Tests for homogeneity of variances

• Introduction
• Preliminary analysis
• Tests for homogeneity of variances
• Description of excel tool
  • Hartley’s test
  • Bartlett’s test
  • Conclusion of the test
  • Failure of homogeneity assumption

Introduction

Most of agricultural research programs are conducted essentially due to two main purposes.

• Explain the way plants and animals respond to changing conditions (basic research)
• Obtain information to solve probelms or to make recommendations (applied research)

Basic research involves a series of single experiments which are carried out in sequence. Such series of experiments are analyzed when completed and due to its sequential nature combined analysis of the experimental data is seldom required.

Combined estimates will be valid only if the sites on which experiments are conducted are random sample of all possible sites in the area

The combined estimate will be unbiased if every possible site has an equal chance of being included.

Several experiments may be repeated over time (years), locations (sites), seasons and harvests.

Preliminary analysis

Before proceeding for combined analysis of variance, certain preliminary measures are required:

• Complete analysis of variance for each experiment (each site or year or season)
• Test heterogeneity in experimental error of series of experiments
• Estimate of interaction of treatment with the repeating factor.

From the output of the above steps the combined analysis of variance may be undertaken. The second step of testing the heterogeneity of variances is the main topic of discussion here. To test whether the variances are heterogeneous or not, an excel tool has been developed to facilitate this preliminary analysis before proceeding for combined analysis of variance.

Tests for homogeneity of variances

There are several methods which are used to test the homogeneity of variances for series of experiments. The most popular methods include:

• Hartley’s test (F-ratio test)
• Levene’s test
• Bartlett’s test
• Brown-Forsythe test

In the excel tool one can perform Hartley’s test and Bartlett’s test for testing homogeneity of variances.

Description of excel tool

The first sheet labeled as homogeneity tests you can choose either F-ratio (Hartley’s test) or Bartlett’s test.

Main page of the excel tool

Hartley’s test

The Hartley’s test is a quick test of homogeneity of variance which is provided by the ratio of largest to the smallest variance (s²) in the set. \[\begin{equation*} F\:test = \frac{s^{2}(max.)} {s^{2}(min.)} \end{equation*}\]

Hartley’s test becomes more useful and simple when error mean squares are arising out of experiments having equal number of replications.

In the main page when you will select the F test another sheet will open where you can enter maximum and minimum variances and error degree of freedom values in Enter Data Here section as shown in figure below.

F test data entry section

The second section of this sheet is F-ratio test which will show the output indicating whether the variances are homogeneous or not. Follow the instructions in conclusion section.

F test results section

As a rule of thumb, if the above ratio be less than 3, then the variances are generally taken as homogeneous otherwise not. 

Bartlett’s test

Another procedure which is more sensitive than the F test is the Bartlett’s test of homogeneity of variance. This test is based on the natural logarithm of the sample variances as describes by Snedecor and Cochran (1980). The calculation for this test is carried out through following formula.

\[M = \gamma\:[\:a(\:ln \:\bar{s}^{2}) \:-\:\sum\limits_{i}\: ln \: s_{i}^{2}\:]\] and

\[C = 1 \:+\:\frac{a + 1}{3a\gamma}\]

Where;

\(\bar{s}^{2} = \frac{\sum_i \: s_{i}^2}{a}\)

\(\gamma =\) error degree of freedom for the individual trials

\(s_i^2 =\) error mean square at location i

\(a =\) number of locations

Then

\[\chi^2 = \frac{M}{C}\]

Bartlett’s test for homogeneity of variance can be performed even when error mean squares are arising out of experiments having different replications.

Conclusion of the test

Based on the probability value in excel tool following conclusions can be drawn.

p-value ≤ 0.05: the variances are heterogenous (see failure of homogeneity assumption).

p-value > 0.05: the variances are homogenous and data can be pooled to carry out combined analysis of variance.

Failure of homogeneity assumption

If the homogeneity test indicates that the variances are heterogeneous then go through the following steps.

1. Data transformation —> \(MSE_{weighted}=\frac{1}{\sqrt{MSE}}\)

2. Define new variable —> \(Old\:variable\:\times\:MSE_{weighted}\)

3. Run a combined analysis of variance by using the new variable

Please comment below if you have any questions

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