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is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">cvar(A,B)</p></text><indexes><phrase>cvar</phrase><phrase>functions, cvar</phrase></indexes></region><region region-id="1419" left="162" top="97.5" width="43.5" height="12.75" align-x="162" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">corr(A,B)</p></text><indexes><phrase>corr</phrase><phrase>functions, corr</phrase></indexes></region><region region-id="796" left="66" top="115.5" width="57.75" height="12.75" align-x="76.5" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Spear(v1,v2)</p></text><indexes><phrase>Spear</phrase><phrase>functions, Spear</phrase></indexes></region><region region-id="1420" left="162" top="115.5" width="72.75" height="12.75" align-x="172.5" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">kendltau(v1,v2)</p></text><indexes><phrase>kendltau</phrase><phrase>functions, kendltau</phrase></indexes></region><region region-id="802" left="66" top="133.5" width="64.5" height="12.75" align-x="76.5" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">contingtb1(M)</p></text><indexes><phrase>contingtb1</phrase><phrase>functions, contingtb1</phrase></indexes></region><region region-id="1421" left="162" top="133.5" width="63" height="12.75" align-x="172.5" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">kendltau2(M)</p></text><indexes><phrase>kendltau2</phrase><phrase>functions, kendltau2</phrase></indexes></region><region region-id="437" left="36" top="158.25" width="318.75" height="120" align-x="36" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The first four of these functions take two arrays of the same length, or in some cases two matrices of the same size, and return the correlation, Pearson's r correlation coefficient, Spearman coefficient, Kendall's tau, or a contingency table. The last two functions take a contingency table matrix and calculate statistics of merit. These quantities give a measure of the association between two data sets, that is, whether one depends on the other. They are measures of whether an independent variable is a significant predictor of a dependent variable, or unrelated entirely. </p></text></region><region region-id="1398" left="36" top="295.5" width="51.75" height="12.75" align-x="36" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Covariance</p></text><indexes><phrase>covariance</phrase></indexes></region><region region-id="1401" left="36" top="320.25" width="300.75" height="12" align-x="43.5" align-y="330" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Consider the linearly related voltage data shown:</p></text></region><region region-id="1402" left="66" top="348.75" width="114" height="161.25" align-x="99.75" align-y="432" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">DATA</ml:id><ml:matrix rows="10" cols="3"><ml:real>1</ml:real><ml:real>2</ml:real><ml:real>3</ml:real><ml:real>4</ml:real><ml:real>5</ml:real><ml:real>6</ml:real><ml:real>7</ml:real><ml:real>8</ml:real><ml:real>9</ml:real><ml:real>10</ml:real><ml:real>0.4</ml:real><ml:real>0.7</ml:real><ml:real>1.075</ml:real><ml:real>1.4</ml:real><ml:real>1.6</ml:real><ml:real>2</ml:real><ml:real>2.4</ml:real><ml:real>2.7</ml:real><ml:real>3.05</ml:real><ml:real>3.4</ml:real><ml:real>0.4</ml:real><ml:real>0.72</ml:real><ml:real>1.1</ml:real><ml:real>1.45</ml:real><ml:real>1.8</ml:real><ml:real>2.17</ml:real><ml:real>2.4</ml:real><ml:real>2.8</ml:real><ml:real>3.2</ml:real><ml:real>3.4</ml:real></ml:matrix></ml:define></math><rendering item-idref="4" /></region><region region-id="829" left="36" top="524.25" width="303.75" height="60" align-x="36" align-y="534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Covariance is the degree to which two variables, x and y<f family="Times New Roman" charset="0"><i><inlineAttr line-through="false"><c val="#000"><sp /></c></inlineAttr></i></f><f family="Verdana" charset="0"><i><inlineAttr line-through="false"><c val="#000">covary</c></inlineAttr></i></f>. Just as <b><u><link href="./dispersion.xmcd#variance" title="Links to section" popup="false">variance</link></u></b> measures how much data deviates from its mean, the covariance measures how much two data sets simultaneously deviate from their respective means.</p></text></region><region region-id="1404" left="66" top="591.75" width="154.5" height="18" align-x="178.5" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">cvar</ml:id><ml:sequence><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>1</ml:real></ml:apply></ml:sequence></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>2.7487500000000002</ml:real></result></ml:eval></math><rendering item-idref="5" /></region><region region-id="1403" left="0" top="618" width="6000" height="6" align-x="0" align-y="618" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1106" left="36" top="626.25" width="309.75" height="24" align-x="36" align-y="636" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">For a least-squares fit line, the covariance is related to the slope of the fit line:</p></text></region><region