omega_partial_ss_bn: omega^2_p (Partial Omega Squared) for Between-Subjects ANOVA...

In MOTE: Effect Size and Confidence Interval Calculator

omega^2_p (Partial Omega Squared) for Between-Subjects ANOVA from F

Description

This function displays \omega^2_p from ANOVA analyses and its non-central confidence interval based on the F distribution. This formula is appropriate for multi-way between-subjects designs.

Usage

omega_partial_ss_bn(dfm, dfe, msm, mse, ssm, n, a = 0.05)

omega.partial.SS.bn(dfm, dfe, msm, mse, ssm, n, a = 0.05)

Arguments

dfm	degrees of freedom for the model/IV/between
dfe	degrees of freedom for the error/residual/within
msm	mean square for the model/IV/between
mse	mean square for the error/residual/within
ssm	sum of squares for the model/IV/between
n	total sample size
a	significance level

Details

Partial omega squared is calculated by subtracting the mean square for the error from the mean square of the model, which is multiplied by degrees of freedom of the model. This is divided by the product of the degrees of freedom for the model are deducted from the sample size, multiplied by the mean square of the error, plus the sum of squares for the model.

\omega^2_p = \frac{df_m (MS_M - MS_E)}{SS_M + (n - df_m) \times MS_E}

Learn more on our example page.

**Note on function and output names:** This effect size is now implemented with the snake_case function name 'omega_partial_ss_bn()' to follow modern R style guidelines. The original dotted version 'omega.partial.SS.bn()' is still available as a wrapper for backward compatibility, and both functions return the same list. The returned object includes both the original element names (e.g., 'omega', 'omegalow', 'omegahigh', 'dfm', 'dfe', 'F', 'p', 'estimate', 'statistic') and newer snake_case aliases (e.g., 'omega_value', 'omega_lower_limit', 'omega_upper_limit', 'df_model', 'df_error', 'f_value', 'p_value'). New code should prefer 'omega_partial_ss_bn()' and the snake_case output names, but existing code using the older names will continue to work.

Value

omega

\omega^2_p effect size (legacy name; see also 'omega_value')

omegalow

lower level confidence interval of \omega^2_p (legacy name; see also 'omega_lower_limit')

omegahigh

upper level confidence interval of \omega^2_p (legacy name; see also 'omega_upper_limit')

dfm

degrees of freedom for the model/IV/between (legacy name; see also 'df_model')

dfe

degrees of freedom for the error/residual/within (legacy name; see also 'df_error')

F

F-statistic (legacy name; see also 'f_value')

p

p-value (legacy name; see also 'p_value')

estimate

the \omega^2_p statistic and confidence interval in APA style for markdown printing

statistic

the F-statistic in APA style for markdown printing

omega_value

\omega^2_p effect size (snake_case alias of 'omega')

omega_lower_limit

lower level confidence interval of \omega^2_p (alias of 'omegalow')

omega_upper_limit

upper level confidence interval of \omega^2_p (alias of 'omegahigh')

df_model

degrees of freedom for the model/IV/between (alias of 'dfm')

df_error

degrees of freedom for the error/residual/within (alias of 'dfe')

f_value

F-statistic (alias of 'F')

p_value

p-value (alias of 'p')

Examples

# The following example is derived from the "bn2_data"
# dataset, included in the MOTE library.

# Is there a difference in athletic spending budget for different sports?
# Does that spending interact with the change in coaching staff?
# This data includes (fake) athletic budgets for baseball,
# basketball, football, soccer, and volleyball teams
# with new and old coaches to determine if there are differences in
# spending across coaches and sports.

# You would calculate one omega value for each F-statistic.
# Here's an example for the interaction using reported ANOVA values.
omega_partial_ss_bn(dfm = 4, dfe = 990,
                    msm = 338057.9 / 4,
                    mse = 32833499 / 990,
                    ssm = 338057.9,
                    n = 1000, a = .05)

# Backwards-compatible dotted name (deprecated)
omega.partial.SS.bn(dfm = 4, dfe = 990,
                    msm = 338057.9 / 4,
                    mse = 32833499 / 990,
                    ssm = 338057.9,
                    n = 1000, a = .05)

# The same analysis can be fit with stats::lm and car::Anova(type = 3).
# This example shows how to obtain the ANOVA table and plug its values
# into omega.partial.SS.bn without relying on ezANOVA.
if (requireNamespace("car", quietly = TRUE)) {

  mod <- stats::lm(money ~ coach * type, data = bn2_data)

  # Type I table (for residual SS and df)
  aov_type1 <- stats::anova(mod)

  # Type III SS table for the effects
  aov_type3 <- car::Anova(mod, type = 3)

  # Extract dfs and sums of squares for the interaction coach:type
  dfm_int <- aov_type3["coach:type", "Df"]
  ssm_int <- aov_type3["coach:type", "Sum Sq"]
  msm_int <- ssm_int / dfm_int

  dfe <- aov_type1["Residuals", "Df"]
  sse <- aov_type1["Residuals", "Sum Sq"]
  mse <- sse / dfe

  omega_partial_ss_bn(dfm = dfm_int,
                      dfe = dfe,
                      msm = msm_int,
                      mse = mse,
                      ssm = ssm_int,
                      n = nrow(bn2_data),
                      a = .05)
}

MOTE documentation built on Dec. 15, 2025, 9:06 a.m.