distChoose.RdPerform a series of goodness-of-fit tests from a (possibly user-specified) set of candidate probability distributions to determine which probability distribution provides the best fit for a data set.
distChoose(y, ...)
# S3 method for class 'formula'
distChoose(y, data = NULL, subset,
na.action = na.pass, ...)
# Default S3 method
distChoose(y, alpha = 0.05, method = "sw",
choices = c("norm", "gamma", "lnorm"), est.arg.list = NULL,
warn = TRUE, keep.data = TRUE, data.name = NULL,
parent.of.data = NULL, subset.expression = NULL, ...)an object containing data for the goodness-of-fit tests. In the default
method, the argument y must be numeric vector of observations.
In the formula method, y must be a formula of the form y ~ 1.
Missing (NA), undefined (NaN), and
infinite (Inf, -Inf) values are allowed but will be
removed.
specifies an optional data frame, list or environment (or object coercible
by as.data.frame to a data frame) containing the variables in the
model. If not found in data, the variables are taken from
environment(formula), typically the environment from which
distChoose is called.
specifies an optional vector specifying a subset of observations to be used.
specifies a function which indicates what should happen when the data contain NAs.
The default is na.pass.
numeric scalar between 0 and 1 specifying the Type I error associated with each
goodness-of-fit test. When method="proucl" the only allowed values for
alpha are 0.01, 0.05, and 0.1. The default value is alpha=0.05.
character string defining which method to use. Possible values are:
"sw". Shapiro-Wilk; the default.
"sf". Shapiro-Francia.
"ppcc". Probability Plot Correlation Coefficient.
"proucl". ProUCL.
See the DETAILS section below.
a character vector denoting the distribution abbreviations of the candidate
distributions. See the help file for Distribution.df for a list
of distributions and their abbreviations.
The default value is choices=c("norm", "gamma", "lnorm"),
indicating the Normal, Gamma, and Lognormal distributions.
This argument is ignored when method="proucl".
a list containing one or more lists of arguments to be passed to the
function(s) estimating the distribution parameters. The name(s) of
the components of the list must be equal to or a subset of the values of the
argument choices. For example, if
choices=c("norm", "gamma", "lnorm"), setting est.arg.list=list(gamma = list(method="bcmle")) indicates
using the bias-corrected maximum-likelihood estimators of shape and scale
for the gamma distribution (see the help file for egamma).
See the help file Estimating Distribution Parameters for a list of
estimating functions.
The default value is est.arg.list=NULL so that all default values for the
estimating functions are used.
When testing for some form of normality (i.e., Normal, Lognormal, Three-Parameter Lognormal, Zero-Modified Normal, or Zero-Modified Lognormal (Delta)), the estimated parameters are provided in the output merely for information, and the choice of the method of estimation has no effect on the goodness-of-fit test statistics or p-values.
This argument is ignored when method="proucl".
logical scalar indicating whether to print a warning message when
observations with NAs, NaNs, or Infs in
y are removed. The default value is TRUE.
logical scalar indicating whether to return the original data. The
default value is keep.data=TRUE.
optional character string indicating the name of the data used for argument y.
character string indicating the source of the data used for the goodness-of-fit test.
character string indicating the expression used to subset the data.
additional arguments affecting the goodness-of-fit test.
The function distChoose returns a list with information on the goodness-of-fit
tests for various distributions and which distribution appears to best fit the
data based on the p-values from the goodness-of-fit tests. This function was written in
order to compare ProUCL's way of choosing the best-fitting distribution (USEPA, 2015) with
other ways of choosing the best-fitting distribution.
Method Based on Shapiro-Wilk, Shapiro-Francia, or Probability Plot Correlation Test
(method="sw", method="sf", or method="ppcc")
For each value of the argument choices, the function distChoose
runs the goodness-of-fit test using the data in y assuming that particular
distribution. For example, if choices=c("norm", "gamma", "lnorm"),
indicating the Normal, Gamma, and Lognormal distributions, and
method="sw", then the usual Shapiro-Wilk test is performed for the Normal
and Lognormal distributions, and the extension of the Shapiro-Wilk test is performed
for the Gamma distribution (see the section
Testing Goodness-of-Fit for Any Continuous Distribution in the help
file for gofTest for an explanation of the latter). The distribution associated
with the largest p-value is the chosen distribution. In the case when all p-values are
less than the value of the argument alpha, the distribution “Nonparametric” is chosen.
