Functions to compute effect size measures for ANOVAs, such as Eta- ($$\eta$$), Omega- ($$\omega$$) and Epsilon- ($$\epsilon$$) squared, and Cohen's f (or their partialled versions) for ANOVA tables. These indices represent an estimate of how much variance in the response variables is accounted for by the explanatory variable(s).

When passing models, effect sizes are computed using the sums of squares obtained from anova(model) which might not always be appropriate. See details.

eta_squared(
model,
partial = TRUE,
generalized = FALSE,
ci = 0.95,
alternative = "greater",
verbose = TRUE,
...
)

omega_squared(
model,
partial = TRUE,
ci = 0.95,
alternative = "greater",
verbose = TRUE,
...
)

epsilon_squared(
model,
partial = TRUE,
ci = 0.95,
alternative = "greater",
verbose = TRUE,
...
)

cohens_f(
model,
partial = TRUE,
ci = 0.95,
alternative = "greater",
squared = FALSE,
verbose = TRUE,
model2 = NULL,
...
)

cohens_f_squared(
model,
partial = TRUE,
ci = 0.95,
alternative = "greater",
squared = TRUE,
verbose = TRUE,
model2 = NULL,
...
)

eta_squared_posterior(
model,
partial = TRUE,
generalized = FALSE,
ss_function = stats::anova,
draws = 500,
verbose = TRUE,
...
)

## Arguments

model A model, ANOVA object, or the result of parameters::model_parameters. If TRUE, return partial indices. If TRUE, returns generalized Eta Squared, assuming all variables are manipulated. Can also be a character vector of observed (non-manipulated) variables, in which case generalized Eta Squared is calculated taking these observed variables into account. For afex_aov model, when generalized = TRUE, the observed variables are extracted automatically from the fitted model, if they were provided then. Confidence Interval (CI) level a character string specifying the alternative hypothesis; Controls the type of CI returned: "greater" (default) or "less" (one-sided CI), or "two.sided" (default, two-sided CI). Partial matching is allowed (e.g., "g", "l", "two"...). See One-Sided CIs in effectsize_CIs. Toggle warnings and messages on or off. Arguments passed to or from other methods. Can be include_intercept = TRUE to include the effect size for the intercept. For Bayesian models, arguments passed to ss_function. Return Cohen's f or Cohen's f-squared? Optional second model for Cohen's f (/squared). If specified, returns the effect size for R-squared-change between the two models. For Bayesian models, the function used to extract sum-of-squares. Uses anova() by default, but can also be car::Anova() for simple linear models. For Bayesian models, an integer indicating the number of draws from the posterior predictive distribution to return. Larger numbers take longer to run, but provide estimates that are more stable.

## Value

A data frame with the effect size(s) between 0-1 (Eta2, Epsilon2, Omega2, Cohens_f or Cohens_f2, possibly with the partial or generalized suffix), and their CIs (CI_low and CI_high).

For eta_squared_posterior(), a data frame containing the ppd of the Eta squared for each fixed effect, which can then be passed to bayestestR::describe_posterior() for summary stats.

A data frame containing the effect size values and their confidence intervals.

## Details

For aov, aovlist and afex_aov models, and for anova objects that provide Sums-of-Squares, the effect sizes are computed directly using Sums-of-Squares (for mlm / maov models, effect sizes are computed for each response separately). For all other model, effect sizes are approximated via test statistic conversion of the omnibus F statistic provided by the appropriate anova() method (see F_to_eta2() for more details.)

### Type of Sums of Squares

The sums of squares (or F statistics) used for the computation of the effect sizes is based on those returned by anova(model) (whatever those may be - for aov and aovlist these are type-1 sums of squares; for lmerMod (and lmerModLmerTest) these are type-3 sums of squares). Make sure these are the sums of squares you are interested in; You might want to pass the result of car::Anova(mode, type = 2) or type = 3 instead of the model itself, or use the afex package to fit ANOVA models.