region-id="1111" left="66" top="657.75" width="163.5" height="18" align-x="183" align-y="672" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">slope</ml:id><ml:sequence><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>1</ml:real></ml:apply></ml:sequence></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.33318181818181819</ml:real></result></ml:eval></math><rendering item-idref="6" /></region><region region-id="1112" left="66" top="687" width="162" height="38.25" align-x="181.5" align-y="708" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:div /><ml:apply><ml:id xml:space="preserve">cvar</ml:id><ml:sequence><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>1</ml:real></ml:apply></ml:sequence></ml:apply><ml:apply><ml:id xml:space="preserve">var</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>0</ml:real></ml:apply></ml:apply></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.33318181818181819</ml:real></result></ml:eval></math><rendering item-idref="7" /></region><region region-id="839" left="36" top="739.5" width="148.5" height="12.75" align-x="36" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Pearson's correlation coefficient</p></text><indexes><phrase>Pearson's correlation coefficient</phrase></indexes></region><region region-id="840" left="36" top="764.25" width="304.5" height="48" align-x="36" align-y="774" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Pearson's correlation coefficient is calculated by <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">corr</c></inlineAttr></b></f>. 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xml:space="preserve">y</ml:id></ml:sequence></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>-1</ml:real></result></ml:eval></math><rendering item-idref="13" /></region><region region-id="1353" left="66" top="897" width="60.75" height="12.75" align-x="109.5" align-y="906" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">cvar</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">y</ml:id></ml:sequence></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>4</ml:real></result></ml:eval></math><rendering item-idref="14" /></region><region region-id="1428" left="144" top="890.25" width="97.5" height="27.75" align-x="224.25" align-y="906" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:div /><ml:apply><ml:id xml:space="preserve">cvar</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">y</ml:id></ml:sequence></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:id xml:space="preserve">stdev</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">stdev</ml:id><ml:id xml:space="preserve">y</ml:id></ml:apply></ml:apply></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.99999999999999978</ml:real></result></ml:eval></math><rendering item-idref="15" /></region><region region-id="1092" left="36" top="926.25" width="231.75" height="12" align-x="36" align-y="936" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The internal calculation represented by <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">corr</c></inlineAttr></b></f> is</p></text></region><region region-id="1093" left="66" top="949.5" width="209.25" height="43.5" align-x="258" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:mult style="default" /><ml:apply><ml:div /><ml:real>1</ml:real><ml:id xml:space="preserve">N</ml:id></ml:apply><ml:parens><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">i</ml:id></ml:boundVars><ml:parens><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:div /><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply><ml:apply><ml:id xml:space="preserve">stdev</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:parens><ml:parens><ml:apply><ml:div /><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id xml:space="preserve">y</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:id xml:space="preserve">y</ml:id></ml:apply></ml:apply><ml:apply><ml:id xml:space="preserve">stdev</ml:id><ml:id xml:space="preserve">y</ml:id></ml:apply></ml:apply></ml:parens></ml:apply></ml:parens></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:apply><ml:minus /><ml:id xml:space="preserve">N</ml:id><ml:real>1</ml:real></ml:apply></ml:bounds></ml:apply></ml:parens></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.99999999999999989</ml:real></result></ml:eval></math><rendering item-idref="16" /></region><region region-id="1094" left="36" top="1004.25" width="308.25" height="72" align-x="36" align-y="1014" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Pearson's coefficient indicates the direction of the correlation (whether the relationship is direct or inversely proportional), while the coefficient of determination gives an equal interval and ratio measure of the <inlineAttr font-style="normal">strength</inlineAttr> of the correlation. You can get the coefficient of determination by squaring the output of <b>corr</b>. For our sample data,</p></text></region><region region-id="1095" left="66" top="1089.75" width="122.25" height="18" align-x="75" align-y="1104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">r</ml:id><ml:apply><ml:id