Goodness-of-Fit Test for a Specified Probability Distribution for Groups
Perform 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, ...)
object |
an object containing data for 2 or more groups to be compared to the
hypothesized distribution specified by |
data |
when |
subset |
when |
na.action |
when |
group |
when |
test |
character string defining which goodness-of-fit test to perform on each group.
Possible values are:
|
distribution |
a character string denoting the distribution abbreviation. See the help file for
When When When When When |
est.arg.list |
a list of arguments to be passed to the function estimating the distribution parameters
for each group of observations.
For example, if When When When When |
n.classes |
for the case when |
cut.points |
for the case when |
param.list |
for the case when |
estimate.params |
for the case when |
n.param.est |
for the case when |
correct |
for the case when |
digits |
a scalar indicating how many significant digits to print out for the parameters
associated with the hypothesized distribution. The default value is |
exact |
for the case when |
ws.method |
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 NOTE: In the case where you are testing whether each group comes from a
Uniform [0,1] distribution (i.e., when you set
|
data.name |
character string indicating the name of the data used for the goodness-of-fit tests.
The default value is |
group.name |
character string indicating the name of the data used to create the groups.
The default value is |
parent.of.data |
character string indicating the source of the data used for the goodness-of-fit tests. |
subset.expression |
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.
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).
Steven P. Millard (EnvStats@ProbStatInfo.com)
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.
# 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
# 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.
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) = 0.2401720
#
#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()Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.