Method Based on ProUCL Algorithm (method="proucl")
When method="proucl", the function distChoose uses the
algorithm that ProUCL (USEPA, 2015) uses to determine the best fitting
distribution. The candidate distributions are the
Normal, Gamma, and Lognormal distributions. The algorithm
used by ProUCL is as follows:
Perform the Shapiro-Wilk and Lilliefors goodness-of-fit tests for the
Normal distribution, i.e., call the function gofTest with
distribution = "norm", test="sw" and distribution = "norm", test="lillie".
If either or both of the associated p-values are greater than or equal to the user-supplied value
of alpha, then choose the Normal distribution. Otherwise, proceed to the next step.
Perform the “ProUCL Anderson-Darling” and “ProUCL Kolmogorov-Smirnov” goodness-of-fit
tests for the Gamma distribution,
i.e., call the function gofTest with distribution="gamma", test="proucl.ad.gamma" and distribution="gamma", test="proucl.ks.gamma".
If either or both of the associated p-values are greater than or equal to the user-supplied value
of alpha, then choose the Gamma distribution. Otherwise, proceed to the next step.
Perform the Shapiro-Wilk and Lilliefors goodness-of-fit tests for the
Lognormal distribution, i.e., call the function gofTest with
distribution="lnorm", test="sw" and distribution="lnorm", test="lillie".
If either or both of the associated p-values are greater than or equal to the user-supplied value
of alpha, then choose the Lognormal distribution. Otherwise, proceed to the next step.
If none of the goodness-of-fit tests above yields a p-value greater than or equal to the user-supplied value
of alpha, then choose the “Nonparametric” distribution.
a list of class "distChoose" containing the results of the goodness-of-fit tests.
Objects of class "distChoose" have a special printing method.
See the help files for distChoose.object for details.
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Gibbons, R.D., D.K. Bhaumik, and S. Aryal. (2009). Statistical Methods for Groundwater Monitoring, Second Edition. John Wiley & Sons, Hoboken.
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Kim, P.J., and R.I. Jennrich. (1973). Tables of the Exact Sampling Distribution of the Two Sample Kolmogorov-Smirnov Criterion. In Harter, H.L., and D.B. Owen, eds. Selected Tables in Mathematical Statistics, Vol. 1. American Mathematical Society, Providence, Rhode Island, pp.79-170.
Kolmogorov, A.N. (1933). Sulla determinazione empirica di una legge di distribuzione. Giornale dell' Istituto Italiano degle Attuari 4, 83-91.
Marsaglia, G., W.W. Tsang, and J. Wang. (2003). Evaluating Kolmogorov's distribution. Journal of Statistical Software, 8(18). doi:10.18637/jss.v008.i18 .
Moore, D.S. (1986). Tests of Chi-Squared Type. In D'Agostino, R.B., and M.A. Stephens, eds. Goodness-of Fit Techniques. Marcel Dekker, New York, pp.63-95.
Pomeranz, J. (1973). Exact Cumulative Distribution of the Kolmogorov-Smirnov Statistic for Small Samples (Algorithm 487). Collected Algorithms from ACM ??, ???-???.
Royston, J.P. (1992a). Approximating the Shapiro-Wilk W-Test for Non-Normality. Statistics and Computing 2, 117-119.
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Royston, P. (1993). A Toolkit for Testing for Non-Normality in Complete and Censored Samples. The Statistician 42, 37-43.
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Shapiro, S.S., and M.B. Wilk. (1965). An Analysis of Variance Test for Normality (Complete Samples). Biometrika 52, 591-611.
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Stephens, M.A. (1986a). Tests Based on EDF Statistics. In D'Agostino, R. B., and M.A. Stevens, eds. Goodness-of-Fit Techniques. Marcel Dekker, New York.
USEPA. (2015). ProUCL Version 5.1.002 Technical Guide. EPA/600/R-07/041, October 2015. Office of Research and Development. U.S. Environmental Protection Agency, Washington, D.C.
