{"id":10895,"date":"2019-11-30T11:16:41","date_gmt":"2019-11-30T09:16:41","guid":{"rendered":"https:\/\/www.datanovia.com\/en\/?post_type=dt_lessons&#038;p=10895"},"modified":"2019-11-30T11:16:41","modified_gmt":"2019-11-30T09:16:41","slug":"transform-data-to-normal-distribution-in-r","status":"publish","type":"dt_lessons","link":"https:\/\/www.datanovia.com\/en\/lessons\/transform-data-to-normal-distribution-in-r\/","title":{"rendered":"Transform Data to Normal Distribution in R"},"content":{"rendered":"<div id=\"rdoc\">\n<p>This chapter describes how to <strong>transform data to normal distribution in R<\/strong>. Parametric methods, such as t-test and ANOVA tests, assume that the dependent (outcome) variable is approximately normally distributed for every groups to be compared.<\/p>\n<p>In the situation where the normality assumption is not met, you could consider transform the data for correcting the non-normal distributions.<\/p>\n<p>When dealing with t-test and ANOVA assumptions, you just need to transform the dependent variable. However, when dealing with the assumptions of linear regression, you can consider transformations of either the independent or dependent variable or both for achieving a linear relationship between variables or to make sure there is homoscedasticity.<\/p>\n<div class=\"warning\">\n<p>Note that, transformation will not always be successful.<\/p>\n<\/div>\n<p>In this article, you will learn:<\/p>\n<ul>\n<li>Common types of non-normal distributions<\/li>\n<li>Methods for transforming the data to correct the non-normal distributions<\/li>\n<\/ul>\n<p>Contents:<\/p>\n<div id=\"TOC\">\n<ul>\n<li><a href=\"#non-normal-distributions\">Non-normal distributions<\/a><\/li>\n<li><a href=\"#transformation-methods\">Transformation methods<\/a><\/li>\n<li><a href=\"#examples-of-transforming-skewed-data\">Examples of transforming skewed data<\/a>\n<ul>\n<li><a href=\"#prerequisites\">Prerequisites<\/a><\/li>\n<li><a href=\"#visualization\">Visualization<\/a><\/li>\n<\/ul>\n<\/li>\n<li><a href=\"#summary-and-discussion\">Summary and discussion<\/a><\/li>\n<\/ul>\n<\/div>\n<div class='dt-sc-hr-invisible-medium  '><\/div>\n<div class='dt-sc-ico-content type1'><div class='custom-icon' ><a href='https:\/\/www.datanovia.com\/en\/product\/practical-statistics-in-r-for-comparing-groups-numerical-variables\/' target='_blank'><span class='fa fa-book'><\/span><\/a><\/div><h4><a href='https:\/\/www.datanovia.com\/en\/product\/practical-statistics-in-r-for-comparing-groups-numerical-variables\/' target='_blank'> Related Book <\/a><\/h4>Practical Statistics in R II - Comparing Groups: Numerical Variables<\/div>\n<div class='dt-sc-hr-invisible-medium  '><\/div>\n<div id=\"non-normal-distributions\" class=\"section level2\">\n<h2>Non-normal distributions<\/h2>\n<p>Skewness is a measure of symmetry for a distribution. The value can be positive, negative or undefined. In a skewed distribution, the central tendency measures (mean, median, mode) will not be equal.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/dn-tutorials\/r-statistics-2-comparing-groups-means\/images\/distribution-shape.jpeg\" alt=\"shape of distributions\" \/><\/p>\n<p>(Image courtesy: <a class=\"uri\" href=\"https:\/\/www.safaribooksonline.com\/library\/view\/clojure-for-data\/9781784397180\/ch01s13.html\">https:\/\/www.safaribooksonline.com\/library\/view\/clojure-for-data\/9781784397180\/ch01s13.html<\/a>)<\/p>\n<ul>\n<li><strong>Positively skewed<\/strong> distribution (or <strong>right skewed<\/strong>): The most frequent values are low; tail is toward the high values (on the right-hand side). Generally, <code>Mode &lt; Median &lt; Mean<\/code>.<\/li>\n<li><strong>Negatively skewed<\/strong> distribution (or <strong>left skewed<\/strong>), the most frequent values are high; tail is toward low values (on the left-hand side). Generally, <code>Mode &gt; Median &gt; Mean<\/code>.<\/li>\n<\/ul>\n<p>The direction of skewness is given by the sign of the skewness coefficient:<\/p>\n<ul>\n<li>A zero means no skewness at all (normal distribution).<\/li>\n<li>A negative value means the distribution is negatively skewed.<\/li>\n<li>A positive value means the distribution is positively skewed.<\/li>\n<\/ul>\n<div class=\"success\">\n<p>The larger the value of skewness, the larger the distribution differs from a normal distribution.