In ANOVA, we always report
- F (the F-value);
- df (degrees of freedom);
- p (statistical significance).
We report these 3 numbers for each effect -possibly just one for one-way ANOVA. Now, p (“Sig.” in SPSS) tells us the likelihood of some effect being zero in our population. A zero effect means that all means are exactly equal for some factor such as gender or experimental group.
However, some effect just being not zero isn't too interesting, is it? What we really want to know is:
how strong is the effect?
We can't conclude that p = 0.05 indicates a stronger effect than p = 0.10 because both are affected by sample sizes and other factors. So how can we quantify how strong effects are for comparing them within or across analyses?
Well, there's several measures of effect size that tell us just that. One that's often used is (partial) eta squared, denoted as η2 (η is the Greek letter eta).
Partial Eta Squared - What Is It?
Partial η2 a proportion of variance accounted for by some effect. If you really really want to know:
$$partial\;\eta^2 = \frac{SS_{effect}}{SS_{effect} + SS_{error}}$$
where SS is short for “sums of squares”, the amount of dispersion in our dependent variable. This means that partial η2 is the variance attributable to an effect divided by the variance that could have been attributable to this effect.
We can easily verify this -and many more calculations- by copy-pasting SPSS’ ANOVA output into this GoogleSheet as shown below.
Note that in one-way ANOVA, we only have one effect. So the variance in our dependent variable is either attributed to the effect or it is error. So for one-way ANOVA
$$partial\;\eta^2 = \frac{SS_{effect}}{SS_{total}}$$
which is equal to (non partial) η2. Let's now go and get (partial) η2 from SPSS.
Example: Happiness Study
A scientist asked 120 people to rate their own happiness on a 100 point scale. Some other questions were employment status, marital status and health. The data thus collected are in happy.sav, part of which is shown below.
We're especially interested in the effect of employment on happiness: (how) are they associated and does the association depend on health or marital status too? Let's first just examine employment with a one-way ANOVA.
Eta Squared for One-Way ANOVA - Option 1
SPSS offers several options for running a one-way ANOVA and many students start off with
η2 is completely absent from this dialog.
We'll therefore use MEANS instead as shown below.
Clicking
results in the syntax below. Since it's way longer than necessary, I prefer just typing a short version that yields identical results. Let's run it.SPSS Syntax for Eta Squared from MEANS
MEANS TABLES=happy BY employed
/CELLS=MEAN COUNT STDDEV
/STATISTICS ANOVA.
*Short version (creates identical output).
means happy by employed
/statistics.
Result
And there we have it: η2 = 0.166: some 17% of all variance in happiness is attributable to employment status. I'd say it's not an awful lot but certainly not negligible.
Note that SPSS mentions “Measures of Association” rather than “effect size”. It could be argued that these are interchangeable but it's somewhat inconsistent anyway.
Eta Squared for One-Way ANOVA - Option 2
Perhaps the best way to run ANOVA in SPSS is from the univariate GLM dialog. The screenshots below guide you through.
This results in the syntax shown below. Let's run it, see what happens.
SPSS Syntax for Eta Squared from UNIANOVA
UNIANOVA happy BY employed
/METHOD=SSTYPE(3)
/INTERCEPT=INCLUDE
/PRINT=ETASQ
/CRITERIA=ALPHA(.05)
/DESIGN=employed.
Result
We find partial η2 = 0.166. It was previously denoted as just η2 but these are identical for one-way ANOVA as already discussed.
Partial Eta Squared for Multiway ANOVA
For multiway ANOVA -involving more than 1 factor- we can get partial η2 from GLM univariate as shown below.
As shown below, we now just add multiple independent variables (“fixed factors”). We then tick
under and we're good to go.Partial Eta Squared Syntax Example
UNIANOVA happy BY employed healthy
/METHOD=SSTYPE(3)
/INTERCEPT=INCLUDE
/PRINT=ETASQ
/CRITERIA=ALPHA(.05)
/DESIGN=employed healthy employed*healthy.
Result
First off, both main effects (employment and health) and the interaction between them are statistically significant. The effect of employment (η2 = .095) is twice as strong as health (η2 = 0.048). And so on.
Note that you couldn't possibly conclude this from their p-values (p = 0.003 for employment and p = 0.018 for health). Although the effects are highly statistically significant, the effect sizes are moderate. We typically see this pattern with larger sample sizes.
Last, we shouldn't really interpret our main effects because the interaction effect is statistically significant: F(2,114) = 4.9, p = 0.009. As explained in SPSS Two Way ANOVA - Basics Tutorial, we'd better inspect simple effects instead of main effects.
Conclusion
We can get (partial) η2 for both one-way and multiway ANOVA from
but it's restricted to one dependent variable at the time. Generally, I'd say this is the way to go for any ANOVA because it's the only option that gets us all the output we generally need -including post hoc tests and Levene's test.
We can run multiple one-way ANOVAs with η2 in one go with
but it lacks important options such as post hoc tests and Levene's test. These -but not η2 - are available from the dialog. This renders both options rather inconvenient unless you need a very basic analysis.
Last, several authors prefer a different measure of effect size called ω2 (“Omega square”). Unfortunately, this seems completely absent from SPSS. For now at least, I guess η2 will have to do...
I hope you found this tutorial helpful. Thanks for reading!
THIS TUTORIAL HAS 24 COMMENTS:
By Ruben Geert van den Berg on November 27th, 2018
Sorry, I'd rather go for
SSeffect / SStotal
to stay in line with r square. By having a common denominator, the effect sizes are comparable. But more importantly, the total model r-square is taken into account.
Btw, there's also Omega squared for ANOVA. But I can't look that up right now, I'm on holiday and I (obviously) didn't bring any books.
By Abdul on November 29th, 2018
Very good post.
By tom on March 31st, 2020
Thanks for this. I've been looking to see if the Eta correlation has a p-value. Since it doesn't give it to me in the cross tabs (nominal by interval correlation) I'm guessing from your description I can use the p-value from the between subjects effect for ANOVA?
By Ruben Geert van den Berg on April 1st, 2020
Yes, that's right.
Getting it from CROSSTABS is a bit tricky because you'll get 2 values, one of which is wrong. That's because SPSS doesn't know which of the 2 variables is quantitative -which matters!- so it presents eta-squared for both scenarios.
I believe you may even get a confidence interval for eta-squared from JASP but I'm not too sure about that...
Hope that helps!
SPSS tutorials
By Farhad, Marağa on April 2nd, 2020
Thank you for all this.
In the example above, the conclusion is that "some 17% of all variance in happiness is attributable to employment". If eta squared indexes can be used to explain how much of variation in the dependent variable can be accounted for by the independent variable, how is eta squared different from R Square in Regression Model Summary?
Both seem to have explanatory power.. how much of variance in dependent variable is accounted for by independent variable.
Thank you