One of the assumptions underlying Analysis of Variance 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
Add
and AddRange
methods of the Samples property 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()
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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"))
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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))
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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 |
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The median of the data is used as the location measure. This works best for normal data. |
|
|
The mean of the data is used as the location measure. This gives better results when the data is
skewed. |
|
|
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 the Add and AddRange()()() methods of the Samples property 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"))
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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))
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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.