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In practice, almost any goodness-of-fit test will not reject the null hypothesis
if the number of observations is relatively small. Conversely, almost any goodness-of-fit
test will reject the null hypothesis if the number of observations is very large,
since “real” data are never distributed according to any theoretical distribution
(Conover, 1980, p.367). For most cases, however, the distribution of “real” data
is close enough to some theoretical distribution that fairly accurate results may be
provided by assuming that particular theoretical distribution. One way to asses the
goodness of the fit is to use goodness-of-fit tests. Another way is to look at
quantile-quantile (Q-Q) plots (see qqPlot).
# Generate 20 observations from a gamma distribution with
# parameters shape = 2 and scale = 3 and:
#
# 1) Call distChoose using the Shapiro-Wilk method.
#
# 2) Call distChoose using the Shapiro-Wilk method and specify
# the bias-corrected method of estimating shape for the Gamma
# distribution.
#
# 3) Compare the results in 2) above with the results using the
# ProUCL method.
#
# Notes: The call to set.seed lets you reproduce this example.
#
# The ProUCL method chooses the Normal distribution, whereas the
# Shapiro-Wilk method chooses the Gamma distribution.
set.seed(47)
dat <- rgamma(20, shape = 2, scale = 3)
# 1) Call distChoose using the Shapiro-Wilk method.
#--------------------------------------------------
distChoose(dat)
#>
#> Results of Choosing Distribution
#> --------------------------------
#>
#> Candidate Distributions: Normal
#> Gamma
#> Lognormal
#>
#> Choice Method: Shapiro-Wilk
#>
#> Type I Error per Test: 0.05
#>
#> Decision: Gamma
#>
#> Estimated Parameter(s): shape = 1.909462
#> scale = 4.056819
#>
#> Estimation Method: MLE
#>
#> Data: dat
#>
#> Sample Size: 20
#>
#> Test Results:
#>
#> Normal
#> Test Statistic: W = 0.9097488
#> P-value: 0.06303695
#>
#> Gamma
#> Test Statistic: W = 0.9834958
#> P-value: 0.970903
#>
#> Lognormal
#> Test Statistic: W = 0.9185006
#> P-value: 0.09271768
#>
#Results of Choosing Distribution
#--------------------------------
#
#Candidate Distributions: Normal
# Gamma
# Lognormal
#
#Choice Method: Shapiro-Wilk
#
#Type I Error per Test: 0.05
#
#Decision: Gamma
#
#Estimated Parameter(s): shape = 1.909462
# scale = 4.056819
#
#Estimation Method: MLE
#
#Data: dat
#
#Sample Size: 20
#
#Test Results:
#
# Normal
# Test Statistic: W = 0.9097488
# P-value: 0.06303695
#
# Gamma
# Test Statistic: W = 0.9834958
# P-value: 0.970903
#
# Lognormal
# Test Statistic: W = 0.9185006
# P-value: 0.09271768
#--------------------
# 2) Call distChoose using the Shapiro-Wilk method and specify
# the bias-corrected method of estimating shape for the Gamma
# distribution.
#---------------------------------------------------------------
distChoose(dat, method = "sw",
est.arg.list = list(gamma = list(method = "bcmle")))
#>
#> Results of Choosing Distribution
#> --------------------------------
#>
#> Candidate Distributions: Normal
#> Gamma
#> Lognormal
#>
#> Choice Method: Shapiro-Wilk
#>
#> Type I Error per Test: 0.05
#>
#> Decision: Gamma
#>
#> Estimated Parameter(s): shape = 1.656376
#> scale = 4.676680
#>
#> Estimation Method: Bias-Corrected MLE
#>
#> Data: dat
#>
#> Sample Size: 20
#>
#> Test Results:
#>
#> Normal
#> Test Statistic: W = 0.9097488
#> P-value: 0.06303695
#>
#> Gamma
#> Test Statistic: W = 0.9834346
#> P-value: 0.9704046
#>
#> Lognormal
#> Test Statistic: W = 0.9185006
#> P-value: 0.09271768
#>
#Results of Choosing Distribution
#--------------------------------
#
#Candidate Distributions: Normal
# Gamma
# Lognormal
#
#Choice Method: Shapiro-Wilk
#
#Type I Error per Test: 0.05
#
#Decision: Gamma
#
#Estimated Parameter(s): shape = 1.656376
# scale = 4.676680
#
#Estimation Method: Bias-Corrected MLE
#
#Data: dat
#
#Sample Size: 20
#
#Test Results:
#
# Normal
# Test Statistic: W = 0.9097488
# P-value: 0.06303695
#
# Gamma
# Test Statistic: W = 0.9834346
# P-value: 0.9704046
#
# Lognormal
# Test Statistic: W = 0.9185006
# P-value: 0.09271768
#--------------------
# 3) Compare the results in 2) above with the results using the
# ProUCL method.
#---------------------------------------------------------------
distChoose(dat, method = "proucl")
#>
#> Results of Choosing Distribution
#> --------------------------------
#>
#> Candidate Distributions: Normal
#> Gamma
#> Lognormal
#>
#> Choice Method: ProUCL
#>
#> Type I Error per Test: 0.05
#>
#> Decision: Normal
#>
#> Estimated Parameter(s): mean = 7.746340
#> sd = 5.432175
#>
#> Estimation Method: mvue
#>
#> Data: dat
#>
#> Sample Size: 20
#>
#> Test Results:
#>
#> Normal
#> Shapiro-Wilk GOF
#> Test Statistic: W = 0.9097488
#> P-value: 0.06303695
#> Lilliefors (Kolmogorov-Smirnov) GOF
#> Test Statistic: D = 0.1547851
#> P-value: 0.238092
#>
#> Gamma
#> ProUCL Anderson-Darling Gamma GOF
#> Test Statistic: A = 0.1853826
#> P-value: >= 0.10
#> ProUCL Kolmogorov-Smirnov Gamma GOF
#> Test Statistic: D = 0.0988692
#> P-value: >= 0.10
#>
#> Lognormal
#> Shapiro-Wilk GOF
#> Test Statistic: W = 0.9185006
#> P-value: 0.09271768
#> Lilliefors (Kolmogorov-Smirnov) GOF
#> Test Statistic: D = 0.149317
#> P-value: 0.2869177
#>
#Results of Choosing Distribution
#--------------------------------
#
#Candidate Distributions: Normal
# Gamma
# Lognormal
#
#Choice Method: ProUCL
#
#Type I Error per Test: 0.05
#
#Decision: Normal
#
#Estimated Parameter(s): mean = 7.746340
# sd = 5.432175
#
#Estimation Method: mvue
#
#Data: dat
#
#Sample Size: 20
#
#Test Results:
#
# Normal
# Shapiro-Wilk GOF
# Test Statistic: W = 0.9097488
# P-value: 0.06303695
# Lilliefors (Kolmogorov-Smirnov) GOF
# Test Statistic: D = 0.1547851
# P-value: 0.238092
#
# Gamma
# ProUCL Anderson-Darling Gamma GOF
# Test Statistic: A = 0.1853826
# P-value: >= 0.10
# ProUCL Kolmogorov-Smirnov Gamma GOF
# Test Statistic: D = 0.0988692
# P-value: >= 0.10
#
# Lognormal
# Shapiro-Wilk GOF
# Test Statistic: W = 0.9185006
# P-value: 0.09271768
# Lilliefors (Kolmogorov-Smirnov) GOF
# Test Statistic: D = 0.149317
# P-value: 0.2869177
#--------------------
# Clean up
#---------
rm(dat)
