| Title: | Exact Multinomial Test: Goodness-of-Fit Test for Discrete Multivariate Data |
|---|---|
| Description: | Goodness-of-fit tests for discrete multivariate data. It is tested if a given observation is likely to have occurred under the assumption of an ab-initio model. Monte Carlo methods are provided to make the package capable of solving high-dimensional problems. |
| Authors: | Uwe Menzel |
| Maintainer: | Uwe Menzel <[email protected]> |
| License: | GPL |
| Version: | 1.3.1 |
| Built: | 2025-02-21 04:44:14 UTC |
| Source: | https://github.com/cran/EMT |
The package provides functions to carry out a Goodness-of-fit test for discrete multivariate data.
It is tested if a given observation is likely to have occurred under the assumption of an ab-initio model.
A p-value can be calculated using different distance measures between observed and expected frequencies.
A Monte Carlo method is provided to make the package capable of solving high-dimensional problems.
The main user functions are multinomial.test and plotMultinom.
| Package: | CCP |
| Type: | Package |
| Version: | 1.3 |
| Date: | 2013-02-06 |
| License: | GPL |
Uwe Menzel
Maintainer: Uwe Menzel <[email protected]>
Internal functions for the EMT package
ExactMultinomialTest(observed, prob, size, groups, numEvents) ExactMultinomialTestChisquare(observed, prob, size, groups, numEvents) MonteCarloMultinomialTest(observed, prob, size, groups, numEvents, ntrial, atOnce) MonteCarloMultinomialTestChisquare(observed, prob, size, groups, numEvents, ntrial, atOnce) chisqStat(observed,expected) findVectors(groups,size)ExactMultinomialTest(observed, prob, size, groups, numEvents) ExactMultinomialTestChisquare(observed, prob, size, groups, numEvents) MonteCarloMultinomialTest(observed, prob, size, groups, numEvents, ntrial, atOnce) MonteCarloMultinomialTestChisquare(observed, prob, size, groups, numEvents, ntrial, atOnce) chisqStat(observed,expected) findVectors(groups,size)
observed |
vector describing the observation: contains the observed numbers of items in each category. |
prob |
vector describing the model: contains the hypothetical probabilities corresponding to each category. |
expected |
vector containing the expected numbers of items in each category under the assumption that the model is valid. |
size |
sample size, sum of the components of the vector |
groups |
number of categories in the experiment. |
numEvents |
number of possible outcomes of the experiment. |
ntrial |
number of simulated samples in the Monte Carlo approach. |
atOnce |
a parameter of more technical nature. Determines how much memory is used for big arrays. |
These functions are not intended to be called by the user.
Goodness-of-fit tests for discrete multivariate data. It is tested if a given observation is likely to have occurred under the assumption of an ab-initio model. Monte Carlo methods are provided to make the function capable of solving high-dimensional problems.
multinomial.test(observed, prob, useChisq = FALSE, MonteCarlo = FALSE, ntrial = 1e6, atOnce = 1e6)multinomial.test(observed, prob, useChisq = FALSE, MonteCarlo = FALSE, ntrial = 1e6, atOnce = 1e6)
observed |
vector describing the observation: contains the observed numbers of items in each category. |
prob |
vector describing the model: contains the hypothetical probabilities corresponding to each category. |
useChisq |
if |
MonteCarlo |
if |
ntrial |
number of simulated samples in the Monte Carlo approach. |
atOnce |
a parameter of more technical nature. Determines how much memory is used for big arrays. |
The Exact Multinomial Test is a Goodness-of-fit test for discrete multivariate data.
It is tested if a given observation is likely to have occurred under the assumption of an ab-initio model.
In the experimental setup belonging to the test, n items fall into k categories with certain probabilities
(sample size n with k categories).
The observation, described by the vector observed, indicates how many items have been observed in each category.
The model, determined by the vector prob, assigns to each category the hypothetical probability that an item falls into it.
Now, if the observation is unlikely to have occurred under the assumption of the model, it is advisible to
regard the model as not valid. The p-value estimates how likely the observation is, given the model.
In particular, low p-values suggest that the model is not valid.
The default approach used by multinomial.test obtains the p-values by
calculating the exact probabilities of all possible outcomes given n and k,
using the multinomial probability distribution function dmultinom provided by R.
Then, by default, the p-value is obtained by summing the probabilities of all outcomes which are less likely
than the observed outcome (or equally likely as the observed outcome), i.e. by summing all
(distance measure based on probabilities).
Alternatively, the p-value can be obtained by summing the probabilities of all outcomes connected with a chisquare no smaller than
the chisquare connected with the actual observation (distance measure based on chisquare).
The latter is triggered by setting useChisq = TRUE.
Having a sample of size n in an experiment with k categories, the number of distinct
possible outcomes is the binomial coefficient choose(n+k-1,k-1). This number grows rapidly with increasing parameters n and
k. If the parameters grow too big, numerical calculation might fail because of time or
memory limitations.