xml:space="preserve">corr</ml:id><ml:sequence><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>1</ml:real></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="17" /></region><region region-id="1349" left="66" top="1119" width="54.75" height="12.75" align-x="74.25" align-y="1128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">r</ml:id><result 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show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Spearman rank correlation</p></text><indexes><phrase>Spearman rank correlation</phrase></indexes></region><region region-id="1117" left="36" top="1178.25" width="296.25" height="36" align-x="36" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The Spearman correlation uses the same formula as Pearson's but applies it instead to the <b><u><link href="./sorting.xmcd#rank" title="Links to section" popup="false">ranks</link></u></b> of the data in each data set</p></text></region><region region-id="1160" left="66" top="1221" width="54.75" height="12.75" align-x="81.75" align-y="1230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Rx</ml:id><ml:apply><ml:id xml:space="preserve">Rank</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:define></math><rendering item-idref="20" /></region><region region-id="1484" left="144" top="1221" width="57.75" height="12.75" align-x="161.25" align-y="1230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Ry</ml:id><ml:apply><ml:id xml:space="preserve">Rank</ml:id><ml:id xml:space="preserve">y</ml:id></ml:apply></ml:define></math><rendering item-idref="21" /></region><region region-id="1168" left="66" top="1245" width="69.75" height="16.5" align-x="79.5" align-y="1254" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve" subscript="s">r</ml:id><ml:apply><ml:id xml:space="preserve">corr</ml:id><ml:sequence><ml:id xml:space="preserve">Rx</ml:id><ml:id xml:space="preserve">Ry</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="22" /></region><region region-id="1485" left="144" top="1245" width="30" height="16.5" align-x="156.75" align-y="1254" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve" subscript="s">r</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>1</ml:real></result></ml:eval></math><rendering item-idref="23" /></region><region region-id="1172" left="66" top="1269.75" width="108" height="18" align-x="99.75" align-y="1284" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Rdata0</ml:id><ml:apply><ml:id xml:space="preserve">Rank</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>0</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="24" /></region><region region-id="1459" left="222" top="1269.75" width="108" height="18" align-x="255.75" align-y="1284" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Rdata1</ml:id><ml:apply><ml:id xml:space="preserve">Rank</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="25" /></region><region region-id="1176" left="36" top="1304.25" width="39" height="12" align-x="36" align-y="1314" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Pearson</p></text></region><region region-id="1497" left="144" top="1299.75" width="157.5" height="18" align-x="255" align-y="1314" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">corr</ml:id><ml:sequence><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>1</ml:real></ml:apply></ml:sequence></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.99906080055306745</ml:real></result></ml:eval></math><rendering item-idref="26" /></region><region region-id="1178" left="36" top="1328.25" width="48.75" height="12" align-x="36" align-y="1338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Spearman</p></text></region><region region-id="1498" left="144" top="1329" width="105.75" height="12.75" align-x="232.5" align-y="1338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">corr</ml:id><ml:sequence><ml:id xml:space="preserve">Rdata0</ml:id><ml:id xml:space="preserve">Rdata1</ml:id></ml:sequence></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>1</ml:real></result></ml:eval></math><rendering item-idref="27" /></region><region region-id="1182" left="36" top="1352.25" width="289.5" height="108" align-x="36" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Since it is based on the Pearson formula, the Spearman coefficient has the same properties, namely, it is always between -1 and +1. Unlike the Pearson, however, the Spearman coefficient can be +1 or -1 without the data lying on a straight line, as is the case for our test voltage data. Spearman's rank coefficient is used in nonparametric hypothesis testing, that is, correlation testing that does not depend on the distribution or form of the data.