gofGroupTest.RdPerform a goodness-of-fit test to determine whether data in a set of groups appear to all come from the same probability distribution (with possibly different parameters for each group).
gofGroupTest(object, ...)
# S3 method for class 'formula'
gofGroupTest(object, data = NULL, subset,
na.action = na.pass, ...)
# Default S3 method
gofGroupTest(object, group, test = "sw",
distribution = "norm", est.arg.list = NULL, n.classes = NULL,
cut.points = NULL, param.list = NULL,
estimate.params = ifelse(is.null(param.list), TRUE, FALSE),
n.param.est = NULL, correct = NULL, digits = .Options$digits,
exact = NULL, ws.method = "normal scores",
data.name = NULL, group.name = NULL, parent.of.data = NULL,
subset.expression = NULL, ...)
# S3 method for class 'data.frame'
gofGroupTest(object, ...)
# S3 method for class 'matrix'
gofGroupTest(object, ...)
# S3 method for class 'list'
gofGroupTest(object, ...)an object containing data for 2 or more groups to be compared to the
hypothesized distribution specified by distribution. In the default method,
the argument object must be a numeric vector.
When object is a data frame, all columns must be numeric.
When object is a matrix, it must be a numeric matrix.
When object is a list, all components must be numeric vectors.
In the formula method, a symbolic specification of the form y ~ g
can be given, indicating the observations in the vector y are to be grouped
according to the levels of the factor g. Missing (NA), undefined (NaN),
and infinite (Inf, -Inf) values are allowed but will be removed.
when object is a formula, data 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 summaryStats is called.
when object is a formula, subset specifies an optional vector specifying
a subset of observations to be used.
when object is a formula, na.action specifies a function which indicates
what should happen when the data contain NAs. The default is na.pass.
when object is a numeric vector, group is a factor or character vector
indicating which group each observation belongs to. When object is a matrix or data frame
this argument is ignored and the columns define the groups. When object is a list
this argument is ignored and the components define the groups. When object is a formula,
this argument is ignored and the right-hand side of the formula specifies the grouping variable.
character string defining which goodness-of-fit test to perform on each group.
Possible values are:
"sw" (Shapiro-Wilk; the default), "sf" (Shapiro-Francia),
"ppcc" (Probability Plot Correlation Coefficient), "skew" (Zero-skew),
"chisq" (Chi-squared), "ks" (Kolmogorov-Smirnov), and
"ws" (Wilk-Shapiro test for Uniform [0, 1] distribution).
a character string denoting the distribution abbreviation. See the help file for
Distribution.df for a list of distributions and their abbreviations.
The default value is distribution="norm" (Normal distribution).
When test="sw", test="sf", or test="ppcc", any continuous
distribuiton is allowed (e.g., "norm" (normal), "lnorm" (lognormal),
"gamma" (gamma), etc.), as well as mixed distributions involving the normal distribution
(i.e., "zmnorm" (zero-modified normal), "zmlnorm" (zero-modified lognormal (delta)),
and "zmlnorm.alt" (zero-modified lognormal with alternative parameterization)).
When test="skew", only the values "norm" (normal), "lnorm" (lognormal),
"lnorm.alt" (lognormal with alternative parameterization),
"zmnorm" (zero-modified normal), "zmlnorm" (zero-modified lognormal (delta)), and "zmlnorm.alt" (zero-modified lognormal with alternative parameterization) are allowed.
When test="ks", any continuous distribution is allowed.
When test="chisq", any distribuiton is allowed.
When test="ws", this argument is ignored.
a list of arguments to be passed to the function estimating the distribution parameters
for each group of observations.
For example, if test="sw" and distribution="gamma", setting est.arg.list=list(method="bcmle")
indicates using the bias-corrected maximum-likelihood estimators of shape and scale
(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 function are used. This argument is ignored if
estimate.params=FALSE.
When test="sw", test="sf", test="ppcc", or test="skew",
and you are 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.
When test="ks", and estimate.params=TRUE,
the estimated parameters are used to specify the null hypothesis of which distribution
the data are assumed to come from.
When test="chisq" and estimate.params=TRUE,
the estimated parameters are used to specify the null hypothesis of which distribution
the data are assumed to come from.
