# Estimate Mean and Coefficient of Variation for a Gamma Distribution Based on Type I Censored Data Estimate the mean and coefficient of variation of a gamma distribution given a sample of data that has been subjected to Type I censoring, and optionally construct a confidence interval for the mean. ```r egammaAltCensored(x, censored, method = "mle", censoring.side = "left", ci = FALSE, ci.method = "profile.likelihood", ci.type = "two-sided", conf.level = 0.95, n.bootstraps = 1000, pivot.statistic = "z", ci.sample.size = sum(!censored)) ``` ## Arguments - `x`: numeric vector of observations. Missing (`NA`), undefined (`NaN`), and infinite (`Inf`, `-Inf`) values are allowed but will be removed. - `censored`: numeric or logical vector indicating which values of `x` are censored. This must be the same length as `x`. If the mode of `censored` is `"logical"`, `TRUE` values correspond to elements of `x` that are censored, and `FALSE` values correspond to elements of `x` that are not censored. If the mode of `censored` is `"numeric"`, it must contain only `1`'s and `0`'s; `1` corresponds to `TRUE` and `0` corresponds to `FALSE`. Missing (`NA`) values are allowed but will be removed. - `method`: character string specifying the method of estimation. Currently, the only available method is maximum likelihood (`method="mle"`). - `censoring.side`: character string indicating on which side the censoring occurs. The possible values are `"left"` (the default) and `"right"`. - `ci`: logical scalar indicating whether to compute a confidence interval for the mean. The default value is `ci=FALSE`. - `ci.method`: character string indicating what method to use to construct the confidence interval for the mean. The possible values are `"profile.likelihood"` (profile likelihood; the default), `"normal.approx"` (normal approximation), and `"bootstrap"` (based on bootstrapping). See the DETAILS section for more information. This argument is ignored if `ci=FALSE`. - `ci.type`: character string indicating what kind of confidence interval to compute. The possible values are `"two-sided"` (the default), `"lower"`, and `"upper"`. This argument is ignored if `ci=FALSE`. - `conf.level`: a scalar between 0 and 1 indicating the confidence level of the confidence interval. The default value is `conf.level=0.95`. This argument is ignored if `ci=FALSE`. - `n.bootstraps`: numeric scalar indicating how many bootstraps to use to construct the confidence interval for the mean when `ci.type="bootstrap"`. This argument is ignored if `ci=FALSE` and/or `ci.method` does not equal `"bootstrap"`. - `pivot.statistic`: character string indicating which pivot statistic to use in the construction of the confidence interval for the mean when `ci.method="normal.approx"` or `ci.method="normal.approx.w.cov"` (see the DETAILS section). The possible values are `pivot.statistic="z"` (the default) and `pivot.statistic="t"`. When `pivot.statistic="t"` you may supply the argument `ci.sample size` (see below). The argument `pivot.statistic` is ignored if `ci=FALSE`. - `ci.sample.size`: numeric scalar indicating what sample size to assume to construct the confidence interval for the mean if `pivot.statistic="t"` and `ci.method="normal.approx"`. The default value is the number of uncensored observations. ## Details If `x` or `censored` contain any missing (`NA`), undefined (`NaN`) or infinite (`Inf`, `-Inf`) values, they will be removed prior to performing the estimation. Let $\underline{x}$ denote a vector of $N$ observations from a gamma distribution with parameters `shape=`$\kappa$ and `scale=`$\theta$. The relationship between these parameters and the mean $\mu$ and coefficient of variation $\tau$ of this distribution is given