---
title: "Effect Sizes: Getting Started"
output: 
  rmarkdown::html_vignette:
    toc: true
    fig_width: 10.08
    fig_height: 6
tags: [r, effect size, rules of thumb, guidelines, conversion]
vignette: >
  \usepackage[utf8]{inputenc}
  %\VignetteIndexEntry{Effect Sizes: Getting Started}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  chunk_output_type: console
bibliography: bibliography.bib
---

```{r, include = FALSE}
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.:

<!-- - **lm.beta** provides standardized regression coefficients for linear models, based on post-hoc model matrix standardization. However, the functionality is available only for a limited number of models (models inheriting from the `lm` class), whereas **effectsize** provides support for many types of models, including (generalized) linear mixed models, Bayesian models, and more. Additionally, in additional to post-hoc model matrix standardization, **effectsize** offers other methods of standardization (see below).   -->
- 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.  
<!-- - Finally, in **base R**, the function `scale()` can be used to standardize vectors, matrices and data frame, which can be used to standardize data prior to model fitting. The coefficients of a linear model fit on such data are in effect standardized regression coefficients. **effectsize** expands an this, allowing for robust standardization (using the median and the MAD, instead of the mean and SD), post-hoc parameter standardization, and more. -->

# 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](https://easystats.github.io/effectsize/reference/index.html), 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].

```{r}
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].

```{r}
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].

```{r}
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):

```{r}
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](https://easystats.github.io/parameters/articles/standardize_parameters_effsize.html).

## 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]:

```{r}
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](https://easystats.github.io/effectsize/articles/anovaES.html).

## 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*). 

```{r}
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:

```{r}
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}
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](https://easystats.github.io/effectsize/articles/interpret.html).

```{r}
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](https://github.com/easystats/effectsize/issues/). See the package's [*Contribution Guidelines*](https://github.com/easystats/effectsize/blob/main/.github/CONTRIBUTING.md/).

# Acknowledgments

**effectsize** is part of the [*easystats*](https://github.com/easystats/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](https://github.com/orgs/easystats/people/) as well as the users.

# References
