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 $(\eta{_p}^2)$, Omega squared $(\omega{_p}^2)$, Epsilon squared ($\epsilon{_p}^2;$ 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, squared = FALSE, ...)
t_to_f(t, df_error, ci = 0.9, squared = FALSE, ...)
F_to_f2(f, df, df_error, ci = 0.9, squared = TRUE, ...)
t_to_f2(t, df_error, ci = 0.9, squared = TRUE, ...)

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.

squared

Return Cohen's f or Cohen's f-squared?

Value

A data frame with the effect size(s) between 0-1 (Eta2_partial, Epsilon2_partial, Omega2_partial, Cohens_f_partial or Cohens_f2_partial), and their CIs (CI_low and CI_high). (Note that for $\omega_p^2$ and $\epsilon_p^2$ 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).

Confidence Intervals

Unless stated otherwise, confidence intervals are estimated using the Noncentrality parameter method; These methods searches for a the best non-central parameters (ncps) of the noncentral t-, F- or Chi-squared distribution for the desired tail-probabilities, and then convert these ncps to the corresponding effect sizes. (See full effectsize-CIs for more.)

CI Contains Zero

Keep in mind that ncp confidence intervals are inverted significance tests, and only inform us about which values are not significantly different than our sample estimate. (They do not inform us about which values are plausible, likely or compatible with our data.) Thus, when CIs contain the value 0, this should not be taken to mean that a null effect size is supported by the data; Instead this merely reflects a non-significant test statistic - i.e. the p-value is greater than alpha (Morey et al., 2016).

For positive only effect sizes (Eta squared, Cramer's V, etc.; Effect sizes associated with Chi-squared and F distributions), this applies also to cases where the lower bound of the CI is equal to 0. Even more care should be taken when the upper bound is equal to 0 - this occurs when p-value is greater than 1-alpha/2 making, the upper bound cannot be estimated, and the upper bound is arbitrarily set to 0 (Steiger, 2004). For example:

eta_squared(aov(mpg ~ factor(gear) + factor(cyl), mtcars[1:7, ]))

## # Effect Size for ANOVA (Type I)
## 
## Parameter    | Eta2 (partial) |       90% CI
## --------------------------------------------
## factor(gear) |           0.58 | [0.00, 0.84]
## factor(cyl)  |           0.46 | [0.00, 0.78]

Details

These functions use the following formulae:

$$\eta_p^2 = \frac{F \times df_{num}}{F \times df_{num} + df_{den}}$$

$$\epsilon_p^2 = \frac{(F - 1) \times df_{num}}{F \times df_{num} + df_{den}}$$

$$\omega_p^2 = \frac{(F - 1) \times df_{num}}{F \times df_{num} + df_{den} + 1}$$

$$f_p = \sqrt{\frac{\eta_p^2}{1-\eta_p^2}}$$

For $t$, the conversion is based on the equality of $t^2 = F$ when $df_{num}=1$.

Choosing an Un-Biased Estimate

Both Omega and Epsilon are unbiased estimators of the population Eta. But which to choose? Though Omega is the more popular choice, it should be noted that:

The formula given above for Omega is only an approximation for complex designs.
Epsilon has been found to be less biased (Carroll & Nordholm, 1975).

References

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
Carroll, R. M., & Nordholm, L. A. (1975). Sampling Characteristics of Kelley's epsilon and Hays' omega. Educational and Psychological Measurement, 35(3), 541-554.
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.
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
Morey, R. D., Hoekstra, R., Rouder, J. N., Lee, M. D., & Wagenmakers, E. J. (2016). The fallacy of placing confidence in confidence intervals. Psychonomic bulletin & review, 23(1), 103-123.
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.

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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