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Testing Homogeneity of Variances
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Testing Homogeneity of Variances
One of the assumptions underlying Analysis of Variances is that
the variances accross groups are identical. This property is called
homogeneity of variances. It is often desirable to verify this
assumption using an appropriate hypothesis test. The Extreme
Optimization Numerical Libraries for .NET provides two such
tests: Bartlett's test and Levene's test.
Bartlett's Test
Bartlett's test is a relatively fast test for homogeneity of
variances. The test is based on the assumption that the samples are
normally distributed. It is sensitive to violations of this
assumption. In practical terms, this means that Bartlett's test
cannot adequately distinguish between violation of homogeneity of
variances and violation of the normality assumption.
The null hypothesis is always that the variances of all groups
are equal. The alternative hypothesis is that at least one of the
variances is different. Bartlett's test is always one-tailed, and
uses a chi-square statistic.
Bartlett's test is implemented by the BartlettTest
class. It has two constructors. The first constructor takes no
arguments. The data and conditions for the test must be specified
by setting properties of the BartlettTest object, and
using the AddSample
and AddSamples
methods to specify samples. The second constructor takes an array
of NumericalVariable
objects, that contain the samples the test is to be applied to.
Example
We start with a collection of measurements of gear diameters
from 10 batches. We want to verify that the variances of the
diameters for the batches are equal. The data comes in two
variables: a CategoricalVariable
that contains the batch numbers, and a NumericalVariable
that contains the corresponding measurements of the diameters. The
first step is to create a CellArray. We can
then use the collection's GetCellVariables
method to return an array of variables with the measurements for
each batch.
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CategoricalVariable batchVariable = new CategoricalVariable("batch", new object[] {...});
NumericalVariable diameterVariable =
new NumericalVariable("diameter", new double[] {...});
CellArray cells = new CellArray(diameterVariable, batchVariable);
NumericalVariable[] variables = cells.GetCellVariables(); |
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Dim batchVariable As CategoricalVariable = _
New CategoricalVariable("batch", New Object() {...})
Dim diameterVariable As NumericalVariable = _
New NumericalVariable("diameter", New Double() {...})
Dim cells As CellArray = _
New CellArray(diameterVariable, batchVariable)
Dim variables As NumericalVariable() = cells.GetCellVariables() |
We can then create the BartlettTest object, and run
the test:
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BartlettTest bartlett = new BartlettTest(variables);
Console.WriteLine("Test statistic: {0:F4}", bartlett.Statistic);
Console.WriteLine("P-value: {0:F4}", bartlett.PValue);
Console.WriteLine("Reject null hypothesis? {0}",
bartlett.Reject() ? "yes" : "no"); |
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Dim bartlett As BartlettTest = New BartlettTest(variables)
Console.WriteLine("Test statistic: {0:F4}", bartlett.Statistic)
Console.WriteLine("P-value: {0:F4}", bartlett.PValue)
Console.WriteLine("Reject null hypothesis? {0}", _
IIf(bartlett.Reject(), "yes", "no")) |
The value of the chi-square statistic is 20.7859 giving a
p-value of 0.0136. As a result, the hypothesis that the variances
are equal is rejected at the 0.05 level.
Once a BartlettTest object has been created, you
can access other properties and methods common to all hypothesis
test classes. For instance, to obtain the critical values for a
significance level of 0.01 and 0.05, the code would be:
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Console.WriteLine("Critical value: {0:F4} at 95%",
bartlett.GetUpperCriticalValue(0.05));
Console.WriteLine("Critical value: {0:F4} at 99%",
bartlett.GetUpperCriticalValue(0.01)); |
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Console.WriteLine("Critical value: {0:F4} at 95%", _
bartlett.GetUpperCriticalValue(0.05))
Console.WriteLine("Critical value: {0:F4} at 99%", _
bartlett.GetUpperCriticalValue(0.01)) |
The values of the critical values (16.9190 at 0.05 and 21.6660
at 0.01) show that the null hypothesis will be rejected at the 0.05
level.
Levene's Test
Levene's test is a slower but more robust test for homogeneity
of variances. Levene's test is much less influenced by departures
from normality that Bartlett's test. For this reason, it is often
the test of choice.
As with Bartlett's test, the null hypothesis is always that the
variances of all groups are equal. The alternative hypothesis is
that at least one of the variances is different. Levene's test is
always one-tailed, and uses an F statistic.
Levene's test comes in three flavors, depending on the measure
of location used in the calculation of the statistic. The options
are enumerated by the LeveneTestLocationMeasure
enumeration:
| Value |
Description |
| Median |
The median of the data is used as the location measure. This
works best for normal data. |
| Mean |
The mean of the data is used as the location measure. This
gives better results when the data is skewed. |
| TrimmedMean |
The 10% trimmed mean is used as the location measure. This
gives better results when the data is heavy-tailed. |
If no value is specified, the median is used.
Levene's test is implemented by the LeveneTest
class. It has three constructors. The first constructor takes no
arguments. The data and conditions for the test must be specified
by setting properties of the LeveneTest object, and
using the AddSample
and AddSamples
methods to specify samples. The second constructor takes an array
of NumericalVariable
objects, that contain the samples the test is to be applied to. The
last constructor takes one additional parameter: a LeveneTestLocationMeasure
value that specifies which measure of location to use in the
calculation of the test statistc. This value can also be accessed
and set through the LocationMeasure
property.
Example
We start from the same data as before: a collection of
measurements of gear diameters from 10 batches. We want to verify
that the variances of the diameters for the batches are equal. See
the example with Bartlett's test for an illustration of how to
prepare the data.
Here, we show how to create the LeveneTest object,
and run the test:
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LeveneTest levene = new LeveneTest(variables);
Console.WriteLine("Test statistic: {0:F4}", levene.Statistic);
Console.WriteLine("P-value: {0:F4}", levene.PValue);
Console.WriteLine("Reject null hypothesis? {0}",
levene.Reject() ? "yes" : "no"); |
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Dim levene As LeveneTest = New LeveneTest(variables)
Console.WriteLine("Test statistic: {0:F4}", levene.Statistic)
Console.WriteLine("P-value: {0:F4}", levene.PValue)
Console.WriteLine("Reject null hypothesis? {0}", _
IIf(levene.Reject(), "yes", "no")) |
The value of the F statistic is 1.7059 giving a p-value of
0.0991. As a result, the hypothesis that the variances are equal is
not rejected at the 0.05 level.
The outcome of Levene's test is clearly different from that of
Bartlett's test for the same data. The reason is most likely that
the data are not distributed normally. Bartlett's test cannot
distinguish non-homogeneity from departure from normality.
Once a LeveneTest object has been created, you can
access other properties and methods common to all hypothesis test
classes. For instance, to obtain the critical values for a
significance level of 0.05 and 0.1, the code would be:
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Console.WriteLine("Critical value: {0:F4} at 95%",
levene.GetUpperCriticalValue(0.05));
Console.WriteLine("Critical value: {0:F4} at 90%",
levene.GetUpperCriticalValue(0.1)); |
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Console.WriteLine("Critical value: {0:F4} at 95%", _
levene.GetUpperCriticalValue(0.05))
Console.WriteLine("Critical value: {0:F4} at 90%", _
levene.GetUpperCriticalValue(0.1)) |
The values of the critical values (1.9856 at 0.05 and 1.7021 at
0.01) show that the null hypothesis will not be rejected at the
0.05 level.
Up: Hypothesis Tests Next: Histograms Previous: Testing Goodness-of-Fit Contents
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