</p></text></region><region region-id="1203" left="36" top="1472.25" width="276" height="24" align-x="36" align-y="1482" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">Spear</c></inlineAttr></b></f> function returns not only this correlation coefficient, but other figures of merit:</p></text><indexes><phrase>Spear</phrase></indexes></region><region region-id="1190" left="42" top="1514.25" width="304.5" height="99" align-x="42" align-y="1524" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="inherit" list-style-type="disc" tabs="inherit">the sum squared difference of ranks, D</p><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="disc" tabs="inherit">the number of standard deviations by which D deviates from the expected value under the null hypothesis</p><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="disc" tabs="inherit">the probability that this value or one larger in absolute value would occur for uncorrelated samples</p><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="inherit" list-style-type="disc" tabs="inherit">Spearman's rank-order correlation coefficient r<f family="Verdana" charset="0"><inlineAttr line-through="false"><c val="#000"><sub>s</sub></c></inlineAttr></f></p><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="disc" tabs="inherit">the probability that this value or one larger in absolute value would occur for uncorrelated samples </p></text></region><region region-id="1407" left="0" top="1620" width="6000" height="6" align-x="0" align-y="1620" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1408" left="36" top="1628.25" width="309.75" height="72" align-x="36" align-y="1638" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Compare the response of a group of patients at a clinic who are given different levels of a medication designed to reduce blood cholesterol. Record the value +1 to indicate lowered cholesterol, 0 to indicate no change, and -1 to indicate higher cholesterol. The medication will be recorded in units administered.</p></text></region><region region-id="1409" left="66" top="1713" width="108.75" height="115.5" align-x="96" align-y="1722" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><component hide-arguments="false" clsid-buddy="a6cfc4d6-3a88-437d-94de-30d1a591cf29" item-idref="28" disable-calc="false"><inputs /><outputs><ml:id xml:space="preserve">Meds</ml:id></outputs></component><rendering item-idref="29" /></region><region region-id="1410" left="204" top="1713" width="105.75" height="115.5" align-x="231" align-y="1722" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><component hide-arguments="false" clsid-buddy="a6cfc4d6-3a88-437d-94de-30d1a591cf29" item-idref="30" disable-calc="false"><inputs /><outputs><ml:id xml:space="preserve">Resp</ml:id></outputs></component><rendering item-idref="31" /></region><region region-id="1411" left="66" top="1842" width="149.25" height="78.75" align-x="149.25" align-y="1884" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">Spear</ml:id><ml:sequence><ml:id xml:space="preserve">Meds</ml:id><ml:id xml:space="preserve">Resp</ml:id></ml:sequence></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="5" cols="1"><ml:real>388</ml:real><ml:real>-0.095127933788104288</ml:real><ml:real>0.92421315282397742</ml:real><ml:real>0.026383741769381061</ml:real><ml:real>0.92866117854999741</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="32" /></region><region region-id="1278" left="36" top="1934.25" width="332.25" height="48" align-x="36" align-y="1944" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The rank-order correlation coefficient is .0264, close to zero, indicating that there is very little evidence for an association between the amount of medication given and the response of the patient.</p></text></region><region region-id="1488" left="174" top="1994.25" width="118.5" height="84" align-x="174" align-y="2004" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here's an example of the correlation of the linear data vector with itself. The correlation is 1, with a small probability (0.0455) of uncorrelated samples.</p></text></region><region region-id="1279" left="66" top="1998" width="99" height="78.75" align-x="113.25" align-y="2040" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">Spear</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">x</ml:id></ml:sequence></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="5" cols="1"><ml:real>0</ml:real><ml:real>-2</ml:real><ml:real>0.0455002655262351</ml:real><ml:real>1</ml:real><ml:real>0</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="33" /></region><region region-id="1412" left="0" top="2094" width="6000" height="6" align-x="0" align-y="2094" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1337" left="36" top="2107.5" width="60.75" height="12.75" align-x="36" align-y="2118" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Kendall's tau</p></text><indexes><phrase>Kendall's tau</phrase></indexes></region><region region-id="1340" left="36" top="2132.25" width="299.25" height="108" align-x="36" align-y="2142" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Kendall's <f family="Symbol" charset="2" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">t</c></inlineAttr></f> uses the relative ranks of the data points in its measure of correlation. Each pair (xi,yi) is compared with each other pair. It then totals the number of comparisons in which the relative ordering of the x's is the same as the relative ordering of the y's (concordant pairs), the number in which the relative orderings are different (discordant pairs), the number in which the x's are the same (extra y pairs), and the number in which the y's are the same (extra x pairs).