{"id":482,"date":"2020-10-13T12:58:59","date_gmt":"2020-10-13T12:58:59","guid":{"rendered":"https:\/\/www.allendowney.com\/blog\/?p=482"},"modified":"2023-10-02T13:27:32","modified_gmt":"2023-10-02T13:27:32","slug":"whatever-the-question-was-correlation-is-not-the-answer","status":"publish","type":"post","link":"https:\/\/www.allendowney.com\/blog\/2020\/10\/13\/whatever-the-question-was-correlation-is-not-the-answer\/","title":{"rendered":"Whatever the question was, correlation is not the answer"},"content":{"rendered":"\n<p>Pearson&#8217;s coefficient of correlation, r, is one of the most widely-reported statistics.  But in my opinion, it is useless; there is no good reason to report it, ever. <\/p>\n\n\n\n<p>Most of the time, what you really care about is either effect size or predictive value:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>To quantify effect size, report the slope of a regression line.<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>To quantify predictive value, report a <a href=\"https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC5570302\/#:~:text=2.3.,observed%20values%20and%20predicted%20values.&amp;text=All%20mean%20absolute%20error%20(MAE,the%20correct%20difference%20%5B18%5D.\" target=\"_blank\" rel=\"noreferrer noopener\">measure of predictive error<\/a> that makes sense in context: MAE, MAPE, RMSE, whatever. <\/li>\n<\/ul>\n\n\n\n<p>If there&#8217;s no reason to prefer one measure over another, report reduction in RMSE, because you can compute it directly from R\u00b2.<\/p>\n\n\n\n<p>If you don&#8217;t care about effect size or predictive value, and you just want to show that there&#8217;s a (linear) relationship between two variables, use R\u00b2, which is more interpretable than r, and exaggerates the strength of the relationship less.<\/p>\n\n\n\n<p>In summary, there is no case where r is the best statistic to report. Most of the time, it answers the wrong question and makes the relationship sound more important than it is.<\/p>\n\n\n\n<p>To explain that second point, let me show an example.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Height and weight<\/h3>\n\n\n\n<p>I&#8217;ll use data from the <a href=\"https:\/\/www.cdc.gov\/brfss\/index.html\">BRFSS<\/a> to quantify the relationship between weight and height.  Here&#8217;s a scatter plot of the data and a regression line:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"398\" height=\"278\" src=\"https:\/\/www.allendowney.com\/blog\/wp-content\/uploads\/2020\/10\/image.png\" alt=\"\" class=\"wp-image-484\" srcset=\"https:\/\/www.allendowney.com\/blog\/wp-content\/uploads\/2020\/10\/image.png 398w, https:\/\/www.allendowney.com\/blog\/wp-content\/uploads\/2020\/10\/image-300x210.png 300w, https:\/\/www.allendowney.com\/blog\/wp-content\/uploads\/2020\/10\/image-387x270.png 387w\" sizes=\"auto, (max-width: 398px) 100vw, 398px\" \/><\/figure>\n\n\n\n<p>The slope of the regression line is 0.9 kg \/ cm, which means that if someone is 1 cm taller, we expect them to be 0.9 kg heavier.  If we care about effect size, that&#8217;s what we should report.<\/p>\n\n\n\n<p>If we care about predictive value, we should compare predictive error with and without the explanatory variable.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Without the model, the estimate that minimizes mean absolute error (MAE) is the median; in that case, the MAE is about 15.9 kg.<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>With the model, MAE is 13.8 kg.<\/li>\n<\/ul>\n\n\n\n<p>So the model reduces MAE by about 13%.<\/p>\n\n\n\n<p>If you don&#8217;t care about effect size or predictive value, you are probably up to no good.  But even in that case, you should report R\u00b2 = 0.22 rather than r = 0.47, because<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>R\u00b2 can be interpreted as the fraction of variance explained by the model; I don&#8217;t love this interpretation because I think the use of &#8220;explained&#8221; is misleading, but it&#8217;s better than r, which has no natural interpretation.<\/li>\n\n\n\n<li>R\u00b2 is generally smaller than r, which means it exaggerates the strength of the relationship less.