by: $$ \kappa = \tau^{-2} \;\;\;\;\;\; (1) $$ $$ \theta = \mu/\kappa \;\;\;\;\;\; (2) $$ $$ \mu = \kappa \; \theta \;\;\;\;\;\; (3) $$ $$ \tau = \kappa^{-1/2} \;\;\;\;\;\; (4) $$ 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 $k$ fixed censoring levels $$ T_1, T_2, \ldots, T_k; \; k \ge 1 \;\;\;\;\;\; (5) $$ For the case when $k \ge 2$, the data are said to be Type I _multiply censored_ . For the case when $k=1$, set $T = T_1$. If the data are left-censored and all $n$ known observations are greater than or equal to $T$, or if the data are right-censored and all $n$ known observations are less than or equal to $T$, then the data are said to be Type I _singly censored_ (Nelson, 1982, p.7), otherwise they are considered to be Type I multiply censored. Let $c_j$ denote the number of observations censored below or above censoring level $T_j$ for $j = 1, 2, \ldots, k$, so that $$ \sum_{i=1}^k c_j = c \;\;\;\;\;\; (6) $$ Let $x_{(1)}, x_{(2)}, \ldots, x_{(N)}$ denote the ordered observations, where now observation means either the actual observation (for uncensored observations) or the censoring level (for censored observations). For right-censored data, if a censored observation has the same value as an uncensored one, the uncensored observation should be placed first. For left-censored data, if a censored observation has the same value as an uncensored one, the censored observation should be placed first. Note that in this case the quantity $x_{(i)}$ does not necessarily represent the $i$'th largest observation from the (unknown) complete sample. Finally, let $\Omega$ (omega) denote the set of $n$ subscripts in the ordered sample that correspond to uncensored observations. Estimation **Maximum Likelihood Estimation** (`method="mle"`) For Type I left censored data, the likelihood function is given by: $$ L(\mu, \tau | \underline{x}) = {N \choose c_1 c_2 \ldots c_k n} \prod_{j=1}^k [F(T_j)]^{c_j} \prod_{i \in \Omega} f[x_{(i)}] \;\;\;\;\;\; (7) $$ where $f$ and $F$ denote the probability density function (pdf) and cumulative distribution function (cdf) of the population (Cohen, 1963; Cohen, 1991, pp.6, 50). That is, $$ f(t) = \frac{t^{\kappa-1} e^{-t/\theta}}{\theta^\kappa \Gamma(\kappa)} \;\;\;\;\;\; (8) $$ (Johnson et al., 1994, p.343), where $\kappa$ and $\theta$ are defined in terms of $\mu$ and $\tau$ by Equations (1) and (2) above. For left singly censored data, Equation (7) simplifies to: $$ L(\mu, \tau | \underline{x}) = {N \choose c} [F(T)]^{c} \prod_{i = c+1}^n f[x_{(i)}] \;\;\;\;\;\; (9) $$ Similarly, for Type I right censored data, the likelihood function is given by: $$ L(\mu, \tau | \underline{x}) = {N \choose c_1 c_2 \ldots c_k n} \prod_{j=1}^k [1 - F(T_j)]^{c_j} \prod_{i \in \Omega} f[x_{(i)}] \;\;\;\;\;\; (10) $$ and for right singly censored data this simplifies to: $$ L(\kappa, \theta | \underline{x}) = {N \choose c} [1 - F(T)]^{c} \prod_{i = 1}^n f[x_{(i)}] \;\;\;\;\;\; (11) $$ The maximum likelihood estimators are computed by minimizing the negative log-likelihood function. Confidence Intervals This section explains how confidence intervals for the mean $\mu$ are computed. **Likelihood Profile** (`ci.method="profile.likelihood"`) This method was proposed by Cox (1970, p.88), and Venzon and Moolgavkar (1988) introduced an efficient method of computation. This method is also discussed by Stryhn and Christensen (2003) and Royston (2007). The idea behind this method is to invert the likelihood-ratio test to obtain a confidence interval for the mean $\mu$ while treating the coefficient of variation $\tau$ as a nuisance parameter. Equation (7) above shows the form of the likelihood function $L(\mu, \tau | \underline{x})$ for multiply left-censored data, where $\mu$ and $\tau$ are defined by Equations (3) and (4), and Equation (10) shows the function for multiply right-censored data. Following Stryhn and Christensen (2003), denote the maximum likelihood estimates of the mean and coefficient of variation by $(\mu^*, \tau^*)$. The likelihood ratio test statistic ($G^2$) of the hypothesis $H_0: \mu = \mu_0$ (where $\mu_0$ is a fixed value) equals the drop in $2 log(L)$ between the full model and the reduced model with $\mu$ fixed at $\mu_0$, i.e., $$ G^2 = 2 \{log[L(\mu^*, \tau^*)] - log[L(\mu_0, \tau_0^*)]\} \;\;\;\;\;\; (12) $$ where $\tau_0^*$ is the maximum likelihood estimate of $\tau$ for the reduced model (i.e., when $\mu = \mu_0$). Under the null hypothesis, the test statistic $G^2$ follows a chi-squared distribution with 1 degree of freedom. Alternatively, we may express the test statistic in terms of the profile likelihood function $L_1$ for the mean $\mu$, which is obtained from the usual likelihood function by maximizing over the parameter $\tau$, i.e., $$ L_1(\mu) = max_{\tau} L(\mu, \tau) \;\;\;\;\;\; (13) $$ Then we have $$ G^2 = 2 \{log[L_1(\mu^*)] - log[L_1(\mu_0)]\} \;\;\;\;\;\; (14) $$ A two-sided $(1-\alpha)100\%$ confidence interval for the mean $\mu$ consists of all values of $\mu_0$ for which the test is not significant at level $alpha$: $$ \mu_0: G^2 \le \chi^2_{1, {1-\alpha}} \;\;\;\;\;\; (15) $$ where $\chi^2_{\nu, p}$ denotes the $p$'th quantile of the chi-squared distribution with $\nu$ degrees of freedom. One-sided lower and one-sided upper confidence intervals are computed in a similar fashion, except that the quantity $1-\alpha$ in Equation (15) is replaced with $1-2\alpha$. **Normal Approximation** (`ci.method="normal.approx"`) This method constructs approximate $(1-\alpha)100\%$ confidence intervals for $\mu$ based on the assumption that the estimator of $\mu$ is approximately normally distributed. That is, a two-sided $(1-\alpha)100\%$ confidence interval for $\mu$ is constructed as: $$ [\hat{\mu} - t_{1-\alpha/2, m-1}\hat{\sigma}_{\hat{\mu}}, \; \hat{\mu} + t_{1-\alpha/2, m-1}\hat{\sigma}_{\hat{\mu}}] \;\;\;\; (16) $$ where $\hat{\mu}$ denotes the estimate of $\mu$, $\hat{\sigma}_{\hat{\mu}}$ denotes the estimated asymptotic standard deviation of the estimator of $\mu$, $m$ denotes the assumed sample size for the confidence interval, and $t_{p,\nu}$ denotes the $p$'th quantile of Student's t-distribuiton with $\nu$ degrees of freedom. One-sided confidence intervals are computed in a similar fashion. The argument `ci.sample.size` determines the value of $m$ and by default is equal to the 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. The standard deviation of the mle of $\mu$ is estimated based on the inverse of the Fisher Information matrix. **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 $(1-\alpha)100\%$ confidence interval for the population mean $\mu$, 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 $\mu$ based on the bootstrap sample created in Step 1, using the same method that was used to estimate $\mu$ using the original observations in $\underline{x}$. Because the bootstrap sample usually does not match the original sample, the estimate of $\mu$ based on the bootstrap sample will usually differ from the original estimate based on $\underline{x}$. 