</p></text></region><region region-id="1342" left="36" top="2264.25" width="318" height="29.25" align-x="45.75" align-y="2280" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:equal /><ml:id xml:space="preserve">τ</ml:id><ml:apply><ml:div /><ml:apply><ml:minus /><ml:id xml:space="preserve">concordant</ml:id><ml:id xml:space="preserve">discordant</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:sqrt /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:id xml:space="preserve">concordant</ml:id><ml:id xml:space="preserve">discordant</ml:id></ml:apply><ml:id xml:space="preserve">extra_y</ml:id></ml:apply></ml:apply><ml:apply><ml:sqrt /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:id xml:space="preserve">concordant</ml:id><ml:id xml:space="preserve">discordant</ml:id></ml:apply><ml:id xml:space="preserve">extra_x</ml:id></ml:apply></ml:apply></ml:apply></ml:apply></ml:apply></math><rendering item-idref="34" /></region><region region-id="1344" left="36" top="2318.25" width="315.75" height="60" align-x="36" align-y="2328" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">In both of these tests, the last entry in the vector gives the probability that a statistic larger in absolute value would occur for uncorrelated samples. This is based on the assumption that the test statistics are approximately normally distributed when the data are uncorrelated.</p></text></region><region region-id="1217" left="36" top="2384.25" width="315" height="60" align-x="36" align-y="2394" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">kendltau</c></inlineAttr></b></f> function yields a vector of length 3 containing: </p><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="disc" tabs="inherit">Kendall's <f family="Symbol" charset="2">t,</f></p><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="disc" tabs="inherit">its number of standard deviations away from zero</p><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="disc" tabs="inherit">the probability that a value this large or larger in absolute value would occur if the samples were uncorrelated<sp count="2" /></p></text><indexes><phrase>kendltau</phrase></indexes></region><region region-id="1287" left="36" top="2456.25" width="318" height="24" align-x="36" align-y="2466" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">We could use the dose/result data, which compares only the relative ordering of ranks to give a measure of correlation.</p></text></region><region region-id="1346" left="66" top="2494.5" width="156" height="45.75" align-x="161.25" align-y="2520" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">kendltau</ml:id><ml:sequence><ml:id xml:space="preserve">Meds</ml:id><ml:id xml:space="preserve">Resp</ml:id></ml:sequence></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="3" cols="1"><ml:real>0.030065841120113151</ml:real><ml:real>0.14978155640099505</ml:real><ml:real>0.88093695915056536</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="35" /></region><region region-id="1289" left="36" top="2558.25" width="327" height="24" align-x="36" align-y="2568" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The correlation measure here is .0301, again providing no support for a relationship between medication and response. </p></text></region><region region-id="1377" left="0" top="2592" width="6000" height="6" align-x="0" align-y="2592" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1366" left="36" top="2605.5" width="90" height="12.75" align-x="61.5" align-y="2616" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Contingency Tables</p></text><indexes><phrase>contingency tables</phrase><phrase>tables, contingency</phrase></indexes></region><region region-id="1309" left="36" top="2630.25" width="334.5" height="60" align-x="36" align-y="2640" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">When there are only a few possible values for each variable, the data may be recorded as a contingency table, with the frequency of responses as entries. For example, if the medication in our example were administered in only 4 doses, we could construct a 3 by 4 contingency table.