What does an eta-squared of 0.06 tell you about a factor in a one-way ANOVA?

## Where eta-squared comes from An analysis of variance decomposes the total variation in an outcome into pieces. In the one-way case, with observations split into groups by a single factor: `SS_total = SS_between + SS_within` - `SS_total` is the summed squared deviation of every observation from the grand mean. - `SS_between` (the factor's sum of squares) measures how far the group means sit from the grand mean, weighted by group size. - `SS_within` (the error sum of squares) measures scatter of observations around their own group's mean. Eta-squared is simply the factor's share of that total: `eta^2 = SS_between / SS_total` So it is a proportion between 0 and 1. A value of 0.06 says the grouping accounts for 6% of the outcome's variation, and 94% is variation the factor does not account for — individual differences, measurement noise, and everything not in the design. ## Why report it next to the F test The F statistic in an ANOVA is a ratio of mean squares, `MS_between / MS_within`, and the test built on it answers a yes/no question: are these group means distinguishable from one another given the within-group scatter? With enough observations per group, an F test will flag differences among means that explain almost none of the outcome's variation. Eta-squared is the magnitude that turns that verdict into something interpretable — it tells you how much of what you actually care about, the spread in the outcome, the factor is responsible for. It is worth being clear about what eta-squared does *not* say. It is not a difference between any two group means, and it carries no direction or units. A large eta-squared with three groups tells you the group means are widely spread relative to within-group scatter, but not which group is highest or by how much. For that you need the mean differences themselves, ideally standardized as a mean-difference effect size for the pairs you care about. ## Benchmarks Cohen's conventional cut-points for variance-explained measures are **0.01 small, 0.06 medium, 0.14 large**. As with all such conventions, they are a fallback for when there is no field-specific reference point, not a law. In a domain where outcomes are dominated by individual variation, a factor explaining 6% may be an important lever; in a tightly controlled physical measurement, 6% may be disappointing. Benchmark against the effects previously observed on that outcome. ## Eta-squared versus partial eta-squared This is the distinction interviewers probe. In a design with more than one factor, two denominators are available: - `eta^2 = SS_factor / SS_total` — the share of *all* the variation. - `partial eta^2 = SS_factor / (SS_factor + SS_error)` — the share of the variation *not already attributed to the other factors*. Partial eta-squared removes the competing factors' sums of squares from the denominator, so it is always at least as large as eta-squared and usually larger. Three consequences: 1. Partial eta-squared values across the factors of a design **do not sum to one** and should never be treated as a partition of the outcome's variance. Plain eta-squared values, together with the error share, do sum to one. 2. Partial eta-squared is **not comparable across studies** with different sets of factors. Adding a strong covariate to a design shrinks the error term, which inflates the partial eta-squared of every other factor without the underlying effect changing at all. 3. Because reports frequently print one and label it the other, always state which denominator was used before quoting a number against Cohen's benchmarks. In a one-way design the two are identical, because the total sum of squares is just the factor plus the error; the distinction only bites once a design has more than one factor. ## Upward bias and omega-squared Eta-squared is computed from sample sums of squares, and the factor's sum of squares absorbs some sampling noise: even when the population group means are identical, sample group means will differ by chance, so `SS_between` is positive and eta-squared is positive. It therefore **overestimates** the population share, and the bias is worst with small groups and many of them. **Omega-squared** subtracts an estimate of that chance contribution from the numerator and adjusts the denominator, giving a less biased estimate of the population variance explained. It is always smaller than eta-squared on the same data, can come out slightly negative when the observed effect is weaker than chance would predict, and is the better number to report from small samples. With large groups the two converge and the choice stops mattering. As with any estimate, report an interval where you can: a 6% share estimated from 20 observations and the same 6% from 2,000 are very different claims.