3. Repeat Steps 1 and 2 $B$ times, where $B$ is some large number. For the function `egammaAltCensored`, 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 $\mu$ to compute the empirical cumulative distribution function of this estimator of $\mu$ (see `ecdfPlot`), and then create a confidence interval for $\mu$ based on this estimated cdf. The two-sided percentile interval (Efron and Tibshirani, 1993, p.170) is computed as: $$ [\hat{G}^{-1}(\frac{\alpha}{2}), \; \hat{G}^{-1}(1-\frac{\alpha}{2})] \;\;\;\;\;\; (17) $$ where $\hat{G}(t)$ denotes the empirical cdf evaluated at $t$ and thus $\hat{G}^{-1}(p)$ denotes the $p$'th empirical quantile, that is, the $p$'th quantile associated with the empirical cdf. Similarly, a one-sided lower confidence interval is computed as: $$ [\hat{G}^{-1}(\alpha), \; \infty] \;\;\;\;\;\; (18) $$ and a one-sided upper confidence interval is computed as: $$ [0, \; \hat{G}^{-1}(1-\alpha)] \;\;\;\;\;\; (19) $$ The function `egammaAltCensored` calls the function `quantile` to compute the empirical quantiles used in Equations (17)-(19). 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 $\mu$ can be off by $k/\sqrt{N}$, 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 $\mu$ may be off by $k/N$ instead of $k/\sqrt{N}$. The two-sided bias-corrected and accelerated confidence interval is computed as: $$ [\hat{G}^{-1}(\alpha_1), \; \hat{G}^{-1}(\alpha_2)] \;\;\;\;\;\; (20) $$ where $$ \alpha_1 = \Phi[\hat{z}_0 + \frac{\hat{z}_0 + z_{\alpha/2}}{1 - \hat{a}(z_0 + z_{\alpha/2})}] \;\;\;\;\;\; (21) $$ $$ \alpha_2 = \Phi[\hat{z}_0 + \frac{\hat{z}_0 + z_{1-\alpha/2}}{1 - \hat{a}(z_0 + z_{1-\alpha/2})}] \;\;\;\;\;\; (22) $$ $$ \hat{z}_0 = \Phi^{-1}[\hat{G}(\hat{\mu})] \;\;\;\;\;\; (23) $$ $$ \hat{a} = \frac{\sum_{i=1}^N (\hat{\mu}_{(\cdot)} - \hat{\mu}_{(i)})^3}{6[\sum_{i=1}^N (\hat{\mu}_{(\cdot)} - \hat{\mu}_{(i)})^2]^{3/2}} \;\;\;\;\;\; (24) $$ where the quantity $\hat{\mu}_{(i)}$ denotes the estimate of $\mu$ using all the values in $\underline{x}$ except the $i$'th one, and $$ \hat{\mu}{(\cdot)} = \frac{1}{N} \sum_{i=1}^N \hat{\mu_{(i)}} \;\;\;\;\;\; (25) $$ A one-sided lower confidence interval is given by: $$ [\hat{G}^{-1}(\alpha_1), \; \infty] \;\;\;\;\;\; (26) $$ and a one-sided upper confidence interval is given by: $$ [0, \; \hat{G}^{-1}(\alpha_2)] \;\;\;\;\;\; (27) $$ where $\alpha_1$ and $\alpha_2$ are computed as for a two-sided confidence interval, except $\alpha/2$ is replaced with $\alpha$ in Equations (21) and (22). 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 $\mu$ with respect to the true value of $\mu$ (Efron and Tibshirani, 1993, p.186). For a normal (Gaussian) distribution, the standard error of the estimate of $\mu$ does not depend on the value of $\mu$, hence the acceleration constant is not really necessary. When `ci.method="bootstrap"`, the function `egammaAltCensored` computes both the percentile method and bias-corrected and accelerated method bootstrap confidence intervals. ## Returns a list of class `"estimateCensored"` containing the estimated parameters and other information. See `estimateCensored.object` for details. ## References Cohen, A.C. (1963). Progressively Censored Samples in Life Testing. **Technometrics** 5 , 327--339 Cohen, A.C. (1991). **Truncated and Censored Samples**. Marcel Dekker, New York, New York, 312pp. Cox, D.R. (1970). **Analysis of Binary Data**. Chapman & Hall, London. 142pp. 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. Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). **Statistical Distributions, Fourth Edition**. John Wiley and Sons, Hoboken, NJ. Helsel, D.R. (2012). **Statistics for Censored Environmental Data Using Minitab and R, Second Edition**. John Wiley & Sons, Hoboken, New Jersey. Johnson, N.L., S. Kotz, and N. Balakrishnan. (1994). **Continuous Univariate Distributions, Volume 1**. Second Edition. John Wiley and Sons, New York, Chapter 17. Millard, S.P., P. Dixon, and N.K. Neerchal. (2014; in preparation). **Environmental Statistics with R**. CRC Press, Boca Raton, Florida. Nelson, W. (1982). **Applied Life Data Analysis**. John Wiley and Sons, New York, 634pp. Royston, P. (2007). Profile Likelihood for Estimation and Confdence Intervals. **The Stata Journal** 7 (3), pp. 376--387. 