For type 3 sum of squares, it is generally recommended to fit models with contr.sum factor weights and centered covariates, for sensible results. See examples and the afex package.

### Un-Biased Estimate of Eta

Both Omega and Epsilon are unbiased estimators of the population's Eta, which is especially important is small samples. But which to choose?

Though Omega is the more popular choice (Albers \& Lakens, 2018), Epsilon is analogous to adjusted R2 (Allen, 2017, p. 382), and has been found to be less biased (Carroll & Nordholm, 1975).

(Note that for Omega- and Epsilon-squared 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.)

### Cohen's f

Cohen's f can take on values between zero, when the population means are all equal, and an indefinitely large number as standard deviation of means increases relative to the average standard deviation within each group.

When comparing two models in a sequential regression analysis, Cohen's f for R-square change is the ratio between the increase in R-square and the percent of unexplained variance.

Cohen has suggested that the values of 0.10, 0.25, and 0.40 represent small, medium, and large effect sizes, respectively.

### Eta Squared from Posterior Predictive Distribution

For Bayesian models (fit with brms or rstanarm), eta_squared_posterior() simulates data from the posterior predictive distribution (ppd) and for each simulation the Eta Squared is computed for the model's fixed effects. This means that the returned values are the population level effect size as implied by the posterior model (and not the effect size in the sample data). See rstantools::posterior_predict() for more info.

## Confidence (Compatibility) Intervals (CIs)

Unless stated otherwise, confidence (compatibility) intervals (CIs) are estimated using the noncentrality parameter method (also called the "pivot method"). This method finds the noncentrality parameter ("ncp") of a noncentral t, F, or χ2 distribution that places the observed t, F, or χ2 test statistic at the desired probability point of the distribution. For example, if the observed t statistic is 2.0, with 50 degrees of freedom, for which cumulative noncentral t distribution is t = 2.0 the .025 quantile (answer: the noncentral t distribution with ncp = .04)? After estimating these confidence bounds on the ncp, they are converted into the effect size metric to obtain a confidence interval for the effect size (Steiger, 2004).

For additional details on estimation and troubleshooting, see effectsize_CIs.

## CIs and Significance Tests

"Confidence intervals on measures of effect size convey all the information in a hypothesis test, and more." (Steiger, 2004). Confidence (compatibility) intervals and p values are complementary summaries of parameter uncertainty given the observed data. A dichotomous hypothesis test could be performed with either a CI or a p value. The 100(1 − α)% confidence interval contains all of the parameter values for which p > α for the current data and model. For example, a 95% confidence interval contains all of the values for which p > .05.

Note that a confidence interval including 0 does not indicate that the null (no effect) is true. Rather, it suggests that the observed data together with the model and its assumptions combined do not provided clear evidence against a parameter value of 0 (same as with any other value in the interval), with the level of this evidence defined by the chosen α level (Rafi & Greenland, 2020; Schweder & Hjort, 2016; Xie & Singh, 2013). To infer no effect, additional judgments about what parameter values are "close enough" to 0 to be negligible are needed ("equivalence testing"; Bauer & Kiesser, 1996).

## 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.

• Allen, R. (2017). Statistics and Experimental Design for Psychologists: A Model Comparison Approach. World Scientific Publishing Company.

• Carroll, R. M., & Nordholm, L. A. (1975). Sampling Characteristics of Kelley's epsilon and Hays' omega. Educational and Psychological Measurement, 35(3), 541-554.

• Kelley, T. (1935) An unbiased correlation ratio measure. Proceedings of the National Academy of Sciences. 21(9). 554-559.

• 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.

• 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.

F_to_eta2()

Other effect size indices: cohens_d(), effectsize(), phi(), rank_biserial(), standardize_parameters()

## Examples

# \donttest{
data(mtcars)
mtcars$am_f <- factor(mtcars$am)
mtcars$cyl_f <- factor(mtcars$cyl)

model <- aov(mpg ~ am_f * cyl_f, data = mtcars)

(eta2 <- eta_squared(model))
#> # Effect Size for ANOVA (Type I)
#>
#> Parameter  | Eta2 (partial) |       95% CI
#> ------------------------------------------
#> am_f       |           0.63 | [0.42, 1.00]
#> cyl_f      |           0.66 | [0.45, 1.00]
#> am_f:cyl_f |           0.10 | [0.00, 1.00]
#>
#> - One-sided CIs: upper bound fixed at (1).

