Wednesday, May 28, 2014

Variance Tests: Levene’s and Brown-Forsythe For 2-Factor ANOVA in Excel 2010 and Excel 2013

This is one of the following seven articles on Two-Factor ANOVA With Replication in Excel

Two-Factor ANOVA With Replication in 5 Steps in Excel 2010 and Excel 2013

Variance Tests: Levene’s and Brown-Forsythe For 2-Factor ANOVA in Excel 2010 and Excel 2013

Shapiro-Wilk Normality Test in Excel For 2-Factor ANOVA With Replication

2-Factor ANOVA With Replication Effect Size in Excel 2010 and Excel 2013

Excel Post Hoc Tukey’s HSD Test For 2-Factor ANOVA With Replication

2-Factor ANOVA With Replication – Test Power With G-Power Utility

Scheirer-Ray-Hare Test Alternative For 2-Factor ANOVA With Replication

 

Variance Tests Levene’s

Test and Brown-Forsythe

Test In Excel For Two-

Factor ANOVA With

Replication

Each of the three F Tests of Two-Factor ANOVA With Replication requires that the variances of all sample groups in the same F Test be similar. Sample groups that have similar variances are said to be homoscedastistic. Sample groups that have significantly different variances are said to be heteroscedastistic.

A rule-of-thumb is as follows: Variances are considered similar if the standard deviation of any one group is no more than twice as large as the standard deviation of any other group. This is equivalent to stating that no data group’s variance can be more than four times the variance of another data group in the same F Test. That is the case here as the following are true for the levels of Factor 1 and Factor 2:

If the sample variance, VAR() in Excel, of the data groups at each factor level are calculated, the results are as follows:

variance,levene,levenes,levelne's,brown-forsythe,anova,two-factor anova,two-way anova,two factor anova, two way anova,excel,excel 2010,excel 2013,statistics
(Click Image To See a Larger Version)

Variances of Factor 1 Levels

Var (Factor 1 Level 1) = 1,597

Var (Factor 1 Level 2) = 1,064

Var (Factor 1 Level 3) = 532

None of the Factor 1 Level data groups have a sample variance that is more than four times as large as the sample variance of another Factor 1 level group. The variance rule-of-thumb indicates that the variances of all data groups that are part of the Factor 1 Main Effects F Test should be considered similar. All of the Factor 1 level data groups are therefore homoscedastistic (have similar variances).

 

Variances of Factor 2 Levels

Var (Factor 2 Level 1) = 852

Var (Factor 2 Level 2) = 1,368

None of the Factor 2 Level data groups have a sample variance that is more than four times as large as the sample variance of another Factor 2 level group. The variance rule-of-thumb indicates that the variances of all data groups that are part of the Factor 2 Main Effects F Test should be considered similar. All of the Factor 2 level data groups are therefore homoscedastistic (have similar variances).

In addition to the variance comparison rule-of-thumb, two statistical tests are commonly performed when it is necessary to evaluate the equality of variances in sample groups. These tests are Levene’s Test and the Brown-Forsythe Test. The Brown-Forsythe Test is more robust against outliers but Levene’s Test is the more popular test.

 

Levene’s Test in Excel For

Sample Variance Comparison

Levene’s Test is a hypothesis test commonly used to test for the equality of variances of two or more sample groups. Levene’s Test is much more robust against non-normality of data than the F Test. That is why Levene’s Test is nearly always preferred over the F Test as a test for variance equality.

The Null Hypothesis of Levene’s Test is average distance to the sample mean is the same for each sample group. Acceptance of this Null Hypothesis implies that the variances of the sampled groups are the same.

Separate Levene’s Test will now be performed on the data groups for Factor 1 Main Effects F Test and for the Factor 2 Main Effects F Test. The absolute value of the distance from each sample point to the sample mean must be calculated. Single-Factor ANOVA in Excel is then run on these data sets.

