intervals are computed in a similar fashion.

The argument ci.sample.size determines the value of m. By default, it is the observed number of uncensored observations. This is simply an ad-hoc method of constructing confidence intervals and is not based on any published theoretical results.

When pivot.statistic="z", the p'th quantile from the standard normal distribution is used in place of the p'th quantile from Student's t-distribution.

Bootstrap and Bias-Corrected Bootstrap Approximation (ci.method="bootstrap")
The bootstrap is a nonparametric method of estimating the distribution (and associated distribution parameters and quantiles) of a sample statistic, regardless of the distribution of the population from which the sample was drawn. The bootstrap was introduced by Efron (1979) and a general reference is Efron and Tibshirani (1993).

In the context of deriving an approximate (1-α)100\% confidence interval for the population mean μ, the bootstrap can be broken down into the following steps:

1. Create a bootstrap sample by taking a random sample of size N from the observations in \underline{x}, where sampling is done with replacement. Note that because sampling is done with replacement, the same element of \underline{x} can appear more than once in the bootstrap sample. Thus, the bootstrap sample will usually not look exactly like the original sample (e.g., the number of censored observations in the bootstrap sample will often differ from the number of censored observations in the original sample).

2. Estimate μ based on the bootstrap sample created in Step 1, using the same method that was used to estimate μ using the original observations in \underline{x}. Because the bootstrap sample usually does not match the original sample, the estimate of μ based on the bootstrap sample will usually differ from the original estimate based on \underline{x}. For the bootstrap-t method (see below), this step also involves estimating the standard error of the estimate of the mean and computing the statistic T = (\hat{μ}_B - \hat{mu}) / \hat{σ}_{\hat{μ}_B} where \hat{μ} denotes the estimate of the mean based on the original sample, and \hat{μ}_B and \hat{σ}_{\hat{μ}_B} denote the estimate of the mean and estimate of the standard error of the estimate of the mean based on the bootstrap sample.

3. Repeat Steps 1 and 2 B times, where B is some large number. For the function enparCensored, the number of bootstraps B is determined by the argument n.bootstraps (see the section ARGUMENTS above). The default value of n.bootstraps is 1000.

4. Use the B estimated values of μ to compute the empirical cumulative distribution function of the estimator of μ or to compute the empirical cumulative distribution function of the statistic T (see ecdfPlot), and then create a confidence interval for μ based on this estimated cdf.

The two-sided percentile interval (Efron and Tibshirani, 1993, p.170) is computed as:

[\hat{G}^{-1}(\frac{α}{2}), \; \hat{G}^{-1}(1-\frac{α}{2})] \;\;\;\;\;\; (10)

where \hat{G}(t) denotes the empirical cdf of \hat{μ}_B evaluated at t and thus \hat{G}^{-1}(p) denotes the p'th empirical quantile of the distribution of \hat{μ}_B, that is, the p'th quantile associated with the empirical cdf. Similarly, a one-sided lower confidence interval is computed as:

[\hat{G}^{-1}(α), \; ∞] \;\;\;\;\;\; (11)

and a one-sided upper confidence interval is computed as:

[-∞, \; \hat{G}^{-1}(1-α)] \;\;\;\;\;\; (12)

The function enparCensored calls the R function quantile to compute the empirical quantiles used in Equations (10)-(12).

The percentile method bootstrap confidence interval is only first-order accurate (Efron and Tibshirani, 1993, pp.187-188), meaning that the probability that the confidence interval will contain the true value of μ can be off by k/√{N}, where k is some constant. Efron and Tibshirani (1993, pp.184–188) proposed a bias-corrected and accelerated interval that is second-order accurate, meaning that the probability that the confidence interval will contain the true value of μ may be off by k/N instead of k/√{N}. The two-sided bias-corrected and accelerated confidence interval is computed as:

[\hat{G}^{-1}(α_1), \; \hat{G}^{-1}(α_2)] \;\;\;\;\;\; (13)

where

α_1 = Φ[\hat{z}_0 + \frac{\hat{z}_0 + z_{α/2}}{1 - \hat{a}(z_0 + z_{α/2})}] \;\;\;\;\;\; (14)