When test="ws", this argument is ignored.
for the case when test="chisq", the number of cells into which the observations
within each group are to be allocated. If the argument cut.points is supplied,
then n.classes is set to length(cut.points)-1. The default value is ceiling(2* (length(x)^(2/5))) and is recommended by Moore (1986).
for the case when test="chisq", a vector of cutpoints that defines the cells for each
group of observations.
The element x[i] is allocated to cell j if cut.points[j] < x[i] \(\le\) cut.points[j+1]. If x[i] is less than or equal to the first cutpoint or
greater than the last cutpoint, then x[i] is treated as missing. If the
hypothesized distribution is discrete, cut.points must be supplied. The default
value is cut.points=NULL, in which case the cutpoints are determined by
n.classes equi-probable intervals.
for the case when test="ks" or test="chisq",
a list with values for the parameters of the specified distribution. See the help file
for Distribution.df for the names and possible values of the parameters
associated with each distribution. The default value is NULL, which forces
estimation of the distribution parameters. This argument is ignored if
estimate.params=TRUE.
for the case when test="ks" or test="chisq",
a logical scalar indicating whether to perform the goodness-of-fit test based on
estimating the distribution parameters (estimate.params=TRUE) or using the
user-supplied distribution parameters specified by param.list
(estimate.params=FALSE). The default value of estimate.params is TRUE if
param.list=NULL, otherwise it is FALSE.
for the case when test="ks" or test="chisq",
an integer indicating the number of parameters estimated from the data.
If estimate.params=TRUE, the default value is the number of parameters associated
with the distribution specified by distribution (e.g., 2 for a normal distribution).
If estimate.params=FALSE, the default value is n.param.est=0.
for the case when test="chisq", a logical scalar indicating whether to use the
continuity correction. The default value is correct=FALSE unless n.classes=2.
a scalar indicating how many significant digits to print out for the parameters
associated with the hypothesized distribution. The default value is .Options$digits.
for the case when test="ks", exact=NULL by default, but can be set to
a logical scalar indicating whether an exact p-value should be computed.
See the help file for ks.test for more information.
character string indicating which method to use when performing the
Wilk-Shapiro test for a Uniform [0,1] distribution
on the p-values from the goodness-of-fit tests on each group. Possible values
are ws.method="normal scores" (the default) or
ws.method="chi-square scores". See the subsection
Wilk-Shapiro goodness-of-fit test for Uniform [0, 1] distribution
under the DETAILS section of the help file for gofTest
for more information.
NOTE: In the case where you are testing whether each group comes from a
Uniform [0,1] distribution (i.e., when you set
test="ws"), the argument ws.method determines which score types
are used for each individual test of the groups as well.
character string indicating the name of the data used for the goodness-of-fit tests.
The default value is data.name=deparse(substitute(object)).
character string indicating the name of the data used to create the groups.
The default value is group.name=deparse(substitute(group)).
character string indicating the source of the data used for the goodness-of-fit tests.
character string indicating the expression used to subset the data.
additional arguments affecting the goodness-of-fit test.
The function gofGroupTest performs a goodness-of-fit test for each group of
data by calling the function gofTest. Using the p-values from these
goodness-of-fit tests, it then calls the function gofTest with the
argument test="ws" to test whether the p-values appear to come from a
Uniform [0,1] distribution.
a list of class "gofGroup" containing the results of the group goodness-of-fit test.
Objects of class "gofGroup" have special printing and plotting methods.
See the help file for gofGroup.object for details.
Gibbons, R.D., D.K. Bhaumik, and S. Aryal. (2009). Statistical Methods for Groundwater Monitoring, Second Edition. John Wiley & Sons, Hoboken.
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. p.17-17.
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.
Wilk, M.B., and S.S. Shapiro. (1968). The Joint Assessment of Normality of Several Independent Samples. Technometrics, 10(4), 825-839.
The Wilk-Shapiro (1968) tests for a Uniform [0, 1] distribution were introduced in the context
of testing whether several independent samples all come from normal distributions, with
possibly different means and variances. The function gofGroupTest extends
this idea to allow you to test whether several independent samples come from the same
distribution (e.g., gamma, extreme value, etc.), with possibly different parameters.
Examples of simultaneously assessing whether several groups come from the same distribution are given in USEPA (2009) and Gibbons et al. (2009).
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).
