{"id":584,"date":"2022-10-28T19:43:13","date_gmt":"2022-10-28T19:43:13","guid":{"rendered":"https:\/\/analysisparalysis.org\/?p=584"},"modified":"2022-11-08T07:43:39","modified_gmt":"2022-11-08T07:43:39","slug":"intransitivity-of-ab-tests","status":"publish","type":"post","link":"https:\/\/analysisparalysis.org\/intransitivity-of-ab-tests\/","title":{"rendered":"The Intransitivity of AB Tests"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"584\" class=\"elementor elementor-584\">\n\t\t\t\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-36cd2db4 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"36cd2db4\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-885f1e4\" data-id=\"885f1e4\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6e528418 elementor-widget elementor-widget-text-editor\" data-id=\"6e528418\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.18.0 - 20-12-2023 *\/\n.elementor-widget-text-editor.elementor-drop-cap-view-stacked .elementor-drop-cap{background-color:#69727d;color:#fff}.elementor-widget-text-editor.elementor-drop-cap-view-framed .elementor-drop-cap{color:#69727d;border:3px solid;background-color:transparent}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap{margin-top:8px}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap-letter{width:1em;height:1em}.elementor-widget-text-editor .elementor-drop-cap{float:left;text-align:center;line-height:1;font-size:50px}.elementor-widget-text-editor .elementor-drop-cap-letter{display:inline-block}<\/style>\t\t\t\t<p><!-- wp:paragraph --><\/p>\n<p>And here we are: the third (and final?) post of our A\/B testing extended universe. After the first post discussed the limits of our conclusions, and the second the greediness of A\/B testing, let\u2019s take up a subject which is more clich\u00e9, I think, but still a point worthy of shining a spotlight on for a minute.<\/p>\n<p>For the previous articles look here:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6bd87db elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6bd87db\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-dce5da0\" data-id=\"dce5da0\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b6581fd elementor-align-center elementor-widget elementor-widget-button\" data-id=\"b6581fd\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-lg\" href=\"#\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">The Greediness of AB Tests<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-deb1afe\" data-id=\"deb1afe\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b991ad4 elementor-widget elementor-widget-button\" data-id=\"b991ad4\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-lg\" href=\"#\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">The Temporal Limitations of AB Tests<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ea3f49e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ea3f49e\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-06244e5\" data-id=\"06244e5\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5aa298f elementor-widget elementor-widget-text-editor\" data-id=\"5aa298f\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<h2>Let\u2019s talk about transitivity of A\/B testing conclusions<\/h2>\n<p>First, let\u2019s talk about transitivity \u2014\u00a0<\/p>\n<h3>What even is that?<\/h3>\n<p>A transitive relation is a relation which is relayed (transits) between comparisons. that is, if R is a transitive relation, and it\u2019s true that x stands in relation R to y, and it\u2019s also true the y stands in relation R to z, than it is true that x stands in relation R to z.<\/p>\n<h3>Where can we find transitivity in real life?<\/h3>\n<p>Well, in all sorts of places.<br \/>Transitivity inherently exists on ordered axes (if ten is bigger than five and five is bigger than one, then ten is bigger than one).<br \/>On interval scales \u2014 if the eighth floor is higher than the sixth floor and the sixth floor is higher than the basement, then the eighth floor is higher than the basement.<br \/>Now that we\u2019ve established where we naturally find transitivity, let\u2019s give an example of A\/B testing.<\/p>\n<p>To reference (not to say blatantly rip off) <a href=\"http:\/\/the-greediness-of-ab-tests\">post no. 2<\/a> \u2014 the button in red (I\u2019m sticking to this joke, even though by this point I\u2019m the only one who finds it funny, and even that just because it was a long day).<\/p>\n<p>We have a red button and hypothesize that if we change it to green, we would get better results. We run the test, find out we were right and all is well.<br \/>The button is green now.<\/p>\n<p>Now, we have a green button and we hypothesize that if we change its color to blue we\u2019d get better results. We run the test, look at that we were right, all is well again.<br \/>Is a blue button better than a red button?<\/p>\n<p>We can\u2019t be certain.<\/p>\n<p>I think the most intuitive example I saw was in the<a href=\"https:\/\/www.haya-data.com\/\"> 2022 hayaData conference<\/a>, where Igal Goldfine described the lack of transitivity of AB testing as a rock, paper, scissors game.<\/p>\n<p>While the game is a closed system where we know precisely what the influences are, a knowledge we don\u2019t have for A\/B tests, the fact remains that until we check \u2014 we can\u2019t assume transitivity exists.<\/p>\n<p>If I wanted to give an example where an A\/B test probably is certainly transitive, I would give the following:<\/p>\n<p>Let\u2019s run a test on a virtual 1980&#8217;s convenience store \u2014 we want to check how the prices of Hubba Bubba gum (you know, the one that comes in a round plastic container) influence the sales of that same bubblegum. Precise definitions are important here \u2014 we\u2019re examining the correlation between price and number of sales (not, say, income).<br \/>The original state is that the price for Hubba Bubba bubble gum is $30 per piece. For our variant group, we\u2019ll lower the price to 5 cents per piece \u2014 quite naturally, the number of gums sold will go up.<\/p>\n<p>Our new base state is 5 cents per piece, and we conduct a new test where we drop the price to precisely zero. Even more naturally, the number of gums \u201csold\u201d will increase again.<\/p>\n<p>So I think the options are pretty clear, and so is the rational behind saying that A\/B tests are not transitive.<\/p>\n<p>The takeaway is fairly straightforward \u2014 when we want to check what improvement (if any) a change has, we don\u2019t have to examine the new variant against every preceding variant, but we can\u2019t assume that the new variant would be better than anything in the past, simply because it\u2019s in the past.<\/p>\n<p>Again, hope that helps with designing you AB tests!<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>And here we are: the third (and final?) post of our A\/B testing extended universe. After the first post discussed the limits of our conclusions, and the second the greediness of A\/B testing, let\u2019s take up a subject which is more clich\u00e9, I think, but still a point worthy of shining a spotlight on for [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":660,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[23],"tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>The Intransitivity of AB Tests - Analysis Paralysis<\/title>\n<meta name=\"description\" content=\"We use AB tests in a consecutive manner, and we often assume transitivity between the results of our AB test, but is that so?\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analysisparalysis.org\/intransitivity-of-ab-tests\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Intransitivity of AB Tests - 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