knitr::opts_chunk$set( collapse = TRUE, comment = "#>", error = TRUE )
library(Rgof) Bsim = c(100, 200) #Number of Simulation Runs
The package Rgof brings together a number of routines for the goodness-of-fit problem for univariate data. We have a data set $\pmb{x}$, and we want to test whether it was generated by the probability distribution F.
The highlights of this package are:
set.seed(123)
Note all runs of the test routine are done with B=1000 and all runs of the power routines with arguments B=c(500, 500), maxProcessor = 2 in order to pass devtools::check().
Continuous Data
Kolmogorov Smirnov (KS) [@massey1951], [@kolmogorov1933], [@smirnov1948]
For all of these tests the distribution of the test statistic under the null hypothesis is found via simulation.
There is a very large literature on chi square tests, the oldest of the goodness of fit tests. For a survey see [@rolke2020].
All the methods above are also implemented for discrete data, except for Zhang's tests, which have no discrete analog.
It is worth noting that these discrete versions are based on the theoretical ideas of the tests and not on the actual formula of calculation for the continuous case. The test statistics can therefore be different even when applied to the same data. For example, the Anderson-Darling test is based on the distance measure
$$A^2=n\int_{-\infty}^{\infty} \frac{(\hat{F}(x)-F(x))^2}{F(x)(1-F(x))}dF(x) $$ where $F$ is the theoretical distribution function under the null hypothesis and $\hat{F}$ is the empirical distribution function. In the case of continuous data it can be shown that
$$A^2=-n-\frac1n\sum_{i=1}^n (2i-1)\left(\log F(x_i) +\log[1-F(x_{n+1-i})\right)$$ However, for discrete data we have
$$A^2=n\sum_{i=1}^k \frac{(\hat{F}(x_i)-F(x_i))^2}{F(x_i)(1-F(x_i))}\left(F(x_i)-F(x_{i-1}\right)$$
with $F(x_0)=0$.
In the continuous case $\hat{F}$ is a step function but $F$ is continuous, and therefore $A^2>0$. In the discrete case however$A^2=0$ is possible. This shows that the two cases are fundamentally different.
As for continuous data null distributions are found using simulation. In fact in the case of discrete data none of the tests has a known distribution for the test statistic under the null hypothesis.
These methods can be used for both discrete and histogram data. The main difference between these two is that discrete data has (a countable) number of possible values whereas histogram data has possible ranges of values (the bins). The only method directly affected by this difference is Wassp1, which requires actual values. All other methods ignore the vals argument.
We generate a data set of size 1000 from a Binomial distribution with n=20 and success probability 0.5, and then test $H_0:F=Bin(20, 0.5)$.
vals=0:20 #possible values of random variable pnull=function() pbinom(0:20, 20, 0.5) # cumulative distribution function (cdf) rnull = function() table(c(0:20, rbinom(1000, 20, 0.5)))-1 # generate data under the null hypothesis, make sure that vector of counts has same length as vals, possibly 0.
x = rnull() # Basic Test gof_test(x, vals, pnull, rnull, B=1000) #Test with adjusted overall p value gof_test_adjusted_pvalue(x, vals, pnull, rnull, B=c(1000, 500))
x = table(c(0:20, rbinom(1000, 20, 0.55)))-1 #true p is 0.55, not 0.5 # Basic Test gof_test(x, vals, pnull, rnull, B=1000, doMethod = "all")$p.value #Test with adjusted overall p value gof_test_adjusted_pvalue(x, vals, pnull, rnull, B=c(1000, 500))
Arguments of gof_test for discrete data/model:
x: vector with counts (histogram heights). Should have a number for each value of vals, possibly 0.
vals: all possible values of discrete random variable, that is all x with $P(X=x)>0$
pnull: function to find values of cumulative distribution function for each value of vals. Function has no arguments.
rnull: function to generate data from true density. Function has no arguments. Function needs to insure that output is a vector with same length as vals.
B=5000: number of simulation runs
w: function to find importance sampling weights, if needed
phat: function to estimate parameters
TS: function to find values of user-supplied test statistics
TSextra: a list that is passed to TS if any additional info is required.
nbins=c(50, 10): number of bins for chi square tests. The first one is already given by the data in the discrete case, the for the second bins are joined.
rate=0, if not 0 sample size is assumed to have come from a Poisson random variable with rate "rate".
minexpcount=5, minimal expected counts for chi square tests.
