adtest
Anderson-Darling test
Description
returns a test
decision for the null hypothesis that the data in vector h
= adtest(x
)x
is
from a population with a normal distribution, using the
Anderson-Darling test. The alternative hypothesis is that
x
is not from a population with a normal distribution. The
result h
is 1
if the test rejects the null
hypothesis at the 5% significance level, or 0
otherwise.
returns
a test decision for the Anderson-Darling test with additional options
specified by one or more name-value pair arguments. For example, you
can specify a null distribution other than normal, or select an alternative
method for calculating the p-value.h
= adtest(x
,Name,Value
)
Examples
Anderson-Darling Test for a Normal Distribution
Load the sample data. Create a vector containing the first column of the students' exam grades data.
load examgrades
x = grades(:,1);
Test the null hypothesis that the exam grades come from a normal distribution. You do not need to specify values for the population parameters.
[h,p,adstat,cv] = adtest(x)
h = logical
0
p = 0.1854
adstat = 0.5194
cv = 0.7470
The returned value of h = 0
indicates that adtest
fails to reject the null hypothesis at the default 5% significance level.
Anderson-Darling Test for Extreme Value Distribution
Load the sample data. Create a vector containing the first column of the students' exam grades data.
load examgrades
x = grades(:,1);
Test the null hypothesis that the exam grades come from an extreme value distribution. You do not need to specify values for the population parameters.
[h,p] = adtest(x,'Distribution','ev')
h = logical
0
p = 0.0714
The returned value of h = 0
indicates that adtest
fails to reject the null hypothesis at the default 5% significance level.
Anderson-Darling Test Using Specified Probability Distribution
Load the sample data. Create a vector containing the first column of the students' exam grades data.
load examgrades
x = grades(:,1);
Create a normal probability distribution object with mean mu = 75
and standard deviation sigma = 10
.
dist = makedist('normal','mu',75,'sigma',10)
dist = NormalDistribution Normal distribution mu = 75 sigma = 10
Test the null hypothesis that x
comes from the hypothesized normal distribution.
[h,p] = adtest(x,'Distribution',dist)
h = logical
0
p = 0.4687
The returned value of h = 0
indicates that adtest
fails to reject the null hypothesis at the default 5% significance level.
Input Arguments
x
— Sample data
vector
Sample data, specified as a vector. Missing observations in x
,
indicated by NaN
, are ignored.
Data Types: single
| double
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: 'Alpha',0.01,'MCTol',0.01
conducts
the hypothesis test at the 1% significance level, and determines the
p-value, p
, using a Monte Carlo simulation with
a maximum Monte Carlo standard error for p
of 0.01.
Distribution
— Hypothesized distribution
'norm'
(default) | 'exp'
| 'ev'
| 'logn'
| 'weibull'
| probability distribution object
Hypothesized distribution of data vector x
,
specified as the comma-separated pair consisting of 'Distribution'
and
one of the following.
'norm' | Normal distribution |
'exp' | Exponential distribution |
'ev' | Extreme value distribution |
'logn' | Lognormal distribution |
'weibull' | Weibull distribution |
In this case, you do not need to specify population parameters.
Instead, adtest
estimates the distribution parameters
from the sample data and tests x
against a composite
hypothesis that it comes from the selected distribution family with
parameters unspecified.
Alternatively, you can specify any continuous probability distribution
object for the null distribution. In this case, you must specify all
the distribution parameters, and adtest
tests x
against
a simple hypothesis that it comes from the given distribution with
its specified parameters.
Example: 'Distribution','exp'
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
MCTol
— Maximum Monte Carlo standard error
positive scalar value
Maximum Monte
Carlo standard error for the p-value, p
,
specified as the comma-separated pair consisting of 'MCTol'
and
a positive scalar value. If you use MCTol
, adtest
determines p
using
a Monte Carlo simulation, and the name-value pair argument Asymptotic
must
have the value false
.
Example: 'MCTol',0.01
Data Types: single
| double
Asymptotic
— Method for calculating p-value
false
(default) | true
Method for calculating the p-value of the
Anderson-Darling test, specified as the comma-separated pair consisting
of 'Asymptotic'
and either true
or false
.
If you specify 'true'
, adtest
estimates
the p-value using the limiting distribution of
the Anderson-Darling test statistic. If you specify false
, adtest
calculates
the p-value based on an analytical formula. For
sample sizes greater than 120, the limiting distribution estimate
is likely to be more accurate than the small sample size approximation
method.
If you specify a distribution family with unknown parameters for the
Distribution
name-value pair,Asymptotic
must befalse
.If you use
MCTol
to calculate the p-value using a Monte Carlo simulation,Asymptotic
must befalse
.
Example: 'Asymptotic',true
Data Types: logical
Output Arguments
h
— Hypothesis test result
1
| 0
Hypothesis test result, returned as a logical value.
If
h
= 1
, this indicates the rejection of the null hypothesis at theAlpha
significance level.If
h
= 0
, this 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 Anderson-Darling 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. p
is
calculated using one of these methods:
If the hypothesized distribution is a fully specified probability distribution object,
adtest
calculatesp
analytically. If'Asymptotic'
istrue
,adtest
uses the asymptotic distribution of the test statistic. If you specify a value for'MCTol'
,adtest
uses a Monte Carlo simulation.If the hypothesized distribution is specified as a distribution family with unknown parameters,
adtest
retrieves the critical value from a table and uses inverse interpolation to determine the p-value. If you specify a value for'MCTol'
,adtest
uses a Monte Carlo simulation.
adstat
— Test statistic
scalar value
Test statistic for the Anderson-Darling test, returned as a scalar value.
If the hypothesized distribution is a fully specified probability distribution object,
adtest
computesadstat
using specified parameters.If the hypothesized distribution is specified as a distribution family with unknown parameters,
adtest
computesadstat
using parameters estimated from the sample data.
cv
— Critical value
scalar value
Critical value for the Anderson-Darling test at the significance
level Alpha
, returned as a scalar value. adtest
determines cv
by
interpolating into a table based on the specified Alpha
significance
level.
More About
Anderson-Darling Test
The Anderson-Darling test is commonly used to test whether a data sample comes from a normal distribution. However, it can be used to test for another hypothesized distribution, even if you do not fully specify the distribution parameters. Instead, the test estimates any unknown parameters from the data sample.
The test statistic belongs to the family of quadratic empirical distribution function statistics, which measure the distance between the hypothesized distribution, F(x) and the empirical cdf, Fn(x) as
over the ordered sample values , where w(x) is a weight function and n is the number of data points in the sample.
The weight function for the Anderson-Darling test is
which places greater weight on the observations in the tails of the distribution, thus making the test more sensitive to outliers and better at detecting departure from normality in the tails of the distribution.
The Anderson-Darling test statistic is
where are the ordered sample data points and n is the number of data points in the sample.
In adtest
, the decision to reject or not
reject the null hypothesis is based on comparing the p-value
for the hypothesis test with the specified significance level, not
on comparing the test statistic with the critical value.
Monte Carlo Standard Error
The Monte Carlo standard error is the error due to simulating the p-value.
The Monte Carlo standard error is calculated as
where is
the estimated p-value of the hypothesis test, and mcreps
is
the number of Monte Carlo replications performed.
adtest
chooses the number of Monte Carlo
replications, mcreps
, large enough to make the
Monte Carlo standard error for less
than the value specified for MCTol
.
Version History
Introduced in R2013a
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