vartest2
Two-sample F-test for equal variances
Syntax
Description
returns
a test decision for the null hypothesis that the data in vectors h
= vartest2(x
,y
)x
and y
comes
from normal distributions with the same variance, using the two-sample F-test.
The alternative hypothesis is that they come from normal distributions
with different variances. 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 two-sample F-test with
additional options specified by one or more name-value pair arguments.
For example, you can change the significance level or conduct a one-sided
test.h
= vartest2(x
,y
,Name,Value
)
Examples
Test for Equal Variances
Load the sample data. Create vectors containing the first and second columns of the data matrix to represent students' grades on two exams.
load examgrades;
x = grades(:,1);
y = grades(:,2);
Test the null hypothesis that the data in x
and y
comes from distributions with the same variance.
[h,p,ci,stats] = vartest2(x,y)
h = 1
p = 0.0019
ci = 2×1
1.2383
2.5494
stats = struct with fields:
fstat: 1.7768
df1: 119
df2: 119
The returned result h = 1
indicates that vartest2
rejects the null hypothesis at the default 5% significance level. ci
contains the lower and upper boundaries of the 95% confidence interval for the true variance ratio. stats
contains the value of the test statistic for the -test and the numerator and denominator degrees of freedom.
One-Sided Hypothesis Test
Load the sample data. Create vectors containing the first and second columns of the data matrix to represent students' grades on two exams.
load examgrades;
x = grades(:,1);
y = grades(:,2);
Test the null hypothesis that the data in x
and y
comes from distributions with the same variance, against the alternative that the population variance of x
is greater than that of y
.
vartest2(x,y,'Tail','right')
ans = 1
The returned result h = 1
indicates that vartest2
rejects the null hypothesis at the default 5% significance level, in favor of the alternative hypothesis that the population variance of x
is greater than that of y
.
Input Arguments
x
— Sample data
vector | matrix | multidimensional array
Sample data, specified as a vector, matrix, or multidimensional array.
If
x
andy
are vectors, they do not need to be the same length.If
x
andy
are matrices, they must have the same number of columns, but do not need to have the same number of rows.vartest2
performs separate tests along each column and returns a vector of the results.If
x
andy
are multidimensional arrays, they must have the same number of dimensions, and the same size along all but the first nonsingleton dimension.
Data Types: single
| double
y
— Sample data
vector | matrix | multidimensional array
Sample data, specified as a vector, matrix, or multidimensional array.
If
x
andy
are vectors, they do not need to be the same length.If
x
andy
are matrices, they must have the same number of columns, but do not need to have the same number of rows.vartest2
performs separate tests along each column and returns a vector of the results.If
x
andy
are multidimensional arrays, they must have the same number of dimensions, and the same size along all but the first nonsingleton dimension.
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: 'Tail','right','Alpha',0.01
specifies
a right-tailed hypothesis test at the 1% significance level.
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
Dim
— Dimension
first nonsingleton dimension (default) | positive integer value
Dimension of the input matrix to test along, specified as the
comma-separated pair consisting of 'Dim'
and a
positive integer value. For example, specifying 'Dim',1
tests
the data in each column for variance equality, while 'Dim',2
tests
the data in each row.
Example: 'Dim',2
Data Types: single
| double
Tail
— Type of alternative hypothesis
'both'
(default) | 'right'
| 'left'
Type of alternative hypothesis to evaluate using the F-test,
specified as the comma-separated pair consisting of 'Tail'
and
one of the following.
'both' | Test the alternative hypothesis that the population variances are not equal. |
'right' | Test the alternative hypothesis that the population variance
of x is greater than that of y . |
'left' | Test the alternative hypothesis that the population variance
of x is less than that of y . |
Example: 'Tail','right'
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.
ci
— Confidence interval
vector
Confidence interval for the true ratio of the population variances,
returned as a two-element vector containing the lower and upper boundaries
of the 100 × (1 – Alpha
)% confidence
interval.
stats
— Test statistics
structure
Test statistics for the hypothesis test, returned as a structure containing:
fstat
— Value of the test statistic.df1
— Numerator degrees of freedom of the test.df2
— Denominator degrees of freedom of the test.
More About
Two-Sample F-Test
The two-sample F-test is used to test if the variances of two populations are equal.
The test statistic is
where s1 and s2 are the sample standard deviations. The test statistic is a ratio of the two sample variances. The further this ratio deviates from 1, the more likely you are to reject the null hypothesis. Under the null hypothesis, the test statistic F has a F-distribution with numerator degrees of freedom equal to N1 – 1 and denominator degrees of freedom equal to N2 – 1, where N1 and N2 are the sample sizes of the two data sets.
Multidimensional Array
A multidimensional array has more than two
dimensions. For example, if x
is a 1-by-3-by-4
array, then x
is a three-dimensional array.
First Nonsingleton Dimension
The first nonsingleton dimension is the first
dimension of an array whose size is not equal to 1. For example, if x
is
a 1-by-2-by-3-by-4 array, then the second dimension is the first nonsingleton
dimension of x
.
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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