## Levene's Test - What Is It?

If we want to compare 3(+) groups on a metric variable, we usually want to know if they have different mean scores. In most cases, we'll run an ANOVA in order to find out. However, this requires the **homogeneity assumption** which states that the population variances are equal for all groups.

You can safely **ignore this assumption** if you have (roughly) **equal sample sizes** for each group. However, if group sizes are (sharply) different, then you do need to make sure that homogeneity of variances is met by your data. A test for finding this out is Levene's test.

## Null Hypothesis

The **null hypothesis** for Levene's test is that
the groups we're comparing all have similar population variances.
If this is true, we'll probably find *slightly* different variances in our *samples* from these populations. However, *very* different sample variances suggests that the population variances weren't equal after all. In this case we'll reject the null hypothesis of equal population variances.

## Levene's Test - Assumptions

Levene's test basically requires two assumptions:

- independent observations and
- metric variables (the test variables are not nominal or ordinal).

## Levene's Test - Example

A fitness company wants to know if 2 **supplements** for stimlating **body fat loss** actually work. They test 2 supplements (a cortisol blocker and a thyroid booster) on 20 people each and another 40 people receive a placebo. All 80 participants have body fat measurements at the start of the experiment (week 11) and weeks 14, 17 and 20. This results in fatloss_unequal.sav, part of which is shown below.

One approach to these data is comparing body fat percentages over the 3 groups (placebo, thyroid, cortisol) for each week separately.Perhaps a better approach to these data is using a single repeated measures ANOVA. Weeks would be the within-subjects factor and supplement would be the between-subjects factor. For now, we'll leave it as an exercise to the reader to carry this out.
We can use an ANOVA for each of the 4 body fat measurements. Since we've **unequal sample sizes**, we need to make sure that each supplement group has the same variance on each of the 4 measurements first.

## Running Levene's test in SPSS

Several SPSS commands contain an option for running Levene's test. The **easiest way** to go -especially for multiple variables- is the **One-Way ANOVA** dialog.The main limitation of the One-Way ANOVA dialog is that it doesn't include any measures of effect size. For more on this, see How to Get (Partial) Eta Squared from SPSS?.

Right, so we navigate to
and fill out the dialog that pops up.

The

under refers to Levene's test.Clicking

results in the syntax below. Let's run it.## SPSS Levene's Test Syntax Example

***SPSS Levene's test syntax as pasted from Analyze - Compare Means - One-Way ANOVA.**

ONEWAY fat11 fat14 fat17 fat20 BY condition

/STATISTICS DESCRIPTIVES HOMOGENEITY

/MISSING ANALYSIS.

## Output for Levene's test

On running our syntax, we get several tables, the second of which is **Test of Homogeneity of Variances**. This holds the results of Levene's test.

As a rule of thumb, we conclude that the **variances are not equal if “Sig.” < 0.05**. The body fat percentages in week 17 and week 20 have unequal population variances over our 3 treatment groups. That is, these 2 variables violate the homogeity of variance assumption needed for an ANOVA.

## Descriptive Statistics Output

Remember that we don't need equal population variances if we have roughly **equal sample sizes**. A sound way for evaluating if this holds is inspecting the Descriptives table in our output.

As we see, our ANOVA is based on **sample sizes of 40, 20 and 20** for all 4 dependent variables. Because they're not (roughly) equal, we *do* need the homogeneity of variance assumption but it's not met by 2 variables.

In this case, we'll report some alternative results (**Welch and Games-Howell**) but these are beyond the scope of this tutorial.

## Reporting Levene's test

Perhaps surprisingly, Levene's test is technically an ANOVA as we'll explain here. We therefore report it like a basic ANOVA too. So we'll write something like Levene's test showed that the variances for body fat percentage in week 20 were not equal, F(2,77) = 4.58, p = 0.013.

## Levene's Test - How Does It Work?

Levene's test works very simply: a larger variance means that -on average- the data values are “further away” from their mean. The figure below illustrates this: watch the histograms become “wider” as the variances increase.

We therefore compute the absolute differences between all scores and their (group) mean. The **means of the absolute differences** should be roughly equal over groups. So technically, Levene's test is an ANOVA on the absolute differences. In other words: we run an ANOVA (on absolute differences) to find out if we can run an ANOVA (on our actual data).

If that just sounds *too weird*, then try running the syntax below. It does exactly what I just explained.

## “Manual” Levene's Test Syntax

***Add group means on fat20 to dataset.**

aggregate outfile * mode addvariables

/break condition

/mfat20 = mean(fat20).

***Compute absolute differences between fat20 and group means.**

compute adfat20 = abs(fat20 - mfat20).

***Run minimal ANOVA on absolute differences. F-test identical to previous Levene's test.**

ONEWAY adfat20 BY condition.

## Result

As we see, these **ANOVA results are identical to Levene's test** in the previous output. I hope this clarifies why we report it as an ANOVA as well.

Thanks for reading!

## This tutorial has 3 comments

## By Fadia abdel-hameed on November 13th, 2017

this is an easy way to understand statistics became an easy science. Thank you very much

## By Ruben Geert van den Berg on September 27th, 2017

Hi Jon!

Technically, I kinda agree although I'm not sure how Welch compares to F with regard to power. However, students are usually expected to follow the steps proposed in this tutorial. Unfortunately, some supervisors may think using Welch directly qualifies as a wrong answer. I recently saw even crazier stuff- a supervisor being unaware of the central limit theorem who claimed that non normally distributed variables can't be analyzed even with huge sample sizes...

There's a ton of things in statistics (and in SPSS too actually) that are outdated but I feel I have insufficient authority to address all such issues. Until then, I'd perhaps better follow the mainstream.

## By Jon Peck on September 26th, 2017

All true, above, but first testing for equality of variances and then choosing a different test is not ideal. In general, why not just bypass the equal variance assumption and use the Brown-Forsythe or Welch's test also available in the Oneway dialog box, although it isn't shown in the Option image above? Those tests are robust to unequal variance and have good statistical properties. If one really does want to test the variances first, probably a signficance value greater than .05 would be a better choice.