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(This article was first published on <strong><a href="https://strengejacke.wordpress.com/2017/07/25/effect-size-statistics-for-anova-tables-rstats/"> R – Strenge Jacke!</a></strong>, and kindly contributed to <a rel="nofollow" href="https://www.r-bloggers.com/">R-bloggers)</a> </div>
<p><a rel="nofollow" href="https://cran.r-project.org/package=sjstats" target="_blank"><img data-orig-file="https://strengejacke.files.wordpress.com/2017/07/logo.png?w=456" data-orig-size="120,139" data-medium-file="https://strengejacke.files.wordpress.com/2017/07/logo.png?w=456?w=120" data-attachment-id="1371" data-image-meta="{" data-image-description="" src="https://strengejacke.files.wordpress.com/2017/07/logo.png?w=456" data-permalink="https://strengejacke.wordpress.com/2017/07/25/effect-size-statistics-for-anova-tables-rstats/logo/" data-large-file="https://strengejacke.files.wordpress.com/2017/07/logo.png?w=456?w=120" alt="" data-image-title="logo" data-comments-opened="1" class="wp-image-1371" data-recalc-dims="1"></a> My <a rel="nofollow" href="https://cran.r-project.org/package=sjstats" target="_blank">sjstats-package</a> has been updated on CRAN. The past updates introduced new functions for various purposes, e.g. predictive accuracy of regression models or improved support for the marvelous <a rel="nofollow" href="https://cran.r-project.org/package=glmmTMB" target="_blank">glmmTMB-package</a>. The current update, however, added some ANOVA tools to the package.</p>
<p>In this post, I want to give a short overview of these new functions, which report different effect size measures. These are useful beyond significance tests (p-values), because they estimate the magnitude of effects, independent from sample size. <em>sjstats</em> provides following functions:</p>
<ul>
<li>eta_sq()</li>
<li>omega_sq()</li>
<li>cohens_f()</li>
<li>anova_stats()</li>
</ul>
<p>First, we need a sample model:</p>
<pre><code>library(sjstats)
<span># load sample data</span>
data(efc)

<span># fit linear model</span>
fit &lt;- aov(
  c12hour ~ as.factor(e42dep) + as.factor(c172code) + c160age,
  data = efc
)</code></pre>
<p>All functions accept objects of class <code>aov</code> or <code>anova</code>, so you can also use model fits from the <em>car-package</em>, which allows fitting Anova’s with different types of sum of squares. Other objects, like <code>lm</code>, will be coerced to <code>anova</code> internally.</p>
<p>The following functions return the effect size statistic as named numeric vector, using the model’s term names.</p>
<h3>Eta Squared</h3>
<p>The eta squared is the proportion of the total variability in the dependent variable that is accounted for by the variation in the independent variable. It is the ratio of the sum of squares for each group level to the total sum of squares. It can be interpreted as percentage of variance accounted for by a variable.</p>
<p>For variables with 1 degree of freedeom (in the numerator), the square root of eta squared is equal to the correlation coefficient r. For variables with more than 1 degree of freedom, eta squared equals R2. This makes eta squared easily interpretable. Furthermore, these effect sizes can easily be converted into effect size measures that can be, for instance, further processed in meta-analyses.</p>
<p>Eta squared can be computed simply with:</p>
<pre><code>eta_sq(fit)
<span>#> as.factor(e42dep) as.factor(c172code) c160age #> 0.266114185 0.005399167 0.048441046</span></code></pre>
<h3>Partial Eta Squared</h3>
<p>The partial eta squared value is the ratio of the sum of squares for each group level to the sum of squares for each group level plus the residual sum of squares. It is more difficult to interpret, because its value strongly depends on the variability of the residuals. Partial eta squared values should be reported with caution, and Levine and Hullett (2002) recommend reporting eta or omega squared rather than partial eta squared.</p>
<p>Use the <code>partial</code>-argument to compute partial eta squared values:</p>
<pre><code>eta_sq(fit, partial = TRUE)
<span>#> as.factor(e42dep) as.factor(c172code) c160age #> 0.281257128 0.007876882 0.066495448</span></code></pre>
<h3>Omega Squared</h3>
<p>While eta squared estimates tend to be biased in certain situations, e.g. when the sample size is small or the independent variables have many group levels, omega squared estimates are corrected for this bias.</p>
<p>Omega squared can be simply computed with:</p>
<pre><code>omega_sq(fit)
<span>#> as.factor(e42dep) as.factor(c172code) c160age #> 0.263453157 0.003765292 0.047586841</span></code></pre>
<h3>Cohen’s F</h3>
<p>Finally, <code>cohens_f()</code> computes Cohen’s F effect size for all independent variables in the model:</p>
<pre><code>cohens_f(fit)
<span>#> as.factor(e42dep) as.factor(c172code) c160age #> 0.62555427 0.08910342 0.26689334</span></code></pre>
<h2>Complete Statistical Table Output</h2>
<p>The <code>anova_stats()</code> function takes a model input and computes a comprehensive summary, including the above effect size measures, returned as tidy data frame (as tibble, to be exact):</p>
<pre><code>anova_stats(fit)
<span>#> # A tibble: 4 x 11
#> term df sumsq meansq statistic p.value etasq partial.etasq omegasq cohens.f power
#> #> 1 as.factor(e42dep) 3 577756.33 192585.444 108.786 0.000 0.266 0.281 0.263 0.626 1.00
#> 2 as.factor(c172code) 2 11722.05 5861.024 3.311 0.037 0.005 0.008 0.004 0.089 0.63
#> 3 c160age 1 105169.60 105169.595 59.408 0.000 0.048 0.066 0.048 0.267 1.00
#> 4 Residuals 834 1476436.34 1770.307 NA NA NA NA NA NA NA</span></code></pre>
<p>Like the other functions, the input may also be an object of class anova, so you can also use model fits from the car package, which allows fitting Anova’s with different types of sum of squares:</p>
<pre><code>anova_stats(car::Anova(fit, type = 3))
<span>#> # A tibble: 5 x 11
#> term sumsq meansq df statistic p.value etasq partial.etasq omegasq cohens.f power
#> #> 1 (Intercept) 26851.070 26851.070 1 15.167 0.000 0.013 0.018 0.012 0.135 0.973
#> 2 as.factor(e42dep) 426461.571 142153.857 3 80.299 0.000 0.209 0.224 0.206 0.537 1.000
#> 3 as.factor(c172code) 7352.049 3676.025 2 2.076 0.126 0.004 0.005 0.002 0.071 0.429
#> 4 c160age 105169.595 105169.595 1 59.408 0.000 0.051 0.066 0.051 0.267 1.000
#> 5 Residuals 1476436.343 1770.307 834 NA NA NA NA NA NA NA</span></code></pre>
<h3>References</h3>
<p>Levine TR, Hullet CR. Eta Squared, Partial Eta Squared, and Misreporting of Effect Size in Communication Research. Human Communication Research 28(4); 2002: 612-625</p>
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