<\/p>\n<\/div>\n<p>The skewness coefficient can be computed using the <code>moments<\/code> R packages:<\/p>\n<pre class=\"r\"><code>library(moments)\r\nskewness(iris$Sepal.Length, na.rm = TRUE)<\/code><\/pre>\n<pre><code>## [1] 0.312<\/code><\/pre>\n<\/div>\n<div id=\"transformation-methods\" class=\"section level2\">\n<h2>Transformation methods<\/h2>\n<p>This section describes different transformation methods, depending to the type of normality violation.<\/p>\n<p>Some common heuristics transformations for non-normal data include:<\/p>\n<ul>\n<li><strong>square-root for moderate skew<\/strong>:\n<ul>\n<li><code>sqrt(x)<\/code> for positively skewed data,<\/li>\n<li><code>sqrt(max(x+1) - x)<\/code> for negatively skewed data<\/li>\n<\/ul>\n<\/li>\n<li><strong>log for greater skew<\/strong>:\n<ul>\n<li><code>log10(x)<\/code> for positively skewed data,<\/li>\n<li><code>log10(max(x+1) - x)<\/code> for negatively skewed data<\/li>\n<\/ul>\n<\/li>\n<li><strong>inverse for severe skew<\/strong>:\n<ul>\n<li><code>1\/x<\/code> for positively skewed data<\/li>\n<li><code>1\/(max(x+1) - x)<\/code> for negatively skewed data<\/li>\n<\/ul>\n<\/li>\n<li><strong>Linearity and heteroscedasticity<\/strong>:\n<ul>\n<li>first try <code>log<\/code> transformation in a situation where the dependent variable starts to increase more rapidly with increasing independent variable values<\/li>\n<li>If your data does the opposite \u2013 dependent variable values decrease more rapidly with increasing independent variable values \u2013 you can first consider a <code>square<\/code> transformation.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<div class=\"warning\">\n<p>Note that, when using a log transformation, a constant should be added to all values to make them all positive before transformation.<\/p>\n<\/div>\n<\/div>\n<div id=\"examples-of-transforming-skewed-data\" class=\"section level2\">\n<h2>Examples of transforming skewed data<\/h2>\n<div id=\"prerequisites\" class=\"section level3\">\n<h3>Prerequisites<\/h3>\n<p>Make sure you have installed the following R packages:<\/p>\n<ul>\n<li><code>ggpubr<\/code> for creating easily publication ready plots<\/li>\n<li><code>moments<\/code> for computing skewness<\/li>\n<\/ul>\n<p>Start by loading the packages:<\/p>\n<pre class=\"r\"><code>library(ggpubr)\r\nlibrary(moments)<\/code><\/pre>\n<p>Demo dataset: Built-in R dataset <code>USJudgeRatings<\/code>.<\/p>\n<pre class=\"r\"><code>data(\"USJudgeRatings\")\r\ndf &lt;- USJudgeRatings\r\nhead(df)<\/code><\/pre>\n<pre><code>##                CONT INTG DMNR DILG CFMG DECI PREP FAMI ORAL WRIT PHYS RTEN\r\n## AARONSON,L.H.   5.7  7.9  7.7  7.3  7.1  7.4  7.1  7.1  7.1  7.0  8.3  7.8\r\n## ALEXANDER,J.M.  6.8  8.9  8.8  8.5  7.8  8.1  8.0  8.0  7.8  7.9  8.5  8.7\r\n## ARMENTANO,A.J.  7.2  8.1  7.8  7.8  7.5  7.6  7.5  7.5  7.3  7.4  7.9  7.8\r\n## BERDON,R.I.     6.8  8.8  8.5  8.8  8.3  8.5  8.7  8.7  8.4  8.5  8.8  8.7\r\n## BRACKEN,J.J.    7.3  6.4  4.3  6.5  6.0  6.2  5.7  5.7  5.1  5.3  5.5  4.8\r\n## BURNS,E.B.      6.2  8.8  8.7  8.5  7.9  8.0  8.1  8.0  8.0  8.0  8.6  8.6<\/code><\/pre>\n<p>In the following examples, we\u2019ll consider two variables:<\/p>\n<ul>\n<li><code>CONT<\/code>: Number of contacts of lawyer with judge. Positively skewed.<\/li>\n<li><code>PHYS<\/code>: Physical ability. Negatively skewed<\/li>\n<\/ul>\n<\/div>\n<div id=\"visualization\" class=\"section level3\">\n<h3>Visualization<\/h3>\n<p>Plot the density distribution of each variable and compare the observed distribution to what we would expect if it were perfectly normal (dashed red line).<\/p>\n<pre class=\"r\"><code># Distribution of CONT variable\r\nggdensity(df, x = \"CONT\", fill = \"lightgray\", title = \"CONT\") +\r\n  scale_x_continuous(limits = c(3, 12)) +\r\n  stat_overlay_normal_density(color = \"red\", linetype = \"dashed\")\r\n\r\n# Distribution of PHYS variable\r\nggdensity(df, x = \"PHYS\", fill = \"lightgray\", title = \"PHYS\") +\r\n  scale_x_continuous(limits = c(3, 12)) +\r\n  stat_overlay_normal_density(color = \"red\", linetype = \"dashed\")<\/code><\/pre>\n<p><img decoding=\"async\" src=\"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/dn-tutorials\/r-statistics-2-comparing-groups-means\/figures\/020-transforming-data-for-normality-density-plots-skewed-data-1.png\" width=\"288\" \/><img decoding=\"async\" src=\"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/dn-tutorials\/r-statistics-2-comparing-groups-means\/figures\/020-transforming-data-for-normality-density-plots-skewed-data-2.png\" width=\"288\" \/><\/p>\n<div class=\"success\">\n<p>The \u201cCONT\u201d variable shows positive skewness. \u201cPHYS\u201d