In this case, usage of a Monte Carlo approach provided by multinomial.test is suggested.
A Monte Carlo approach, activated by setting MonteCarlo = TRUE,
simulates withdrawal of ntrial samples of size n from the hypothetical distribution specified by the vector prob.
The default value for ntrial is 100000 but might be incremented for big n and k.
The advantage of the Monte Carlo approach is that memory requirements and running time are essentially determined by ntrial
but not by n or k.
By default, the p-value is then obtained by summing the relative frequencies of occurrence of unusual outcomes, i.e. of
outcomes occurring less frequently than the observed one (or equally frequent as the observed one).
Alternatively, as above, Pearson's chisquare can be used as a distance measure by setting useChisq = TRUE.
The parameter atOnce is of more technical nature, with a default value of . This value should be decremented
for computers with low memory to avoid overflow, and can be incremented for large-CPU computers to speed up calculations.
The parameter is only effective for Monte Carlo calculations.
id |
textual description of the method used. |
size |
sample size n, equals the sum of the components of the vector |
groups |
number of categories k in the experiment, equals the number of components of the vector |
numEvents |
number of different events for the model considered. |
stat |
textual description of the distance measure used. |
allProb |
vector containing the probabilities (rel. frequencies for the Monte Carlo approach) of all possible outcomes (might be huge for big n and k). |
criticalValue |
the critical value of the hypothesis test. |
ntrial |
number of trials if the Monte Carlo approach was used, |
p.value |
the calculated p-value rounded to four significant digits. |
For two categories (), the test is called Exact Binomial Test.
Uwe Menzel <[email protected]>
H. Bayo Lawal (2003) Categorical data analysis with SAS and SPSS applications, Volume 1, Chapter 3 ISBN: 978-0-8058-4605-8
Read, T. R. C. and Cressie, N. A. C. (1988). Goodness-of-fit statistics for discrete multivariate data. Springer, New York.
The Multinomial Distribution: dmultinom
## Load the EMT package: library(EMT) ## Input data for a three-dimensional case: observed <- c(5,2,1) prob <- c(0.25, 0.5, 0.25) ## Calculate p-value using default options: out <- multinomial.test(observed, prob) # p.value = 0.0767 ## Plot the probabilities for each event: plotMultinom(out) ## Calculate p-value for the same input using Pearson's chisquare: out <- multinomial.test(observed, prob, useChisq = TRUE) # p.value = 0.0596 ; not the same! ## Test the hypothesis that all sides of a dice have the same probabilities: prob <- rep(1/6, 6) observed <- c(4, 5, 2, 7, 0, 1) out <- multinomial.test(observed, prob) # p.value = 0.0357 -> better get another dice! # the same problem using a Monte Carlo approach: ## Not run: out <- multinomial.test(observed, prob, MonteCarlo = TRUE, ntrial = 5e+6) ## End(Not run)## Load the EMT package: library(EMT) ## Input data for a three-dimensional case: observed <- c(5,2,1) prob <- c(0.25, 0.5, 0.25) ## Calculate p-value using default options: out <- multinomial.test(observed, prob) # p.value = 0.0767 ## Plot the probabilities for each event: plotMultinom(out) ## Calculate p-value for the same input using Pearson's chisquare: out <- multinomial.test(observed, prob, useChisq = TRUE) # p.value = 0.0596 ; not the same! ## Test the hypothesis that all sides of a dice have the same probabilities: prob <- rep(1/6, 6) observed <- c(4, 5, 2, 7, 0, 1) out <- multinomial.test(observed, prob) # p.value = 0.0357 -> better get another dice! # the same problem using a Monte Carlo approach: ## Not run: out <- multinomial.test(observed, prob, MonteCarlo = TRUE, ntrial = 5e+6) ## End(Not run)
This function takes the results of multinomial.test as input and plots the calculated probability
distribution.
plotMultinom(listMultinom)plotMultinom(listMultinom)
listMultinom |
a list created by running the function |
The function plotMultinom displays a barplot of the probabilities for the individual events.
The probabilities are shown in descending order from the left to the right.
Events contributing to the p-value are marked red.
Plots are only made if the number of different events is lower than or equal to 100 and for low number of trials in Monte Carlo simulations.
The first argument (listMultinom) is returned without modification.
Uwe Menzel <[email protected]>
The Multinomial Distribution: multinomial.test
## Load the EMT package: library(EMT) ## input and calculation of p-values: observed <- c(5,2,1) prob <- c(0.25, 0.5, 0.25) out <- multinomial.test(observed, prob) # p.value = 0.0767 ## Plot the probability distribution: plotMultinom(out)## Load the EMT package: library(EMT) ## input and calculation of p-values: observed <- c(5,2,1) prob <- c(0.25, 0.5, 0.25) out <- multinomial.test(observed, prob) # p.value = 0.0767 ## Plot the probability distribution: plotMultinom(out)