Effect Sizes: Getting StartedSource:
In both theoretical and applied research, it is often of interest to assess the strength of an observed association. This is typically done to allow the judgment of the magnitude of an effect [especially when units of measurement are not meaningful, e.g., in the use of estimated latent variables; Bollen (1989)], to facilitate comparing between predictors’ importance within a given model, or both. Though some indices of effect size, such as the correlation coefficient (itself a standardized covariance coefficient) are readily available, other measures are often harder to obtain. effectsize is an R package (R Core Team 2020) that fills this important gap, providing utilities for easily estimating a wide variety of standardized effect sizes (i.e., effect sizes that are not tied to the units of measurement of the variables of interest) and their confidence intervals (CIs), from a variety of statistical models. effectsize provides easy-to-use functions, with full documentation and explanation of the various effect sizes offered, and is also used by developers of other R packages as the back-end for effect size computation, such as parameters (Lüdecke et al. 2020), ggstatsplot (Patil 2018), gtsummary (Sjoberg et al. 2020) and more.
effectsize’s functionality is in part comparable to packages like lm.beta (Behrendt 2014), MOTE (Buchanan et al. 2019), and MBESS (K. Kelley 2020). Yet, there are some notable differences, e.g.:
- Both MOTE and MBESS provide
functions for computing effect sizes such as Cohen’s d and
effect sizes for ANOVAs (Cohen 1988), and
their confidence intervals. However, both require manual input of
F- or t-statistics, degrees of freedom, and
sums of squares for the computation the effect sizes, whereas
effectsize can automatically extract this information
from the provided models, thus allowing for better ease-of-use as well
as reducing any potential for error.
effectsize provides various functions for extracting and estimating effect sizes and their confidence intervals [estimated using the noncentrality parameter method; Steiger (2004)]. In this article, we provide basic usage examples for estimating some of the most common effect size. A comprehensive overview, including in-depth examples and a full list of features and functions, are accessible via a dedicated website (https://easystats.github.io/effectsize/).
effectsize provides functions for estimating the
common indices of standardized differences such as Cohen’s d
cohens_d()), Hedges’ g (
for both paired and independent samples (Cohen
1988; Hedges and Olkin 1985), and Glass’ \(\Delta\) (
independent samples with different variances (Hedges and Olkin 1985).
phi()) and Cramér’s V (
can be used to estimate the strength of association between two
categorical variables (Cramér 1946), while
Cohen’s g (
cohens_g()) estimates the deviance
between paired categorical variables (Cohen
Note: this functionality has been moved to the
Standardizing parameters (i.e., coefficients) can allow for their
comparison within and between models, variables and studies. To this
end, two functions are available:
returns an updated model, re-fit with standardized data, and
standardize_parameters(), which returns a table of
standardized coefficients from a provided model [for a list of supported
models, see the insight package; Lüdecke, Waggoner, and Makowski (2019)].
model <- lm(mpg ~ cyl * am, data = mtcars ) datawizard::standardize(model) #> #> Call: #> lm(formula = mpg ~ cyl * am, data = data_std) #> #> Coefficients: #> (Intercept) cyl am cyl:am #> -0.0977 -0.7426 0.1739 -0.1930 parameters::standardize_parameters(model) #> # Standardization method: refit #> #> Parameter | Std. Coef. | 95% CI #> ----------------------------------------- #> (Intercept) | -0.10 | [-0.30, 0.11] #> cyl | -0.74 | [-0.95, -0.53] #> am | 0.17 | [-0.04, 0.39] #> cyl:am | -0.19 | [-0.41, 0.02]
Standardized parameters can also be produced for generalized linear models (GLMs; where only the predictors are standardized):
model <- glm(am ~ cyl + hp, family = "binomial", data = mtcars ) parameters::standardize_parameters(model, exponentiate = TRUE) #> # Standardization method: refit #> #> Parameter | Std_Odds_Ratio | 95% CI #> -------------------------------------------- #> (Intercept) | 0.53 | [0.18, 1.32] #> cyl | 0.05 | [0.00, 0.29] #> hp | 6.70 | [1.32, 61.54] #> #> - Response is unstandardized.
standardize_parameters() provides several
standardization methods, such as robust standardization, or
pseudo-standardized coefficients for (generalized) linear mixed
models (Hoffman 2015). A full review of
these methods can be found in the Parameter
and Model Standardization vignette.
