DMAR (version 1.0.0)

omega_squared: Omega Squared (Effect Size for ANOVA)

Description

Computes the sample omega squared (\(\omega^2\)), Hays' (1994) bias-corrected estimator of the proportion of variance in the dependent variable accounted for by a fixed effect. Accepts either the raw ANOVA summary (F, the effect and error degrees of freedom, and total N) or a fitted aov or lm object, in which case the function returns one row per effect (partial \(\omega^2\) in factorial designs).

Usage

omega_squared(
  object = NULL,
  F_value = NULL,
  df_effect = NULL,
  df_error = NULL,
  N = NULL
)

Value

A data.frame with one row per effect. The columns are effect, omega_squared (point estimate),

F_value, df_effect, df_error, and N. When the raw-argument interface is used, effect is

"overall".

Arguments

object

Optional. A fitted aov or lm object. When supplied, the function loops over the non-Residuals rows of anova(object) and returns one row per effect.

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).

N

Total sample size (ignored if object is supplied; nobs(object) is used instead).

Author

Ken Kelley [email protected]

Details

The confidence interval is provided by the separate ci_omega_squared, paralleling the existing smd/ci_smd and eta_squared/ci_eta_squared pairings.

Point estimate. The reported value is Hays' (1994) sample omega squared, which for a one-way design is $$\hat{\omega}^2 = \frac{df_{\text{effect}} (F - 1)}{df_{\text{effect}} (F - 1) + N}.$$ For factorial designs the same formula applied per effect yields partial omega squared (Olejnik & Algina, 2003); negative values are truncated to zero. This is the same point-estimate convention used by ci_omega_squared, so the two functions agree on the point estimate row by row.

Hand-in-hand with ci_omega_squared(). Pair this function with ci_omega_squared when reporting effect sizes: omega_squared() returns the point estimate(s), and ci_omega_squared() returns the same point estimate plus its noncentrality-based confidence limits (Steiger, 2004). The columns shared by the two functions (effect, omega_squared, F_value, df_effect, df_error, N) are aligned so the outputs compose cleanly with merge() or a join.

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.

References

Hays, W. L. (1994). Statistics (5th ed.). Fort Worth, TX: Harcourt Brace College Publishers.

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")

Keppel, G. (1991). Design and analysis: A researcher's handbook (3rd ed.). Prentice Hall.

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")

See Also

ci_omega_squared, eta_squared, ci_eta_squared, ci_pvaf

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

Examples

# 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.
omega_squared(F_value = 11.221, df_effect = 4, df_error = 50, N = 55)

# 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)
omega_squared(fit_one)

# 3. Two-factor ANOVA: partial omega squared per effect for the
#        manipulated expectancy treatment and the measured grade
#        classification (pygmalion data, unequal cell sizes,
#        N = 310). The treatment by grade interaction is weak here
#        (F = 1.19), so the additive model is used.
fit_additive <- aov(iq_8 ~ treatment + factor(grade), data = pygmalion)
omega_squared(fit_additive)

# 4. omega_squared() and ci_omega_squared() compose: the point
#        estimates agree row-by-row.
pt  <- omega_squared(fit_additive)
ci  <- ci_omega_squared(fit_additive)
merge(pt, ci, by = c("effect", "omega_squared",
                     "F_value", "df_effect", "df_error", "N"))