chi2gof
Chi-square goodness-of-fit test
Description
returns
a test decision for the null hypothesis that the data in vector h
= chi2gof(x
)x
comes
from a normal distribution with a mean and variance estimated from x
,
using the chi-square goodness-of-fit
test. The alternative hypothesis is that the data does not
come from such a distribution. The result h
is 1
if
the test rejects the null hypothesis at the 5% significance level,
and 0
otherwise.
returns
a test decision for the chi-square goodness-of-fit test with additional
options specified by one or more name-value pair arguments. For example,
you can test for a distribution other than normal, or change the significance
level of the test.h
= chi2gof(x
,Name,Value
)
Examples
Test for Normal Distribution
Create a standard normal probability distribution object. Generate a data vector x
using random numbers from the distribution.
pd = makedist('Normal'); rng default; % for reproducibility x = random(pd,100,1);
Test the null hypothesis that the data in x
comes from a population with a normal distribution.
h = chi2gof(x)
h = 0
The returned value h = 0
indicates that chi2gof
does not reject the null hypothesis at the default 5% significance level.
Test Hypothesis at Different Significance Level
Create a standard normal probability distribution object. Generate a data vector x
using random numbers from the distribution.
pd = makedist('Normal'); rng default; % for reproducibility x = random(pd,100,1);
Test the null hypothesis that the data in x
comes from a population with a normal distribution at the 1% significance level.
[h,p] = chi2gof(x,'Alpha',0.01)
h = 0
p = 0.3775
The returned value h = 0
indicates that chi2gof
does not reject the null hypothesis at the 1% significance level.
Test for Weibull Distribution Using Probability Distribution Object
Load the light bulb lifetime sample data.
load lightbulb
Create a vector from the first column of the data matrix, which contains the lifetime in hours of the light bulbs.
x = lightbulb(:,1);
Test the null hypothesis that the data in x
comes from a population with a Weibull distribution. Use fitdist
to create a probability distribution object with A
and B
parameters estimated from the data.
pd = fitdist(x,'Weibull'); h = chi2gof(x,'CDF',pd)
h = 1
The returned value h = 1
indicates that chi2gof
rejects the null hypothesis at the default 5% significance level.
Test for Poisson Distribution
Create six bins, numbered 0 through 5, to use for data pooling.
bins = 0:5;
Create a vector containing the observed counts for each bin and compute the total number of observations.
obsCounts = [6 16 10 12 4 2]; n = sum(obsCounts);
Fit a Poisson probability distribution object to the data and compute the expected count for each bin. Use the transpose operator .'
to transform bins
and obsCounts
from row vectors to column vectors.
pd = fitdist(bins','Poisson','Frequency',obsCounts'); expCounts = n * pdf(pd,bins);
Test the null hypothesis that the data in obsCounts
comes from a Poisson distribution with a lambda parameter equal to lambdaHat
.
[h,p,st] = chi2gof(bins,'Ctrs',bins,... 'Frequency',obsCounts, ... 'Expected',expCounts,... 'NParams',1)
h = 0
p = 0.4654
st = struct with fields:
chi2stat: 2.5550
df: 3
edges: [-0.5000 0.5000 1.5000 2.5000 3.5000 5.5000]
O: [6 16 10 12 6]
E: [7.0429 13.8041 13.5280 8.8383 6.0284]
The returned value h = 0
indicates that chi2gof
does not reject the null hypothesis at the default 5% significance level. The vector E
contains the expected counts for each bin under the null hypothesis, and O
contains the observed counts for each bin.
Test for Normal Distribution Using Function Handle
Use the probability distribution function normcdf
as a function handle in the chi-square goodness-of-fit test (chi2gof
).
Test the null hypothesis that the sample data in the input vector x
comes from a normal distribution with parameters µ and σ equal to the mean (mean
) and standard deviation (std
) of the sample data, respectively.
rng('default') % For reproducibility x = normrnd(50,5,100,1); h = chi2gof(x,'cdf',{@normcdf,mean(x),std(x)})
h = 0
The returned result h = 0
indicates that chi2gof
does not reject the null hypothesis at the default 5% significance level.
Input Arguments
x
— Sample data
vector
Sample data for the hypothesis test, specified as a vector.
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: 'NBins',8,'Alpha',0.01
pools the
data into eight bins and conducts the hypothesis test at the 1% significance
level.
NBins
— Number of bins
10
(default) | positive integer value
Number of bins to use for the data pooling, specified as the
comma-separated pair consisting of 'NBins'
and
a positive integer value. If you specify a value for NBins
,
do not specify a value for Ctrs
or Edges
.
Example: 'NBins',8
Data Types: single
| double
Ctrs
— Bin centers
vector
Bin centers, specified as the comma-separated pair consisting
of 'Ctrs'
and a vector of center values for each
bin. If you specify a value for Ctrs
, do not specify
a value for NBins
or Edges
.
Example: 'Ctrs',[1 2 3 4 5]
Data Types: single
| double
Edges
— Bin edges
vector
Bin edges, specified as the comma-separated pair consisting
of 'Edges'
and a vector of edge values for each
bin. If you specify a value for Edges
, do not specify
a value for NBins
or Ctrs
.
