{"id":259,"date":"2021-01-27T19:37:20","date_gmt":"2021-01-27T19:37:20","guid":{"rendered":"http:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/matthew-davison\/?p=259"},"modified":"2021-04-28T19:03:30","modified_gmt":"2021-04-28T19:03:30","slug":"is-yawning-contagious","status":"publish","type":"post","link":"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/matthew-davison\/2021\/01\/27\/is-yawning-contagious\/","title":{"rendered":"Is Yawning Contagious?"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Over the last month I have reignited my love for the television show <a href=\"https:\/\/www.imdb.com\/title\/tt0383126\/\" data-type=\"URL\" data-id=\"https:\/\/www.imdb.com\/title\/tt0383126\/\">&#8220;Mythbusters&#8221;<\/a>. If you haven&#8217;t seen it before, it consists of a team of special effects artists who try to prove if certain urban legends are fact or fiction.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In this blog post we turn our attention to Episode 4 Season 3 (according to IMDB). In this they test if yawning is contagious. That is, if you see someone else yawn then you yourself will feel the need to yawn. You can watch the experiment below.<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-4-3 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<div class=\"nv-iframe-embed\"><iframe loading=\"lazy\" title=\"Is Yawning Contagious   Second Test\" width=\"1200\" height=\"900\" src=\"https:\/\/www.youtube.com\/embed\/GjHFvOzNiCo?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/div>\n<\/div><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">From the percentages, the sample size, and the fact they give the &#8220;yawn stimulus&#8221; to 2 out of 3 people it can be determined that in the &#8220;control&#8221; group 4 out of 16 people yawned, and in the &#8220;stimulus&#8221; group 10 out of 34 people yawned.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This is a 4% increase in yawns therefore yawning is contagious, right?<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\"><p>Given how large our sample was I would say it&#8217;s confirmed.<\/p><cite>Adam Savage<\/cite><\/blockquote>\n\n\n\n<p class=\"wp-block-paragraph\">In each group, a person either yawns, or they don&#8217;t. We can model this as fifty <a href=\"https:\/\/en.wikipedia.org\/wiki\/Bernoulli_distribution\" data-type=\"URL\" data-id=\"https:\/\/en.wikipedia.org\/wiki\/Bernoulli_distribution\">Bernoulli random variables<\/a> (1 if yawn occurs, 0 otherwise). We will assume that members of the &#8220;control&#8221; group have a probability <span class=\"wp-katex-eq\" data-display=\"false\">p_1<\/span> of yawning, while the &#8220;stimulus&#8221; group have probability <span class=\"wp-katex-eq\" data-display=\"false\">p_2<\/span>. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Our question, &#8220;Is yawning contagious?&#8221;, is now a mathematical one. Is <span class=\"wp-katex-eq\" data-display=\"false\">p_1 \\neq p_2<\/span>?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In hypothesis testing, <span class=\"wp-katex-eq\" data-display=\"false\">p_1 \\neq p_2<\/span> is called our &#8220;alternative hypothesis&#8221; and it is often denoted as <span class=\"wp-katex-eq\" data-display=\"false\">H_A<\/span>. On the other hand, <span class=\"wp-katex-eq\" data-display=\"false\">p_1=p_2<\/span> is called our &#8220;null hypothesis&#8221; and it is often denoted by <span class=\"wp-katex-eq\" data-display=\"false\">H_0<\/span>. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We will perform a <a href=\"https:\/\/en.wikipedia.org\/wiki\/Likelihood-ratio_test\" data-type=\"URL\" data-id=\"https:\/\/en.wikipedia.org\/wiki\/Likelihood-ratio_test\">Likelihood-Ratio test<\/a> to see if there is enough evidence to reject our null hypothesis.  Rejecting the null hypothesis means it has been determined that if it were true the two probabilities were equal, then we would have a very low chance of getting the data we have, implying that our assumption is untrue since we got that data.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For example, if a child comes up to you and claims they rolled a fair dice 100 times and got a 3 every time then they are either lying to you, or the dice isn&#8217;t fair (probability of getting each number is not the same).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By using calculus we find values for <span class=\"wp-katex-eq\" data-display=\"false\">p_1<\/span> and <span class=\"wp-katex-eq\" data-display=\"false\">p_2<\/span> that maximise the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Likelihood_function\" data-type=\"URL\" data-id=\"https:\/\/en.wikipedia.org\/wiki\/Likelihood_function\">likelihood function<\/a>. This is where the data that the Mythbusters gathered gets introduced into the problem, giving the following likelihood function:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><span class=\"wp-katex-eq\" data-display=\"false\">L(p_1,p_2)=p_1^4 p_2^{10}(1-p_1)^{12}(1-p_2)^{24}<\/span>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If we assume that the probabilities are not equal (as in the alternative hypothesis), then the values that maximise the above likelihood function are <span class=\"wp-katex-eq\" data-display=\"false\">{p}_1^A = 4\/16<\/span> and <span class=\"wp-katex-eq\" data-display=\"false\">{p}_2^A=10\/34<\/span>. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If we assume that the probabilities are equal (as in the null hypothesis), then the values that maximise the likelihood function are  <span class=\"wp-katex-eq\" data-display=\"false\">{p}_1^N = {p}_2^N = 14\/50<\/span>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We now construct what is called a &#8220;test statistic&#8221;, denoted as <span class=\"wp-katex-eq\" data-display=\"false\">T<\/span>, and then compare this with an appropriate distribution to see how likely we are to observe this test statistic. If we are not likely to see it, then this suggests we need to reject <span class=\"wp-katex-eq\" data-display=\"false\">H_0<\/span> (myth confirmed).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since we are performing a Likelihood-Ratio test, the following test statistic is used:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><span class=\"wp-katex-eq\" data-display=\"false\">T = 2\\log\\left( L({p}_1^A,{p}_2^A) \\div L({p}_1^N,{p}_2^N) \\right) <\/span><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By inserting the values that were found above <span class=\"wp-katex-eq\" data-display=\"false\">T = 0.1063772 <\/span>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The probability of seeing this test statistic value is 0.74. This means that the data the Mythbusters collected doesn&#8217;t suggest that yawning is more likely if you are stimulus group. For us to reject the null hypothesis we would want the test statistic value to be less than approximately 0.05 (also known as a 5% significance level).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In summary, the 4% increase in yawns that the Mythbusters found isn&#8217;t enough evidence to confidently say that yawning is contagious. My recommendation would be to change the status of this myth from &#8220;confirmed&#8221; to &#8220;plausible&#8221;. <\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Further Reading<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">For the further reading I leave two resources. The first for the reader who is interested in hypothesis testing, the second for the reader who is interested in Mythbusters.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/statisticsbyjim.com\/hypothesis-testing\/hypothesis-testing-mythbusters-yawns-contagious\/\" data-type=\"URL\" data-id=\"https:\/\/statisticsbyjim.com\/hypothesis-testing\/hypothesis-testing-mythbusters-yawns-contagious\/\">Jim Frost, a statistician, explains this myth on his blog and explains further concepts of &#8220;power&#8221; in hypothesis testing<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/mythresults.com\/episode28\" data-type=\"URL\" data-id=\"https:\/\/mythresults.com\/episode28\">The blog &#8220;Mythresults&#8221; who tracked the outcomes of every myth in every episode, already aware of the maths and have changed their status of the myth to &#8220;plausible&#8221;<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Over the last month I have reignited my love for the television show &#8220;Mythbusters&#8221;. If you haven&#8217;t seen it before, it consists of a team of special effects artists who try to prove if certain urban legends are fact or fiction. In this blog post we turn our attention to Episode 4 Season 3 (according&hellip;&nbsp;<a href=\"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/matthew-davison\/2021\/01\/27\/is-yawning-contagious\/\" rel=\"bookmark\">Read More &raquo;<span class=\"screen-reader-text\">Is Yawning Contagious?<\/span><\/a><\/p>\n","protected":false},"author":24,"featured_media":285,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"neve_meta_sidebar":"","neve_meta_container":"","neve_meta_enable_content_width":"","neve_meta_content_width":0,"neve_meta_title_alignment":"","neve_meta_author_avatar":"","neve_post_elements_order":"","neve_meta_disable_header":"","neve_meta_disable_footer":"","neve_meta_disable_title":"","footnotes":""},"categories":[3,6],"tags":[],"class_list":["post-259","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-misc","category-popular"],"_links":{"self":[{"href":"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/matthew-davison\/wp-json\/wp\/v2\/posts\/259","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/matthew-davison\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/matthew-davison\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/matthew-davison\/wp-json\/wp\/v2\/users\/24"}],"replies":[{"embeddable":true,"href":"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/matthew-davison\/wp-json\/wp\/v2\/comments?post=259"}],"version-history":[{"count":53,"href":"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/matthew-davison\/wp-json\/wp\/v2\/posts\/259\/revisions"}],"predecessor-version":[{"id":628,"href":"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/matthew-davison\/wp-json\/wp\/v2\/posts\/259\/revisions\/628"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/matthew-davison\/wp-json\/wp\/v2\/media\/285"}],"wp:attachment":[{"href":"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/matthew-davison\/wp-json\/wp\/v2\/media?parent=259"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/matthew-davison\/wp-json\/wp\/v2\/categories?post=259"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/matthew-davison\/wp-json\/wp\/v2\/tags?post=259"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}