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,
...
)
model  A model, ANOVA object, or the result of 

partial  If 
generalized  If TRUE, returns generalized Eta Squared, assuming all
variables are manipulated. Can also be a character vector of observed
(nonmanipulated) variables, in which case generalized Eta Squared is
calculated taking these observed variables into account. For 
ci  Confidence Interval (CI) level 
alternative  a character string specifying the alternative hypothesis;
Controls the type of CI returned: 
verbose  Toggle warnings and messages on or off. 
...  Arguments passed to or from other methods.

squared  Return Cohen's f or Cohen's fsquared? 
model2  Optional second model for Cohen's f (/squared). If specified, returns the effect size for Rsquaredchange between the two models. 
ss_function  For Bayesian models, the function used to extract
sumofsquares. Uses 
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. 
A data frame with the effect size(s) between 01 (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.
For aov
, aovlist
and afex_aov
models, and for anova
objects that
provide SumsofSquares, the effect sizes are computed directly using
SumsofSquares (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.)
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 type1 sums of squares; for
lmerMod
(and lmerModLmerTest
) these are type3 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.
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 Epsilonsquared 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 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
Rsquare change is the ratio between the increase in Rsquare
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.
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.
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
χ^{2}
distribution that places the observed t, F, or
χ^{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.
"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).
Albers, C., \& Lakens, D. (2018). When power analyses based on pilot data are biased: Inaccurate effect size estimators and followup bias. Journal of experimental social psychology, 74, 187195.
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), 541554.
Kelley, T. (1935) An unbiased correlation ratio measure. Proceedings of the National Academy of Sciences. 21(9). 554559.
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, 164182.
Other effect size indices:
cohens_d()
,
effectsize()
,
phi()
,
rank_biserial()
,
standardize_parameters()
# \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]
#>
#>  Onesided 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]
#>
#>  Onesided 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
#>  Onesided 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]
#>
#>  Onesided 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]
#>
#>  Onesided 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]
#>
#>  Onesided 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
#>
#>  Onesided 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
#>
#>  Onesided CIs: upper bound fixed at (1).
#> (Interpretation rule: cohen1992)
# Recommended: Type3 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
#> meancentered 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]
#>
#>  Onesided CIs: upper bound fixed at (1).
# afex takes care of both type3 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  4.25e03  [0.00, 1.00]
#> treatment:gender:phase  0.02  [0.00, 1.00]
#>
#>  Observed variabels: gender
#>  Onesided 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]
#>
#>  Onesided 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)
}
}
# }