</p></text></region><region region-id="1494" left="36" top="2702.25" width="119.25" height="12" align-x="36" align-y="2712" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"><b><c val="#800000">Change in Cholesterol</c></b></p></text></region><region region-id="1330" left="108" top="2720.25" width="91.5" height="24" align-x="108" align-y="2730" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"><b><c val="#800000">Medication Level</c></b></p><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"><b>0 150 250 500</b></p></text></region><region region-id="1331" left="168" top="2756.25" width="51" height="12" align-x="168" align-y="2766" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"><b><c val="#800000">Decrease</c></b></p></text></region><region region-id="1332" left="66" top="2758.5" width="97.5" height="45.75" align-x="94.5" align-y="2784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Table</ml:id><ml:matrix rows="3" cols="4"><ml:real>0</ml:real><ml:real>2</ml:real><ml:real>3</ml:real><ml:real>2</ml:real><ml:real>5</ml:real><ml:real>2</ml:real><ml:real>13</ml:real><ml:real>7</ml:real><ml:real>1</ml:real><ml:real>4</ml:real><ml:real>10</ml:real><ml:real>0</ml:real></ml:matrix></ml:define></math><rendering item-idref="36" /></region><region region-id="1333" left="168" top="2774.25" width="58.5" height="12" align-x="168" align-y="2784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"><b><c val="#800000">No Change</c></b></p></text></region><region region-id="1334" left="168" top="2792.25" width="48" height="12" align-x="168" align-y="2802" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"><b><c val="#800000">Increase</c></b></p></text></region><region region-id="1335" left="66" top="2818.5" width="140.25" height="45.75" align-x="140.25" align-y="2844" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">kendltau2</ml:id><ml:id xml:space="preserve">Table</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="3" cols="1"><ml:real>-0.2326626443580524</ml:real><ml:real>-2.3584841559891681</ml:real><ml:real>0.018349744776367988</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="37" /></region><region region-id="1486" left="36" top="2876.25" width="313.5" height="72" align-x="36" align-y="2886" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">This time we get -.2327 as the measure of correlation, with probability .0183 that this value or one larger than this in absolute value could occur for uncorrelated samples. Interpretation of these results is not always obvious, but we would likely conclude that in this case an increase in the medication is related to a decrease in cholesterol.</p></text></region><region region-id="1384" left="36" top="2966.25" width="292.5" height="72" align-x="36" align-y="2976" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The output of the <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">contingtbl</c></inlineAttr></b></f> function is a vector of length 5 containing <f family="Symbol" charset="2" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">c</c></inlineAttr></f>2, degrees of freedom, the probability a <f family="Symbol" charset="2" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">c</c></inlineAttr></f>2 this large or larger would occur if variables had no association, Cramer's V, and the contingency coefficient C. Small values of the probability indicate a significant association.</p></text><indexes><phrase>contingtbl</phrase></indexes></region><region region-id="1492" left="66" top="3054" width="117.75" height="79.5" align-x="105" align-y="3096" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:matrix rows="5" cols="1"><ml:id xml:space="preserve">χ2</ml:id><ml:id xml:space="preserve">df</ml:id><ml:id xml:space="preserve">prob</ml:id><ml:id xml:space="preserve">V</ml:id><ml:id xml:space="preserve">C</ml:id></ml:matrix><ml:apply><ml:id xml:space="preserve">contingtbl</ml:id><ml:id xml:space="preserve">Table</ml:id></ml:apply></ml:define></math><rendering item-idref="38" /></region><region region-id="1390" left="204" top="3051" width="126.75" height="84.75" align-x="242.25" align-y="3096" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:matrix rows="5" cols="1"><ml:id xml:space="preserve">χ2</ml:id><ml:id xml:space="preserve">df</ml:id><ml:id xml:space="preserve">prob</ml:id><ml:id xml:space="preserve">V</ml:id><ml:id xml:space="preserve">C</ml:id></ml:matrix><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="5" cols="1"><ml:real>21.218323586744638</ml:real><ml:real>6</ml:real><ml:real>0.0016760301799672267</ml:real><ml:real>0.46531011807953432</ml:real><ml:real>0.54970578287515715</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="39" /></region><region region-id="1387" left="36" top="3143.25" width="292.5" height="51" align-x="36" align-y="3156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The probability that the computed value of<f family="Symbol" charset="2"><sp /></f><f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false">c</inlineAttr></f><f family="Verdana" charset="0" size="10"><inlineAttr font-weight="normal" line-through="false"><c val="#000"><sup>2</sup></c></inlineAttr></f> or a larger value would occur if the two variables were independent is very small, so we reject the null hypothesis of independence.</p></text></region><region region-id="1487" left="0" top="3198" width="6000" height="6" align-x="0" align-y="3198" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1388" left="36" top="3206.25" width="314.25" height="99" align-x="36" align-y="3216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The strength of this association is given by the last two elements of the vector, Cramer's V and the contingency coefficient. 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