Goodness-of-Fit Test for a Specified Probability Distribution for Groups

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Description

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).

Usage

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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, ...)

Arguments

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.

data

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.

subset

when object is a formula, subset specifies an optional vector specifying a subset of observations to be used.

na.action

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.

group

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.

test

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).

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.

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 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.

n.classes

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).

cut.points

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] 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.

param.list

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.

estimate.params

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.

n.param.est

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.

correct

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.

digits

a scalar indicating how many significant digits to print out for the parameters associated with the hypothesized distribution. The default value is
.Options$digits.

exact

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.

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 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.

data.name

character string indicating the name of the data used for the goodness-of-fit tests. The default value is data.name=deparse(substitute(object)).

group.name

character string indicating the name of the data used to create the groups. The default value is group.name=deparse(substitute(group)).

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.

Details

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.

Value

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.

Note

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).

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

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.

See Also

gofTest, gofGroup.object, print.gofGroup, plot.gofGroup, qqPlot.

Examples

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  # 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()

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