Levene’s Test is performed on the Factor 1 level groups as follows:

variance,levene,levenes,levelne's,brown-forsythe,anova,two-factor anova,two-way anova,two factor anova, two way anova,excel,excel 2010,excel 2013,statistics (Click Image To See a Larger Version)

variance,levene,levenes,levelne's,brown-forsythe,anova,two-factor anova,two-way anova,two factor anova, two way anova,excel,excel 2010,excel 2013,statistics (Click Image To See a Larger Version)

α was set at 0.05 for this ANOVA test. The p Value of 0.2526 is larger than 0.05. This indicates that the average distances to the sample mean for each Factor 1 level data group are not significantly different. This result of Levene’s test is interpreted to mean that the Factor 1 level data groups have similar variances and are therefore homoscedstistic.

Levene’s Test is performed on the Factor 2 level groups as follows:

variance,levene,levenes,levelne's,brown-forsythe,anova,two-factor anova,two-way anova,two factor anova, two way anova,excel,excel 2010,excel 2013,statistics

variance,levene,levenes,levelne's,brown-forsythe,anova,two-factor anova,two-way anova,two factor anova, two way anova,excel,excel 2010,excel 2013,statistics (Click Image To See a Larger Version)

α was set at 0.05 for this ANOVA test. The p Value of 0.2519 is larger than 0.05. This indicates that the average distances to the sample mean for each Factor 2 level data group are not significantly different. This result of Levene’s test is interpreted to mean that the Factor 2 level data groups have similar variances and are therefore homoscedastistic.

We therefore conclude as a result of Levene’s Test that the group variances for each F Test are the same or, at least, that we don’t have enough evidence to state that the variances of either of the F Tests are different. Levene’s Test is sensitive to outliers because relies on the sample mean, which can be unduly affected by outliers. A very similar nonparametric test called the Brown-Forsythe Test relies on sample medians and is therefore much less affected by outliers as Levene’s Test is or by non-normality as the F Test is.

 

Brown-Forsythe Test in Excel

For Sample Variance Comparison

The Brown-Forsythe Test is a hypothesis test commonly used to test for the equality of variances of two or more sample groups. The Null Hypothesis of the Brown-Forsythe Test is average distance to the sample median is the same for each sample group. Acceptance of this Null Hypothesis implies that the variances of the sampled groups are similar. The distance to the median for each data point of the three sample groups is shown as follows:

Separate Brown-Forsythe Test will now be performed on the data groups for Factor 1 Main Effects F Test and for the Factor 2 Main Effects F Test. The absolute value of the distance from each sample point to the sample median must be calculated. Single-Factor ANOVA in Excel is then run on these data sets.

The Brown-Forsythe Test is performed on the Factor 1 level groups as follows:

variance,levene,levenes,levelne's,brown-forsythe,anova,two-factor anova,two-way anova,two factor anova, two way anova,excel,excel 2010,excel 2013,statistics (Click Image To See a Larger Version)

variance,levene,levenes,levelne's,brown-forsythe,anova,two-factor anova,two-way anova,two factor anova, two way anova,excel,excel 2010,excel 2013,statistics (Click Image To See a Larger Version)

α was set at 0.05 for this ANOVA test. The p Value of 0.2530 is larger than 0.05. This indicates that the average distances to the sample mean for each Factor 1 level data group are not significantly different. This result of this Brown-Forsythe test is interpreted to mean that the Factor 1 level data groups have similar variances and are therefore homoscedastistic.

The Brown-Forsythe Test is performed on the Factor 2 level groups as follows:

variance,levene,levenes,levelne's,brown-forsythe,anova,two-factor anova,two-way anova,two factor anova, two way anova,excel,excel 2010,excel 2013,statistics (Click Image To See a Larger Version)

variance,levene,levenes,levelne's,brown-forsythe,anova,two-factor anova,two-way anova,two factor anova, two way anova,excel,excel 2010,excel 2013,statistics (Click Image To See a Larger Version)

α was set at 0.05 for this ANOVA test. The p Value of 0.3065 is larger than 0.05. This indicates that the average distances to the sample mean for each Factor 2 level data group are not significantly different. This result of this Brown-Forsythe test is interpreted to mean that the Factor 2 level data groups have similar variances and are therefore homoscedastistic.

We therefore conclude as a result of this Brown-Forsythe Test that the group variances for each F Test are the same or, at least, that we don’t have enough evidence to state that the variances of either of the F Tests are different.

Each of these two variance tests, Levene’s Test and the Brown-Forsythe Test, can be considered relatively equivalent to the other.

 

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