α_2 = Φ[\hat{z}_0 + \frac{\hat{z}_0 + z_{1-α/2}}{1 - \hat{a}(z_0 + z_{1-α/2})}] \;\;\;\;\;\; (15)

\hat{z}_0 = Φ^{-1}[\hat{G}(\hat{μ})] \;\;\;\;\;\; (16)

\hat{a} = \frac{∑_{i=1}^N (\hat{μ}_{(\cdot)} - \hat{μ}_{(i)})^3}{6[∑_{i=1}^N (\hat{μ}_{(\cdot)} - \hat{μ}_{(i)})^2]^{3/2}} \;\;\;\;\;\; (17)

where the quantity \hat{μ}_{(i)} denotes the estimate of μ using all the values in \underline{x} except the i'th one, and

\hat{μ}{(\cdot)} = \frac{1}{N} ∑_{i=1}^N \hat{μ_{(i)}} \;\;\;\;\;\; (18)

A one-sided lower confidence interval is given by:

[\hat{G}^{-1}(α_1), \; ∞] \;\;\;\;\;\; (19)

and a one-sided upper confidence interval is given by:

[-∞, \; \hat{G}^{-1}(α_2)] \;\;\;\;\;\; (20)

where α_1 and α_2 are computed as for a two-sided confidence interval, except α/2 is replaced with α in Equations (14) and (15).

The constant \hat{z}_0 incorporates the bias correction, and the constant \hat{a} is the acceleration constant. The term “acceleration” refers to the rate of change of the standard error of the estimate of μ with respect to the true value of μ (Efron and Tibshirani, 1993, p.186). For a normal (Gaussian) distribution, the standard error of the estimate of μ does not depend on the value of μ, hence the acceleration constant is not really necessary.

For the bootstrap-t method, the two-sided confidence interval (Efron and Tibshirani, 1993, p.160) is computed as:

[\hat{μ} - t_{1-α/2}\hat{σ}_{\hat{μ}}, \; \hat{μ} - t_{α/2}\hat{σ}_{\hat{μ}}] \;\;\;\;\;\; (21)

where \hat{μ} and \hat{σ}_{\hat{μ}} denote the estimate of the mean and standard error of the estimate of the mean based on the original sample, and t_p denotes the p'th empirical quantile of the bootstrap distribution of the statistic T. Similarly, a one-sided lower confidence interval is computed as:

[\hat{μ} - t_{1-α}\hat{σ}_{\hat{μ}}, \; ∞] \;\;\;\;\;\; (22)

and a one-sided upper confidence interval is computed as:

[-∞, \; \hat{μ} - t_{α}\hat{σ}_{\hat{μ}}] \;\;\;\;\;\; (23)

When ci.method="bootstrap", the function enparCensored computes the percentile method, bias-corrected and accelerated method, and bootstrap-t bootstrap confidence intervals. The percentile method is transformation respecting, but not second-order accurate. The bootstrap-t method is second-order accurate, but not transformation respecting. The bias-corrected and accelerated method is both transformation respecting and second-order accurate (Efron and Tibshirani, 1993, p.188).

Value

a list of class "estimateCensored" containing the estimated parameters and other information. See estimateCensored.object for details.

Note

A sample of data contains censored observations if some of the observations are reported only as being below or above some censoring level. In environmental data analysis, Type I left-censored data sets are common, with values being reported as “less than the detection limit” (e.g., Helsel, 2012). Data sets with only one censoring level are called singly censored; data sets with multiple censoring levels are called multiply or progressively censored.

Statistical methods for dealing with censored data sets have a long history in the field of survival analysis and life testing. More recently, researchers in the environmental field have proposed alternative methods of computing estimates and confidence intervals in addition to the classical ones such as maximum likelihood estimation.

Helsel (2012, Chapter 6) gives an excellent review of past studies of the properties of various estimators based on censored environmental data.

In practice, it is better to use a confidence interval for the mean or a joint confidence region for the mean and standard deviation, rather than rely on a single point-estimate of the mean. Since confidence intervals and regions depend on the properties of the estimators for both the mean and standard deviation, the results of studies that simply evaluated the performance of the mean and standard deviation separately cannot be readily extrapolated to predict the performance of various methods of constructing confidence intervals and regions. Furthermore, for several of the methods that have been proposed to estimate the mean based on type I left-censored data, standard errors of the estimates are not available, hence it is not possible to construct confidence intervals (El-Shaarawi and Dolan, 1989).

