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* (`hedges_g()`

)
for both paired and independent samples (Cohen
1988; Hedges and Olkin 1985), and Glass’ \(\Delta\) (`glass_delta()`

) for
independent samples with different variances (Hedges and Olkin 1985).

```
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)
#> Cohen's d | 95% CI
#> --------------------------
#> -1.48 | [-2.27, -0.67]
#>
#> - Estimated using pooled SD.
```

Pearson’s \(\phi\)
(`phi()`

) and Cramér’s *V* (`cramers_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
1988).

```
<- rbind(
M c(150, 130, 35, 55),
c(100, 50, 10, 40),
c(165, 65, 2, 25)
)
cramers_v(M)
#> Cramer's V (adj.) | 95% CI
#> --------------------------------
#> 0.17 | [0.11, 1.00]
#>
#> - One-sided CIs: upper bound fixed at [1.00].
```

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; Lüdecke, Waggoner, and Makowski (2019)].

```
<- lm(mpg ~ cyl * am,
model data = mtcars
)
::standardize(model)
datawizard#>
#> Call:
#> lm(formula = mpg ~ cyl * am, data = data_std)
#>
#> Coefficients:
#> (Intercept) cyl am cyl:am
#> -0.0977 -0.7426 0.1739 -0.1930
::standardize_parameters(model)
parameters#> # 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):

```
<- glm(am ~ cyl + hp,
model family = "binomial",
data = mtcars
)
::standardize_parameters(model, exponentiate = TRUE)
parameters#> # 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
<- subset(ChickWeight, ave(weight, Chick, FUN = length) == 12)
ChickWeight $Time <- factor(ChickWeight$Time)
ChickWeight
<- aov(weight ~ Diet * Time + Error(Chick / Time),
model 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 `effectsize()`

convenience
function for estimating effect sizes from R’s `htest`

-type
objects. For example:

```
data(hardlyworking, package = "effectsize")
<- oneway.test(salary ~ n_comps,
aov1 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].
<- rbind(c(762, 327, 468), c(484, 239, 477), c(484, 239, 477))
xtab <- chisq.test(xtab)
Xsq 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)].

```
r_to_d(0.7)
#> [1] 1.96
d_to_oddsratio(1.96)
#> [1] 35
oddsratio_to_riskratio(34.99, p0 = 0.4)
#> [1] 2.4
oddsratio_to_r(34.99)
#> [1] 0.7
```

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")
#> [1] "very small" "medium" "large"
#> (Rules: cohen1988)
```

**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.

Behrendt, Stefan. 2014. *lm.beta: Add
Standardized Regression Coefficients to Lm-Objects*. https://CRAN.R-project.org/package=lm.beta/.

Bollen, Kenneth A. 1989. *Structural Equations with Latent
Variables*. New York: John Wiley & Sons. https://doi.org/10.1002/9781118619179.

Borenstein, Michael, Larry V Hedges, JPT Higgins, and Hannah R
Rothstein. 2009. “Converting Among Effect Sizes.”
*Introduction to Meta-Analysis*, 45–49.

Buchanan, Erin M., Amber Gillenwaters, John E. Scofield, and K. D.
Valentine. 2019. *MOTE: Measure of the
Effect: Package to Assist in Effect Size Calculations and Their
Confidence Intervals*. https://github.com/doomlab/MOTE/.

Cohen, J. 1988. *Statistical Power Analysis for the Behavioral
Sciences, 2nd Ed.* New York: Routledge.

Cramér, Harald. 1946. *Mathematical Methods of Statistics*.
Princeton: Princeton University Press.

Friedman, Herbert. 1982. “Simplified Determinations of Statistical
Power, Magnitude of Effect and Research Sample Sizes.”
*Educational and Psychological Measurement* 42 (2): 521–26.

Grant, Robert L. 2014. “Converting an Odds Ratio to a Range of
Plausible Relative Risks for Better Communication of Research
Findings.” *Bmj* 348: f7450.

Hedges, L, and I Olkin. 1985. *Statistical Methods for
Meta-Analysis*. San Diego: Academic Press.

Hoffman, Lesa. 2015. *Longitudinal Analysis: Modeling Within-Person
Fluctuation and Change*. Routledge.

Kelley, Ken. 2020. *MBESS: The MBESS R Package*. https://CRAN.R-project.org/package=MBESS/.

Kelley, Truman L. 1935. “An Unbiased Correlation Ratio
Measure.” *Proceedings of the National Academy of Sciences of
the United States of America* 21 (9): 554–59. https://doi.org/10.1073/pnas.21.9.554.

Lüdecke, Daniel, Mattan S Ben-Shachar, Indrajeet Patil, and Dominique
Makowski. 2020. “Extracting, Computing and Exploring the
Parameters of Statistical Models Using R.”
*Journal of Open Source Software* 5 (53): 2445. https://doi.org/10.21105/joss.02445.

Lüdecke, Daniel, Philip Waggoner, and Dominique Makowski. 2019.
“insight: A Unified Interface to
Access Information from Model Objects in R.”
*Journal of Open Source Software* 4 (38): 1412. https://doi.org/10.21105/joss.01412.

Olejnik, Stephen, and James Algina. 2003. “Generalized Eta and
Omega Squared Statistics: Measures of Effect Size for Some Common
Research Designs.” *Psychological Methods* 8 (4): 434.

Patil, Indrajeet. 2018. “ggstatsplot:
’Ggplot2’ Based Plots with Statistical Details.” *CRAN*.
https://doi.org/10.5281/zenodo.2074621.

R Core Team. 2020. *R: A Language and Environment for
Statistical Computing*. Vienna, Austria: R Foundation for
Statistical Computing. https://www.R-project.org/.

Sjoberg, Daniel D., Michael Curry, Margie Hannum, Karissa Whiting, and
Emily C. Zabor. 2020. *gtsummary:
Presentation-Ready Data Summary and Analytic Result Tables*. https://CRAN.R-project.org/package=gtsummary/.

Steiger, James 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
(2): 164–82. https://doi.org/10.1037/1082-989X.9.2.164.