Eta^{2}
In the context of ANOVA-like tests, it is common to report ANOVA-like effect sizes. These effect sizes represent the amount of variance explained by each of the model’s terms, where each term can be represented by 1 or more parameters.
For example, in the following case, the parameters for the
treatment
term represent specific contrasts between the
factor’s levels (treatment groups) - the difference between each level
and the reference level (obk.long == 'control'
).
data(obk.long, package = "afex")
# modify the data slightly for the demonstration:
obk.long <- obk.long[1:240 %% 3 == 0, ]
obk.long$id <- seq_len(nrow(obk.long))
m <- lm(value ~ treatment, data = obk.long)
parameters::model_parameters(m)
> Parameter | Coefficient | SE | 95% CI | t(77) | p
> ------------------------------------------------------------------
> (Intercept) | 4.28 | 0.36 | [3.56, 5.00] | 11.85 | < .001
> treatment [A] | 1.97 | 0.54 | [0.89, 3.05] | 3.64 | < .001
> treatment [B] | 2.09 | 0.47 | [1.15, 3.03] | 4.42 | < .001
>
> Uncertainty intervals (equal-tailed) and p-values (two-tailed) computed
> using a Wald t-distribution approximation.
But we can also ask about the overall effect of
treatment
- how much of the variation in our dependent
variable value
can be predicted by (or explained by) the
variation between the treatment
groups. Such a question can
be answered with an ANOVA test:
parameters::model_parameters(anova(m))
> Parameter | Sum_Squares | df | Mean_Square | F | p
> -----------------------------------------------------------
> treatment | 72.23 | 2 | 36.11 | 11.08 | < .001
> Residuals | 250.96 | 77 | 3.26 | |
>
> Anova Table (Type 1 tests)
As we can see, the variance in value
(the
sums-of-squares, or SS) has been split into
pieces:
- The part associated with
treatment
. - The unexplained part (The Residual-SS).
We can now ask what is the percent of the total variance in
value
that is associated with treatment
. This
measure is called Eta-squared (written as \(\eta^2\)):
\[ \eta^2 = \frac{SS_{effect}}{SS_{total}} = \frac{72.23}{72.23 + 250.96} = 0.22 \]
and can be accessed via the eta_squared()
function:
library(effectsize)
options(es.use_symbols = TRUE) # get nice symbols when printing! (On Windows, requires R >= 4.2.0)
eta_squared(m, partial = FALSE)
> # Effect Size for ANOVA (Type I)
>
> Parameter | η² | 95% CI
> -------------------------------
> treatment | 0.22 | [0.09, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
Adding More Terms
When we add more terms to our model, we can ask two different questions about the percent of variance explained by a predictor - how much variance is accounted by the predictor in total, and how much is accounted when controlling for any other predictors. The latter questions is answered by the partial-Eta squared (\(\eta^2_p\)), which is the percent of the partial variance (after accounting for other predictors in the model) associated with a term:
\[
\eta^2_p = \frac{SS_{effect}}{SS_{effect} + SS_{error}}
\] which can also be accessed via the eta_squared()
function:
m <- lm(value ~ gender + phase + treatment, data = obk.long)
eta_squared(m, partial = FALSE)
> # Effect Size for ANOVA (Type I)
>
> Parameter | η² | 95% CI
> -----------------------------------
> gender | 0.03 | [0.00, 1.00]
> phase | 9.48e-03 | [0.00, 1.00]
> treatment | 0.25 | [0.11, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
eta_squared(m) # partial = TRUE by default
> # Effect Size for ANOVA (Type I)
>
> Parameter | η² (partial) | 95% CI
> ---------------------------------------
> gender | 0.04 | [0.00, 1.00]
> phase | 0.01 | [0.00, 1.00]
> treatment | 0.26 | [0.12, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
(phase
is a repeated-measures variable, but for
simplicity it is not modeled as such.)
In the calculation above, the SSs were computed sequentially
- that is the SS for phase
is computed after
controlling for gender
, and the SS for
treatment
is computed after controlling for both
gender
and phase
. This method of sequential
SS is called also type-I test. If this is what you
want, that’s great - however in many fields (and other statistical
programs) it is common to use “simultaneous” sums of squares
(type-II or type-III tests), where each SS is
computed controlling for all other predictors, regardless of order. This
can be done with car::Anova(type = ...)
