Chi-square goodness-of-fit test
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.
h = chi2gof(x,Name,Value)
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.
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.
Load the light bulb lifetime sample data.
load lightbulbCreate 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.
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.
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.
x — Sample dataSample data for the hypothesis test, specified as a vector.
Specify optional
comma-separated pairs of Name,Value arguments. Name is
the argument name and Value is the corresponding value.
Name must appear inside quotes. You can specify several name and value
pair arguments in any order as
Name1,Value1,...,NameN,ValueN.
'NBins',8,'Alpha',0.01 pools the
data into eight bins and conducts the hypothesis test at the 1% significance
level.'NBins' — Number of bins10 (default) | positive integer valueNumber 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 centersBin 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 edgesBin 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 distributionThe 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 using fitdist or specify
them using makedist.
If CDF is a function handle, the
distribution function must take x 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 take x 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 countsExpected 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 parametersNumber 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 of NParams is 0.
If you specify CDF as a cell
array, the default value of NParams is the number
of parameters in the array.
If you specify Expected, the
default value of NParams is 0.
Example: 'NParams',1
Data Types: single | double
'EMin' — Minimum expected count per bin5 (default) | nonnegative integer valueMinimum 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' — FrequencyFrequency 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 level0.05 (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
h — Hypothesis test result1 | 0Hypothesis test result, returned as 1 or 0.
If h = 1,
this indicates the rejection of the null hypothesis at the Alpha significance
level.
If h = 0,
this indicates a failure to reject the null hypothesis at the Alpha significance
level.
p — p-valuep-value of the test, returned as a scalar
value in the range [0,1]. p is the probability
of observing a test statistic as extreme as, or more extreme than,
the observed value under the null hypothesis. Small values of p cast
doubt on the validity of the null hypothesis.
stats — Test statisticsTest 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.
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.
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.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
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