Effect Sizes: Getting Started"

In effectsize: Indices of Effect Size

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

Aims of the Package

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; @bollen1989structural], 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 [@rcore] 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 [@ludecke2020extracting], ggstatsplot [@patil2020ggstatsplot], gtsummary [@sjoberg2020gtsummary] and more.

Comparison to Other Packages

effectsize's functionality is in part comparable to packages like lm.beta [@behrendt2014lmbeta], MOTE [@buchanan2019MOTE], and MBESS [@kelley2020MBESS]. 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 [@cohen1988statistical], 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.
  

Examples of Features

effectsize provides various functions for extracting and estimating effect sizes and their confidence intervals [estimated using the noncentrality parameter method; @steiger2004beyond]. 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/).

Indices of Effect Size

Standardized Differences

effectsize provides functions for estimating the common indices of standardized differences such as Cohen's d (cohens_d()), Hedges' g (hedges_g()) for both paired and independent samples [@cohen1988statistical; @hedges1985statistical], and Glass' $\Delta$ (glass_delta()) for independent samples with different variances [@hedges1985statistical].

library(effectsize)
options(es.use_symbols = TRUE) # get nice symbols when printing! (On Windows, requires R >= 4.2.0)

cohens_d(mpg ~ am, data = mtcars)

Contingency Tables

Pearson's $\phi$ (phi()) and Cramér's V (cramers_v()) can be used to estimate the strength of association between two categorical variables [@cramer1946mathematical], while Cohen's g (cohens_g()) estimates the deviance between paired categorical variables [@cohen1988statistical].

M <- rbind(
  c(150, 130, 35, 55),
  c(100, 50, 10, 40),
  c(165, 65, 2, 25)
)

cramers_v(M)

Parameter and Model Standardization

Note: this functionality has been moved to the parameters and datawizard packages.

Standardizing parameters (i.e., coefficients) can allow for their comparison within and between models, variables and studies. To this end, two functions are available: standardize(), which 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; @luedecke2019insight].

model <- lm(mpg ~ cyl * am,
  data = mtcars
)

datawizard::standardize(model)

parameters::standardize_parameters(model)

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)

standardize_parameters() provides several standardization methods, such as robust standardization, or pseudo-standardized coefficients for (generalized) linear mixed models [@hoffman2015longitudinal]. A full review of these methods can be found in the Parameter and Model Standardization vignette.

Effect Sizes for ANOVAs

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 $\eta^2_G$ [@cohen1988statistical; @olejnik2003generalized]:

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)

eta_squared(model, generalized = "Time")

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 [@kelley1935unbiased; @olejnik2003generalized]. For more details about the various effect size measures and their applications, see the Effect sizes for ANOVAs vignette.

Effect Size Conversion

From Test Statistics

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 [@friedman1982simplified]. 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)
)

t_to_d(t = -5.14, df_error = 22)

t_to_r(t = -5.14, df_error = 22)

These functions also power the effectsize() convenience function for estimating effect sizes from R's htest-type objects. For example:

data(hardlyworking, package = "effectsize")

aov1 <- oneway.test(salary ~ n_comps,
  data = hardlyworking, var.equal = TRUE
)
effectsize(aov1)

xtab <- rbind(c(762, 327, 468), c(484, 239, 477), c(484, 239, 477))
Xsq <- chisq.test(xtab)
effectsize(Xsq)

These functions also power our Effect Sizes From Test Statistics shiny app (https://easystats4u.shinyapps.io/statistic2effectsize/).

Between Effect Sizes

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; @borenstein2009converting; @grant2014converting].

r_to_d(0.7)

d_to_oddsratio(1.96)

oddsratio_to_riskratio(34.99, p0 = 0.4)

oddsratio_to_r(34.99)

Effect Size Interpretation

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.

interpret_cohens_d(c(0.02, 0.52, 0.86), rules = "cohen1988")

Licensing and Availability

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.

Acknowledgments

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.

References

effectsize documentation built on Sept. 14, 2023, 5:07 p.m.