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.

## Usage

```
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`

.- partial
If

`TRUE`

, return partial indices.- generalized
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.- ci
Confidence Interval (CI) level

- alternative
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.- verbose
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 (when it is included in the ANOVA table).For Bayesian models, arguments passed to

`ss_function`

.

- squared
Return Cohen's

*f*or Cohen's*f*-squared?- model2
Optional second model for Cohen's f (/squared). If specified, returns the effect size for R-squared-change between the two models.

- ss_function
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.- draws
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

*are unbiased estimators of the population's*

**Epsilon***, which is especially important is small samples. But which to choose?*

**Eta**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 \(\chi^2\) distribution that places the observed
*t*, *F*, or \(\chi^2\) 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 - \(\alpha\))% confidence
interval contains all of the parameter values for which *p* > \(\alpha\)
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 \(\alpha\) 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.

## See also

Other effect size indices:
`cles()`

,
`cohens_d()`

,
`effectsize.BFBayesFactor()`

,
`phi()`

,
`rank_biserial()`

## Examples

```
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 [1.00].
# 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 [1.00].
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 variables: cyl_f
#> - One-sided CIs: upper bound fixed at [1.00].
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 [1.00].
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 [1.00].
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 [1.00].
#> - Interpretation rule: cohen1992
# Recommended: Type-2 or -3 effect sizes + effects coding
# -------------------------------------------------------
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 [1.00].
# afex takes care of both type-3 effects and effects coding:
data(obk.long, package = "afex")
model <- afex::aov_car(value ~ treatment * gender + Error(id / (phase)),
data = obk.long, observed = "gender")
#> Warning: More than one observation per design cell, aggregating data using `fun_aggregate = mean`.
#> To turn off this warning, pass `fun_aggregate = mean` explicitly.
#> Contrasts set to contr.sum for the following variables: treatment, gender
eta_squared(model)
#> # Effect Size for ANOVA (Type III)
#>
#> Parameter | Eta2 (partial) | 95% CI
#> ------------------------------------------------------
#> treatment | 0.44 | [0.00, 1.00]
#> gender | 0.27 | [0.00, 1.00]
#> treatment:gender | 0.36 | [0.00, 1.00]
#> phase | 0.62 | [0.35, 1.00]
#> treatment:phase | 0.49 | [0.14, 1.00]
#> gender:phase | 0.03 | [0.00, 1.00]
#> treatment:gender:phase | 0.11 | [0.00, 1.00]
#>
#> - One-sided CIs: upper bound fixed at [1.00].
epsilon_squared(model)
#> # Effect Size for ANOVA (Type III)
#>
#> Parameter | Epsilon2 (partial) | 95% CI
#> ----------------------------------------------------------
#> treatment | 0.33 | [0.00, 1.00]
#> gender | 0.19 | [0.00, 1.00]
#> treatment:gender | 0.24 | [0.00, 1.00]
#> phase | 0.58 | [0.30, 1.00]
#> treatment:phase | 0.39 | [0.02, 1.00]
#> gender:phase | -0.07 | [0.00, 1.00]
#> treatment:gender:phase | -0.06 | [0.00, 1.00]
#>
#> - One-sided CIs: upper bound fixed at [1.00].
omega_squared(model)
#> # Effect Size for ANOVA (Type III)
#>
#> Parameter | Omega2 (partial) | 95% CI
#> --------------------------------------------------------
#> treatment | 0.31 | [0.00, 1.00]
#> gender | 0.18 | [0.00, 1.00]
#> treatment:gender | 0.22 | [0.00, 1.00]
#> phase | 0.26 | [0.00, 1.00]
#> treatment:phase | 0.15 | [0.00, 1.00]
#> gender:phase | -0.02 | [0.00, 1.00]
#> treatment:gender:phase | -0.02 | [0.00, 1.00]
#>
#> - One-sided CIs: upper bound fixed at [1.00].
eta_squared(model, generalized = TRUE) # observed vars are pulled from the afex model.
#> # 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 variables: gender
#> - One-sided CIs: upper bound fixed at [1.00].
## Approx. effect sizes for mixed models
## -------------------------------------
model <- lme4::lmer(mpg ~ am_f * cyl_f + (1 | vs), data = mtcars)
#> boundary (singular) fit: see help('isSingular')
omega_squared(model)
#> # 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 [1.00].
## Bayesian Models (PPD)
## ---------------------
if (FALSE) {
fit_bayes <- rstanarm::stan_glm(
mpg ~ factor(cyl) * wt + qsec,
data = mtcars, family = gaussian(),
refresh = 0
)
es <- eta_squared_posterior(fit_bayes, verbose = FALSE,
ss_function = car::Anova, type = 3)
bayestestR::describe_posterior(es, test = NULL)
# compare to:
fit_freq <- lm(mpg ~ factor(cyl) * wt + qsec,
data = mtcars)
aov_table <- car::Anova(fit_freq, type = 3)
eta_squared(aov_table)
}
```