# More types:
eta_squared(model, partial = FALSE)
#> # Effect Size for ANOVA (Type I)
#>
#> Parameter  | Eta2 |       95% CI
#> --------------------------------
#> am_f       | 0.36 | [0.13, 1.00]
#> cyl_f      | 0.41 | [0.14, 1.00]
#> am_f:cyl_f | 0.02 | [0.00, 1.00]
#>
#> - One-sided CIs: upper bound fixed at (1).
eta_squared(model, generalized = "cyl_f")
#> # Effect Size for ANOVA (Type I)
#>
#> Parameter  | Eta2 (generalized) |       95% CI
#> ----------------------------------------------
#> am_f       |               0.36 | [0.13, 1.00]
#> cyl_f      |               0.63 | [0.42, 1.00]
#> am_f:cyl_f |               0.04 | [0.00, 1.00]
#>
#> - Observed variabels: cyl_f
#> - One-sided CIs: upper bound fixed at (1).
omega_squared(model)
#> # Effect Size for ANOVA (Type I)
#>
#> Parameter  | Omega2 (partial) |       95% CI
#> --------------------------------------------
#> am_f       |             0.57 | [0.35, 1.00]
#> cyl_f      |             0.60 | [0.37, 1.00]
#> am_f:cyl_f |             0.02 | [0.00, 1.00]
#>
#> - One-sided CIs: upper bound fixed at (1).
epsilon_squared(model)
#> # Effect Size for ANOVA (Type I)
#>
#> Parameter  | Epsilon2 (partial) |       95% CI
#> ----------------------------------------------
#> am_f       |               0.61 | [0.40, 1.00]
#> cyl_f      |               0.63 | [0.41, 1.00]
#> am_f:cyl_f |               0.03 | [0.00, 1.00]
#>
#> - One-sided CIs: upper bound fixed at (1).
cohens_f(model)
#> # Effect Size for ANOVA (Type I)
#>
#> Parameter  | Cohen's f (partial) |           95% CI
#> ---------------------------------------------------
#> am_f       |                1.30 | [0.86,      Inf]
#> cyl_f      |                1.38 | [0.90,      Inf]
#> am_f:cyl_f |                0.33 | [0.00,      Inf]
#>
#> - One-sided CIs: upper bound fixed at (Inf).

if (require(see)) plot(eta2)

model0 <- aov(mpg ~ am_f + cyl_f, data = mtcars) # no interaction
cohens_f_squared(model0, model2 = model)
#> Cohen's f2 (partial) |           95% CI | R2_delta
#> --------------------------------------------------
#> 0.11                 | [0.00,      Inf] |     0.02
#>
#> - One-sided CIs: upper bound fixed at (Inf).

## Interpretation of effect sizes
## -------------------------------------

interpret_omega_squared(0.10, rules = "field2013")
#> [1] "medium"
#> (Rules: field2013)
#>
interpret_eta_squared(0.10, rules = "cohen1992")
#> [1] "small"
#> (Rules: cohen1992)
#>
interpret_epsilon_squared(0.10, rules = "cohen1992")
#> [1] "small"
#> (Rules: cohen1992)
#>

interpret(eta2, rules = "cohen1992")
#> # Effect Size for ANOVA (Type I)
#>
#> Parameter  | Eta2 (partial) |       95% CI | Interpretation
#> -----------------------------------------------------------
#> am_f       |           0.63 | [0.42, 1.00] |          large
#> cyl_f      |           0.66 | [0.45, 1.00] |          large
#> am_f:cyl_f |           0.10 | [0.00, 1.00] |          small
#>
#> - One-sided CIs: upper bound fixed at (1).
#> (Interpretation rule: cohen1992)

