// Numbas version: exam_results_page_options
{"name": "Calculating the Spearman rank correlation coefficient and p-value", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "<p>A researcher has developed a questionnaire, which has been completed by 8 married couples (A-H), to find out how interest in sports plays a role in interpersonal attraction. The questions sought to place each partner&nbsp;in the marriage on a 20 point scale in which low scores represent little interest in sports, and high scores represent very interested in sports.</p>\n<p>The data were:</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<th align=\"left\">Couple</th>\n<th>$\\var{obj[0]}$</th>\n<th>$\\var{obj[1]}$</th>\n<th>$\\var{obj[2]}$</th>\n<th>$\\var{obj[3]}$</th>\n<th>$\\var{obj[4]}$</th>\n<th>$\\var{obj[5]}$</th>\n<th>$\\var{obj[6]}$</th>\n<th>$\\var{obj[7]}$</th>\n</tr>\n<tr>\n<th align=\"left\">Partner $X$</th>\n<td>$\\var{r1[0]}$</td>\n<td>$\\var{r1[1]}$</td>\n<td>$\\var{r1[2]}$</td>\n<td>$\\var{r1[3]}$</td>\n<td>$\\var{r1[4]}$</td>\n<td>$\\var{r1[5]}$</td>\n<td>$\\var{r1[6]}$</td>\n<td>$\\var{r1[7]}$</td>\n</tr>\n<tr>\n<th align=\"left\">Partner $Y$</th>\n<td>$\\var{r2[0]}$</td>\n<td>$\\var{r2[1]}$</td>\n<td>$\\var{r2[2]}$</td>\n<td>$\\var{r2[3]}$</td>\n<td>$\\var{r2[4]}$</td>\n<td>$\\var{r2[5]}$</td>\n<td>$\\var{r2[6]}$</td>\n<td>$\\var{r2[7]}$</td>\n</tr>\n</tbody>\n</table>\n<p>In this exercise you will find Spearman's correlation coefficient for this data and test it's significance in order to test the following null hypothesis:</p>\n<p>\\[ H_0: \\text{There is no association between the attitudes of wives and husbands.} \\]</p>", "variables": {"n": {"definition": "8", "templateType": "anything", "group": "Data given in question", "name": "n", "description": ""}, "one": {"definition": "0.881", "templateType": "anything", "group": "Critical values of the spearman correlation coefficient", "name": "one", "description": ""}, "ss": {"definition": "[ssq[0]-t[0]^2/n,ssq[1]-t[1]^2/n]", "templateType": "anything", "group": "Ungrouped variables", "name": "ss", "description": ""}, "m": {"definition": "20", "templateType": 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"<p>Husband's data</p>"}, "five": {"definition": "0.738", "templateType": "anything", "group": "Critical values of the spearman correlation coefficient", "name": "five", "description": ""}, "spcoef": {"definition": "precround(1-6*ssd/(n*(n^2-1)),3)", "templateType": "anything", "group": "Spearman correlation coeffcient", "name": "spcoef", "description": "<p>Spearman correlation coefficient</p>"}, "aspcoef": {"definition": "abs(spcoef)", "templateType": "anything", "group": "Spearman correlation coeffcient", "name": "aspcoef", "description": "<p>Absolute value of the spearman correlation coefficient</p>"}, "tsqovern": {"definition": "[t[0]^2/n,t[1]^2/n]", "templateType": "anything", "group": "Ungrouped variables", "name": "tsqovern", "description": ""}, "ssd": {"definition": "sum(map(x^2,x,d))", "templateType": "anything", "group": "Spearman correlation coeffcient", "name": "ssd", "description": "<p>The difference between the wife's and husband's score.</p>"}, "ssq": {"definition": 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"variableReplacementStrategy": "originalfirst", "maxValue": "d[7]", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "correctAnswerStyle": "plain"}], "prompt": "<p>To find the Spearman correlation coefficient for the original score data, you need to&nbsp;rank the data, the lowest rank has rank $1$ and the highest score has rank $8$.</p>\n<p>Also calculate the differences in the ranks, i.e. for each couple&nbsp;calculate</p>\n<p>\\[&nbsp; d_i = (\\text{partner X score}) -&nbsp;(\\text{partner Y score}).