ChiUsePhat=TRUE, if TRUE uses phat for parameter estimation. If false uses method of minimum chi square
maxProcessors=1 if greater than 1 number of cores for parallel processing.
doMethods="all" names of methods to include
The arguments of gof_test_adjusted_pvalue for discrete data/model are the same, except that the number of simulation runs B is two numbers. The first is used for estimating the individual p values, the second for the adjustment.
In some fields like high energy physics it is common that the sample size is not fixed but a random variable drawn from a Poisson distribution with a known rate. Our package easily accomodates that:
rnull = function() table(c(0:20, rbinom(rpois(1, 650), 20, 0.5)))-1 x = rnull() gof_test(x, vals, pnull, rnull, rate=650, B=1000)$p.value
We generate a data set of size 1000 from a binomial distribution with n=20 and success probability p, and then test F=Bin(20, .). p is estimated from data.
vals=0:20 pnull=function(p=0.5) pbinom(0:20, 20, ifelse(p>0&&p<1, p, 0.5)) rnull = function(p=0.5) table(c(0:20, rbinom(1000, 20, p)))-1 phat = function(x) sum(0:20*x)/sum(x)/20
x = table(c(0:20, rbinom(1000, 20, 0.5)))-1 gof_test(x, vals, pnull, rnull, phat=phat, B=1000)$p.value
x = table(c(0:20, rbinom(1000, 20, 0.55)))-1 # p is not 0.5, but data is still from a binomial distribution with n=20 gof_test(x, vals, pnull, rnull, phat=phat, B=1000)$p.value
x = table(c(rep(0:20, 5), rbinom(1000-21*5, 20, 0.53))) # data has to many small and large values to be from a binomial gof_test(x, vals, pnull, rnull, phat=phat, B=1000)$p.value
The estimation of the parameter(s) in the case of the chi square tests is done either by using the function phat or via the minimum chi square method. The routine uses a general function minimizer. If there are values of the parameter that are not possible this can lead to warnings. It is best to put a check into the pnull function to avoid this issue. As an example the function pnull above checks that the success probability p is in the interval $(0,1)$.
A variant of discrete data sometimes encountered is data given in the form of a histogram, that is as a set of bins and their counts. The main distinction is that discrete data has specific values, for example the non-negative integers for a Poisson distribution, whereas histogram data has ranges of numbers, the bins. It turns out that, though, that the only method that requires actual values is Wassp1, and for that method one can use the midpoint of the intervals.
As an example consider the following case: we have histogram data and we want to test whether it comes from an exponential rate 1 distribution, truncated to the interval 0-2:
rnull = function() { y = rexp(2500, 1) # Exp(1) data y = y[y<2][1:1500] # 1500 events on 0-2 bins = 0:40/20 # binning hist(y, bins, plot=FALSE)$counts # find bin counts } x = rnull() bins = 0:40/20 vals = (bins[-1]+bins[-21])/2 pnull = function() { bins = 1:40/20 pexp(bins, 1)/pexp(2, 1) } gof_test(x, vals, pnull, rnull)$p.value
pnull = function(x) pnorm(x) rnull = function() rnorm(1000) TSextra = list(qnull=function(x) qnorm(x)) #optional quantile function used by chi square tests and Wassp1 test.