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. Stryhn, H., and J. Christensen. (2003). **Confidence Intervals by the Profile Likelihood Method, with Applications in Veterinary Epidemiology**. Contributed paper at ISVEE X (November 2003, Chile). [https://gilvanguedes.com/wp-content/uploads/2019/05/Profile-Likelihood-CI.pdf](https://gilvanguedes.com/wp-content/uploads/2019/05/Profile-Likelihood-CI.pdf). Venzon, D.J., and S.H. Moolgavkar. (1988). A Method for Computing Profile-Likelihood-Based Confidence Intervals. **Journal of the Royal Statistical Society, Series C (Applied Statistics)** 37 (1), pp. 87--94. ## Author(s) Steven P. Millard (EnvStats@ProbStatInfo.com ) ## 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 for parameters of a normal or lognormal distribution 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 (or coefficient of variation), rather than rely on a single point-estimate of the mean. 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 coefficient of variation of a gamma distribution when data are subjected to single or multiple censoring. See, for example, Singh et al. (2006). ## See Also `egammaCensored`, GammaDist , `egamma`, `estimateCensored.object`. ## Examples ```r # Chapter 15 of USEPA (2009) gives several examples of estimating the mean # and standard deviation of a lognormal distribution on the log-scale using # manganese concentrations (ppb) in groundwater at five background wells. # In EnvStats these data are stored in the data frame # EPA.09.Ex.15.1.manganese.df. # Here we will estimate the mean and coefficient of variation # ON THE ORIGINAL SCALE using the MLE and # assuming a gamma distribution. # First look at the data: #----------------------- EPA.09.Ex.15.1.manganese.df # Sample Well Manganese.Orig.ppb Manganese.ppb Censored #1 1 Well.1 <5 5.0 TRUE #2 2 Well.1 12.1 12.1 FALSE #3 3 Well.1 16.9 16.9 FALSE #... #23 3 Well.5 3.3 3.3 FALSE #24 4 Well.5 8.4 8.4 FALSE #25 5 Well.5 <2 2.0 TRUE longToWide(EPA.09.Ex.15.1.manganese.df, "Manganese.Orig.ppb", "Sample", "Well", paste.row.name = TRUE) # Well.1 Well.2 Well.3 Well.4 Well.5 #Sample.1 <5 <5 <5 6.3 17.9 #Sample.2 12.1 7.7 5.3 11.9 22.7 #Sample.3 16.9 53.6 12.6 10 3.3 #Sample.4 21.6 9.5 106.3 <2 8.4 #Sample.5 <2 45.9 34.5 77.2 <2 # Now estimate the mean and coefficient of variation # using the MLE, and compute a confidence interval # for the mean using the profile-likelihood method. #--------------------------------------------------- with(EPA.09.Ex.15.1.manganese.df, egammaAltCensored(Manganese.ppb, Censored, ci = TRUE)) #Results of Distribution Parameter Estimation #Based on Type I Censored Data #-------------------------------------------- # #Assumed Distribution: Gamma # #Censoring Side: left # #Censoring Level(s): 2 5 # #Estimated Parameter(s): mean = 19.664797 # cv = 1.252936 # #Estimation Method: MLE # #Data: Manganese.ppb # #Censoring Variable: Censored # #Sample Size: 25 # #Percent Censored: 24% # #Confidence Interval for: mean # #Confidence Interval Method: Profile Likelihood # #Confidence Interval Type: two-sided # #Confidence Level: 95% # #Confidence Interval: LCL = 12.25151 # UCL = 34.35332 #---------- # Compare the confidence interval for the mean # based on assuming a lognormal distribution versus # assuming a gamma distribution. with(EPA.09.Ex.15.1.manganese.df, elnormAltCensored(Manganese.ppb, Censored, ci = TRUE))$interval$limits # LCL UCL #12.37629 69.87694 with(EPA.09.Ex.15.1.manganese.df, egammaAltCensored(Manganese.ppb, Censored, ci = TRUE))$interval$limits # LCL UCL #12.25151 34.35332 ```

egammaAltCensored function