\\]</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<th align=\"left\">Couple</th>\n<th>$\\var{obj[0]}$</th>\n<th>$\\var{obj[1]}$</th>\n<th>$\\var{obj[2]}$</th>\n<th>$\\var{obj[3]}$</th>\n<th>$\\var{obj[4]}$</th>\n<th>$\\var{obj[5]}$</th>\n<th>$\\var{obj[6]}$</th>\n<th>$\\var{obj[7]}$</th>\n</tr>\n<tr>\n<th align=\"left\">Partner&nbsp;$X$</th>\n<td>[[0]]</td>\n<td>[[1]]</td>\n<td>[[2]]</td>\n<td>[[3]]</td>\n<td>[[4]]</td>\n<td>[[5]]</td>\n<td>[[6]]</td>\n<td>[[7]]</td>\n</tr>\n<tr>\n<th align=\"left\">Partner&nbsp;$Y$</th>\n<td>[[8]]</td>\n<td>[[9]]</td>\n<td>[[10]]</td>\n<td>[[11]]</td>\n<td>[[12]]</td>\n<td>[[13]]</td>\n<td>[[14]]</td>\n<td>[[15]]</td>\n</tr>\n<tr>\n<th align=\"left\">Differences $(d_i)$</th>\n<td>[[16]]</td>\n<td>[[17]]</td>\n<td>[[18]]</td>\n<td>[[19]]</td>\n<td>[[20]]</td>\n<td>[[21]]</td>\n<td>[[22]]</td>\n<td>[[23]]</td>\n</tr>\n</tbody>\n</table>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true}, {"type": "gapfill", "marks": 0, "showCorrectAnswer": true, "scripts": {}, "unitTests": [], "steps": [{"type": "information", "marks": 0, "showCorrectAnswer": true, "scripts": {}, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "variableReplacements": [], "prompt": "<p>If there are two sets of ranks $x_1,x_2,\\ldots,x_n$ and &nbsp;$y_1,y_2,\\ldots,y_n$ where both sets have no ties, and differences are $d_i=x_i-y_i$ then if $\\sum d_i^2=D$ we have:</p>\n<p>\\[r_s=1 - \\frac{6 \\times D}{n(n^2-1)}\\]</p>", "extendBaseMarkingAlgorithm": true}], "sortAnswers": false, "customMarkingAlgorithm": "", "variableReplacements": [], "stepsPenalty": 0, "prompt": "<p>Hence calculate the Spearman correlation coefficient:</p>\n<p>$r_s=\\;$&nbsp;[[0]]</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"mustBeReduced": false, "allowFractions": false, "showCorrectAnswer": true, "scripts": {}, "correctAnswerStyle": "plain", "precisionMessage": "You have not given your answer to the correct precision.", "customMarkingAlgorithm": "", "precision": "3", "minValue": "spcoef", "showPrecisionHint": true, "precisionPartialCredit": 0, "variableReplacementStrategy": "originalfirst", "strictPrecision": 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"<p>State the degrees of freedom of the significance test.</p>\n<p>[[0]]</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true}, {"type": "gapfill", "marks": 0, "showCorrectAnswer": true, "scripts": {}, "unitTests": [], "sortAnswers": false, "customMarkingAlgorithm": "", "variableReplacements": [], "gaps": [{"type": "numberentry", "mustBeReduced": false, "marks": 0.25, "showCorrectAnswer": true, "scripts": {}, "unitTests": [], "allowFractions": false, "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "minValue": "ten", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "maxValue": "ten", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "correctAnswerStyle": "plain"}, {"type": "numberentry", "mustBeReduced": false, "marks": 0.25, "showCorrectAnswer": true, "scripts": {}, "unitTests": [], "allowFractions": false, "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "minValue": "five", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "maxValue": "five", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "correctAnswerStyle": "plain"}, {"type": "numberentry", "mustBeReduced": false, "marks": 0.25, "showCorrectAnswer": true, "scripts": {}, "unitTests": [], "allowFractions": false, "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "minValue": "one", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "maxValue": "one", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "correctAnswerStyle": "plain"}, {"type": "numberentry", "mustBeReduced": false, "marks": 0.25, "showCorrectAnswer": true, "scripts": {}, "unitTests": [], "allowFractions": false, "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "minValue": "point2", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "maxValue": "point2", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "correctAnswerStyle": "plain"}], "prompt": "<p>Using the rounded value&nbsp;of the Spearman correlation coefficient that you have found, find the the significance level by looking up the critical values in a table.