Thursday, May 29, 2014

Levene’s and Brown-Forsythe Tests in Excel For Single-Factor ANOVA Sample Group Variance Comparison

This is one of the following sixteen articles on Single-Factor ANOVA in Excel

Overview of Single-Factor ANOVA

Single-Factor ANOVA in 5 Steps in Excel 2010 and Excel 2013

Shapiro-Wilk Normality Test in Excel For Each Single-Factor ANOVA Sample Group

Kruskal-Wallis Test Alternative For Single Factor ANOVA in 7 Steps in Excel 2010 and Excel 2013

Levene’s and Brown-Forsythe Tests in Excel For Single-Factor ANOVA Sample Group Variance Comparison

Single-Factor ANOVA - All Excel Calculations

Overview of Post-Hoc Testing For Single-Factor ANOVA

Tukey-Kramer Post-Hoc Test in Excel For Single-Factor ANOVA

Games-Howell Post-Hoc Test in Excel For Single-Factor ANOVA

Overview of Effect Size For Single-Factor ANOVA

ANOVA Effect Size Calculation Eta Squared in Excel 2010 and Excel 2013

ANOVA Effect Size Calculation Psi – RMSSE – in Excel 2010 and Excel 2013

ANOVA Effect Size Calculation Omega Squared in Excel 2010 and Excel 2013

Power of Single-Factor ANOVA Test Using Free Utility G*Power

Welch’s ANOVA Test in 8 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar

Brown-Forsythe F-Test in 4 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar

Levene’s and Brown- Forsythe Tests in Excel For Single-Factor ANOVA Sample Group Variance Comparison

Single-Factor ANOVA requires that the variances of all sample groups 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. That is the case here as the following are true:

s1 = Group1 standard deviation = 1.495

s2 = Group2 standard deviation = 1.514

s3 = Group3 standard deviation = 1.552

The variances of all three groups are very similar. A quick look at the box plot of the data would have confirmed that as well.

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 Group 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. The distance to the mean for each data point of both samples is shown as follows:

Levene’s Test involves performing Single-Factor ANOVA on the groups of distances to the mean. This can be easily implemented in Excel by applying the Excel data analysis tool ANOVA: Single Factor. Here is the completed dialogue box for this test:

Applying this tool on the above data produces the following output:

The Null Hypothesis of Levene’s Test states that the average distances to the mean for the two groups are the same. Acceptance of this Null Hypothesis would imply that the sample groups have the similar variances. The p Value shown in the Excel ANOVA output equals 0.9566. This is much larger than the Alpha (0.05) that is typically used for an ANOVA Test so the Null Hypothesis cannot be rejected.

We therefore conclude as a result of Levene’s Test that the variances are the same or, at least, that we don’t have enough evidence to state that the variances 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 Group 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:

The Brown-Forsythe Test involves performing Single-Factor ANOVA on the groups of distances to the median. This can be easily implemented in Excel by applying the Excel data analysis tool ANOVA: Single Factor. Here is the completed dialogue box for this test:

Applying this tool on the above data produces the following output:

The Null Hypothesis of the Brown-Forsythe Test states that the average distances to the median for the three groups are the same. Acceptance of this Null Hypothesis would imply that the sample groups have similar variances. The p Value shown in the Excel ANOVA output equals 0.9582. This is much larger than the Alpha (0.05) that is typically used for an ANOVA Test so the Null Hypothesis cannot be rejected.

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

Each of these two variance tests can be considered relatively equivalent to the others.

Alternative Tests To Single-Factor ANOVA When Sample Group Variances Are Not Similarity

When groups cannot be shown to have homogeneous (similar) variances, either Welch’s ANOVA or the Brown-Forsythe F test should be used in place of Single-Factor ANOVA. Both of these tests will be performed on this example’s data in blog articles that follow this one.

Excel Master Series Blog Directory

Statistical Topics and Articles In Each Topic

• Histograms in Excel
• Bar Chart in Excel
• Combinations & Permutations in Excel
• Normal Distribution in Excel
• t-Distribution in Excel
• Binomial Distribution in Excel
• z-Tests in Excel
• t-Tests in Excel
• Hypothesis Tests of Proportion in Excel
• Chi-Square Independence Tests in Excel
• Chi-Square Goodness-Of-Fit Tests in Excel
• F Tests in Excel
• Correlation in Excel
• Pearson Correlation in Excel
• Spearman Correlation in Excel
• Confidence Intervals in Excel
• Simple Linear Regression in Excel
• Multiple Linear Regression in Excel
• Logistic Regression in Excel
• Single-Factor ANOVA in Excel
• Two-Factor ANOVA With Replication in Excel
• Two-Factor ANOVA Without Replication in Excel
• Randomized Block Design ANOVA in Excel
• Repeated-Measures ANOVA in Excel
• ANCOVA in Excel
• Normality Testing in Excel
• Nonparametric Testing in Excel
• Post Hoc Testing in Excel
• Creating Interactive Graphs of Statistical Distributions in Excel
• Solving Problems With Other Distributions in Excel
• Optimization With Excel Solver
• Chi-Square Population Variance Test in Excel
• Analyzing Data With Pivot Tables
• SEO Functions in Excel
• Time Series Analysis in Excel
• VLOOKUP