A variant of rank-transformation is 'quantile normalization' in which a further transformation is applied to the ranks such that the resulting values have some defined distribution (often a normal distribution with a specified mean and variance). Further analyses of quantile-normalized data may then assume that distribution to compute significance values. However, two specific types of secondary transformations, the random normal scores and expected normal scores transformation, have been shown to greatly inflate Type I errors and severely reduce statistical power (Sawilowsky, 1985a, 1985b).

Several standardized measures of effect are used within the context of ANOVA to describe the degree of relationship between a predictor or set of predictors and the dependent variable. Effect size estimates are reported to allow researchers to compare findings in studies and across disciplines. Common effect size estimates reported in bivariate (e.g. ANOVA) and multivariate (MANOVA, MANCOVA, Multiple Discriminant Analysis) statistical analysis includes eta-squared, partial eta-squared, omega, and intercorrelation (Strang, 2009).

η2 ( eta-squared ): Eta-squared describes the ratio of variance explained in the dependent variable by a predictor while controlling for other predictors. Eta-squared is a biased estimator of the variance explained by the model in the population (it only estimates effect size in the sample). On average it overestimates the variance explained in the population. As the sample size gets larger the amount of bias gets smaller. It is, however, an easily calculated estimator of the proportion of the variance in a population explained by the treatment. Note that earlier versions of statistical software (such as SPSS) incorrectly reports Partial eta squared under the misleading title "Eta squared".

Partial η2 (Partial eta-squared): Partial eta-squared describes the "proportion of total variation attributable to the factor, partialling out (excluding) other factors from the total nonerror variation" (Pierce, Block & Aguinis, 2004, p. 918). Partial eta squared is normally higher than eta squared (except in simple one-factor models).

Several variations of benchmarks exist.

The generally accepted regression benchmark for effect size comes from (Cohen, 1992; 1988): 0.20 is a minimal solution (but significant in social science research); 0.50 is a medium effect; anything equal to or greater than 0.80 is a large effect size (Keppel & Wickens, 2004; Cohen, 1992).

Because this common interpretation of effect size has been repeated from Cohen (1988) over the years with no change or comment to validity for contemporary experimental research, it is questionable outside of psychological/behavioural studies, and more so questionable even then without a full understanding of the limitations ascribed by Cohen. Note: The use of specific partial eta-square values for large medium or small as a "rule of thumb" should be avoided.

Nevertheless, alternative rules of thumb have emerged in certain disciplines: Small = 0.01; medium = 0.06; large = 0.14 (Kittler, Menard & Phillips, 2007).

Omega Squared Omega squared provides a relatively unbiased estimate of the variance explained in the population by a predictor variable. It takes random error into account more so than eta squared, which is incredibly biased to be too large. The calculations for omega squared differ depending on the experimental design. For a fixed experimental design (in which the categories are explicitly set), omega squared is calculated as follows: