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eta_squared  R Documentation 
\eta^2
and Other Effect Size for ANOVAFunctions 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,
generalized = FALSE,
squared = FALSE,
method = c("eta", "omega", "epsilon"),
model2 = NULL,
ci = 0.95,
alternative = "greater",
verbose = TRUE,
...
)
cohens_f_squared(
model,
partial = TRUE,
generalized = FALSE,
squared = TRUE,
method = c("eta", "omega", "epsilon"),
model2 = NULL,
ci = 0.95,
alternative = "greater",
verbose = TRUE,
...
)
eta_squared_posterior(
model,
partial = TRUE,
generalized = FALSE,
ss_function = stats::anova,
draws = 500,
verbose = TRUE,
...
)
model 
An ANOVA table (or an ANOVAlike table, e.g., outputs from

partial 
If 
generalized 
A character vector of observed (nonmanipulated) variables
to be used in the estimation of a generalized Eta Squared. Can also be

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? 
method 
What effect size should be used as the basis for Cohen's f? 
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. 
For aov
(or lm
), aovlist
and afex_aov
models, and for anova
objects
that provide SumsofSquares, the effect sizes are computed directly using
SumsofSquares. (For maov
(or mlm
) models, effect sizes are computed for
each response separately.)
For other ANOVA tables and models (converted to ANOVAlike tables via
anova()
methods), effect sizes are approximated via test statistic
conversion of the omnibus F statistic provided by the (see F_to_eta2()
for more details.)
When model
is a statistical model, the sums of squares (or F statistics)
used for the computation of the effect sizes are based on those returned by
anova(model)
. Different models have different default output type. For
example, for aov
and aovlist
these are type1 sums of squares, but 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
convert your model to an ANOVA(like) table yourself and then pass the result
to eta_squared()
. See examples below for use of car::Anova()
and the
afex
package.
For type 3 sum of squares, it is generally recommended to fit models with
orthogonal factor weights (e.g., contr.sum
) 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 and Lakens, 2018), Epsilon is
analogous to adjusted R2 (Allen, 2017, p. 382), and has been found to be less
biased (Carroll & Nordholm, 1975).
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.
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.
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.
"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).
see
The see
package contains relevant plotting functions. See the plotting vignette in the see
package.
Albers, C., and 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.
F_to_eta2()
Other effect sizes for ANOVAs:
rank_epsilon_squared()
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))
# More types:
eta_squared(model, partial = FALSE)
eta_squared(model, generalized = "cyl_f")
omega_squared(model)
epsilon_squared(model)
cohens_f(model)
model0 < aov(mpg ~ am_f + cyl_f, data = mtcars) # no interaction
cohens_f_squared(model0, model2 = model)
## Interpretation of effect sizes
## 
interpret_omega_squared(0.10, rules = "field2013")
interpret_eta_squared(0.10, rules = "cohen1992")
interpret_epsilon_squared(0.10, rules = "cohen1992")
interpret(eta2, rules = "cohen1992")
plot(eta2) # Requires the {see} package
# Recommended: Type2 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)
epsilon_squared(model_anova)
# afex takes care of both type3 effects and effects coding:
data(obk.long, package = "afex")
model < afex::aov_car(value ~ gender + Error(id / (phase * hour)),
data = obk.long, observed = "gender"
)
omega_squared(model)
eta_squared(model, generalized = TRUE) # observed vars are pulled from the afex model.
## Approx. effect sizes for mixed models
## 
model < lme4::lmer(mpg ~ am_f * cyl_f + (1  vs), data = mtcars)
omega_squared(model)
## Bayesian Models (PPD)
## 
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)
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