eta_squared: Eta Squared (Effect Size for ANOVA)

Description

Computes the sample eta squared ($\eta^2$), the proportion of variance in the dependent variable accounted for by a fixed effect. Accepts either the raw ANOVA summary (F and the effect and error degrees of freedom) or a fitted model object. Supports both between-subjects designs (single-stratum aov or lm fits) and within-subjects / mixed designs (aovlist fits produced by aov with an Error() term in the formula). For factorial and within-subjects designs the function returns partial $\eta^2$ per effect (one row per non-Residuals effect across all strata), each computed against its own stratum's error term.

Usage

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

Value

A data.frame with one row per effect. For single-stratum (aov/lm) fits and the raw-argument interface the columns are effect, eta_squared,

F_value, df_effect, df_error. For multi-stratum (aovlist) fits an additional stratum column reports which error stratum each effect's F test came from. When the raw-argument interface is used, effect is "overall".

Arguments

object: Optional. A fitted model object of class aov, lm, or aovlist (a multi-stratum aov fit such as aov(y ~ A + Error(subject/A), data = d)). When supplied, the function loops over the non-Residuals effects across all error strata and returns one row per effect, with the stratum reported.
F_value: Observed F-value from the fixed-effects ANOVA (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).

Author

Ken Kelley [email protected]

Details

The confidence interval is provided by the separate ci_eta_squared, paralleling the existing smd/ci_smd pairing.

Point estimate. The function uses the algebraically equivalent F-and-df form $$\hat{\eta}^2 = \frac{df_{\text{effect}} \cdot F}{df_{\text{effect}} \cdot F + df_{\text{error}}},$$ which equals $\mathit{SS}_{\text{effect}} / (\mathit{SS}_{\text{effect}} + \mathit{SS}_{\text{error}})$. In a one-way ANOVA this is also $\mathit{SS}_{\text{effect}} / \mathit{SS}_{\text{total}}$, the conventional total $\eta^2$. In a factorial design the same expression yields partial $\eta^2$ for each effect, because $\mathit{SS}_{\text{error}}$ appears in the denominator instead of $\mathit{SS}_{\text{total}}$; this matches the convention used by ci_omega_squared.

Designs supported.

Between-subjects ANOVA (one-way or factorial): supply either a fitted aov/lm model or the raw F and degrees of freedom for a single effect.
Within-subjects or mixed ANOVA: supply a fitted aovlist model produced with an Error() term, e.g.\ aov(y ~ time + Error(subject/time), data = d). The function walks every error stratum returned by summary(object) and reports each effect with the stratum's residual df, so the $\eta^2$ value uses the stratum's specific error term. The reported stratum column tells you which one.

For more advanced model classes (lmerMod, lme, etc.) the fitted-model interface is not yet supported; supply the relevant F and degrees of freedom via the raw interface.

Sums of squares in factorial designs. anova() on an aov/lm uses Type I (sequential) sums of squares. For balanced designs all three types agree; for unbalanced designs they differ. If Type II or III F-values are required, compute them with e.g.\ car::Anova(object, type = 3) and pass the relevant F and degrees of freedom into the raw-argument interface.

Generalized eta squared, comparable across designs. The basic $\eta^2$ (and partial $\eta^2$) returned here are not comparable across studies that differ in factor structure. See eta_squared_generalized for a comparable alternative (Olejnik & Algina, 2003; Bakeman, 2005).

References

Bakeman, R. (2005). Recommended effect size statistics for repeated measures designs. Behavior Research Methods, 37(3), 379--384. tools:::Rd_expr_doi("10.3758/BF03192707")

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.

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

Kelley, K., & Preacher, K. J. (2012). On effect size. Psychological Methods, 17, 137--152. tools:::Rd_expr_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.)

Olejnik, S., & Algina, J. (2003). Generalized eta and omega squared statistics: Measures of effect size for some common research designs. Psychological Methods, 8(4), 434--447. tools:::Rd_expr_doi("10.1037/1082-989X.8.4.434")

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. tools:::Rd_expr_doi("10.1037/1082-989X.9.2.164")

Examples

Run this code

# 1. Raw-argument interface. Bargman's (1970) 5-group one-way ANOVA,
#        also used in Venables (1975), Fleishman (1980), and Steiger (2004):
#        11 subjects per group, observed F = 11.221.
eta_squared(F_value = 11.221, df_effect = 4, df_error = 50)

# 2. One way ANOVA from a fitted model (depression_bdi: three
#        treatment arms, 10 per arm, N = 30).
fit_one <- aov(bdi_post ~ condition, data = depression_bdi)
eta_squared(fit_one)

# 3. 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_factorial <- aov(iq_8 ~ treatment * factor(grade), data = pygmalion)
eta_squared(fit_factorial)

# 4. Within-subjects (repeated measures) ANOVA. Simulated 20-subject
#        x 3-time design; each subject is measured at Pre, Mid, Post.
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(fit_rm)   # 'stratum' column identifies the within-subjects error

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