:
eta_squared(car::Anova(m, type = 2), partial = FALSE)
> # Effect Size for ANOVA (Type II)
>
> Parameter | η² | 95% CI
> -----------------------------------
> gender | 0.05 | [0.00, 1.00]
> phase | 9.22e-03 | [0.00, 1.00]
> treatment | 0.24 | [0.11, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
eta_squared(car::Anova(m, type = 3)) # partial = TRUE by default
> # Effect Size for ANOVA (Type III)
>
> Parameter | η² (partial) | 95% CI
> ---------------------------------------
> gender | 0.07 | [0.01, 1.00]
> phase | 0.01 | [0.00, 1.00]
> treatment | 0.26 | [0.12, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
\(\eta^2_p\) will always be larger than \(\eta^2\). The idea is to simulate the effect size in a design where only the term of interest was manipulated. This terminology assumes some causal relationship between the predictor and the outcome, which reflects the experimental world from which these analyses and measures hail; However, \(\eta^2_p\) can also simply be seen as a signal-to-noise- ratio, as it only uses the term’s SS and the error-term’s SS.[^in repeated-measure designs the term-specific residual-SS is used for the computation of the effect size].
(Note that in a one-way fixed-effect designs \(\eta^2 = \eta^2_p\).)
Adding Interactions
Type II and type III treat interaction differently. Without going
into the weeds here, keep in mind that when using type III SS,
it is important to center all of the predictors; for numeric
variables this can be done by mean-centering the predictors; for factors
this can be done by using orthogonal coding (such as
contr.sum
for effects-coding) for the dummy
variables (and NOT treatment coding, which is the default in
R). This unfortunately makes parameter interpretation harder, but
only when this is does do the SSs associated with each
lower-order term (or lower-order interaction) represent the
SS of the main effect (with
treatment coding they represent the SS of the simple
effects).
# compare
m_interaction1 <- lm(value ~ treatment * gender, data = obk.long)
# to:
m_interaction2 <- lm(
value ~ treatment * gender,
data = obk.long,
contrasts = list(
treatment = "contr.sum",
gender = "contr.sum"
)
)
eta_squared(car::Anova(m_interaction1, type = 3))
> 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 | η² (partial) | 95% CI
> ----------------------------------------------
> treatment | 0.12 | [0.02, 1.00]
> gender | 9.11e-03 | [0.00, 1.00]
> treatment:gender | 0.20 | [0.07, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
eta_squared(car::Anova(m_interaction2, type = 3))
> 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 | η² (partial) | 95% CI
> ----------------------------------------------
> treatment | 0.27 | [0.13, 1.00]
> gender | 0.12 | [0.03, 1.00]
> treatment:gender | 0.20 | [0.07, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
If all of this type-III-effects-coding seems like a hassle, you can
use the afex
package, which takes care of all of this
behind the scenes:
> Loading required package: lme4
> Loading required package: Matrix
> ************
> Welcome to afex. For support visit: http://afex.singmann.science/
> - Functions for ANOVAs: aov_car(), aov_ez(), and aov_4()
> - Methods for calculating p-values with mixed(): 'S', 'KR', 'LRT', and 'PB'
> - 'afex_aov' and 'mixed' objects can be passed to emmeans() for follow-up tests
> - Get and set global package options with: afex_options()
> - Set sum-to-zero contrasts globally: set_sum_contrasts()
> - For example analyses see: browseVignettes("afex")
> ************
>
> Attaching package: 'afex'
> The following object is masked _by_ '.GlobalEnv':
>
> obk.long
> The following object is masked from 'package:lme4':
>
> lmer
m_afex <- aov_car(value ~ treatment * gender + Error(id), data = obk.long)
> Contrasts set to contr.sum for the following variables: treatment, gender
eta_squared(m_afex)
> # Effect Size for ANOVA (Type III)
>
> Parameter | η² (partial) | 95% CI
> ----------------------------------------------
> treatment | 0.27 | [0.13, 1.00]
> gender | 0.12 | [0.03, 1.00]
> treatment:gender | 0.20 | [0.07, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
Other Measures of Effect Size
Unbiased Effect Sizes
These effect sizes are unbiased estimators of the population’s \(\eta^2\):
- Omega Squared (\(\omega^2\))
- Epsilon Squared (\(\epsilon^2\)), also referred to as Adjusted Eta Squared.
omega_squared(m_afex)
> # Effect Size for ANOVA (Type III)
>
> Parameter | ω² (partial) | 95% CI
> ----------------------------------------------
> treatment | 0.24 | [0.10, 1.00]
> gender | 0.10 | [0.02, 1.00]
> treatment:gender | 0.17 | [0.05, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
epsilon_squared(m_afex)
> # Effect Size for ANOVA (Type III)
>
> Parameter | ε² (partial) | 95% CI
> ----------------------------------------------
> treatment | 0.25 | [0.11, 1.00]
> gender | 0.11 | [0.02, 1.00]
> treatment:gender | 0.18 | [0.06, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
Both \(\omega^2\) and \(\epsilon^2\) (and their partial counterparts, \(\omega^2_p\) & \(\epsilon^2_p\)) are unbiased estimators of the population’s \(\eta^2\) (or \(\eta^2_p\), respectively), which is especially important is small samples. Though \(\omega^2\) is the more popular choice (Albers and Lakens 2018), \(\epsilon^2\) is analogous to adjusted-\(R^2\) (Allen 2017, 382), and has been found to be less biased (Carroll and Nordholm 1975).