<\/li>\n<\/ul>\n\n\n\n<p>[UPDATE: <a href=\"https:\/\/twitter.com\/katiecorker\/status\/1316020933828644864\">Katie Corker<\/a> corrected my claim that r has no natural interpretation: it is the standardized slope.  In this example, we expect someone who is one standard deviation taller than the mean to be 0.47 standard deviations heavier than the mean.  <a href=\"https:\/\/sebastianraschka.com\/faq\/docs\/pearson-r-vs-linear-regr.html\">Sebastian Raschka does a nice job explaining this here<\/a>.]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">In general&#8230;<\/h3>\n\n\n\n<p>This dataset is not unusual.\u00a0<em>R\u00b2<\/em>\u00a0and\u00a0r generally overstate the predictive value of the model.<\/p>\n\n\n\n<p>The following figure shows the relationship between r, <em>R\u00b2<\/em>, and the reduction in RMSE.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"372\" height=\"293\" src=\"https:\/\/www.allendowney.com\/blog\/wp-content\/uploads\/2020\/10\/image-1.png\" alt=\"\" class=\"wp-image-487\" srcset=\"https:\/\/www.allendowney.com\/blog\/wp-content\/uploads\/2020\/10\/image-1.png 372w, https:\/\/www.allendowney.com\/blog\/wp-content\/uploads\/2020\/10\/image-1-300x236.png 300w, https:\/\/www.allendowney.com\/blog\/wp-content\/uploads\/2020\/10\/image-1-343x270.png 343w\" sizes=\"auto, (max-width: 372px) 100vw, 372px\" \/><\/figure>\n\n\n\n<p>Values of r that sound impressive correspond to values of R\u00b2 that are more modest and to reductions in RMSE which are substantially less impressive.<\/p>\n\n\n\n<p>This inflation is particularly hazardous when r is small.  For example, if you see r = 0.25, you might think you&#8217;ve found an important relationship.  But that only &#8220;explains&#8221; 6% of the variance, and in terms of predictive value, only decreases RMSE by 3%.<\/p>\n\n\n\n<p>In some contexts, that predictive value might be useful, but it is substantially more modest than r=0.25 might lead you to believe.<\/p>\n\n\n\n<p><a href=\"https:\/\/colab.research.google.com\/github\/AllenDowney\/ElementsOfDataScience\/blob\/master\/examples\/correlation.ipynb\">The details of this example are in this Jupyter notebook.<\/a><\/p>\n\n\n\n<p><a href=\"https:\/\/colab.research.google.com\/github\/AllenDowney\/ElementsOfDataScience\/blob\/master\/examples\/correlation2.ipynb\">And the analysis I used to generate the last figure is in this notebook.<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Pearson&#8217;s coefficient of correlation, r, is one of the most widely-reported statistics. But in my opinion, it is useless; there is no good reason to report it, ever. Most of the time, what you really care about is either effect size or predictive value: If there&#8217;s no reason to prefer one measure over another, report reduction in RMSE, because you can compute it directly from R\u00b2. If you don&#8217;t care about effect size or predictive value, and you just want&#8230;<\/p>\n<p class=\"read-more\"><a class=\"btn btn-default\" href=\"https:\/\/www.allendowney.com\/blog\/2020\/10\/13\/whatever-the-question-was-correlation-is-not-the-answer\/\"> Read More<span class=\"screen-reader-text\">  Read More<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[1],"tags":[78,67,28],"class_list":["post-482","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-brfss","tag-correlation","tag-statistics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Whatever the question was, correlation is not the answer - Probably Overthinking It<\/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.allendowney.com\/blog\/2020\/10\/13\/whatever-the-question-was-correlation-is-not-the-answer\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Whatever the question was, correlation is not the answer - Probably Overthinking It\" \/>\n<meta property=\"og:description\" content=\"Pearson&#8217;s coefficient of correlation, r, is one of the most widely-reported statistics. 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It is also related to a new course at Brilliant.org, Explaining Variation. Suppose you find a correlation of 0.36. How would you characterize it? 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