heplots (version 1.3-8)

etasq: Measures of Partial Association (Eta-squared) for Linear Models


Calculates partial eta-squared for linear models or multivariate analogs of eta-squared (or R^2), indicating the partial association for each term in a multivariate linear model. There is a different analog for each of the four standard multivariate test statistics: Pillai's trace, Hotelling-Lawley trace, Wilks' Lambda and Roy's maximum root test.


etasq(x, ...)

# S3 method for lm etasq(x, anova = FALSE, partial = TRUE, ...)

# S3 method for mlm etasq(x, ...)

# S3 method for Anova.mlm etasq(x, anova = FALSE, ...)



A lm, mlm or Anova.mlm object


A logical, indicating whether the result should also contain the test statistics produced by Anova().


A logical, indicating whether to calculate partial or classical eta^2.

Other arguments passed down to Anova.


When anova=FALSE, a one-column data frame containing the eta-squared values for each term in the model.

When anova=TRUE, a 5-column (lm) or 7-column (mlm) data frame containing the eta-squared values and the test statistics produced by print.Anova() for each term in the model.


For univariate linear models, classical \(\eta^2\) = SSH / SST and partial \(\eta^2\) = SSH / (SSH + SSE). These are identical in one-way designs.

Partial eta-squared describes the proportion of total variation attributable to a given factor, partialing out (excluding) other factors from the total nonerror variation. These are commonly used as measures of effect size or measures of (non-linear) strength of association in ANOVA models.

All multivariate tests are based on the \(s=min(p, df_h)\) latent roots of \(H E^{-1}\). The analogous multivariate partial \(\eta^2\) measures are calculated as:

Pillai's trace (V)

\(\eta^2 = V/s\)

Hotelling-Lawley trace (T)

\(\eta^2 = T/(T+s)\)

Wilks' Lambda (L)

\(\eta^2 = L^{1/s}\)

Roy's maximum root (R)

\(\eta^2 = R/(R+1)\)


Muller, K. E. and Peterson, B. L. (1984). Practical methods for computing power in testing the Multivariate General Linear Hypothesis Computational Statistics and Data Analysis, 2, 143-158.

Muller, K. E. and LaVange, L. M. and Ramey, S. L. and Ramey, C. T. (1992). Power Calculations for General Linear Multivariate Models Including Repeated Measures Applications. Journal of the American Statistical Association, 87, 1209-1226.

See Also



data(Soils)  # from car package
soils.mod <- lm(cbind(pH,N,Dens,P,Ca,Mg,K,Na,Conduc) ~ Block + Contour*Depth, data=Soils)
etasq(soils.mod) # same
etasq(Anova(soils.mod), anova=TRUE)

etasq(soils.mod, test="Wilks")
etasq(soils.mod, test="Hotelling")
# }