# Example 10-4 of USEPA (2009, page 10-20) gives an example of
# simultaneously testing the assumption of normality for nickel
# concentrations (ppb) in groundwater collected at 4 monitoring
# wells over 5 months. 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
# Test for a normal distribution at each well:
#--------------------------------------------
gofGroup.list <- gofGroupTest(Nickel.ppb ~ Well,
data = EPA.09.Ex.10.1.nickel.df)
gofGroup.list
#>
#> Results of Group Goodness-of-Fit Test
#> -------------------------------------
#>
#> Test Method: Wilk-Shapiro GOF (Normal Scores)
#>
#> Hypothesized Distribution: Normal
#>
#> Data: Nickel.ppb
#>
#> Grouping Variable: Well
#>
#> Data Source: EPA.09.Ex.10.1.nickel.df
#>
#> Number of Groups: 4
#>
#> Sample Sizes: Well.1 = 5
#> Well.2 = 5
#> Well.3 = 5
#> Well.4 = 5
#>
#> Test Statistic: z (G) = -3.658696
#>
#> P-values for
#> Individual Tests: Well.1 = 0.03510747
#> Well.2 = 0.02385344
#> Well.3 = 0.01120775
#> Well.4 = 0.10681461
#>
#> P-value for
#> Group Test: 0.0001267509
#>
#> Alternative Hypothesis: At least one group
#> does not come from a
#> Normal Distribution.
#Results of Group Goodness-of-Fit Test
#-------------------------------------
#
#Test Method: Wilk-Shapiro GOF (Normal Scores)
#
#Hypothesized Distribution: Normal
#
#Data: Nickel.ppb
#
#Grouping Variable: Well
#
#Data Source: EPA.09.Ex.10.1.nickel.df
#
#Number of Groups: 4
#
#Sample Sizes: Well.1 = 5
# Well.2 = 5
# Well.3 = 5
# Well.4 = 5
#
#Test Statistic: z (G) = -3.658696
#
#P-values for
#Individual Tests: Well.1 = 0.03510747
# Well.2 = 0.02385344
# Well.3 = 0.01120775
# Well.4 = 0.10681461
#
#P-value for
#Group Test: 0.0001267509
#
#Alternative Hypothesis: At least one group
# does not come from a
# Normal Distribution.
dev.new()
plot(gofGroup.list)
#----------
# Test for a lognormal distribution at each well:
#-----------------------------------------------
gofGroupTest(Nickel.ppb ~ Well, data = EPA.09.Ex.10.1.nickel.df,
dist = "lnorm")
#>
#> Results of Group Goodness-of-Fit Test
#> -------------------------------------
#>
#> Test Method: Wilk-Shapiro GOF (Normal Scores)
#>
#> Hypothesized Distribution: Lognormal
#>
#> Data: Nickel.ppb
#>
#> Grouping Variable: Well
#>
#> Data Source: EPA.09.Ex.10.1.nickel.df
#>
#> Number of Groups: 4
#>
#> Sample Sizes: Well.1 = 5
#> Well.2 = 5
#> Well.3 = 5
#> Well.4 = 5
#>
#> Test Statistic: z (G) = 0.240172
#>
#> P-values for
#> Individual Tests: Well.1 = 0.6898164
#> Well.2 = 0.6700394
#> Well.3 = 0.3208299
#> Well.4 = 0.5041375
#>
#> P-value for
#> Group Test: 0.5949015
#>
#> Alternative Hypothesis: At least one group
#> does not come from a
#> Lognormal Distribution.
#Results of Group Goodness-of-Fit Test
#-------------------------------------
#
#Test Method: Wilk-Shapiro GOF (Normal Scores)
#
#Hypothesized Distribution: Lognormal
#
#Data: Nickel.ppb
#
#Grouping Variable: Well
#
#Data Source: EPA.09.Ex.10.1.nickel.df
#
#Number of Groups: 4
#
#Sample Sizes: Well.1 = 5
# Well.2 = 5
# Well.3 = 5
# Well.4 = 5
#
#Test Statistic: z (G) = 0.2401720
#
#P-values for
#Individual Tests: Well.1 = 0.6898164
# Well.2 = 0.6700394
# Well.3 = 0.3208299
# Well.4 = 0.5041375
#
#P-value for
#Group Test: 0.5949015
#
#Alternative Hypothesis: At least one group
# does not come from a
# Lognormal Distribution.
#----------
# Clean up
rm(gofGroup.list)
graphics.off()