Unlike standardized parameters, the effect sizes reported in the
context of ANOVAs (analysis of variance) or ANOVA-like tables represent
the amount of variance explained by each of the model’s terms, where
each term can be represented by one or more parameters.
eta_squared() can produce such popular effect sizes as
Eta-squared (\(\eta^2\)), its partial
version (\(\eta^2_p\)), as well as the
generalized \(\eta^2_G\) (Cohen 1988; Olejnik and Algina 2003):
options(contrasts = c("contr.sum", "contr.poly")) data("ChickWeight") # keep only complete cases and convert `Time` to a factor ChickWeight <- subset(ChickWeight, ave(weight, Chick, FUN = length) == 12) ChickWeight$Time <- factor(ChickWeight$Time) model <- aov(weight ~ Diet * Time + Error(Chick / Time), data = ChickWeight ) eta_squared(model, partial = TRUE) #> # Effect Size for ANOVA (Type I) #> #> Group | Parameter | η² (partial) | 95% CI #> ---------------------------------------------------- #> Chick | Diet | 0.27 | [0.06, 1.00] #> Chick:Time | Time | 0.87 | [0.85, 1.00] #> Chick:Time | Diet:Time | 0.22 | [0.11, 1.00] #> #> - One-sided CIs: upper bound fixed at [1.00]. eta_squared(model, generalized = "Time") #> # Effect Size for ANOVA (Type I) #> #> Group | Parameter | η² (generalized) | 95% CI #> -------------------------------------------------------- #> Chick | Diet | 0.04 | [0.00, 1.00] #> Chick:Time | Time | 0.74 | [0.71, 1.00] #> Chick:Time | Diet:Time | 0.03 | [0.00, 1.00] #> #> - Observed variables: Time #> - One-sided CIs: upper bound fixed at [1.00].
effectsize also offers \(\epsilon^2_p\)
epsilon_squared()) and \(\omega^2_p\)
omega_squared()), which are less biased estimates of the
variance explained in the population (T. L.
Kelley 1935; Olejnik and Algina 2003). For more details about the
various effect size measures and their applications, see the Effect
sizes for ANOVAs vignette.
In many real world applications there are no straightforward ways of obtaining standardized effect sizes. However, it is possible to get approximations of most of the effect size indices (d, r, \(\eta^2_p\)…) with the use of test statistics (Friedman 1982). These conversions are based on the idea that test statistics are a function of effect size and sample size (or more often of degrees of freedom). Thus it is possible to reverse-engineer indices of effect size from test statistics (F, t, \(\chi^2\), and z).
F_to_eta2( f = c(40.72, 33.77), df = c(2, 1), df_error = c(18, 9) ) #> η² (partial) | 95% CI #> --------------------------- #> 0.82 | [0.66, 1.00] #> 0.79 | [0.49, 1.00] #> #> - One-sided CIs: upper bound fixed at [1.00]. t_to_d(t = -5.14, df_error = 22) #> d | 95% CI #> ---------------------- #> -2.19 | [-3.23, -1.12] t_to_r(t = -5.14, df_error = 22) #> r | 95% CI #> ---------------------- #> -0.74 | [-0.85, -0.49]
These functions also power the
function for estimating effect sizes from R’s
objects. For example:
data(hardlyworking, package = "effectsize") aov1 <- oneway.test(salary ~ n_comps, data = hardlyworking, var.equal = TRUE ) effectsize(aov1) #> η² | 95% CI #> ------------------- #> 0.20 | [0.14, 1.00] #> #> - One-sided CIs: upper bound fixed at [1.00]. xtab <- rbind(c(762, 327, 468), c(484, 239, 477), c(484, 239, 477)) Xsq <- chisq.test(xtab) effectsize(Xsq) #> Cramer's V (adj.) | 95% CI #> -------------------------------- #> 0.07 | [0.05, 1.00] #> #> - One-sided CIs: upper bound fixed at [1.00].
These functions also power our Effect Sizes From Test Statistics shiny app (https://easystats4u.shinyapps.io/statistic2effectsize/).
For comparisons between different types of designs and analyses, it is useful to be able to convert between different types of effect sizes [d, r, Odds ratios and Risk ratios; Borenstein et al. (2009); Grant (2014)].
Finally, effectsize provides convenience functions to apply existing or custom interpretation rules of thumb, such as for instance Cohen’s (1988). Although we strongly advocate for the cautious and parsimonious use of such judgment-replacing tools, we provide these functions to allow users and developers to explore and hopefully gain a deeper understanding of the relationship between data values and their interpretation. More information is available in the Automated Interpretation of Indices of Effect Size vignette.
effectsize is licensed under the GNU General Public License (v3.0), with all source code stored at GitHub (https://github.com/easystats/effectsize), and with a corresponding issue tracker for bug reporting and feature enhancements. In the spirit of honest and open science, we encourage requests/tips for fixes, feature updates, as well as general questions and concerns via direct interaction with contributors and developers, by filing an issue. See the package’s Contribution Guidelines.
effectsize is part of the easystats ecosystem, a collaborative project created to facilitate the usage of R for statistical analyses. Thus, we would like to thank the members of easystats as well as the users.