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Partial Eta Squared (Effect Size for ANOVA)

Source: R/eta_squared_partial.R
eta_squared_partial.Rd

Computes the sample partial eta squared (\(\eta^2_p\)), the proportion of variance accounted for by a fixed effect after the variance attributable to the other effects in the model has been removed: $$\hat{\eta}^2_p = \frac{\mathit{SS}_{\text{effect}}}{\mathit{SS}_{\text{effect}} + \mathit{SS}_{\text{error}}} = \frac{df_{\text{effect}} \cdot F}{df_{\text{effect}} \cdot F + df_{\text{error}}}.$$ Accepts either the raw ANOVA summary (F, effect df, error df) or a fitted aov/lm/aovlist object, in which case the function returns one row per effect (with stratum identification for within-subjects fits).

Usage

eta_squared_partial(
  object = NULL,
  F_value = NULL,
  df_effect = NULL,
  df_error = NULL
)

Arguments

object

Optional. A fitted model object of class aov, lm, or aovlist (multi-stratum aov fit, e.g.\ aov(y ~ A + Error(subject/A), data = d)).

F_value

Observed F-value (ignored if object is supplied).

df_effect

Numerator degrees of freedom for the effect (ignored if object is supplied).

df_error

Error (residual) degrees of freedom (ignored if object is supplied).

Value

A data.frame with one row per effect. Single-stratum fits and the raw interface return columns effect, eta_squared_partial, F_value, df_effect, df_error. aovlist (within-subjects / mixed) fits additionally include a stratum column identifying which error term each effect's F test came from. With the raw-argument interface effect is "overall".

Details

This function is the explicitly-named counterpart of eta_squared. The two share the same point-estimate formula, in a one-way ANOVA they coincide with total \(\eta^2\); in a factorial or within-subjects ANOVA both functions return the per-effect partial value computed against that effect's own error stratum. eta_squared_partial is provided so that user code that explicitly intends partial \(\eta^2\) carries that meaning in its name.

Designs supported. Single-stratum aov/lm fits and multi-stratum aovlist fits (within-subjects and mixed designs) are both handled by the model interface. For multi-stratum fits, each effect uses its own stratum's residual df, so a within-subjects factor's partial \(\eta^2\) is computed against the within-subjects error and a between-subjects factor's is computed against the between-subjects error.

References

Cohen, J. (1973). Eta-squared and partial eta-squared in fixed factor ANOVA designs. Educational and Psychological Measurement, 33(1), 107–112.

Kelley, K. (2007). Confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20(8), 1–24. doi:10.18637/jss.v020.i08

Kelley, K., & Preacher, K. J. (2012). On effect size. Psychological Methods, 17, 137–152. doi:10.1037/a0028086

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 3 on \(\eta^2\), Chapter 7 on factorial designs, and Chapter 11 on generalized \(\eta^2\) for within-subjects designs.)

Steiger, J. H. (2004). Beyond the F test: Effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis. Psychological Methods, 9(2), 164–182. doi:10.1037/1082-989X.9.2.164

See also

ci_eta_squared_partial, eta_squared

Other effect size estimates: cles(), cliff_delta(), correction_for_attenuation(), eta_squared(), eta_squared_generalized(), expected_partial_r(), expected_r(), expected_smd(), nnt_from_smd(), omega_squared(), omega_squared_partial(), probability_of_superiority_paired(), proportion_of_superiority(), responder_analysis(), smd_trimmed()

Author

Ken Kelley kkelley@nd.edu

Examples

# Raw-argument interface.
eta_squared_partial(F_value = 11.221, df_effect = 4, df_error = 50)
#>  effect  eta_squared_partial F_value df_effect df_error
#>  overall 0.473               11.2    4         50      

# Factorial ANOVA: partial eta squared per effect (pygmalion data:
# expectancy treatment x grade, 2 x 6 with unequal cell sizes,
# N = 310). The treatment is manipulated; grade is a measured
# classification of the pupils.
fit <- aov(iq_8 ~ treatment * factor(grade), data = pygmalion)
eta_squared_partial(fit)
#>  effect                  eta_squared_partial F_value df_effect df_error
#>  treatment               0.0215              6.54    1         298     
#>  factor(grade)           0.0448              2.8     5         298     
#>  treatment:factor(grade) 0.0196              1.19    5         298     

# Within-subjects ANOVA: per-effect partial eta squared with stratum.
set.seed(113)
n <- 20
rm_data <- data.frame(
  subject = factor(rep(seq_len(n), each = 3)),
  time    = factor(rep(c("Pre", "Mid", "Post"), n),
                   levels = c("Pre", "Mid", "Post")),
  y       = rnorm(n, sd = 1.5)[rep(seq_len(n), each = 3)] +
            0.7 * rep(1:3, n) + rnorm(n * 3, sd = 1.2)
)
fit_rm <- aov(y ~ time + Error(subject/time), data = rm_data)
eta_squared_partial(fit_rm)
#>  effect eta_squared_partial stratum      F_value df_effect df_error
#>  time   0.231               subject:time 5.7     2         38