Example: 'Edges',[-2.5 -1.5 -0.5 0.5 1.5 2.5]
Data Types: single
| double
CDF
— cdf of hypothesized distribution
probability distribution object | function handle | cell array
The cdf of the hypothesized distribution, specified as the comma-separated
pair consisting of 'CDF'
and a probability distribution
object, function handle, or cell array.
If
CDF
is a probability distribution object, the degrees of freedom account for whether you estimate the parameters usingfitdist
or specify them usingmakedist
.If
CDF
is a function handle, the distribution function must takex
as its only argument.If
CDF
is a cell array, the first element must be a function handle, and the remaining elements must be parameter values, one per cell. The function must takex
as its first argument, and the other parameters in the array as later arguments.
If you specify a value for CDF
, do not specify
a value for Expected
.
Example: 'CDF',pd_object
Data Types: single
| double
Expected
— Expected counts
vector of nonnegative values
Expected counts for each bin, specified as the comma-separated
pair of 'Expected'
and a vector of nonnegative
values. If Expected
depends on estimated parameters,
use NParams
to ensure that chi2gof
correctly
calculates the degrees of freedom. If you specify a value for Expected
,
do not specify a value for CDF
.
Example: 'Expected',[19.1446 18.3789 12.3224 8.2432
4.1378]
Data Types: single
| double
NParams
— Number of estimated parameters
positive integer value
Number of estimated parameters used to describe the null distribution,
specified as the comma-separated pair consisting of 'NParams'
and
a positive integer value. This value adjusts the degrees of freedom
of the test based on the number of estimated parameters used to compute
the cdf or expected counts.
The default value for NParams
depends on
how you specify the null distribution:
If you specify
CDF
as a probability distribution object,NParams
is equal to the number of estimated parameters used to create the object.If you specify
CDF
as a function name or handle, the default value ofNParams
is0
.If you specify
CDF
as a cell array, the default value ofNParams
is the number of parameters in the array.If you specify
Expected
, the default value ofNParams
is0
.
Example: 'NParams',1
Data Types: single
| double
EMin
— Minimum expected count per bin
5
(default) | nonnegative integer value
Minimum expected count per bin, specified as the comma-separated
pair consisting of 'EMin'
and a nonnegative integer
value. If the bin at the extreme end of either tail has an expected
value less than EMin
, it is combined with a neighboring
bin until the count in each extreme bin is at least 5. If any interior
bins have a count less than 5, chi2gof
displays
a warning, but does not combine the interior bins. In that case, you
should use fewer bins, or provide bin centers or edges, to increase
the expected counts in all bins. Specify EMin
as 0
to
prevent the combining of bins.
Example: 'EMin',0
Data Types: single
| double
Frequency
— Frequency
vector of nonnegative integer values
Frequency of data values, specified as the comma-separated pair
consisting of 'Frequency'
and a vector of nonnegative
integer values that is the same length as the vector x
.
Example: 'Frequency',[20 16 13 10 8]
Data Types: single
| double
Alpha
— Significance level
0.05
(default) | scalar value in the range (0,1)
Significance level of the hypothesis test, specified as the
comma-separated pair consisting of 'Alpha'
and
a scalar value in the range (0,1).
Example: 'Alpha',0.01
Data Types: single
| double
Output Arguments
h
— Hypothesis test result
1
| 0
Hypothesis test result, returned as 1
or 0
.
A value of
1
indicates the rejection of the null hypothesis at theAlpha
significance level.A value of
0
indicates a failure to reject the null hypothesis at theAlpha
significance level.
p
— p-value
scalar value in the range [0,1]
p-value of the test, returned as a scalar value in the range [0,1].
p
is the probability of observing a test statistic that is as
extreme as, or more extreme than, the observed value under the null hypothesis. A small
value of p
indicates that the null hypothesis might not be
valid.
stats
— Test statistics
structure
Test statistics, returned as a structure containing the following:
chi2stat
— Value of the test statistic.df
— Degrees of freedom of the test.edges
— Vector of bin edges after pooling.O
— Vector of observed counts for each bin.E
— Vector of expected counts for each bin.
More About
Chi-Square Goodness-of-Fit Test
The chi-square goodness-of-fit test determines if a data sample comes from a specified probability distribution, with parameters estimated from the data.
The test groups the data into bins, calculating the observed and expected counts for those bins, and computing the chi-square test statistic
where Oi are the observed counts and Ei are the expected counts based on the hypothesized distribution. The test statistic has an approximate chi-square distribution when the counts are sufficiently large.
Algorithms
chi2gof
compares the value of the test statistic
to a chi-square distribution with degrees of freedom equal to nbins -
1 - nparams, where nbins is
the number of bins used for the data pooling and nparams is
the number of estimated parameters used to determine the expected
counts. If there are not enough degrees of freedom to conduct the
test, chi2gof
returns the p-value
as NaN
.
Extended Capabilities
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced before R2006a
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