Generalized Eta^{2}
Partial Eta squared aims at estimating the effect size in a design where only the term of interest was manipulated, assuming all other terms are have also manipulated. However, not all predictors are always manipulated - some can only be observed. For such cases, we can use generalized Eta squared (\(\eta^2_G\)), which like \(\eta^2_p\) estimating the effect size in a design where only the term of interest was manipulated, accounting for the fact that some terms cannot be manipulated (and so their variance would be present in such a design).
eta_squared(m_afex, generalized = "gender")
> # Effect Size for ANOVA (Type III)
>
> Parameter | η² (generalized) | 95% CI
> --------------------------------------------------
> treatment | 0.21 | [0.08, 1.00]
> gender | 0.10 | [0.02, 1.00]
> treatment:gender | 0.18 | [0.06, 1.00]
>
> - Observed variables: gender
> - One-sided CIs: upper bound fixed at [1.00].
\(\eta^2_G\) is useful in repeated-measures designs, as it can estimate what a within-subject effect size would have been had that predictor been manipulated between-subjects (Olejnik and Algina 2003).
Cohen’s f
Finally, we have the forgotten child - Cohen’s \(f\). Cohen’s \(f\) is a transformation of \(\eta^2_p\), and is the ratio between the term-SS and the error-SS.
\[\text{Cohen's} f_p = \sqrt{\frac{\eta^2_p}{1-\eta^2_p}} = \sqrt{\frac{SS_{effect}}{SS_{error}}}\]
It can take on values between zero, when the population means are all equal, and an indefinitely large number as the means are further and further apart. It is analogous to Cohen’s \(d\) when there are only two groups.
cohens_f(m_afex)
> # Effect Size for ANOVA (Type III)
>
> Parameter | Cohen's f (partial) | 95% CI
> ----------------------------------------------------
> treatment | 0.61 | [0.38, Inf]
> gender | 0.37 | [0.17, Inf]
> treatment:gender | 0.50 | [0.28, Inf]
>
> - One-sided CIs: upper bound fixed at [Inf].
When Sum-of-Squares are Hard to Come By
Until now we’ve discusses effect sizes in fixed-effect linear model and repeated-measures ANOVA’s - cases where the SSs are readily available, and so the various effect sized presented can easily be estimated. How ever this is not always the case.
For example, in linear mixed models (LMM/HLM/MLM), the estimation of all required SSs is not straightforward. However, we can still approximate these effect sizes (only their partial versions) based on the test-statistic approximation method (learn more in the Effect Size from Test Statistics vignette).
>
> Attaching package: 'lmerTest'
> The following object is masked from 'package:lme4':
>
> lmer
> The following object is masked from 'package:stats':
>
> step
fit_lmm <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy)
anova(fit_lmm) # note the type-3 errors
> Type III Analysis of Variance Table with Satterthwaite's method
> Sum Sq Mean Sq NumDF DenDF F value Pr(>F)
> Days 30031 30031 1 17 45.9 3.3e-06 ***
> ---
> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
F_to_eta2(45.8, df = 1, df_error = 17)
> η² (partial) | 95% CI
> ---------------------------
> 0.73 | [0.51, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
Or directly with `eta_squared() and co.:
eta_squared(fit_lmm)
> # Effect Size for ANOVA (Type III)
>
> Parameter | η² (partial) | 95% CI
> ---------------------------------------
> Days | 0.73 | [0.51, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
epsilon_squared(fit_lmm)
> # Effect Size for ANOVA (Type III)
>
> Parameter | ε² (partial) | 95% CI
> ---------------------------------------
> Days | 0.71 | [0.48, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
omega_squared(fit_lmm)
> # Effect Size for ANOVA (Type III)
>
> Parameter | ω² (partial) | 95% CI
> ---------------------------------------
> Days | 0.70 | [0.47, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
Another case where SSs are not available is when using Bayesian models…
For Bayesian Models
An alternative route to obtaining effect sizes of explained variance, is via the use of the posterior predictive distribution (PPD). The PPD is the Bayesian expected distribution of possible unobserved values. Thus, after observing some data, we can estimate not just the expected mean values (the conditional marginal means), but also the full distribution of data around these values (Gelman et al. 2014, chap. 7).
By sampling from the PPD, we can decompose the sample to the various SSs needed for the computation of explained variance measures. By repeatedly sampling from the PPD, we can generate a posterior distribution of explained variance estimates. But note that these estimates are conditioned not only on the location-parameters of the model, but also on the scale-parameters of the model! So it is vital to validate the PPD before using it to estimate explained variance measures.