# Recommended: Type-3 effect sizes + effects coding
# -------------------------------------------------
if (require(car, quietly = TRUE)) {
contrasts(mtcars$am_f) <- contr.sum contrasts(mtcars$cyl_f) <- contr.sum

model <- aov(mpg ~ am_f * cyl_f, data = mtcars)
model_anova <- car::Anova(model, type = 3)

eta_squared(model_anova)
}
#> Type 3 ANOVAs only give sensible and informative results when covariates are
#>   mean-centered and factors are coded with orthogonal contrasts (such as those
#>   produced by 'contr.sum', 'contr.poly', or 'contr.helmert', but *not* by the
#>   default 'contr.treatment').
#> # Effect Size for ANOVA (Type III)
#>
#> Parameter  | Eta2 (partial) |       95% CI
#> ------------------------------------------
#> am_f       |           0.11 | [0.00, 1.00]
#> cyl_f      |           0.63 | [0.42, 1.00]
#> am_f:cyl_f |           0.10 | [0.00, 1.00]
#>
#> - One-sided CIs: upper bound fixed at (1).

# afex takes care of both type-3 effects and effects coding:
if (require(afex)) {
data(obk.long, package = "afex")
model <- aov_car(value ~ treatment * gender + Error(id / (phase)),
data = obk.long, observed = "gender"
)
eta_squared(model)
epsilon_squared(model)
omega_squared(model)
eta_squared(model, partial = FALSE)
epsilon_squared(model, partial = FALSE)
omega_squared(model, partial = FALSE)
eta_squared(model, generalized = TRUE) # observed vars are pulled from the afex model.
}
#> Warning: More than one observation per cell, aggregating the data using mean (i.e, fun_aggregate = mean)!
#> Contrasts set to contr.sum for the following variables: treatment, gender
#> # Effect Size for ANOVA (Type III)
#>
#> Parameter              | Eta2 (generalized) |       95% CI
#> ----------------------------------------------------------
#> treatment              |               0.25 | [0.00, 1.00]
#> gender                 |               0.16 | [0.00, 1.00]
#> treatment:gender       |               0.24 | [0.00, 1.00]
#> phase                  |               0.20 | [0.00, 1.00]
#> treatment:phase        |               0.13 | [0.00, 1.00]
#> gender:phase           |           4.25e-03 | [0.00, 1.00]
#> treatment:gender:phase |               0.02 | [0.00, 1.00]
#>
#> - Observed variabels: gender
#> - One-sided CIs: upper bound fixed at (1).

## Approx. effect sizes for mixed models
## -------------------------------------
if (require(lmerTest, quietly = TRUE)) {
model <- lmer(mpg ~ am_f * cyl_f + (1 | vs), data = mtcars)
omega_squared(model)
}
#> boundary (singular) fit: see ?isSingular
#> # Effect Size for ANOVA (Type III)
#>
#> Parameter  | Omega2 (partial) |       95% CI
#> --------------------------------------------
#> am_f       |             0.07 | [0.00, 1.00]
#> cyl_f      |             0.60 | [0.36, 1.00]
#> am_f:cyl_f |             0.03 | [0.00, 1.00]
#>
#> - One-sided CIs: upper bound fixed at (1).

## Bayesian Models (PPD)
## ---------------------
if (FALSE) {
if (require(rstanarm) && require(bayestestR) && require(car)) {
fit_bayes <- stan_glm(mpg ~ factor(cyl) * wt + qsec,
data = mtcars,
family = gaussian(),
refresh = 0
)

es <- eta_squared_posterior(fit_bayes,
ss_function = car::Anova, type = 3
)
bayestestR::describe_posterior(es)

# compare to:
fit_freq <- lm(mpg ~ factor(cyl) * wt + qsec,
data = mtcars
)
aov_table <- car::Anova(fit_freq, type = 3)
eta_squared(aov_table)
}
}
# }