x = rnorm(1000) #Basic Tests gof_test(x, NA, pnull, rnull, B=1000, TSextra=TSextra)$p.value #Adjusted p value gof_test_adjusted_pvalue(x, NA, pnull, rnull, B=c(1000,500), TSextra=TSextra)
x = rnorm(1000, 0.5) gof_test(x, NA, pnull, rnull, B=1000, TSextra=TSextra)$p.value
pnull = function(x, p=0) pnorm(x, p) TSextra = list(qnull = function(x, p=0) qnorm(x, p)) rnull = function(p) rnorm(1000, p) phat = function(x) mean(x)
x = rnorm(1000) gof_test(x, NA, pnull, rnull, phat=phat, TSextra=TSextra, B=1000)$p.value
x = rnorm(1000, 0.5) gof_test(x, NA, pnull, rnull, phat=phat, TSextra=TSextra)$p.value
x = rnorm(1000, 0.5, 2) gof_test(x, NA, pnull, rnull, phat=phat, TSextra=TSextra, B=1000)$p.value
pnull = function(x, p=c(0, 1)) pnorm(x, p[1], ifelse(p[2]>0, p[2], 0.001)) TSextra = list(qnull = function(x, p=c(0, 1)) qnorm(x, p[1], ifelse(p[2]>0, p[2], 0.001))) rnull = function(p=c(0, 1)) rnorm(1000, p[1], ifelse(p[2]>0, p[2], 0.001)) phat = function(x) c(mean(x), sd(x))
x = rnorm(1000) gof_test(x, NA, pnull, rnull, phat=phat, TSextra=TSextra, B=1000)$p.value
x = rnorm(1000, 0.5) gof_test(x, NA, pnull, rnull, phat=phat, TSextra=TSextra, B=1000)$p.value
x = rnorm(1000, 0.5, 2) gof_test(x, NA, pnull, rnull, phat=phat, TSextra=TSextra, B=1000)$p.value
x = rt(1000, 2) gof_test(x, NA, pnull, rnull, phat=phat, TSextra=TSextra, B=1000)$p.value
For estimating the power of the various tests one also has to provide the routine ralt, which generates data under the alternative hypothesis:
vals = 0:10 pnull = function() pbinom(0:10, 10, 0.5) rnull =function () table(c(0:10, rbinom(100, 10, 0.5)))-1 ralt =function (p=0.5) table(c(0:10, rbinom(100, 10, p)))-1 P=gof_power(pnull, vals, rnull, ralt, param_alt=seq(0.5, 0.6, 0.02), B=Bsim, nbins=c(11, 5)) plot_power(P, "p", Smooth=FALSE)
In all cases the arguments are the same as for gof_test. In addition we now have
ralt: a routine with one parameter that generates data under some alternative hypothesis.
param_alt: values to be passed to ralt. This allows the calculation of the power for many different values.
alpha=0.05 type I error probability for tests.
B=c(1000, 1000) the first number is the number of simulation runs for power estimation and the second the number of runs to be used to find the null distribution.
vals = 0:10 pnull = function(p=0.5) pbinom(0:10, 10, ifelse(0<p&p<1,p,0.001)) rnull = function (p=0.5) table(c(0:10, rbinom(100, 10, ifelse(0<p&p<1,p,0.001))))-1 phat = function(x) sum(0:10*x)/1000
ralt =function (p=0.5) table(c(0:10, rbinom(100, 10, p)))-1 gof_power(pnull, vals, rnull, ralt, c(0.5, 0.6), phat=phat, B=Bsim, nbins=c(11, 5), maxProcessors = 2)
Note that power estimation in the case of a composite hypothesis (aka with parameters estimated) is much slower than the simple hypothesis case.
ralt =function (p=0.5) table(c(rep(0:10, 2), rbinom(100, 10, p))) gof_power(pnull, vals, rnull, ralt, 0.5, phat=phat, B=Bsim, nbins=c(11, 5), maxProcessors = 2)
pnull = function(x) pnorm(x) TSextra = list(qnull = function(x) qnorm(x)) rnull = function() rnorm(100) ralt = function(mu=0) rnorm(100, mu) gof_power(pnull, NA, rnull, ralt, c(0, 1), TSextra=TSextra, B=Bsim)
pnull = function(x, p=c(0,1)) pnorm(x, p[1], ifelse(p[2]>0, p[2], 0.01)) TSextra = list(qnull = function(x, p=c(0,1)) qnorm(x, p[1], ifelse(p[2]>0, p[2], 0.01))) rnull = function(p=c(0,1)) rnorm(500, p[1], p[2]) ralt = function(mu=0) rnorm(100, mu) phat = function(x) c(mean(x), sd(x)) gof_power(pnull, NA, rnull, ralt, c(0, 1), phat= phat, TSextra=TSextra, B=Bsim, maxProcessor=2)
ralt = function(df=1) { # t distribution truncated at +- 5 x=rt(1000, df) x=x[abs(x)<5] x[1:100] } gof_power(pnull, NA, rnull, ralt, c(2, 50), phat=phat, Range=c(-5,5), TSextra=TSextra, B=Bsim, maxProcessor=2)
Its is very easy for a user to add other goodness-of-fit tests to the package. This can be done by editing the routines TS_cont and/or TS_disc, which are located in the folder inst/examples in the Rgof library folder. Or a user can write their own version of these files.
Example
Say we wish to use a test that is a variant of the Cramer-vonMises test, using the integrated absolute difference of the empirical and the theoretical distribution function:
$$\int_{-\infty}^{\infty} \vert F(x) - \hat{F}(x) \vert dF(x)$$ For continuous data we have the routine
newTScont = function(x, Fx) { Fx=sort(Fx) n=length(x) out = sum(abs( (2*1:n-1)/2/n-Fx )) names(out) = "CvM alt" out }
This routine has to have two arguments x and Fx. Note that the return object has to be a named vector. The object TSextra can be used to provide further information to the TS routine, if necessary.
Then we can run this test with
pnull = function(x) punif(x) rnull = function() runif(500) x = rnull() Rgof::gof_test(x, NA, pnull, rnull, TS=newTScont)
Say we want to find the power of this test when the true distribution is a linear:
ralt = function(slope=0) { if(slope==0) y=runif(500) else y=(slope-1+sqrt((1-slope)^2+4*slope* runif(500)))/2/slope }
gof_power(pnull, NA, rnull, ralt, TS=newTScont, param_alt=round(seq(0, 0.5, length=5), 3), Range=c(0,1), B=Bsim)
for discrete data we write will the routine using Rcpp:
#include <Rcpp.h> using namespace Rcpp; // [[Rcpp::export]] NumericVector newTSdisc(IntegerVector x, NumericVector Fx, NumericVector vals) { Rcpp::CharacterVector methods=CharacterVector::create("CvM alt"); int const nummethods=methods.size(); int k=x.size(), n, i; NumericVector TS(nummethods), ecdf(k); double tmp; TS.names() = methods; n=0; for(i=0;i<k;++i) n = n + x[i]; ecdf(0) = double(x(0))/double(n); for(i=1;i<k;++i) { ecdf(i) = ecdf(i-1) + x(i)/double(n); } tmp = std::abs(ecdf[0]-Fx(0))*Fx(0); for(i=1;i<k;++i) tmp = tmp + std::abs(ecdf(i)-Fx(i))*(Fx(i)-Fx(i-1)); TS(0) = tmp; return TS; }
Again the routine has to have three arguments x, Fx and vals and the output vector has to have names.
Note that one drawback of writing the routine in Rcpp is that it is then not possible to use multiple processors.
vals=1:50/51 pnull = function() (1:50)/50 rnull = function() c(rmultinom(1, 500, rep(1/50,50))) x = rnull() gof_test(x, vals, pnull, rnull, TS=newTSdisc)
and for power calculations we can run
ralt = function(slope=0) { if(slope==0) p=rep(1/50, 50) else p=diff(slope * (0:50/50)^2 + (1 - slope) * 0:50/50) c(rmultinom(1, 500, p)) } gof_power(pnull, vals, rnull, ralt, TS=newTSdisc, param_alt=round(seq(0, 0.5, length=5), 3), B=Bsim)
Sometimes the data/model uses importance sampling weights. This can be done as follows. Say we want to test whether the data comes from a standard normal distribution, truncated to [-3,3] and with weights from a t distribution with 3 degrees of freedom:
$H_0: F=N(0,1)$, $X\sim t(3)$
df=3 pnull=function(x) pnorm(x)/(2*pnorm(3)-1) rnull=function() {x=rt(2000, df);x=x[abs(x)<3];sort(x[1:1000])} w=function(x) (dnorm(x)/(2*pnorm(3)-1))/(dt(x,df)/(2*pt(3,df)-1)) x=sort(rnull()) plot(x, w(x), type="l", ylim=c(0, 2*max(w(x)))) ralt=function(m=0) {x=rt(2000,df)+m;x=x[abs(x)<3];sort(x[1:1000])} set.seed(111) gof_power(pnull, NA, rnull, ralt, w=w, param_alt = c(0,0.2), Range=c(-3,3),B=Bsim)
It should be noted that these tests are quite sensitive to the size of the weights and to the sample size, so one should always do simulation study to verify that they work in the case under consideration.
Any scripts or data that you put into this service are public.
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.