Let’s fit our model:
> Loading required package: Rcpp
> This is rstanarm version 2.32.1
> - See https://mc-stan.org/rstanarm/articles/priors for changes to default priors!
> - Default priors may change, so it's safest to specify priors, even if equivalent to the defaults.
> - For execution on a local, multicore CPU with excess RAM we recommend calling
> options(mc.cores = parallel::detectCores())
m_bayes <- stan_glm(value ~ gender + phase + treatment,
data = obk.long, family = gaussian(),
refresh = 0
)
We can use eta_squared_posterior()
to get the posterior
distribution of \(eta^2\) or \(eta^2_p\) for each effect. Like an ANOVA
table, we must make sure to use the right effects-coding and
SS-type:
pes_posterior <- eta_squared_posterior(m_bayes,
draws = 500, # how many samples from the PPD?
partial = TRUE, # partial eta squared
# type 3 SS
ss_function = car::Anova, type = 3,
verbose = FALSE
)
head(pes_posterior)
> gender phase treatment
> 1 0.194 0.1694 0.065
> 2 0.260 0.0268 0.248
> 3 0.004 0.0161 0.186
> 4 0.149 0.1023 0.392
> 5 0.014 0.1156 0.406
> 6 0.025 0.0097 0.364
bayestestR::describe_posterior(pes_posterior,
rope_range = c(0, 0.1), test = "rope"
)
> Summary of Posterior Distribution
>
> Parameter | Median | 95% CI | ROPE | % in ROPE
> ------------------------------------------------------------
> gender | 0.07 | [0.00, 0.27] | [0.00, 0.10] | 64.56%
> phase | 0.05 | [0.00, 0.20] | [0.00, 0.10] | 83.54%
> treatment | 0.26 | [0.04, 0.47] | [0.00, 0.10] | 7.17%
Compare to:
m_ML <- lm(value ~ gender + phase + treatment, data = obk.long)
eta_squared(car::Anova(m_ML, type = 3))
> # Effect Size for ANOVA (Type III)
>
> Parameter | η² (partial) | 95% CI
> ---------------------------------------
> gender | 0.07 | [0.01, 1.00]
> phase | 0.01 | [0.00, 1.00]
> treatment | 0.26 | [0.12, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
For Ordinal Outcomes
When our outcome is not a numeric variable, the effect sizes described above cannot be used - measured based on sum-of-squares are ill suited for such outcomes. Instead, we must use effect sizes for ordinal ANOVAs.
In R
, there are two functions for running
ordinal one way ANOVAs: kruskal.test()
for
differences between independent groups, and friedman.test()
for differences between dependent groups.
For the one-way ordinal ANOVA, the Rank-Epsilon-Squared (\(E^2_R\)) and Rank-Eta-Squared (\(\eta^2_H\)) are measures of association similar to their non-rank counterparts: values range between 0 (no relative superiority between any of the groups) to 1 (complete separation - with no overlap in ranks between the groups).
group_data <- list(
g1 = c(2.9, 3.0, 2.5, 2.6, 3.2), # normal subjects
g2 = c(3.8, 2.7, 4.0, 2.4), # with obstructive airway disease
g3 = c(2.8, 3.4, 3.7, 2.2, 2.0) # with asbestosis
)
kruskal.test(group_data)
>
> Kruskal-Wallis rank sum test
>
> data: group_data
> Kruskal-Wallis chi-squared = 0.8, df = 2, p-value = 0.7
rank_epsilon_squared(group_data)
> ε²(R) | 95% CI
> --------------------
> 0.06 | [0.02, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
rank_eta_squared(group_data)
> η²(H) | 95% CI
> --------------------
> 0.13 | [0.08, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].
For an ordinal repeated measures one-way ANOVA, Kendall’s W is a measure of agreement on the effect of condition between various “blocks” (the subjects), or more often conceptualized as a measure of reliability of the rating / scores of observations (or “groups”) between “raters” (“blocks”).
# Subjects are COLUMNS
(ReactionTimes <- matrix(
c(
398, 338, 520,
325, 388, 555,
393, 363, 561,
367, 433, 470,
286, 492, 536,
362, 475, 496,
253, 334, 610
),
nrow = 7, byrow = TRUE,
dimnames = list(
paste0("Subject", 1:7),
c("Congruent", "Neutral", "Incongruent")
)
))
> Congruent Neutral Incongruent
> Subject1 398 338 520
> Subject2 325 388 555
> Subject3 393 363 561
> Subject4 367 433 470
> Subject5 286 492 536
> Subject6 362 475 496
> Subject7 253 334 610
friedman.test(ReactionTimes)
>
> Friedman rank sum test
>
> data: ReactionTimes
> Friedman chi-squared = 11, df = 2, p-value = 0.004
kendalls_w(ReactionTimes)
> Kendall's W | 95% CI
> --------------------------
> 0.80 | [0.76, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].