Statistics Definitions > Eta Squared / Partial Eta Squared

## What is Eta Squared?

Eta squared is the proportion of variance associated with one or more main effects, errors or interactions in ANOVA.

## Calculation

The formula is:

Eta

^{2}= SS_{effect}/ SS_{total}, where:

SS_{effect} is the sums of squares for the effect you are studying.

SS_{total} is the total sums of squares for *all* effects, errors and interactions in the ANOVA study.

You might also see the formula written, equivalently, as:

Eta^{2} = SS_{between} / SS_{total}, where:

Eta squared is easy to calculate from ANOVA output.

Eta squared is easy to calculate from ANOVA output.

For example, let’s say you were studying depression with main effects that include general anxiety, sleep disorders and major illness. You perform an ANOVA and get the following results:

- Total SS: 62.29
- Anxiety SS: 4.08
- Sleep disorders SS: 9.2
- Major illness SS: 19.54.

Dividing each individual SS by the total, we find Eta^{2} for each main effect as follows:

Eta^{2}_{Anxiety}: SS_{Anxiety} = 4.08/62.29 = 0.066 = 6.6%.

Eta^{2}_{Sleep disorders}: SS_{Sleep disorders} = 9.2/62.29 = 0.148 = 14.8%.

Eta^{2}_{Major illness}: SS_{Major illness} = 19.54/62.29 = 0.314 = 31.4%.

These results show that 6.6% of variance is associated with anxiety, 14.5% of variance is associated with sleep disorders and 31.5% of variance is associated with major illness. This leads to the conclusion that major illness is the most important main effect.

## Partial Eta Squared

Partial eta squared is the ratio of variance associated with an effect, plus that effect and its associated error variance. The formula is similar to eta^{2}:

Partial eta

^{2}= SS_{effect}/ SS_{effect}+ SS_{error}.

In fact, when you only have one independent variable, partial eta^{2} is the same as eta^{2}

Partial etas are usually used when a person appears in more than one cell (i.e. the cells are not independent). Non independent cells show up in MANOVA studies. The results show the percentage of variance in each effect or interaction, and the error that is accounted for by that effect.

**Caution**: values labeled eta squared on some printouts from SPSS are actually partial eta^{2}. For more details, see eta squared hcr.

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

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