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.

Usage

etasq(x, ...)

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

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

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

Arguments

x: A lm, mlm or Anova.mlm object
...: Other arguments passed down to Anova.
anova: A logical, indicating whether the result should also contain the test statistics produced by Anova().
partial: A logical, indicating whether to calculate partial or classical eta^2.

Value

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.

Details

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, partialling 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): \(\eta^2 = R/(R+1)\)

References

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.

Author

Michael Friendly

Examples


library(car)
data(Soils, package="carData")
soils.mod <- lm(cbind(pH,N,Dens,P,Ca,Mg,K,Na,Conduc) ~ Block + Contour*Depth, data=Soils)
#Anova(soils.mod)
etasq(Anova(soils.mod))
#>                   eta^2
#> Block         0.5585973
#> Contour       0.6692989
#> Depth         0.5983772
#> Contour:Depth 0.2058495
etasq(soils.mod) # same
#>                   eta^2
#> Block         0.5585973
#> Contour       0.6692989
#> Depth         0.5983772
#> Contour:Depth 0.2058495
etasq(Anova(soils.mod), anova=TRUE)
#> 
#> Type II MANOVA Tests: Pillai test statistic
#>                 eta^2 Df test stat approx F num Df den Df    Pr(>F)    
#> Block         0.55860  3    1.6758   3.7965     27     81 1.777e-06 ***
#> Contour       0.66930  2    1.3386   5.8468     18     52 2.730e-07 ***
#> Depth         0.59838  3    1.7951   4.4697     27     81 8.777e-08 ***
#> Contour:Depth 0.20585  6    1.2351   0.8640     54    180    0.7311    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

etasq(soils.mod, test="Wilks")
#>                   eta^2
#> Block         0.5701385
#> Contour       0.7434504
#> Depth         0.8294239
#> Contour:Depth 0.2250388
etasq(soils.mod, test="Hotelling")
#>                   eta^2
#> Block         0.5823516
#> Contour       0.8009753
#> Depth         0.9421533
#> Contour:Depth 0.2456774