&nbsp;</p>\n<p>Input the&nbsp;table values in the table below:</p>\n<table border=\"1\">\n<tbody>\n<tr align=\"centre\">\n<td style=\"width: 58px;\">$10\\%$</td>\n<td style=\"width: 53px;\">$5\\%$</td>\n<td style=\"width: 53px;\">$1\\%$</td>\n<td style=\"width: 63px;\">$0.2\\%$</td>\n</tr>\n<tr align=\"centre\">\n<td style=\"width: 58px;\">[[0]]</td>\n<td style=\"width: 53px;\">[[1]]</td>\n<td style=\"width: 53px;\">[[2]]</td>\n<td style=\"width: 63px;\">[[3]]</td>\n</tr>\n</tbody>\n</table>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true}, {"type": "1_n_2", "showCellAnswerState": true, "marks": 0, "showCorrectAnswer": true, "scripts": {}, "unitTests": [], "minMarks": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "shuffleChoices": false, "displayColumns": "0", "maxMarks": 0, "prompt": "<p>By comparing the spearman correlation coefficient to the table values found in part&nbsp;<strong>(d)</strong>, choose the option that best describes the outcome of the $p$-value&nbsp;</p>", "showFeedbackIcon": true, "choices": ["$p \\leq 0.002$", "$0.002 \\lt p \\leq 0.01$", "$0.01 \\lt p \\leq 0.05$", "$0.05 \\lt p \\leq 0.1$", "$p \\gt 0.1$"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "extendBaseMarkingAlgorithm": true, "matrix": "vs"}, {"type": "1_n_2", "showCellAnswerState": true, "marks": 0, "showCorrectAnswer": true, "scripts": {}, "unitTests": [], "minMarks": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "shuffleChoices": false, "displayColumns": "1", "maxMarks": 0, "prompt": "<p>What decision can you come to as to the hypothesis that the&nbsp;partners in these married couples have the same&nbsp;interest in sports?</p>", "showFeedbackIcon": true, "choices": ["Very Strong Evidence of an association between $X$ and $Y$", "Strong evidence of an association between $X$ and $Y$", "Evidence of an association between $X$ and $Y$", "Weak evidence of an association between $X$ and $Y$", "No evidence of an association between $X$ and $Y$"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "extendBaseMarkingAlgorithm": true, "matrix": "vs"}], "tags": [], "functions": {"darr": {"type": "list", "parameters": [["n", "number"], ["m", "number"], ["a", "list"]], "definition": "if(n=1,a,darr(n-1,m,[random(1..m except a)]+a))", "language": "jme"}, "pstdev": {"type": "number", "parameters": [["l", "list"]], "definition": "sqrt(abs(l)/(abs(l)-1))*stdev(l)", "language": "jme"}, "rk": {"type": "list", "parameters": [["a", "list"]], "definition": "\n      /*This gives the ranking of the entries in a, c counts the ties */\n        var out = [];\n      for(var j=0;j<a.length;j++){\n        var s=0;\n        var c=0;\n        for(var i=0;i<a.length;i++){\n      if(a[j]>a[i]){s+=1;}\n      else\n        if(a[j]==a[i]){c+=1;}\n        }\n      out[j]=(2*s+c+1)/2;\n      }\n      return out;\n      ", "language": "javascript"}, "marr": {"type": "number", "parameters": [["a", "list"]], "definition": "if(length(a)=2,max(a[0],a[1]),max(a[0],marr(a[1..length(a)])))", "language": "jme"}, "tesarr": {"type": "list", "parameters": [["a", "list"], ["b", "list"], ["d", "number"], ["m", "number"]], "definition": "if(marr(map(abs(x),x,list(vector(a)-vector(b))))<d,b,tesarr(a,darr(length(a),m,[random(1..m)]),d,m))", "language": "jme"}}, "ungrouped_variables": ["ssq", "tsqovern", "ss", "m", "t"], "extensions": ["stats"], "advice": "<h4>(a)</h4>\n<p>To rank the couples scores, we allocate the lowest numerical score a value of $1$ and repeat this for numbers $1..8$.