variable is negatively skewed<\/p>\n<\/div>\n<p>Compute skewness:<\/p>\n<pre class=\"r\"><code>skewness(df$CONT, na.rm = TRUE)<\/code><\/pre>\n<pre><code>## [1] 1.09<\/code><\/pre>\n<pre class=\"r\"><code>skewness(df$PHYS, na.rm = TRUE)<\/code><\/pre>\n<pre><code>## [1] -1.56<\/code><\/pre>\n<p>Log transformation of the skewed data:<\/p>\n<pre class=\"r\"><code>df$CONT &lt;- log10(df$CONT)\r\ndf$PHYS &lt;- log10(max(df$CONT+1) - df$CONT)<\/code><\/pre>\n<pre class=\"r\"><code># Distribution of CONT variable\r\nggdensity(df, x = \"CONT\", fill = \"lightgray\", title = \"CONT\") +\r\n  stat_overlay_normal_density(color = \"red\", linetype = \"dashed\")\r\n\r\n# Distribution of PHYS variable\r\nggdensity(df, x = \"PHYS\", fill = \"lightgray\", title = \"PHYS\") +\r\n  stat_overlay_normal_density(color = \"red\", linetype = \"dashed\")<\/code><\/pre>\n<p><img decoding=\"async\" src=\"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/dn-tutorials\/r-statistics-2-comparing-groups-means\/figures\/020-transforming-data-for-normality-density-plots-log-transformed-data-1.png\" width=\"288\" \/><img decoding=\"async\" src=\"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/dn-tutorials\/r-statistics-2-comparing-groups-means\/figures\/020-transforming-data-for-normality-density-plots-log-transformed-data-2.png\" width=\"288\" \/><\/p>\n<p>Compute skewness on the transformed data:<\/p>\n<pre class=\"r\"><code>skewness(df$CONT, na.rm = TRUE)<\/code><\/pre>\n<pre><code>## [1] 0.656<\/code><\/pre>\n<pre class=\"r\"><code>skewness(df$PHYS, na.rm = TRUE)<\/code><\/pre>\n<pre><code>## [1] -0.818<\/code><\/pre>\n<div class=\"success\">\n<p>The log10 transformation improves the distribution of the data to normality.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"summary-and-discussion\" class=\"section level2\">\n<h2>Summary and discussion<\/h2>\n<p>This article describes how to transform data for normality, an assumption required for parametric tests such as t-tests and ANOVA tests.<\/p>\n<p>In the situation where the normality assumption is not met, you could consider running the statistical tests (t-test or ANOVA) on the transformed and non-transformed data to see if there are any meaningful differences.<\/p>\n<p>If both tests lead you to the same conclusions, you might not choose to transform the outcome variable and carry on with the test outputs on the original data.<\/p>\n<p>Note that transformation makes the interpretation of the analysis much more difficult. For example, if you run a t-test for comparing the mean of two groups after transforming the data, you cannot simply say that there is a difference in the two groups\u2019 means. Now, you have the added step of interpreting the fact that the difference is based on the log transformation. For this reason, transformations are usually avoided unless necessary for the analysis to be valid.<\/p>\n<p>For analyses like the F or t family of tests (i.e., independent and dependent sample t-tests, ANOVAs, MANOVAs, and regressions), violations of normality are not usually a death sentence for validity. With large enough sample sizes (&gt; 30 or 40), there\u2019s a pretty good chance that the data will be normally distributed; or at least close enough to normal that you can get away with using parametric tests (central limit theorem).<\/p>\n<\/div>\n<\/div>\n<p><!--end rdoc--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Parametric methods, such as t-test and ANOVA tests, assume that the dependent (outcome) variable is approximately normally distributed for every groups to be compared. This chapter describes how to transform data to normal distribution in R.<\/p>\n","protected":false},"author":1,"featured_media":9116,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"","class_list":["post-10895","dt_lessons","type-dt_lessons","status-publish","has-post-thumbnail","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Transform Data to Normal Distribution in R: Easy Guide - Datanovia<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.datanovia.com\/en\/lessons\/transform-data-to-normal-distribution-in-r\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Transform Data to Normal Distribution in R: Easy Guide - Datanovia\" \/>\n<meta property=\"og:description\" content=\"Parametric methods, such as t-test and ANOVA tests, assume that the dependent (outcome) variable is approximately normally distributed for every groups to be compared. 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