F_to_eta2: Convert test statistics (F, t) to indices of partial variance explained (partial Eta / Omega / Epsilon squared and Cohen's f)

Description

These functions are convenience functions to convert F and t test statistics to partial Eta squared, ( $η {_{p}}^{2}$ ), Omega squared ( $ω {_{p}}^{2}$ ), Epsilon squared ( $ϵ {_{p}}^{2}$ ; an alias for the adjusted Eta squared) and Cohen's f. These are useful in cases where the various Sum of Squares and Mean Squares are not easily available or their computation is not straightforward (e.g., in liner mixed models, contrasts, etc.). For test statistics derived from lm and aov models, these functions give exact results. For all other cases, they return close approximations.

See Effect Size from Test Statistics vignette.

Usage

F_to_eta2(f, df, df_error, ci = 0.9, ...)
t_to_eta2(t, df_error, ci = 0.9, ...)
F_to_epsilon2(f, df, df_error, ci = 0.9, ...)
t_to_epsilon2(t, df_error, ci = 0.9, ...)
F_to_eta2_adj(f, df, df_error, ci = 0.9, ...)
t_to_eta2_adj(t, df_error, ci = 0.9, ...)
F_to_omega2(f, df, df_error, ci = 0.9, ...)
t_to_omega2(t, df_error, ci = 0.9, ...)
F_to_f(f, df, df_error, ci = 0.9, ...)
t_to_f(t, df_error, ci = 0.9, ...)

Arguments

df, df_error

Degrees of freedom of numerator or of the error estimate (i.e., the residuals).

Confidence Interval (CI) level

...

Arguments passed to or from other methods.

t, f

The t or the F statistics.

Value

A data frame with the effect size(s) between 0-1, and confidence interval(s) (Note that for $ω_{p}^{2}$ and $ϵ_{p}^{2}$ it is possible to compute a negative number; even though this doesn't make any practical sense, it is recommended to report the negative number and not a 0).

Details

These functions use the following formulae:

$η_{p}^{2} = \frac{F \times d f_{n u m}}{F \times d f_{n u m} + d f_{d e n}}$

$ϵ_{p}^{2} = \frac{(F - 1) \times d f_{n u m}}{F \times d f_{n u m} + d f_{d e n}}$

$ω_{p}^{2} = \frac{(F - 1) \times d f_{n u m}}{F \times d f_{n u m} + d f_{d e n} + 1}$

$f_{p} = \sqrt{\frac{η_{p}^{2}}{1 - η_{p}^{2}}}$

For $t$ , the conversion is based on the equality of $t^{2} = F$ when $d f_{n u m} = 1$ .

Confidence Intervals

Confidence intervals are estimated using the Noncentrality parameter method; These methods searches for a the best ncp (non-central parameters) for of the noncentral F distribution for the desired tail-probabilities, and then convert these ncps to the corresponding effect sizes.

References

Friedman, H. (1982). Simplified determinations of statistical power, magnitude of effect and research sample sizes. Educational and Psychological Measurement, 42(2), 521-526. 10.1177/001316448204200214
Mordkoff, J. T. (2019). A Simple Method for Removing Bias From a Popular Measure of Standardized Effect Size: Adjusted Partial Eta Squared. Advances in Methods and Practices in Psychological Science, 2(3), 228-232. 10.1177/2515245919855053
Albers, C., & Lakens, D. (2018). When power analyses based on pilot data are biased: Inaccurate effect size estimators and follow-up bias. Journal of experimental social psychology, 74, 187-195. 10.31234/osf.io/b7z4q
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, 164-182.
Cumming, G., & Finch, S. (2001). A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions. Educational and Psychological Measurement, 61(4), 532-574.

Examples

Run this code

# NOT RUN {
if (require("afex")) {
  data(md_12.1)
  aov_ez("id", "rt", md_12.1,
    within = c("angle", "noise"),
    anova_table = list(correction = "none", es = "pes")
  )
}
# compare to:
(etas <- F_to_eta2(
  f = c(40.72, 33.77, 45.31),
  df = c(2, 1, 2),
  df_error = c(18, 9, 18)
))

if(require(see)) plot(etas)


if (require("lmerTest")) { # for the df_error
  fit <- lmer(extra ~ group + (1 | ID), sleep)
  anova(fit)
  # Type III Analysis of Variance Table with Satterthwaite's method
  #       Sum Sq Mean Sq NumDF DenDF F value   Pr(>F)
  # group 12.482  12.482     1     9  16.501 0.002833 **
  # ---
  # Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
  F_to_eta2(16.501, 1, 9)
  F_to_omega2(16.501, 1, 9)
  F_to_epsilon2(16.501, 1, 9)
  F_to_f(16.501, 1, 9)
}

## Use with emmeans based contrasts
if (require(emmeans)) {
  warp.lm <- lm(breaks ~ wool * tension, data = warpbreaks)

  jt <- joint_tests(warp.lm, by = "wool")
  F_to_eta2(jt$F.ratio, jt$df1, jt$df2)
}
# }

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