SGB (version 1.0.1.1)

GoodnessFit: Goodness of fit tests on the marginal distributions of each part in a SGB model

Description

Kolmogorov-Smirnov goodness of fit tests
Cramer-von-Mises goodness of fit tests.

Usage

ks.SGB(u,shape1,shape2,scale,alpha=0.05)
cvm.SGB(u,shape1,shape2,scale,alpha=0.05)
# S3 method for testSGB
print(x,...)

Value

A list of class 'testSGB' with the following components:

method

either "One-sample Kolmogorov-Smirnov test" or "Cramer-von Mises test of goodness-of-fit"

Compositions

name of the dataset u

tests

data frame with \(D\) rows and 3 columns: test statistics for each part against the beta distribution and corresponding p-values and cutoff. Any p-value smaller than the cutoff means that the assumption of the beta distribution for all the margins is rejected.

A print method exists for the class "testSGB".

Arguments

u

data matrix of compositions (independent variables) \((N \times D)\); \(D\): number of parts

shape1

positive number, overall shape parameter of the SGB distribution. See SGBdistrib.

shape2

vector of length \( D\), Dirichlet shape parameters of the SGB distribution. See SGBdistrib.

scale

matrix of the same dimensions as u, containing the shape compositions, or positive number if the scales of all parts are identical. See SGBdistrib.

alpha

overall level of the test, default 0.05.

x

an object of class "testSGB".

...

further arguments passed to or from other methods.

Details

ks.SGB calls ks.test and cvm.SGB calls cvm.test.
Consider \(z=C[(u/scale)^{shape1}]\), where \(C[.]\) is the closure operation. The scale compositions \(scale\) may be modelled with auxiliary variables. Under the SGB hypothesis, the components of \( z\) should be marginally beta-distributed. The functions provide \( D\) tests, one for each part.
Theoretically, the parameters should be known and not estimated on the data. Thus the test using estimated parameters is conservative.
The cutoff value is based on the false discovery rate for multiple comparisons (Benjamini and Hochberg, 1995), which is simply alpha*i/D for the i-th ordered p-value, i=1,...,D (number of tests). Reject the null hypothesis if at least one p-value is smaller than the cutoff. The overall level is then alpha. The proof of the result does not use an independence assumption between the tests.

References

Benjamini, Y. and Y. Hochberg (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. Journal of the Royal Statistical Society. Series B (Methodological) 57 (1), 289-300.

Birnbaum, Z. W. and Fred H. Tingey (1951), One-sided confidence contours for probability distribution functions. The Annals of Mathematical Statistics, 22/4, 592-596.

Conover, William J. (1971), Practical Nonparametric Statistics. New York: John Wiley & Sons. Pages 295-301 (one-sample Kolmogorov test), 309-314 (two-sample Smirnov test).

Csorgo, S. and Faraway, J.J. (1996) The exact and asymptotic distributions of Cramer-von Mises statistics. Journal of the Royal Statistical Society, Series B 58, 221-234.

Durbin, J. (1973), Distribution theory for tests based on the sample distribution function. SIAM.

Marsaglia, G., Wai Wan Tsang and Jingbo Wang (2003), Evaluating Kolmogorov's distribution. Journal of Statistical Software, 8/18.

See Also

SGBdistrib for the theoretical distribution, oilr for the regression results.

Examples

## Generate 1000 random variates according to SGB(shape1,rep(1/3,3),shape2)
shape1 <- 0.6
shape2 <- c(10,20, 30)
rnum <- rSGB(1000,shape1,rep(1,3)/3,shape2)
ks.SGB(rnum,shape1=shape1, shape2=shape2,scale=1)
## same result as
ks.SGB(rnum,shape1=shape1,scale= matrix(rep(1/3,3000),ncol=3), shape2=shape2)
library(goftest)
cvm.SGB(rnum,shape1=shape1,scale= matrix(rep(1/3,3000),ncol=3), shape2=shape2)

## Arctic lake data

# oilr is a SGB regression object, see \code{\link{oilr}}.
data(oilr)       # regSGB object
data(arc)
ua <- arc[1:3]   # compositions

## Kolmogorov-Smirnov goodness of fit test
ks.SGB(ua,shape1=oilr[["par"]][1],shape2=oilr[["par"]][4:6],scale=oilr[["scale"]])
## Rounding shape1 affects the results less than rounding shape2.
ks.SGB(ua,shape1=round(oilr[["par"]][1],3),shape2=round(oilr[["par"]][4:6],1),
	 scale=oilr[["scale"]])
ks.SGB(ua,shape1=round(oilr[["par"]][1],1),shape2=round(oilr[["par"]][4:6],3),
	 scale=oilr[["scale"]])

## Cramer-von-Mises goodness of fit test
library(goftest)
cvm.SGB(ua,shape1=oilr[["par"]][1],shape2=oilr[["par"]][4:6],scale=oilr[["scale"]])