&nbsp;</p>\n<p>To calculate the differences, we subract the partner $Y$'s rank from&nbsp;partner $X$'s, for each couple.</p>\n<h4>(b)</h4>\n<p>To calculate Spearman's correlation coefficient, we use the following the formula:</p>\n<p>\\[r_s=1 - \\frac{6 \\times D}{n(n^2-1)}\\]</p>\n<p>where $ D = \\sum{d_i}^2.$</p>\n<p>Firstly we calculate $D$,</p>\n<p>\\begin{align}<br/>D = \\sum{d_i}^2 &amp;= (\\var{d[0]})^2 +(\\var{d[1]})^2+(\\var{d[2]})^2+(\\var{d[3]})^2+(\\var{d[4]})^2+(\\var{d[5]})^2+(\\var{d[6]})^2+(\\var{d[7]})^2\\\\<br/>&amp; = \\var{ssd}.<br/>\\end{align}</p>\n<p>\\begin{align}<br/>r_s &amp;= 1 - \\frac{ 6 \\times&nbsp;\\var{ssd}}{\\var{n} \\times (\\var{n}^2-1)}\\\\<br/>&amp; = \\var{spcoef}<br/>\\end{align}</p>\n<h4>(c)</h4>\n<p>The sample size is $ n=\\var{n}$, so we look up the critical values with $ n =\\var{n} $ and thus the degrees of freedom are $\\var{n}.$</p>\n<h4>(d)</h4>\n<p>Using the statistical table for the critical values for the spearman correlation coefficient in your notes, with $ n = \\var{n}$, we find the following values:</p>\n<table border=\"1\">\n<tbody>\n<tr align=\"centre\">\n<td style=\"width: 88px; text-align: center;\">$10\\%$</td>\n<td style=\"width: 90px; text-align: center;\">$5\\%$</td>\n<td style=\"width: 91px; text-align: center;\">$1\\%$</td>\n<td style=\"width: 109px; text-align: center;\">$0.2\\%$</td>\n</tr>\n<tr align=\"centre\">\n<td style=\"width: 88px; text-align: center;\">$\\var{ten}$</td>\n<td style=\"width: 90px; text-align: center;\">$\\var{five}$</td>\n<td style=\"width: 91px; text-align: center;\">$\\var{one}$</td>\n<td style=\"width: 109px; text-align: center;\">$\\var{point2}$</td>\n</tr>\n</tbody>\n</table>\n<h4>(e)</h4>\n<p>When $r_s&nbsp;&lt;&nbsp;\\var{ten}$, then $p &gt; 0.1 $.</p>\n<p>When $\\var{ten}&lt; r_s&nbsp;&lt;\\var{five} $, then $0.05&lt; p&nbsp;&lt; 0.1 $.</p>\n<p>When $\\var{five}&lt; r_s&nbsp;&lt;\\var{one} $, then $0.01&lt; p&nbsp;&lt; 0.05$.</p>\n<p>When $\\var{one}&lt; r_s&nbsp;&lt;\\var{point2} $, then $0.002&lt; p&nbsp;&lt; 0.01$.</p>\n<p>When $r_s&nbsp;&gt; \\var{point2}$, then $p&nbsp;&lt; 0.002 $.</p>\n<h4>(f)</h4>\n<p>When $p&gt;0.1$,&nbsp; we have no evidence of an association between $x$ and $y$</p>\n<p>When&nbsp;$0.05&lt; p&nbsp;&lt; 0.1$, we have weak evidence of an association between $x$ and $y$</p>\n<p>When&nbsp;$0.01&lt; p&nbsp;&lt; 0.05$, there is evidence of an association between $x$ and $y$</p>\n<p>When $0.002&lt;p&lt;0.01$, there is strong evidence of an association between $x$ and $y$</p>\n<p>When $p\\le 0.002$, there is very strong evidence of an association between $x$ and $y$</p>\n<p></p>", "variable_groups": [{"variables": ["r1", "r2", "n", "obj"], "name": "Data given in question"}, {"variables": ["spcoef", "aspcoef", "d", "ssd", "vs"], "name": "Spearman correlation coeffcient"}, {"variables": ["ten", "five", "one", "point2"], "name": "Critical values of the spearman correlation coefficient"}, {"variables": ["rr1", "rr2"], "name": "Ranking the data"}], "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "<p>Spearman rank correlation calculated. 8 paired observations.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"css": "", "js": ""}, "type": "question", "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Lauren Frances Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2490/"}]}]}], "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Lauren Frances Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2490/"}]}