{"id":115,"date":"2024-08-21T12:05:21","date_gmt":"2024-08-21T12:05:21","guid":{"rendered":"http:\/\/localhost\/researchmethods\/?post_type=content&#038;p=115"},"modified":"2026-08-26T12:49:15","modified_gmt":"2026-08-26T12:49:15","slug":"chapter-23-multi-factorial-anova-designs","status":"publish","type":"content","link":"https:\/\/routledgelearning.com\/researchmethods\/student-resources\/chapter-23-multi-factorial-anova-designs\/","title":{"rendered":"Chapter 23 &#8211; Multi-factorial ANOVA designs"},"content":{"rendered":"\n<div class=\"wp-block-group has-base-background-color has-background has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\" style=\"margin-top:0;margin-bottom:0;padding-top:var(--wp--preset--spacing--70);padding-bottom:var(--wp--preset--spacing--70)\">\n<div class=\"wp-block-group alignwide is-layout-flow wp-block-group-is-layout-flow\"><nav style=\"--separator: &quot;\/&quot;\" class=\"wp-block-breadcrumbs\" aria-label=\"Breadcrumbs\"><ol><li><a href=\"https:\/\/routledgelearning.com\/researchmethods\/\">Home<\/a><\/li><li><a href=\"https:\/\/routledgelearning.com\/researchmethods\/student-resources\/\">Student Resources<\/a><\/li><li><span aria-current=\"page\">Chapter 23 &#8211; Multi-factorial ANOVA designs<\/span><\/li><\/ol><\/nav>\n\n<h1 class=\"alignwide wp-block-post-title has-large-font-size\">Chapter 23 &#8211; Multi-factorial ANOVA designs<\/h1>\n\n\n<p class=\"wp-block-paragraph\">This chapter deals with between groups multi-factorial ANOVA, where more than one independent variable is manipulated or observed.&nbsp;<\/p>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-group has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\" style=\"margin-top:0;margin-bottom:0;padding-top:var(--wp--preset--spacing--60);padding-bottom:var(--wp--preset--spacing--60)\">\n<div class=\"wp-block-columns alignwide is-layout-flex wp-container-core-columns-is-layout-7387b849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:75%\">\n<h2 id=\"aioseo-exercises\" class=\"wp-block-heading\">Exercises<\/h2>\n\n\n\n<h3 id=\"aioseo-exercise-23-1\" class=\"wp-block-heading\">Exercise 23.1<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Calculating two way unrelated ANOVA on a new data set<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The data set used to calculate the example of a two-way unrelated ANOVA in this chapter is provided below and is named two way unrelated (book).sav. An Excel file with the same name is also provided.<\/p>\n\n\n\n<div class=\"wp-block-buttons is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link has-background wp-element-button\" href=\"https:\/\/routledgelearning.com\/researchmethods\/wp-content\/uploads\/sites\/21\/2024\/09\/Two-way-unrelated-book.zip\" style=\"background-color:#ebd7c8;padding-top:var(--wp--preset--spacing--20);padding-right:var(--wp--preset--spacing--40);padding-bottom:var(--wp--preset--spacing--20);padding-left:var(--wp--preset--spacing--40);box-shadow:none\">Download &#8211; Two way unrelated (book) Data Sets<\/a><\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The data set provided below (two-way unrelated ex) is one of fictitious data from a research project on leadership styles. Each participant has an LPC score, which stands for \u2018least preferred co-worker\u2019. People with high scores on this variable are able to get along with and accept relatively uncritically even those workers whom they least prefer to interact with. Such people make good leaders when situations at work are difficult (they are \u2018people oriented\u2019). By contrast low LPC people make good task leaders and are particularly effective when working conditions are good but tend to do poorly as leaders when conditions are a little difficult.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The variables in the file are sitfav with levels of highly favourable and moderately favourable (work conditions) and lpclead with levels of high and low being the categories of high and low LPC scorers. Hence in these results we would expect to find an interaction between situation favourability and LPC leadership category. High LPC people should do well in moderately favourable conditions whereas low LPC people should do well in highly favourable conditions. How well the leaders do is the dependent variable and is measured on a scale of 1-10 as assessed by a panel of independent raters. Let\u2019s see what the spoof data says. Conduct a two-way unrelated ANOVA analysis, including relevant means and standard deviations, and checking for homogeneity of variance and for effect sizes and power for each test.<\/p>\n\n\n\n<div class=\"wp-block-buttons is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link has-background wp-element-button\" href=\"https:\/\/routledgelearning.com\/researchmethods\/wp-content\/uploads\/sites\/21\/2024\/09\/2way-unrelated-ex.sav_.zip\" style=\"background-color:#ebd7c8;padding-top:var(--wp--preset--spacing--20);padding-right:var(--wp--preset--spacing--40);padding-bottom:var(--wp--preset--spacing--20);padding-left:var(--wp--preset--spacing--40);box-shadow:none\">Download &#8211; 2 way unrelated EX Data Sets<\/a><\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The answers I got are revealed when you select the button below.<\/p>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>Show answer<\/summary>\n<p class=\"wp-block-paragraph\">The main effect for LPC leadership is not significant (overall one type of leader did no better than the other),<em> F<\/em>(1,20) = 0.220, <em>p<\/em> = .644. The main effect for situation was also not significant (leadership performances overall were similar for highly and moderately favourable conditions), <em>F<\/em>(1,20) = 0.220, <em>p<\/em> = .644). However, there was a significant interaction between situation and leadership type. In highly favourable conditions, High LPC leaders (M = 5.33, SD = 1.03) scored lower than low LPC leaders (M = 6.5, SD = 1.64), whereas in moderately favourable conditions they scored higher (M = 7.0, SD = 1.41) than low LPC leaders (M = 5.33, SD = 1.03), <em>F<\/em>(1,20) = 7.049, <em>p<\/em> = .015. Levene\u2019s test for homogeneity of variance was not significant so homogeneity was assumed. Partial eta-squared for the interaction was .261 with power estimated at .714.<\/p>\n<\/details>\n\n\n\n<h3 id=\"aioseo-exercise-23-2\" class=\"wp-block-heading\">Exercise 23.2<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Interpreting an SPSS output for a two-way unrelated analysis.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here is part of the SPSS output data for a quasi-experiment in which participants were grouped according to their attitude towards students. This is the \u2018attitude group\u2019 variable in the display below. Each group was exposed to just one of several sets of information about a fictitious person including their position on reintroducing government grants to students. Participants were later asked to rate the person on several characteristics including \u2018liking\u2019. It can be assumed for instance that participants who were pro students would show a higher liking for someone who wanted to introduce grants than someone who didn\u2019t. Study the print out and try to answer the questions below.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Levene&#8217;s test of equality of error variancesa<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Dependent Variable: liking<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td>F<\/td><td>df1<\/td><td>df2<\/td><td>Sig.<\/td><\/tr><tr><td>2.757<\/td><td>5<\/td><td>41<\/td><td>.031<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Tests of between-subjects effects<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Dependent Variable: liking<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td>Source<\/td><td>Type III sum of squares<\/td><td>df<\/td><td>Mean square<\/td><td>F<\/td><td>Sig.<\/td><\/tr><tr><td>Corrected Model<\/td><td>114.601a<\/td><td>5<\/td><td>22.920<\/td><td>7.947<\/td><td>.000<\/td><\/tr><tr><td>Intercept<\/td><td>1880.558<\/td><td>1<\/td><td>1880.558<\/td><td>652.033<\/td><td>.000<\/td><\/tr><tr><td>Information<\/td><td>3.670<\/td><td>2<\/td><td>1.835<\/td><td>.636<\/td><td>.534<\/td><\/tr><tr><td>Attitudegroup<\/td><td>15.953<\/td><td>1<\/td><td>15.953<\/td><td>5.531<\/td><td>.024<\/td><\/tr><tr><td>Information * attitudegroup<\/td><td>93.557<\/td><td>2<\/td><td>46.778<\/td><td>16.219<\/td><td>.000<\/td><\/tr><tr><td>Error<\/td><td>118.250<\/td><td>41<\/td><td>2.884<\/td><td>&nbsp;<\/td><td>&nbsp;<\/td><\/tr><tr><td>Total<\/td><td>2135.000<\/td><td>47<\/td><td>&nbsp;<\/td><td>&nbsp;<\/td><td>&nbsp;<\/td><\/tr><tr><td>Corrected total<\/td><td>232.851<\/td><td>46<\/td><td>&nbsp;<\/td><td>&nbsp;<\/td><td>&nbsp;<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ol style=\"list-style-type:lower-alpha\" class=\"wp-block-list\">\n<li>R Squared = .492 (Adjusted R Squared = .430)<\/li>\n<\/ol>\n\n\n\n<div class=\"wp-block-group has-base-element-background-color has-background has-global-padding is-layout-constrained wp-container-core-group-is-layout-0910e091 wp-block-group-is-layout-constrained\" style=\"border-width:1px;padding-top:var(--wp--preset--spacing--40);padding-right:var(--wp--preset--spacing--40);padding-bottom:var(--wp--preset--spacing--40);padding-left:var(--wp--preset--spacing--40)\">\n<div class=\"wp-block-group alignwide is-layout-flow wp-block-group-is-layout-flow\"><script>\n                            if (window.qmn_quiz_data === undefined) {\n                                    window.qmn_quiz_data = new Object();\n                            }\n                    <\/script><script>window.qmn_quiz_data[\"23\"] = {\"quiz_id\":\"23\",\"quiz_name\":\"exercise232\",\"disable_answer\":0,\"ajax_show_correct\":0,\"progress_bar\":\"0\",\"contact_info_location\":\"0\",\"qpages\":{\"2\":{\"id\":\"2\",\"quizID\":\"23\",\"pagekey\":\"QDTjfHam\",\"hide_prevbtn\":\"0\"}},\"skip_validation_time_expire\":0,\"timer_limit_val\":0,\"disable_scroll_next_previous_click\":0,\"disable_scroll_on_result\":0,\"disable_first_page\":\"0\",\"enable_result_after_timer_end\":0,\"enable_quick_result_mc\":0,\"end_quiz_if_wrong\":0,\"form_disable_autofill\":0,\"disable_mathjax\":0,\"enable_quick_correct_answer_info\":\"0\",\"quick_result_correct_answer_text\":\"Correct! You have selected correct answer.\",\"quick_result_wrong_answer_text\":\"Wrong! You have selected wrong answer.\",\"quiz_processing_message\":\"\",\"quiz_limit_choice\":\"Limit of choice is reached.\",\"not_allow_after_expired_time\":0,\"scheduled_time_end\":false,\"prevent_reload\":0,\"limit_email_based_submission\":0,\"total_user_tries\":0,\"is_logged_in\":false,\"pagination\":{\"amount\":1,\"section_comments\":\"\",\"total_questions\":5,\"previous_text\":\"Previous\",\"next_text\":\"Next\",\"start_quiz_survey_text\":\"Start Quiz\",\"submit_quiz_text\":\"Submit\"},\"error_messages\":{\"email_error_text\":\"Not a valid e-mail address!\",\"number_error_text\":\"This field must be a number!\",\"incorrect_error_text\":\"The entered text is not correct!\",\"empty_error_text\":\"Please complete all required fields!\",\"contact_field_required_error_text\":\"Please complete all required fields!\",\"url_error_text\":\"The entered URL is not valid!\",\"minlength_error_text\":\"Required atleast %minlength% characters.\",\"maxlength_error_text\":\"Maximum %maxlength% characters allowed.\",\"recaptcha_error_text\":\"ReCaptcha is missing\",\"phone_error_text\":\"Phone number is invalid\",\"empty_grid_radio_checkbox_text\":\"Please complete all  rows required fields!\",\"empty_audio_text\":\"Please upload audio!\"},\"question_list\":{\"151\":{\"question_id\":\"151\",\"quiz_id\":\"23\",\"question_name\":\"The number of information conditions was:\",\"answer_array\":\"a:4:{i:0;a:3:{i:0;s:1:\\\"2\\\";i:1;d:0;i:2;i:0;}i:1;a:3:{i:0;s:1:\\\"3\\\";i:1;d:0;i:2;i:1;}i:2;a:3:{i:0;s:1:\\\"4\\\";i:1;d:0;i:2;i:0;}i:3;a:3:{i:0;s:1:\\\"5\\\";i:1;d:0;i:2;i:0;}}\",\"answer_one\":\"\",\"answer_one_points\":\"0\",\"answer_two\":\"\",\"answer_two_points\":\"0\",\"answer_three\":\"\",\"answer_three_points\":\"0\",\"answer_four\":\"\",\"answer_four_points\":\"0\",\"answer_five\":\"\",\"answer_five_points\":\"0\",\"answer_six\":\"\",\"answer_six_points\":\"0\",\"correct_answer\":\"0\",\"question_answer_info\":\"<p><em>df<\\\/em> for Information = 2 so there must have been 3 conditions. (<em>df<\\\/em> = conds-1)<\\\/p>\",\"comments\":\"1\",\"hints\":\"\",\"question_order\":\"1\",\"question_type\":\"0\",\"question_type_new\":\"0\",\"question_settings\":\"a:16:{s:8:\\\"required\\\";i:0;s:12:\\\"answerEditor\\\";s:4:\\\"text\\\";s:14:\\\"question_title\\\";s:10:\\\"Question 1\\\";s:14:\\\"featureImageID\\\";s:0:\\\"\\\";s:15:\\\"featureImageSrc\\\";s:0:\\\"\\\";s:11:\\\"matchAnswer\\\";s:6:\\\"random\\\";s:16:\\\"placeholder_text\\\";s:0:\\\"\\\";s:14:\\\"case_sensitive\\\";s:0:\\\"\\\";s:16:\\\"image_size-width\\\";s:0:\\\"\\\";s:17:\\\"image_size-height\\\";s:0:\\\"\\\";s:8:\\\"autofill\\\";s:0:\\\"\\\";s:10:\\\"limit_text\\\";s:1:\\\"0\\\";s:23:\\\"limit_multiple_response\\\";s:1:\\\"0\\\";s:17:\\\"file_upload_limit\\\";s:1:\\\"4\\\";s:16:\\\"file_upload_type\\\";s:21:\\\"image,application\\\/pdf\\\";s:15:\\\"min_text_length\\\";s:0:\\\"\\\";}\",\"category\":\"\",\"linked_question\":\"\",\"deleted\":\"0\",\"deleted_question_bank\":\"0\",\"answers\":[[\"2\",0,0],[\"3\",0,1],[\"4\",0,0],[\"5\",0,0]]},\"152\":{\"question_id\":\"152\",\"quiz_id\":\"23\",\"question_name\":\"The number of attitude groups was:\",\"answer_array\":\"a:4:{i:0;a:3:{i:0;s:1:\\\"2\\\";i:1;d:0;i:2;i:1;}i:1;a:3:{i:0;s:1:\\\"3\\\";i:1;d:0;i:2;i:0;}i:2;a:3:{i:0;s:1:\\\"4\\\";i:1;d:0;i:2;i:0;}i:3;a:3:{i:0;s:1:\\\"5\\\";i:1;d:0;i:2;i:0;}}\",\"answer_one\":\"\",\"answer_one_points\":\"0\",\"answer_two\":\"\",\"answer_two_points\":\"0\",\"answer_three\":\"\",\"answer_three_points\":\"0\",\"answer_four\":\"\",\"answer_four_points\":\"0\",\"answer_five\":\"\",\"answer_five_points\":\"0\",\"answer_six\":\"\",\"answer_six_points\":\"0\",\"correct_answer\":\"0\",\"question_answer_info\":\"<p><em>df<\\\/em> for attitudegroup = 1 so there must have been 2 conditions. (<em>df<\\\/em> = conds-1)<\\\/p>\",\"comments\":\"1\",\"hints\":\"\",\"question_order\":\"1\",\"question_type\":\"0\",\"question_type_new\":\"0\",\"question_settings\":\"a:16:{s:8:\\\"required\\\";i:0;s:12:\\\"answerEditor\\\";s:4:\\\"text\\\";s:14:\\\"question_title\\\";s:10:\\\"Question 2\\\";s:14:\\\"featureImageID\\\";s:0:\\\"\\\";s:15:\\\"featureImageSrc\\\";s:0:\\\"\\\";s:11:\\\"matchAnswer\\\";s:6:\\\"random\\\";s:16:\\\"placeholder_text\\\";s:0:\\\"\\\";s:14:\\\"case_sensitive\\\";s:0:\\\"\\\";s:16:\\\"image_size-width\\\";s:0:\\\"\\\";s:17:\\\"image_size-height\\\";s:0:\\\"\\\";s:8:\\\"autofill\\\";s:0:\\\"\\\";s:10:\\\"limit_text\\\";s:1:\\\"0\\\";s:23:\\\"limit_multiple_response\\\";s:1:\\\"0\\\";s:17:\\\"file_upload_limit\\\";s:1:\\\"4\\\";s:16:\\\"file_upload_type\\\";s:21:\\\"image,application\\\/pdf\\\";s:15:\\\"min_text_length\\\";s:0:\\\"\\\";}\",\"category\":\"\",\"linked_question\":\"\",\"deleted\":\"0\",\"deleted_question_bank\":\"0\",\"answers\":[[\"2\",0,1],[\"3\",0,0],[\"4\",0,0],[\"5\",0,0]]},\"153\":{\"question_id\":\"153\",\"quiz_id\":\"23\",\"question_name\":\"Which one of these statements is true?\",\"answer_array\":\"a:4:{i:0;a:3:{i:0;s:76:\\\"The main effect for information and the interaction effect were significant.\\\";i:1;d:0;i:2;i:0;}i:1;a:3:{i:0;s:44:\\\"Only the interaction effect was significant.\\\";i:1;d:0;i:2;i:0;}i:2;a:3:{i:0;s:79:\\\"The main effect for attitude group and the interaction effect were significant.\\\";i:1;d:0;i:2;i:1;}i:3;a:3:{i:0;s:56:\\\"Only the main effect for attitude group was significant.\\\";i:1;d:0;i:2;i:0;}}\",\"answer_one\":\"\",\"answer_one_points\":\"0\",\"answer_two\":\"\",\"answer_two_points\":\"0\",\"answer_three\":\"\",\"answer_three_points\":\"0\",\"answer_four\":\"\",\"answer_four_points\":\"0\",\"answer_five\":\"\",\"answer_five_points\":\"0\",\"answer_six\":\"\",\"answer_six_points\":\"0\",\"correct_answer\":\"0\",\"question_answer_info\":\"looking at the \\u2018significant\\u2019 column in the effects table we see that only attitudegroup and the interaction of attitudegroup with information had a <em>p<\\\/em> value less than .05.\",\"comments\":\"1\",\"hints\":\"\",\"question_order\":\"1\",\"question_type\":\"0\",\"question_type_new\":\"0\",\"question_settings\":\"a:16:{s:8:\\\"required\\\";i:0;s:12:\\\"answerEditor\\\";s:4:\\\"text\\\";s:14:\\\"question_title\\\";s:10:\\\"Question 3\\\";s:14:\\\"featureImageID\\\";s:0:\\\"\\\";s:15:\\\"featureImageSrc\\\";s:0:\\\"\\\";s:11:\\\"matchAnswer\\\";s:6:\\\"random\\\";s:16:\\\"placeholder_text\\\";s:0:\\\"\\\";s:14:\\\"case_sensitive\\\";s:0:\\\"\\\";s:16:\\\"image_size-width\\\";s:0:\\\"\\\";s:17:\\\"image_size-height\\\";s:0:\\\"\\\";s:8:\\\"autofill\\\";s:0:\\\"\\\";s:10:\\\"limit_text\\\";s:1:\\\"0\\\";s:23:\\\"limit_multiple_response\\\";s:1:\\\"0\\\";s:17:\\\"file_upload_limit\\\";s:1:\\\"4\\\";s:16:\\\"file_upload_type\\\";s:21:\\\"image,application\\\/pdf\\\";s:15:\\\"min_text_length\\\";s:0:\\\"\\\";}\",\"category\":\"\",\"linked_question\":\"\",\"deleted\":\"0\",\"deleted_question_bank\":\"0\",\"answers\":[[\"The main effect for information and the interaction effect were significant.\",0,0],[\"Only the interaction effect was significant.\",0,0],[\"The main effect for attitude group and the interaction effect were significant.\",0,1],[\"Only the main effect for attitude group was significant.\",0,0]]},\"154\":{\"question_id\":\"154\",\"quiz_id\":\"23\",\"question_name\":\"Which one of these statements is true?\",\"answer_array\":\"a:4:{i:0;a:3:{i:0;s:56:\\\"The information provided shows that the data are skewed.\\\";i:1;d:0;i:2;i:0;}i:1;a:3:{i:0;s:79:\\\"The information provided shows that there is a problem of kurtosis in the data.\\\";i:1;d:0;i:2;i:0;}i:2;a:3:{i:0;s:60:\\\"In this analysis there is no problem with unequal variances.\\\";i:1;d:0;i:2;i:0;}i:3;a:3:{i:0;s:51:\\\"In this analysis equal variances cannot be assumed.\\\";i:1;d:0;i:2;i:1;}}\",\"answer_one\":\"\",\"answer_one_points\":\"0\",\"answer_two\":\"\",\"answer_two_points\":\"0\",\"answer_three\":\"\",\"answer_three_points\":\"0\",\"answer_four\":\"\",\"answer_four_points\":\"0\",\"answer_five\":\"\",\"answer_five_points\":\"0\",\"answer_six\":\"\",\"answer_six_points\":\"0\",\"correct_answer\":\"0\",\"question_answer_info\":\"Levene tests for equality of variances and it shows a <em>p<\\\/em> value of .031. It is significant so equal variances cannot be assumed.\",\"comments\":\"1\",\"hints\":\"\",\"question_order\":\"1\",\"question_type\":\"0\",\"question_type_new\":\"0\",\"question_settings\":\"a:16:{s:8:\\\"required\\\";i:0;s:12:\\\"answerEditor\\\";s:4:\\\"text\\\";s:14:\\\"question_title\\\";s:10:\\\"Question 4\\\";s:14:\\\"featureImageID\\\";s:0:\\\"\\\";s:15:\\\"featureImageSrc\\\";s:0:\\\"\\\";s:11:\\\"matchAnswer\\\";s:6:\\\"random\\\";s:16:\\\"placeholder_text\\\";s:0:\\\"\\\";s:14:\\\"case_sensitive\\\";s:0:\\\"\\\";s:16:\\\"image_size-width\\\";s:0:\\\"\\\";s:17:\\\"image_size-height\\\";s:0:\\\"\\\";s:8:\\\"autofill\\\";s:0:\\\"\\\";s:10:\\\"limit_text\\\";s:1:\\\"0\\\";s:23:\\\"limit_multiple_response\\\";s:1:\\\"0\\\";s:17:\\\"file_upload_limit\\\";s:1:\\\"4\\\";s:16:\\\"file_upload_type\\\";s:21:\\\"image,application\\\/pdf\\\";s:15:\\\"min_text_length\\\";s:0:\\\"\\\";}\",\"category\":\"\",\"linked_question\":\"\",\"deleted\":\"0\",\"deleted_question_bank\":\"0\",\"answers\":[[\"The information provided shows that the data are skewed.\",0,0],[\"The information provided shows that there is a problem of kurtosis in the data.\",0,0],[\"In this analysis there is no problem with unequal variances.\",0,0],[\"In this analysis equal variances cannot be assumed.\",0,1]]},\"155\":{\"question_id\":\"155\",\"quiz_id\":\"23\",\"question_name\":\"Which one of these statements is true?\",\"answer_array\":\"a:4:{i:0;a:3:{i:0;s:63:\\\"The result for the interaction should be written F1,41 = 16.219\\\";i:1;d:0;i:2;i:0;}i:1;a:3:{i:0;s:63:\\\"The result for the interaction should be written F2,41 = 16.219\\\";i:1;d:0;i:2;i:1;}i:2;a:3:{i:0;s:63:\\\"The result for the interaction should be written F2,47 = 16.219\\\";i:1;d:0;i:2;i:0;}i:3;a:3:{i:0;s:63:\\\"The result for the interaction should be written F1,47 = 16.219\\\";i:1;d:0;i:2;i:0;}}\",\"answer_one\":\"\",\"answer_one_points\":\"0\",\"answer_two\":\"\",\"answer_two_points\":\"0\",\"answer_three\":\"\",\"answer_three_points\":\"0\",\"answer_four\":\"\",\"answer_four_points\":\"0\",\"answer_five\":\"\",\"answer_five_points\":\"0\",\"answer_six\":\"\",\"answer_six_points\":\"0\",\"correct_answer\":\"0\",\"question_answer_info\":\"The interaction effect has 2 <em>df<\\\/em> and the error <em>df<\\\/em> are 41, hence we write <em>F<\\\/em>(2,41) = 16.219 (from the table\\u2019s <em>F<\\\/em> column).\",\"comments\":\"1\",\"hints\":\"\",\"question_order\":\"1\",\"question_type\":\"0\",\"question_type_new\":\"0\",\"question_settings\":\"a:16:{s:8:\\\"required\\\";i:0;s:12:\\\"answerEditor\\\";s:4:\\\"text\\\";s:14:\\\"question_title\\\";s:10:\\\"Question 5\\\";s:14:\\\"featureImageID\\\";s:0:\\\"\\\";s:15:\\\"featureImageSrc\\\";s:0:\\\"\\\";s:11:\\\"matchAnswer\\\";s:6:\\\"random\\\";s:16:\\\"placeholder_text\\\";s:0:\\\"\\\";s:14:\\\"case_sensitive\\\";s:0:\\\"\\\";s:16:\\\"image_size-width\\\";s:0:\\\"\\\";s:17:\\\"image_size-height\\\";s:0:\\\"\\\";s:8:\\\"autofill\\\";s:0:\\\"\\\";s:10:\\\"limit_text\\\";s:1:\\\"0\\\";s:23:\\\"limit_multiple_response\\\";s:1:\\\"0\\\";s:17:\\\"file_upload_limit\\\";s:1:\\\"4\\\";s:16:\\\"file_upload_type\\\";s:21:\\\"image,application\\\/pdf\\\";s:15:\\\"min_text_length\\\";s:0:\\\"\\\";}\",\"category\":\"\",\"linked_question\":\"\",\"deleted\":\"0\",\"deleted_question_bank\":\"0\",\"answers\":[[\"The result for the interaction should be written F1,41 = 16.219\",0,0],[\"The result for the interaction should be written F2,41 = 16.219\",0,1],[\"The result for the interaction should be written F2,47 = 16.219\",0,0],[\"The result for the interaction should be written F1,47 = 16.219\",0,0]]}},\"first_page\":false,\"questions_settings\":[]}\n                    <\/script>\t\t<div class='qsm-quiz-container qsm-quiz-container-23 qmn_quiz_container mlw_qmn_quiz qsm_auto_pagination_enabled quiz_theme_default  '>\n\t\t\t\t\t\t\t\t<form name=\"quizForm23\" id=\"quizForm23\" action=\"\/researchmethods\/wp-json\/wp\/v2\/content\/115\/\" method=\"POST\" class=\"qsm-quiz-form qmn_quiz_form mlw_quiz_form\" novalidate enctype=\"multipart\/form-data\">\n\t\t\t\t<input type=\"hidden\" name=\"qsm_hidden_questions\" id=\"qsm_hidden_questions\" value=\"\">\n\t\t\t\t<input type=\"hidden\" name=\"qsm_nonce\" id=\"qsm_nonce_23\" value=\"65f5ae0de1\">\n\t\t\t\t<input type=\"hidden\" name=\"qsm_unique_key\" id=\"qsm_unique_key_23\" value=\"6a8f8c8136756\">\n\t\t\t\t<div id=\"mlw_error_message\" class=\"qsm-error-message qmn_error_message_section\"><\/div>\n\t\t\t\t<span id=\"mlw_top_of_quiz\"><\/span>\n\t\t\t\t\t\t\t\t\t<div class=\"qsm-auto-page-row qsm-question-page qsm-apc-1\" data-apid=\"1\" data-qpid=\"1\" style=\"display: none;\">\n\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-0  question-section-id-151 slide0 \">\n\t\t\t\t\t\t<div class='mlw_qmn_new_question'>Question 1 <\/div>\n\t\t\t<div class='mlw_qmn_question  qsm_remove_bold' >\n\t\t<p>The number of information conditions was:<\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class='qmn_radio_answers mlwRequiredRadio'>\n\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question151-1 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question151\" id=\"question151_1\" value=\"0\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question151_1\">\n\t\t\t\t\t2\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question151-2 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question151\" id=\"question151_2\" value=\"1\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question151_2\">\n\t\t\t\t\t3\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question151-3 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question151\" id=\"question151_3\" value=\"2\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question151_3\">\n\t\t\t\t\t4\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question151-4 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question151\" id=\"question151_4\" value=\"3\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question151_4\">\n\t\t\t\t\t5\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t \t\t\t<label style=\"display: none !important;\" for=\"question151_none\">None<\/label>\n\t\t\t<input type=\"radio\" style=\"display: none;\" name=\"question151\" id=\"question151_none\" checked=\"checked\" value=\"\" \/>\n\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_151\" value=\"\" \/>\n\t\t\t\t<\/div><!-- .quiz_section -->\n\t\t\t\t\t\t\t\t<\/div><!-- .qsm-auto-page-row -->\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm-auto-page-row qsm-question-page qsm-apc-2\" data-apid=\"2\" data-qpid=\"2\" style=\"display: none;\">\n\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-0  question-section-id-152 slide1 \">\n\t\t\t\t\t\t<div class='mlw_qmn_new_question'>Question 2 <\/div>\n\t\t\t<div class='mlw_qmn_question  qsm_remove_bold' >\n\t\t<p>The number of attitude groups was:<\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class='qmn_radio_answers mlwRequiredRadio'>\n\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question152-1 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question152\" id=\"question152_1\" value=\"0\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question152_1\">\n\t\t\t\t\t2\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question152-2 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question152\" id=\"question152_2\" value=\"1\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question152_2\">\n\t\t\t\t\t3\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question152-3 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question152\" id=\"question152_3\" value=\"2\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question152_3\">\n\t\t\t\t\t4\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question152-4 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question152\" id=\"question152_4\" value=\"3\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question152_4\">\n\t\t\t\t\t5\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t \t\t\t<label style=\"display: none !important;\" for=\"question152_none\">None<\/label>\n\t\t\t<input type=\"radio\" style=\"display: none;\" name=\"question152\" id=\"question152_none\" checked=\"checked\" value=\"\" \/>\n\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_152\" value=\"\" \/>\n\t\t\t\t<\/div><!-- .quiz_section -->\n\t\t\t\t\t\t\t\t<\/div><!-- .qsm-auto-page-row -->\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm-auto-page-row qsm-question-page qsm-apc-3\" data-apid=\"3\" data-qpid=\"3\" style=\"display: none;\">\n\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-0  question-section-id-153 slide2 \">\n\t\t\t\t\t\t<div class='mlw_qmn_new_question'>Question 3 <\/div>\n\t\t\t<div class='mlw_qmn_question  qsm_remove_bold' >\n\t\t<p>Which one of these statements is true?<\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class='qmn_radio_answers mlwRequiredRadio'>\n\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question153-1 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question153\" id=\"question153_1\" value=\"0\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question153_1\">\n\t\t\t\t\tThe main effect for information and the interaction effect were significant.\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question153-2 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question153\" id=\"question153_2\" value=\"1\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question153_2\">\n\t\t\t\t\tOnly the interaction effect was significant.\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question153-3 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question153\" id=\"question153_3\" value=\"2\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question153_3\">\n\t\t\t\t\tThe main effect for attitude group and the interaction effect were significant.\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question153-4 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question153\" id=\"question153_4\" value=\"3\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question153_4\">\n\t\t\t\t\tOnly the main effect for attitude group was significant.\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t \t\t\t<label style=\"display: none !important;\" for=\"question153_none\">None<\/label>\n\t\t\t<input type=\"radio\" style=\"display: none;\" name=\"question153\" id=\"question153_none\" checked=\"checked\" value=\"\" \/>\n\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_153\" value=\"\" \/>\n\t\t\t\t<\/div><!-- .quiz_section -->\n\t\t\t\t\t\t\t\t<\/div><!-- .qsm-auto-page-row -->\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm-auto-page-row qsm-question-page qsm-apc-4\" data-apid=\"4\" data-qpid=\"4\" style=\"display: none;\">\n\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-0  question-section-id-154 slide3 \">\n\t\t\t\t\t\t<div class='mlw_qmn_new_question'>Question 4 <\/div>\n\t\t\t<div class='mlw_qmn_question  qsm_remove_bold' >\n\t\t<p>Which one of these statements is true?<\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class='qmn_radio_answers mlwRequiredRadio'>\n\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question154-1 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question154\" id=\"question154_1\" value=\"0\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question154_1\">\n\t\t\t\t\tThe information provided shows that the data are skewed.\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question154-2 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question154\" id=\"question154_2\" value=\"1\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question154_2\">\n\t\t\t\t\tThe information provided shows that there is a problem of kurtosis in the data.\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question154-3 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question154\" id=\"question154_3\" value=\"2\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question154_3\">\n\t\t\t\t\tIn this analysis there is no problem with unequal variances.\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question154-4 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question154\" id=\"question154_4\" value=\"3\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question154_4\">\n\t\t\t\t\tIn this analysis equal variances cannot be assumed.\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t \t\t\t<label style=\"display: none !important;\" for=\"question154_none\">None<\/label>\n\t\t\t<input type=\"radio\" style=\"display: none;\" name=\"question154\" id=\"question154_none\" checked=\"checked\" value=\"\" \/>\n\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_154\" value=\"\" \/>\n\t\t\t\t<\/div><!-- .quiz_section -->\n\t\t\t\t\t\t\t\t<\/div><!-- .qsm-auto-page-row -->\n\t\t\t\t\t\t\t\t\t\t<div class=\"qsm-auto-page-row qsm-question-page qsm-apc-5\" data-apid=\"5\" data-qpid=\"5\" style=\"display: none;\">\n\t\t\t\t\t\t\t\t<div class=\"quiz_section qsm-question-wrapper question-type-0  question-section-id-155 slide4 \">\n\t\t\t\t\t\t<div class='mlw_qmn_new_question'>Question 5 <\/div>\n\t\t\t<div class='mlw_qmn_question  qsm_remove_bold' >\n\t\t<p>Which one of these statements is true?<\/p>\n\t<\/div>\n\t\t<fieldset>\n\t\t<legend><\/legend>\n\t<div class='qmn_radio_answers mlwRequiredRadio'>\n\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question155-1 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question155\" id=\"question155_1\" value=\"0\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question155_1\">\n\t\t\t\t\tThe result for the interaction should be written F1,41 = 16.219\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question155-2 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question155\" id=\"question155_2\" value=\"1\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question155_2\">\n\t\t\t\t\tThe result for the interaction should be written F2,41 = 16.219\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question155-3 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question155\" id=\"question155_3\" value=\"2\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question155_3\">\n\t\t\t\t\tThe result for the interaction should be written F2,47 = 16.219\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"qmn_mc_answer_wrap  mrq_checkbox_class\" id=\"question155-4 \">\n\t\t\t\t\t\t\t\t\t\t\t<input type='radio' class='qmn_quiz_radio qmn-multiple-choice-input ' name=\"question155\" id=\"question155_4\" value=\"3\" \/>\n\t\t\t\t\t<label class=\"qsm-input-label\" for=\"question155_4\">\n\t\t\t\t\tThe result for the interaction should be written F1,47 = 16.219\t\t\t\t\t<\/label>\n\t\t\t\t\t \t\t\t\t<\/div>\n\t\t\t\t\t \t\t\t<label style=\"display: none !important;\" for=\"question155_none\">None<\/label>\n\t\t\t<input type=\"radio\" style=\"display: none;\" name=\"question155\" id=\"question155_none\" checked=\"checked\" value=\"\" \/>\n\t\t\t\t<\/div>\n\t<\/fieldset>\n\t<input type=\"hidden\" name=\"answer_limit_keys_155\" value=\"\" \/>\n\t\t\t\t<\/div><!-- .quiz_section -->\n\t\t\t\t\t\t\t\t<\/div><!-- .qsm-auto-page-row -->\n\t\t\t\t\t\t\t<input type=\"hidden\" name=\"qmn_question_list\" value=\"151Q152Q153Q154Q155Q\" \/>\n\t\t\t\t\t<div class=\"qsm-auto-page-row quiz_section quiz_end empty_quiz_end qsm-d-none\" >\n\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<div id=\"mlw_error_message_bottom\" class=\"qsm-error-message qmn_error_message_section\"><\/div>\n\t\t\t\t\t<input type=\"hidden\" name=\"qmn_all_questions_count\" id=\"qmn_all_questions_count\" value=\"5\" \/>\n\t\t\t\t\t<input type=\"hidden\" name=\"total_questions\" id=\"total_questions\" value=\"5\" \/>\n\t\t\t\t\t<input type=\"hidden\" name=\"timer\" id=\"timer\" value=\"0\" \/>\n\t\t\t\t\t<input type=\"hidden\" name=\"timer_ms\" id=\"timer_ms\" value=\"0\"\/>\n\t\t\t\t\t<input type=\"hidden\" class=\"qmn_quiz_id\" name=\"qmn_quiz_id\" id=\"qmn_quiz_id\" value=\"23\" \/>\n\t\t\t\t\t<input type='hidden' name='complete_quiz' value='confirmation' \/>\n\t\t\t\t\t\t\t\t\t<\/form>\n\t\t\t\t\t\t<\/div>\n\t\t<div style=\"display: none;\" class=\"qsm-popup qsm-popup-slide\" id=\"modal-4\" aria-hidden=\"false\"><div class=\"qsm-popup__overlay\" tabindex=\"-1\" data-micromodal-close=\"\"><div class=\"qsm-popup__container qmn_quiz_container\" role=\"dialog\" aria-modal=\"true\"><div class=\"qsm-popup__content\"><img decoding=\"async\" src=\"https:\/\/routledgelearning.com\/researchmethods\/wp-content\/plugins\/quiz-master-next\/assets\/clock.png\" alt=\"clock.png\"\/><p class=\"qsm-time-up-text\"> Time&#8217;s up<\/p><\/div><footer class=\"qsm-popup__footer\"><button class=\"qsm-popup-secondary-button qmn_btn\" data-micromodal-close=\"\" aria-label=\"Close this dialog window\" onclick=\"location.reload();\">Cancel<\/button><\/footer><\/div><\/div><\/div><\/div>\n<\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 id=\"aioseo-weblinks\" class=\"wp-block-heading\">Weblinks<\/h2>\n\n\n\n<h3 id=\"aioseo-multi-factorial-anova-designs-weblinks\" class=\"wp-block-heading\">Multi-factorial ANOVA designs<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Laerd statistics pages for two-way unrelated ANOVA.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/statistics.laerd.com\/spss-tutorials\/two-way-anova-using-spss-statistics-2.php\">Two-way ANOVA Output and Interpretation in SPSS Statistics &#8211; Including Simple Main Effects | Laerd Statistics<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The Hyperstat pages for between subjects factorial designs:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/davidmlane.com\/hyperstat\/factorial_ANOVA.html\">HyperStat Online: Factorial Between-Subjects ANOVA (davidmlane.com)<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Two sites on the argument about interpreting main effects when there is a significant interaction effect:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/stats.stackexchange.com\/questions\/93667\/how-does-the-interpretation-of-main-effects-in-a-two-way-anova-change-depending\">How does the interpretation of main effects in a Two-Way ANOVA change depending on whether the interaction effect is significant? &#8211; Cross Validated (stackexchange.com)<\/a><br><br><a><\/a><a href=\"https:\/\/www.theanalysisfactor.com\/interpret-main-effects-interaction\/\">Actually, you can interpret some main effects in the presence of an interaction &#8211; The Analysis Factor<\/a><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 id=\"aioseo-further-information\" class=\"wp-block-heading\">Further Information<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">In Chapter 21 of the book we calculated a two-way ANOVA on the data that are provided here in Exercise 1. The book tells you that you can obtain effect size and power using SPSS or G*Power, but that the by-hand calculations are rather complex. For the sake of completeness though I will give the detail here.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Remember we are dealing with an experiment where participants consume either strong coffee, decaffeinated coffee or nothing. These conditions are referred to as caffeine, decaff and none. Two groups of participants are tested, those who have just had five hours\u2019 sleep and those who have been awake for a full 24 hours. We therefore have a 2 (sleep) x 3 (caffeine) design. The results Table 21.2 from the book is: <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Skill scores<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"645\" height=\"500\" src=\"https:\/\/routledgelearning.com\/researchmethods\/wp-content\/uploads\/sites\/21\/2024\/08\/Table-21.2.png\" alt=\"\" class=\"wp-image-292\" srcset=\"https:\/\/routledgelearning.com\/researchmethods\/wp-content\/uploads\/sites\/21\/2024\/08\/Table-21.2.png 645w, https:\/\/routledgelearning.com\/researchmethods\/wp-content\/uploads\/sites\/21\/2024\/08\/Table-21.2-300x233.png 300w\" sizes=\"auto, (max-width: 645px) 100vw, 645px\" \/><figcaption class=\"wp-element-caption\">Table 21.2<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">* Overlong decimal figures are used here in order that our figures come close to those given by SPSS. With sensible rounding, our ANOVA results would be more different from the SPSS result than they are.<br>Table 21.2: Driving skill scores by caffeine and sleep conditions<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Main effects<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The general rules for calculating effect size and power for main effects in a two-way design are as follows, where we will refer to one factor with the general term A and the other with the term B.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For factor A, <math display=\"block\" class=\"tml-display\" style=\"display:block math;\">\n  <mrow>\n    <msubsup>\n      <mrow>\n        <mi mathvariant=\"normal\">\u03a6<\/mi>\n      <\/mrow>\n      <mi>A<\/mi>\n      <mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo>\n    <\/msubsup>\n    <mo>=<\/mo>\n    <msqrt>\n      <mfrac>\n        <mrow>\n          <mo stretchy=\"true\" lspace=\"0em\" rspace=\"0em\">\u2211<\/mo>\n          <msub>\n            <mi>\u03b1<\/mi>\n            <mi>i<\/mi>\n          <\/msub>\n          <msup>\n            <mrow><\/mrow>\n            <mn>2<\/mn>\n          <\/msup>\n        <\/mrow>\n        <mrow>\n          <mi>a<\/mi>\n          <msub>\n            <mi>\u03c3<\/mi>\n            <mi>e<\/mi>\n          <\/msub>\n          <msup>\n            <mrow><\/mrow>\n            <mn>2<\/mn>\n          <\/msup>\n        <\/mrow>\n      <\/mfrac>\n    <\/msqrt>\n  <\/mrow>\n<\/math> = where \u03b1i is the difference between the grand mean and each mean for factor A ignoring any difference across factor B. Ideally, these means would be the true population means if known but in calculating power after an experiment we use the sample means and assume these are good estimates of the population means. For our caffeine\/sleep study, then, the grand mean is 4.48 and the \u03b1 values would be the differences between 4.48 and each value, that is, 5.13 (caffeine), 4.25 (decaff) and 4.06 (none). Each of these differences is squared, the results added and this result for the top of the equation is divided by a, the number of levels of A (three in the caffeine case) multiplied by the mean square for error (MSE) from the ANOVA calculation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Calculating for the caffeine effect we get:<\/p>\n\n\n\n<math display=\"block\" class=\"tml-display\" style=\"display:block math;\">\n  <mrow>\n    <msubsup>\n      <mrow>\n        <mi mathvariant=\"normal\">\u03a6<\/mi>\n      <\/mrow>\n      <mi>A<\/mi>\n      <mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo>\n    <\/msubsup>\n    <mo>=<\/mo>\n    <msqrt>\n      <mfrac>\n        <mrow>\n          <mo form=\"prefix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">(<\/mo>\n          <mn>4.48<\/mn>\n          <mo>\u2212<\/mo>\n          <mn>5.125<\/mn>\n          <msup>\n            <mo form=\"postfix\" stretchy=\"false\">)<\/mo>\n            <mn>2<\/mn>\n          <\/msup>\n          <mo>+<\/mo>\n          <mo form=\"prefix\" stretchy=\"false\">(<\/mo>\n          <mn>4.48<\/mn>\n          <mo>\u2212<\/mo>\n          <mn>4.25<\/mn>\n          <msup>\n            <mo form=\"postfix\" stretchy=\"false\">)<\/mo>\n            <mn>2<\/mn>\n          <\/msup>\n          <mo>+<\/mo>\n          <mo form=\"prefix\" stretchy=\"false\">(<\/mo>\n          <mn>4.48<\/mn>\n          <mo>\u2212<\/mo>\n          <mn>4.0625<\/mn>\n          <msup>\n            <mo form=\"postfix\" stretchy=\"false\">)<\/mo>\n            <mn>2<\/mn>\n          <\/msup>\n        <\/mrow>\n        <mrow>\n          <mn>3<\/mn>\n          <mo>\u00d7<\/mo>\n          <mn>2.074<\/mn>\n        <\/mrow>\n      <\/mfrac>\n    <\/msqrt>\n  <\/mrow>\n<\/math>\n\n\n\n<p class=\"wp-block-paragraph\">= 0.321<\/p>\n\n\n\nOur effect size, also referred to (by Cohen, 1988) as f, is 0.321. If we want to consult tables then we now need <math display=\"block\" class=\"tml-display\" style=\"display:block math;\">\n  <mrow>\n    <msup>\n      <mrow>\n        <mi mathvariant=\"normal\">\u03a6<\/mi>\n      <\/mrow>\n      <mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo>\n    <\/msup>\n    <msub>\n      <mrow><\/mrow>\n      <mrow>\n        <mi mathvariant=\"normal\">A<\/mi>\n      <\/mrow>\n    <\/msub>\n    <msqrt>\n      <mi>n<\/mi>\n    <\/msqrt>\n  <\/mrow>\n<\/math> which =   in general terms but with a factorial ANOVA we substitute n&#8217; for n where , <math display=\"block\" class=\"tml-display\" style=\"display:block math;\">\n  <mfrac>\n    <mrow>\n      <mi>d<\/mi>\n      <msub>\n        <mi>f<\/mi>\n        <mtext>error\u00a0<\/mtext>\n      <\/msub>\n    <\/mrow>\n    <mrow>\n      <mi>d<\/mi>\n      <msub>\n        <mi>f<\/mi>\n        <mtext>effect\u00a0<\/mtext>\n      <\/msub>\n      <mo>+<\/mo>\n      <mn>1<\/mn>\n    <\/mrow>\n  <\/mfrac>\n<\/math> . Our df are broken down like this:\n\n\n\n<p class=\"wp-block-paragraph\">Total df = N \u2013 1 = 47<br>Main effect (caffeine) df = 2<br>Main effect (sleep) df = 1<br>Interaction df = 2 x 1 = 2<br>Error df = 47 \u2013 2 \u2013 1 \u2013 2 = 42<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence<\/p>\n\n\n\n<math display=\"block\" class=\"tml-display\" style=\"display:block math;\">\n  <mrow>\n    <msup>\n      <mi>n<\/mi>\n      <mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo>\n    <\/msup>\n    <mo>=<\/mo>\n    <mfrac>\n      <mn>42<\/mn>\n      <mrow>\n        <mn>2<\/mn>\n        <mo>+<\/mo>\n        <mn>1<\/mn>\n      <\/mrow>\n    <\/mfrac>\n    <mo>+<\/mo>\n    <mn>1<\/mn>\n    <mo>=<\/mo>\n    <mn>15<\/mn>\n  <\/mrow>\n<\/math>\n\n\n\n<p class=\"wp-block-paragraph\">and = 0.321\u221a15 = 1.24<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We go to appendix Table 13 with \u03a6 = 1.24, df1 = 2 and dfe = 42 and \u03b1 = 0.05. With a bit of extrapolation, we find that power is around 0.48 (don&#8217;t forget that power = 1 \u2013 \u03b2). This is close enough to SPSS and G*Power which both agree on 0.471.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">As explained in the book SPSS provides effect size and power if you select this before your analysis and in G*Power select F tests, ANOVA: Fixed effects, special, main effects and interactions and Post hoc: Compute achieved power &#8211; given \u03b1, sample size and effect size. The values to enter are 0.321 for effect size, .05 for \u03b1 err probability, 48 for total sample size, 2 for numerator df and 3 for number of groups. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Interaction effects<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For the interaction things are a bit tricky. The formula is<\/p>\n\n\n\n<math display=\"block\" class=\"tml-display\" style=\"display:block math;\">\n  <mrow>\n    <msubsup>\n      <mrow>\n        <mi mathvariant=\"normal\">\u03a6<\/mi>\n      <\/mrow>\n      <mrow>\n        <mtext><\/mtext>\n        <mi>AB<\/mi>\n      <\/mrow>\n      <mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo>\n    <\/msubsup>\n    <mo>=<\/mo>\n    <msqrt>\n      <mfrac>\n        <mrow>\n          <mo stretchy=\"true\" lspace=\"0em\" rspace=\"0em\">\u2211<\/mo>\n          <mrow>\n            <mo fence=\"true\" form=\"prefix\">(<\/mo>\n            <msub>\n              <mi>\u03b1<\/mi>\n              <mi>i<\/mi>\n            <\/msub>\n            <msubsup>\n              <mi>\u03b2<\/mi>\n              <mi>j<\/mi>\n              <mn>2<\/mn>\n            <\/msubsup>\n            <mo fence=\"true\" form=\"postfix\">)<\/mo>\n          <\/mrow>\n        <\/mrow>\n        <mrow>\n          <mi>a<\/mi>\n          <mi>b<\/mi>\n          <msubsup>\n            <mi>\u03c3<\/mi>\n            <mi>e<\/mi>\n            <mn>2<\/mn>\n          <\/msubsup>\n        <\/mrow>\n      <\/mfrac>\n    <\/msqrt>\n  <\/mrow>\n<\/math>\n\n\n\n<p class=\"wp-block-paragraph\">The calculation of is carried out for each individual cell of the data table. The calculation is:<\/p>\n\n\n\n<math display=\"block\" class=\"tml-display\" style=\"display:block math\"><mrow><msub><mover><mi>x<\/mi><mo stretchy=\"false\" class=\"tml-xshift\" style=\"math-style:normal;math-depth:0\">\u203e<\/mo><\/mover><mrow><mi>a<\/mi><mn>1<\/mn><mi>b<\/mi><mn>1<\/mn><\/mrow><\/msub><mo>\u2212<\/mo><msub><mover><mi>x<\/mi><mo stretchy=\"false\" class=\"tml-xshift\" style=\"math-style:normal;math-depth:0\">\u203e<\/mo><\/mover><mrow><mi>a<\/mi><mn>1<\/mn><\/mrow><\/msub><mo>\u2212<\/mo><msub><mover><mi>x<\/mi><mo stretchy=\"false\" class=\"tml-xshift\" style=\"math-style:normal;math-depth:0\">\u203e<\/mo><\/mover><mrow><mi>b<\/mi><mn>1<\/mn><\/mrow><\/msub><mo>+<\/mo><msub><mover><mi>x<\/mi><mo stretchy=\"false\" class=\"tml-xshift\" style=\"math-style:normal;math-depth:0\">\u203e<\/mo><\/mover><mi>g<\/mi><\/msub><\/mrow><\/math>where<math display=\"block\" class=\"tml-display\" style=\"display:block math\"><msub><mover><mi>x<\/mi><mo stretchy=\"false\" class=\"tml-xshift\" style=\"math-style:normal;math-depth:0\">\u203e<\/mo><\/mover><mrow><mi>a<\/mi><mn>1<\/mn><mi>b<\/mi><mn>1<\/mn><\/mrow><\/msub><\/math>is the cell mean (5 hours\/caffeine if working from the top left in table 1 above),<math display=\"block\" class=\"tml-display\" style=\"display:block math\"><msub><mover><mi>x<\/mi><mo stretchy=\"false\" class=\"tml-xshift\" style=\"math-style:normal;math-depth:0\">\u203e<\/mo><\/mover><mrow><mi>a<\/mi><mn>1<\/mn><\/mrow><\/msub><\/math>is the mean of the caffeine condition which that cell is in (caffeine and 5.125 in this case),<math display=\"block\" class=\"tml-display\" style=\"display:block math\"><msub><mover><mi>x<\/mi><mo stretchy=\"false\" class=\"tml-xshift\" style=\"math-style:normal;math-depth:0\">\u203e<\/mo><\/mover><mrow><mi>b<\/mi><mn>1<\/mn><\/mrow><\/msub><\/math>is the mean for the sleep condition of that cell (5 hours and 4.7917) and finally<math display=\"block\" class=\"tml-display\" style=\"display:block math\"><msub><mover><mi>x<\/mi><mo stretchy=\"false\" class=\"tml-xshift\" style=\"math-style:normal;math-depth:0\">\u203e<\/mo><\/mover><mi>g<\/mi><\/msub><\/math>is the grand mean. Each of these six values (one for each of the cells in the table) is squared and the results are added together. This sum is divided by n&#8217; times the MSE where n&#8217; is calculated as above.\n\n\n\n<p class=\"wp-block-paragraph\">Put perhaps more simply for each cell you subtract from the cell mean the mean of the row it is in and the mean of the column it is in and then add the grand mean. Square the result and divide by n&#8217; times the MSE. Let&#8217;s do this now. The calculation may look horrific but if you stick to the rule just stated you should be able to follow each step:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">1. top line of the fraction inside the square root sign is:<br>(6.25 \u2013 5.125 \u2013 4.7917 + 4.48)2 + (4 \u2013 5.125 \u2013 4.1667 + 4.48)2 + (4.125 \u2013 4.25 \u2013 4.7917 + 4.48)2 +<br>(4.375 \u2013 4.25 \u2013 4.1667 + 4.48)2 + (4 \u2013 4.0625 \u2013 4.7917 + 4.48)2 + (4.125 \u2013 4.0625 \u2013 4.1667 + 4.48)2<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Which comes to 1.983!<\/p>\n\n\n\n<p>Dividing this by<math display=\"block\" class=\"tml-display\" style=\"display:block math\"><mrow><mi>a<\/mi><mi>b<\/mi><msubsup><mi>\u03c3<\/mi><mi>e<\/mi><mn>2<\/mn><\/msubsup><\/mrow><\/math>we get: 1.983\/(2 x 3 x 2.074) = 0.159<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The square root of 0.159 is 0.399. This is our value for \u03a6&#8217; (or f)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We need to calculate n&#8217; and we use<\/p>\n\n\n\n<math display=\"block\" class=\"tml-display\" style=\"display:block math;\">\n  <mrow>\n    <msup>\n      <mi>n<\/mi>\n      <mo lspace=\"0em\" rspace=\"0em\" class=\"tml-prime\">\u2032<\/mo>\n    <\/msup>\n    <mo>=<\/mo>\n    <mfrac>\n      <mrow>\n        <mi>d<\/mi>\n        <msub>\n          <mi>f<\/mi>\n          <mrow>\n            <mi>e<\/mi>\n            <mi>r<\/mi>\n            <mi>r<\/mi>\n            <mi>o<\/mi>\n            <mi>r<\/mi>\n          <\/mrow>\n        <\/msub>\n      <\/mrow>\n      <mrow>\n        <mi>d<\/mi>\n        <msub>\n          <mi>f<\/mi>\n          <mrow>\n            <mi>e<\/mi>\n            <mi>f<\/mi>\n            <mi>f<\/mi>\n            <mi>e<\/mi>\n            <mi>c<\/mi>\n            <mi>t<\/mi>\n          <\/mrow>\n        <\/msub>\n        <mo>+<\/mo>\n        <mn>1<\/mn>\n      <\/mrow>\n    <\/mfrac>\n    <mo>=<\/mo>\n    <mn>1<\/mn>\n  <\/mrow>\n<\/math>\n\n\n\n<p class=\"wp-block-paragraph\">as above but here is for the interaction and hence dfcaffeine x dfsleep which is 2 x 1 = 2. Hence n&#8217; will again be 15.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u03a6 then is \u03a6&#8217; x \u221an&#8217; = 0.399 x \u221a15 = 1.54 Going to appendix Table 13 with \u03a6 = 1.54, df for the interaction being 2, dferror = 42 and \u03b1 = 0.05 we need (again) to do some extrapolation between the table values but I get a table value of around 0.35 and, remembering that power = 1 \u2013 \u03b2 this gives power at 0.65 which is pretty close to the SPSS and G*Power values of 0.662. To use G*Power proceed exactly as given above but for the number of groups enter 6.<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:25%\">\n<div class=\"wp-block-group is-vertical is-nowrap is-layout-flex wp-container-core-group-is-layout-b3ff8204 wp-block-group-is-layout-flex wp-container-1 is-position-sticky\">\n<p class=\"wp-block-paragraph\" id=\"on-this-page\"><strong>On this page<\/strong><\/p>\n\n\n\n<nav class=\"wp-block-table-of-contents\"><ol><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/researchmethods\/student-resources\/chapter-23-multi-factorial-anova-designs\/#aioseo-exercises\">Exercises<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/researchmethods\/student-resources\/chapter-23-multi-factorial-anova-designs\/#aioseo-weblinks\">Weblinks<\/a><\/li><li><a class=\"wp-block-table-of-contents__entry\" href=\"https:\/\/routledgelearning.com\/researchmethods\/student-resources\/chapter-23-multi-factorial-anova-designs\/#aioseo-further-information\">Further Information<\/a><\/li><\/ol><\/nav>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>This chapter deals with between groups multi-factorial ANOVA, where more than one independent variable is manipulated or observed.&nbsp; Exercises Exercise 23.1 Calculating two way unrelated ANOVA on a new data set The data set used to calculate the example of a two-way unrelated ANOVA in this chapter is provided below and is named two way [&hellip;]<\/p>\n","protected":false},"featured_media":0,"parent":121,"menu_order":0,"template":"single-chapter","format":"standard","meta":{"_acf_changed":false},"categories":[],"class_list":["post-115","content","type-content","status-publish","format-standard","hentry"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO Pro 5.0.0.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"This chapter deals with between groups multi-factorial ANOVA, where more than one independent variable is manipulated or observed. 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