#====================================================================
# Example 10-2 of USEPA (2009, page 10-14) gives an example of
# using the Shapiro-Wilk test to test the assumption of normality
# for nickel concentrations (ppb) in groundwater collected over
# 4 years. The data for this example are stored in
# EPA.09.Ex.10.1.nickel.df.
EPA.09.Ex.10.1.nickel.df
#> Month Well Nickel.ppb
#> 1 1 Well.1 58.8
#> 2 3 Well.1 1.0
#> 3 6 Well.1 262.0
#> 4 8 Well.1 56.0
#> 5 10 Well.1 8.7
#> 6 1 Well.2 19.0
#> 7 3 Well.2 81.5
#> 8 6 Well.2 331.0
#> 9 8 Well.2 14.0
#> 10 10 Well.2 64.4
#> 11 1 Well.3 39.0
#> 12 3 Well.3 151.0
#> 13 6 Well.3 27.0
#> 14 8 Well.3 21.4
#> 15 10 Well.3 578.0
#> 16 1 Well.4 3.1
#> 17 3 Well.4 942.0
#> 18 6 Well.4 85.6
#> 19 8 Well.4 10.0
#> 20 10 Well.4 637.0
# Month Well Nickel.ppb
#1 1 Well.1 58.8
#2 3 Well.1 1.0
#3 6 Well.1 262.0
#4 8 Well.1 56.0
#5 10 Well.1 8.7
#6 1 Well.2 19.0
#7 3 Well.2 81.5
#8 6 Well.2 331.0
#9 8 Well.2 14.0
#10 10 Well.2 64.4
#11 1 Well.3 39.0
#12 3 Well.3 151.0
#13 6 Well.3 27.0
#14 8 Well.3 21.4
#15 10 Well.3 578.0
#16 1 Well.4 3.1
#17 3 Well.4 942.0
#18 6 Well.4 85.6
#19 8 Well.4 10.0
#20 10 Well.4 637.0
# Use distChoose with the probability plot correlation method,
# and for the lognormal distribution specify the
# mean and CV parameterization:
#------------------------------------------------------------
distChoose(Nickel.ppb ~ 1, data = EPA.09.Ex.10.1.nickel.df,
choices = c("norm", "gamma", "lnormAlt"), method = "ppcc")
#>
#> Results of Choosing Distribution
#> --------------------------------
#>
#> Candidate Distributions: Normal
#> Gamma
#> Lognormal
#>
#> Choice Method: PPCC
#>
#> Type I Error per Test: 0.05
#>
#> Decision: Lognormal
#>
#> Estimated Parameter(s): mean = 213.415628
#> cv = 2.809377
#>
#> Estimation Method: mvue
#>
#> Data: Nickel.ppb
#>
#> Data Source: EPA.09.Ex.10.1.nickel.df
#>
#> Sample Size: 20
#>
#> Test Results:
#>
#> Normal
#> Test Statistic: r = 0.8199825
#> P-value: 5.753418e-05
#>
#> Gamma
#> Test Statistic: r = 0.9749044
#> P-value: 0.317334
#>
#> Lognormal
#> Test Statistic: r = 0.9912528
#> P-value: 0.9187852
#>
#Results of Choosing Distribution
#--------------------------------
#
#Candidate Distributions: Normal
# Gamma
# Lognormal
#
#Choice Method: PPCC
#
#Type I Error per Test: 0.05
#
#Decision: Lognormal
#
#Estimated Parameter(s): mean = 213.415628
# cv = 2.809377
#
#Estimation Method: mvue
#
#Data: Nickel.ppb
#
#Data Source: EPA.09.Ex.10.1.nickel.df
#
#Sample Size: 20
#
#Test Results:
#
# Normal
# Test Statistic: r = 0.8199825
# P-value: 5.753418e-05
#
# Gamma
# Test Statistic: r = 0.9749044
# P-value: 0.317334
#
# Lognormal
# Test Statistic: r = 0.9912528
# P-value: 0.9187852
#--------------------
# Repeat the above example using the ProUCL method.
#--------------------------------------------------
distChoose(Nickel.ppb ~ 1, data = EPA.09.Ex.10.1.nickel.df,
method = "proucl")
#>
#> Results of Choosing Distribution
#> --------------------------------
#>
#> Candidate Distributions: Normal
#> Gamma
#> Lognormal
#>
#> Choice Method: ProUCL
#>
#> Type I Error per Test: 0.05
#>
#> Decision: Gamma
#>
#> Estimated Parameter(s): shape = 0.5198727
#> scale = 326.0894272
#>
#> Estimation Method: MLE
#>
#> Data: Nickel.ppb
#>
#> Data Source: EPA.09.Ex.10.1.nickel.df
#>
#> Sample Size: 20
#>
#> Test Results:
#>
#> Normal
#> Shapiro-Wilk GOF
#> Test Statistic: W = 0.6788888
#> P-value: 2.17927e-05
#> Lilliefors (Kolmogorov-Smirnov) GOF
#> Test Statistic: D = 0.3267052
#> P-value: 5.032807e-06
#>
#> Gamma
#> ProUCL Anderson-Darling Gamma GOF
#> Test Statistic: A = 0.5076725
#> P-value: >= 0.10
#> ProUCL Kolmogorov-Smirnov Gamma GOF
#> Test Statistic: D = 0.1842904
#> P-value: >= 0.10
#>
#> Lognormal
#> Shapiro-Wilk GOF
#> Test Statistic: W = 0.978946
#> P-value: 0.9197735
#> Lilliefors (Kolmogorov-Smirnov) GOF
#> Test Statistic: D = 0.08405167
#> P-value: 0.9699648
#>
#Results of Choosing Distribution
#--------------------------------
#
#Candidate Distributions: Normal
# Gamma
# Lognormal
#
#Choice Method: ProUCL
#
#Type I Error per Test: 0.05
#
#Decision: Gamma
#
#Estimated Parameter(s): shape = 0.5198727
# scale = 326.0894272
#
#Estimation Method: MLE
#
#Data: Nickel.ppb
#
#Data Source: EPA.09.Ex.10.1.nickel.df
#
#Sample Size: 20
#
#Test Results:
#
# Normal
# Shapiro-Wilk GOF
# Test Statistic: W = 0.6788888
# P-value: 2.17927e-05
# Lilliefors (Kolmogorov-Smirnov) GOF
# Test Statistic: D = 0.3267052
# P-value: 5.032807e-06
#
# Gamma
# ProUCL Anderson-Darling Gamma GOF
# Test Statistic: A = 0.5076725
# P-value: >= 0.10
# ProUCL Kolmogorov-Smirnov Gamma GOF
# Test Statistic: D = 0.1842904
# P-value: >= 0.10
#
# Lognormal
# Shapiro-Wilk GOF
# Test Statistic: W = 0.978946
# P-value: 0.9197735
# Lilliefors (Kolmogorov-Smirnov) GOF
# Test Statistic: D = 0.08405167
# P-value: 0.9699648
#====================================================================
if (FALSE) { # \dontrun{
# 1) Simulate 1000 trials where for each trial you:
# a) Generate 20 observations from a Gamma distribution with
# parameters mean = 10 and CV = 1.
# b) Use distChoose with the Shapiro-Wilk method.
# c) Use distChoose with the ProUCL method.
#
# 2) Compare the proportion of times the
# Normal vs. Gamma vs. Lognormal vs. Nonparametric distribution
# is chosen for b) and c) above.
#------------------------------------------------------------------
set.seed(58)
N <- 1000
Choose.fac <- factor(rep("", N), levels = c("Normal", "Gamma", "Lognormal", "Nonparametric"))
Choose.df <- data.frame(SW = Choose.fac, ProUCL = Choose.fac)
for(i in 1:N) {
dat <- rgammaAlt(20, mean = 10, cv = 1)
Choose.df[i, "SW"] <- distChoose(dat, method = "sw")$decision
Choose.df[i, "ProUCL"] <- distChoose(dat, method = "proucl")$decision
}
summaryStats(Choose.df, digits = 0)
# ProUCL(N) ProUCL(Pct) SW(N) SW(Pct)
#Normal 443 44 41 4
#Gamma 546 55 733 73
#Lognormal 9 1 215 22
#Nonparametric 2 0 11 1
#Combined 1000 100 1000 100
#--------------------
# Repeat above example for the Lognormal Distribution with mean=10 and CV = 1.
#-----------------------------------------------------------------------------
set.seed(297)
N <- 1000
Choose.fac <- factor(rep("", N), levels = c("Normal", "Gamma", "Lognormal", "Nonparametric"))
Choose.df <- data.frame(SW = Choose.fac, ProUCL = Choose.fac)
for(i in 1:N) {
dat <- rlnormAlt(20, mean = 10, cv = 1)
Choose.df[i, "SW"] <- distChoose(dat, method = "sw")$decision
Choose.df[i, "ProUCL"] <- distChoose(dat, method = "proucl")$decision
}
summaryStats(Choose.df, digits = 0)
# ProUCL(N) ProUCL(Pct) SW(N) SW(Pct)
#Normal 313 31 15 2
#Gamma 556 56 254 25
#Lognormal 121 12 706 71
#Nonparametric 10 1 25 2
#Combined 1000 100 1000 100
#--------------------
# Clean up
#---------
rm(N, Choose.fac, Choose.df, i, dat)
} # }