Levene’s test examines if 2+ populations all have

equal variances on some variable.

## Levene's Test - What Is It?

If we want to compare 2(+) groups on a quantitative variable, we usually want to know if they have equal mean scores. For finding out if that's the case, we often use

- an independent samples t-test for
**2 groups**or - a one-way ANOVA for
**3+ groups**.

Both tests require the **homogeneity assumption**: the population variances of the dependent variable must equal for all groups.

You can safely ignore this assumption if you've roughly **equal** sample sizes for all groups you're comparing. However, if you've sharply **different** sample sizes, then you need to make sure that homogeneity is met by your data.

Now, we usually don't know the population variances. However, we *do* know the sample variances. And if these don't differ too much, then the population variances being equal seems credible.

But at what point do we no longer believe the population variances to be all equal? **Levene’s test** tells us precisely that.

## Null Hypothesis

The null hypothesis for Levene's test is that
the groups we're comparing all have equal population variances.
If this is true, we'll probably find *slightly* different variances in samples from these populations. However, *very* different sample variances suggest 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
- the test variable is quantitative -that is, not nominal or ordinal.

## Levene's Test - Example

A fitness company wants to know if 2 supplements for stimulating body fat loss actually work. They test 2 supplements (a cortisol blocker and a thyroid booster) on 20 people each. An additional 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 mixed 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.
This can be done with an ANOVA for each of the 4 body fat measurements. However, since we've **unequal sample sizes**, we first need to make sure that our supplement groups have equal variances.

## 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.This dialog was greatly improved in SPSS version 27 and now includes measures of effect size such as (partial) eta squared. So let's navigate to
and fill out the dialog that pops up.

As shown below, the Homogeneity of variance test under Options refers to Levene's test.

Clicking syntax below. Let's run it.

results in the## 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 -shown below- is the **Test of Homogeneity of Variances**. This holds the results of Levene's test.

As a rule of thumb, we conclude that
**population variances are not equal if “Sig.” or p < 0.05**. For the first 2 variables, p > 0.05: for fat percentage in weeks 11 and 14 we

*don't*reject the null hypothesis of equal population variances.

For the last 2 variables, p < 0.05: for fat percentages in weeks 17 and 20, we reject the null hypothesis of equal population variances. So 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 alternative measures (Welch and Games-Howell) that don't require the homogeneity assumption. How to run and interpret these is covered in SPSS ANOVA - Levene’s Test “Significant”.

## Reporting Levene's test

Perhaps surprisingly, Levene's test is technically an ANOVA as we'll explain here. We therefore report it like just 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* = .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) means. The means of these *absolute* differences should be roughly equal over groups. So technically,
Levene's test is an ANOVA on the absolute difference scores.
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 confuses you, 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 37 COMMENTS:

## By Jeff Boggs on March 30th, 2020

With regard to Levene's Test, you say that it takes into account the absolute differences between each observation and its mean. Does this replace the sum of squared deviations?

I ask because I'm trying to figure out why the F-statistics are different but the p-values are the same when I run both Levene's Test and a One-Way ANOVA. Your explanation is the clearest I have found so far, but I want to make sure I understand it.

## By Ruben Geert van den Berg on March 30th, 2020

Hi Jef!

There's no reason whatsoever why p-values should be equal for Levene's test and the actual ANOVA. The only thing that should be similar between the 2 tests are the df1 (#group - 1) and df2 (#observations - #groups).

Levene's test is technically an ANOVA on absolute differences between scores and group means. You could compute these manually and then run an ANOVA on them -which

doesinvolve sums of squares like any other ANOVA. The result should be identical to the SPSS Levene's test results.This is shown in the final syntax example of this tutorial. If you've any doubts, rerun that first and then rerun it on you own data.

Hope that helps!

SPSS tutorials

## By Jeff Boggs on March 30th, 2020

Hello Rueben,

Thank you for the clear response. I will re-run my analysis with this in mind. Best,

Jeff

## By Dominika Chudá on April 10th, 2020

Super! Thank you.

## By Robin on April 16th, 2020

Thank you for this, very helpful!!

Quick question - when I run my one-way ANOVA and get a Levene's test output, it gives me separate Levene's statistics based on mean, median, median with adjusted df, and trimmed mean. Which would you suggest I use for basic judgement of whether the homogeneity of variance assumption is met please?

Thank you!