Few studies have been done to evaluate the performance of methods for constructing confidence intervals for the mean or joint confidence regions for the mean and standard deviation when data are subjected to single or multiple censoring. See, for example, Singh et al. (2006).

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

Barker, C. (2009). The Mean, Median, and Confidence Intervals of the Kaplan-Meier Survival Estimate – Computations and Applications. The American Statistician 63(1), 78–80.

Beal, D. (2010). A Macro for Calculating Summary Statistics on Left Censored Environmental Data Using the Kaplan-Meier Method. Paper SDA-09, presented at Southeast SAS Users Group 2010, September 26-28, Savannah, GA.

Efron, B. (1979). Bootstrap Methods: Another Look at the Jackknife. The Annals of Statistics 7, 1–26.

Efron, B., and R.J. Tibshirani. (1993). An Introduction to the Bootstrap. Chapman and Hall, New York, 436pp.

El-Shaarawi, A.H., and D.M. Dolan. (1989). Maximum Likelihood Estimation of Water Quality Concentrations from Censored Data. Canadian Journal of Fisheries and Aquatic Sciences 46, 1033–1039.

Gillespie, B.W., Q. Chen, H. Reichert, A. Franzblau, E. Hedgeman, J. Lepkowski, P. Adriaens, A. Demond, W. Luksemburg, and D.H. Garabrant. (2010). Estimating Population Distributions When Some Data Are Below a Limit of Detection by Using a Reverse Kaplan-Meier Estimator. Epidemiology 21(4), S64–S70.

Helsel, D.R. (2012). Statistics for Censored Environmental Data Using Minitab and R, Second Edition. John Wiley \& Sons, Hoboken, New Jersey.

Irwin, J.O. (1949). The Standard Error of an Estimate of Expectation of Life, with Special Reference to Expectation of Tumourless Life in Experiments with Mice. Journal of Hygiene 47, 188–189.

Kaplan, E.L., and P. Meier. (1958). Nonparametric Estimation From Incomplete Observations. Journal of the American Statistical Association 53, 457-481.

Klein, J.P., and M.L. Moeschberger. (2003). Survival Analysis: Techniques for Censored and Truncated Data, Second Edition. Springer, New York, 537pp.

Lee, E.T., and J.W. Wang. (2003). Statistical Methods for Survival Data Analysis, Third Edition. John Wiley \& Sons, Hoboken, New Jersey, 513pp.

Meier, P., T. Karrison, R. Chappell, and H. Xie. (2004). The Price of Kaplan-Meier. Journal of the American Statistical Association 99(467), 890–896.

Miller, R.G. (1981). Survival Analysis. John Wiley and Sons, New York.

Nelson, W. (1982). Applied Life Data Analysis. John Wiley and Sons, New York, 634pp.

Singh, A., R. Maichle, and S. Lee. (2006). On the Computation of a 95% Upper Confidence Limit of the Unknown Population Mean Based Upon Data Sets with Below Detection Limit Observations. EPA/600/R-06/022, March 2006. Office of Research and Development, U.S. Environmental Protection Agency, Washington, D.C.

Singh, A., N. Armbya, and A. Singh. (2010). ProUCL Version 4.1.00 Technical Guide (Draft). EPA/600/R-07/041, May 2010. Office of Research and Development, U.S. Environmental Protection Agency, Washington, D.C.

USEPA. (2009). Statistical Analysis of Groundwater Monitoring Data at RCRA Facilities, Unified Guidance. EPA 530/R-09-007, March 2009. Office of Resource Conservation and Recovery Program Implementation and Information Division. U.S. Environmental Protection Agency, Washington, D.C.

USEPA. (2010). Errata Sheet - March 2009 Unified Guidance. EPA 530/R-09-007a, August 9, 2010. Office of Resource Conservation and Recovery, Program Information and Implementation Division. U.S. Environmental Protection Agency, Washington, D.C.

See Also

ecdfPlotCensored, qqPlotCensored, estimateCensored.object.

Examples