{"id":1379,"date":"2023-06-22T02:20:44","date_gmt":"2023-06-22T02:20:44","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/pair-wise-comparisons-for-anova-learn-it-4\/"},"modified":"2025-05-16T23:00:52","modified_gmt":"2025-05-16T23:00:52","slug":"pair-wise-comparisons-for-anova-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/pair-wise-comparisons-for-anova-learn-it-4\/","title":{"raw":"Pair-wise Comparisons for ANOVA \u2013 Learn It 4","rendered":"Pair-wise Comparisons for ANOVA \u2013 Learn It 4"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Complete an ANOVA hypothesis test for pair-wise comparisons&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12801,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;arial&quot;,&quot;16&quot;:10}\">Complete pair-wise comparisons for ANOVA<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate a confidence interval and p-value for pair-wise comparisons and explain what it means&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:13057,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:3,&quot;12&quot;:0,&quot;15&quot;:&quot;arial&quot;,&quot;16&quot;:10}\">Calculate a confidence interval and p-value for pair-wise comparisons and explain what it means<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Type I Error<\/h2>\r\n<p>Recall that sometimes, due to chance, the result of the hypothesis test does not align with reality. If we reject a correct null hypothesis, we have made a <strong>type I error<\/strong>.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>type I error<\/h3>\r\n<p>The probability of committing a <strong>type I error<\/strong> is equal to the significance level: [latex]P(\\text{Type I Error}) = \\alpha[\/latex].<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2248[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2249[\/ohm2_question]<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>family-wise error rate<\/h3>\r\n<p>Suppose we perform [latex]m[\/latex] independent hypothesis tests.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The probability of making a type I error (at least one false rejection) is: [latex]1-(1-\\alpha)^m[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>In our example, we have six comparisons, so the probability of committing a type I error is: [latex]1-(1-0.05)^6 = 0.265 = 26.5\\%[\/latex]. This is likely too high and definitely not [latex]0.05[\/latex]. To avoid this problem, we need a method to maintain an overall level of significance even when several tests are performed.We call this the family-wise error rate.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The <strong>family-wise error rate<\/strong> is defined as the probability of rejecting at least one of the true null hypotheses.<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Complete an ANOVA hypothesis test for pair-wise comparisons&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12801,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;arial&quot;,&quot;16&quot;:10}\">Complete pair-wise comparisons for ANOVA<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate a confidence interval and p-value for pair-wise comparisons and explain what it means&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:13057,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:3,&quot;12&quot;:0,&quot;15&quot;:&quot;arial&quot;,&quot;16&quot;:10}\">Calculate a confidence interval and p-value for pair-wise comparisons and explain what it means<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Type I Error<\/h2>\n<p>Recall that sometimes, due to chance, the result of the hypothesis test does not align with reality. If we reject a correct null hypothesis, we have made a <strong>type I error<\/strong>.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>type I error<\/h3>\n<p>The probability of committing a <strong>type I error<\/strong> is equal to the significance level: [latex]P(\\text{Type I Error}) = \\alpha[\/latex].<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2248\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2248&theme=lumen&iframe_resize_id=ohm2248&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2249\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2249&theme=lumen&iframe_resize_id=ohm2249&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>family-wise error rate<\/h3>\n<p>Suppose we perform [latex]m[\/latex] independent hypothesis tests.<\/p>\n<p>&nbsp;<\/p>\n<p>The probability of making a type I error (at least one false rejection) is: [latex]1-(1-\\alpha)^m[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>In our example, we have six comparisons, so the probability of committing a type I error is: [latex]1-(1-0.05)^6 = 0.265 = 26.5\\%[\/latex]. This is likely too high and definitely not [latex]0.05[\/latex]. To avoid this problem, we need a method to maintain an overall level of significance even when several tests are performed.We call this the family-wise error rate.<\/p>\n<p>&nbsp;<\/p>\n<p>The <strong>family-wise error rate<\/strong> is defined as the probability of rejecting at least one of the true null hypotheses.<\/p>\n<\/section>\n","protected":false},"author":8,"menu_order":30,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1348,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1379"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1379\/revisions"}],"predecessor-version":[{"id":6852,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1379\/revisions\/6852"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1348"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1379\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1379"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1379"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1379"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1379"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}