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list-style-type="inherit" tabs="inherit"><f family="Arial" charset="0"><u><inlineAttr line-through="false"><c val="#808000"><link href="./examples.xmcd" title="Links to Section Contents" popup="false"><sp count="6" />EXAMPLES</link></c></inlineAttr></u></f></p></text></region><region region-id="114" left="36" top="72.75" width="239.25" height="13.5" align-x="36" align-y="84" 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="Sous-titre" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Stabilizing and Normalizing the Error Variance</p></text><indexes><phrase>stabilizing and normalizing the error variance</phrase><phrase>preconditioning data, error variance</phrase><phrase>data, preconditioning</phrase><phrase>data, error variance</phrase><phrase>error variance</phrase><phrase>variance, error</phrase><phrase>normalizing, error variance</phrase><phrase>random error</phrase><phrase>nonnormality, correcting, transforming</phrase><phrase>correcting, errors, nonnormality</phrase></indexes></region><region region-id="1766" left="36" top="86.25" width="77.25" height="12" align-x="36" align-y="96" 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">by Paul Lorczak</p></text></region><region region-id="1697" left="36" top="122.25" width="298.5" height="39" align-x="36" align-y="132" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit"><c val="#000">Suppose we have a multivariate data set where the value of </c><f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false"><c val="#000">s</c></inlineAttr></f><c val="#000"><sup>2</sup></c><c val="#000">, the random error, is increasing with respect to one of the independent variables:</c></p></text></region><region region-id="1934" left="36" top="170.25" width="60.75" height="12" align-x="36" align-y="180" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">sample size:</p></text></region><region region-id="1935" left="144" top="171" width="27.75" height="12.75" align-x="155.25" align-y="180" 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">n</ml:id><ml:real>75</ml:real></ml:define></math><rendering item-idref="4" /></region><region region-id="1701" left="66" top="195.75" width="312" height="224.25" align-x="99.75" align-y="210" 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">σINCR</ml:id><ml:program><ml:localDefine><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:id xml:space="preserve">runif</ml:id><ml:sequence><ml:id xml:space="preserve">n</ml:id><ml:real>0</ml:real><ml:real>1</ml:real></ml:sequence></ml:apply></ml:localDefine><ml:localDefine><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:id xml:space="preserve">runif</ml:id><ml:sequence><ml:id xml:space="preserve">n</ml:id><ml:real>0</ml:real><ml:real>1</ml:real></ml:sequence></ml:apply></ml:localDefine><ml:str xml:space="preserve">Two columns of random data for x1 and x2</ml:str><ml:localDefine><ml:id xml:space="preserve">σincr</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>0</ml:real></ml:apply></ml:localDefine><ml:str xml:space="preserve">create a column of errors that depend on the first data column</ml:str><ml:localDefine><ml:id xml:space="preserve">ε</ml:id><ml:apply><ml:vectorize /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">σincr</ml:id><ml:apply><ml:id xml:space="preserve">rnorm</ml:id><ml:sequence><ml:id xml:space="preserve">n</ml:id><ml:real>0</ml:real><ml:real>1</ml:real></ml:sequence></ml:apply></ml:apply></ml:apply></ml:localDefine><ml:localDefine><ml:id 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disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:apply><ml:matcol /><ml:id xml:space="preserve">X</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:matcol /><ml:id xml:space="preserve">σINCR</ml:id><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="9" /></region><region region-id="1986" left="0" top="492" width="6000" height="6" align-x="0" align-y="492" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1703" left="36" top="500.25" width="285.75" height="24" align-x="36" align-y="510" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit"><c val="#000">The error term </c><f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false"><c val="#000">e</c></inlineAttr></f><c val="#000"> is from a normal distribution with a mean of zero</c><c val="#000">. </c></p></text></region><region region-id="1705" left="36" top="536.25" width="291" height="36" align-x="36" align-y="546" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit"><c val="#000">To check model assumptions, do a </c><b><u><link href="./residuals.xmcd" title="Links to section" popup="false">multivariate polynomial fit</link></u></b><c val="#000"> with the two columns of independent variables to the single column of dependent variables:</c></p></text></region><region region-id="1866" left="66" top="585" width="105" height="12.75" align-x="100.5" 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height="84" align-x="36" align-y="726" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit"><f family="Verdana" charset="0" size="10"><inlineAttr line-through="false"><c val="#000">If scatter plots of the </c></inlineAttr></f><b><u><link href="./residuals.xmcd" title="Links to section" popup="false">standardized residuals</link></u></b><f family="Verdana" charset="0" size="10"><b><inlineAttr font-style="normal" line-through="false"><c val="#000"><sp /></c></inlineAttr></b></f><f family="Verdana" charset="0" size="10"><c val="#000"><inlineAttr font-style="normal" line-through="false">versus each of the independent variables or versus the predicted values </inlineAttr></c></f>show no discernible pattern, then the variance of the error terms is most likely constant. On the plot of the residuals versus <c val="#000"><f family="Verdana" charset="0"><inlineAttr font-style="normal" line-through="false">x1</inlineAttr></f></c> (shown in red), the points spread out increasingly from left to right, indicating that <f family="Symbol" charset="2" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">e</c></inlineAttr></f> increases with <f family="Verdana" charset="0"><inlineAttr font-style="normal" line-through="false"><c val="#000">x1</c></inlineAttr></f>.</p></text></region><region region-id="1711" left="36" top="810" width="258" height="141" align-x="36" align-y="810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="15" /><rendering item-idref="16" /></region><region region-id="1712" left="36" top="962.25" width="248.25" height="12" align-x="36" align-y="972" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit"><c val="#000">On a similar graph for </c><f family="Verdana" charset="0"><inlineAttr font-weight="normal" font-style="normal" line-through="false"><c val="#000">x2</c></inlineAttr></f><c val="#000">, the pattern is not clear.</c></p></text></region><region region-id="1713" left="36" top="984" width="264" height="147" align-x="36" align-y="984" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="17" /><rendering item-idref="18" /></region><region region-id="1714" left="0" top="1146" width="6000" height="6" align-x="0" align-y="1146" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1715" left="36" top="1160.25" width="294.75" height="24" align-x="36" align-y="1170" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The pattern, however, is again evident on the graph of the residuals versus the predicted values: </p></text></region><region region-id="1716" left="36" top="1200" width="268.5" height="143.25" align-x="36" align-y="1200" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="19" /><rendering item-idref="20" /></region><region region-id="1717" left="36" top="1364.25" width="311.25" height="72" align-x="36" align-y="1374" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">To perform a valid regression, we need to counteract the increasing error variance by <b><u><link href="./scale.xmcd" title="Links to section" popup="false">transforming</link></u></b> the dependent data. The variance stabilizing transformations are listed below in order of increasing severity. The first equation is enabled. To see the effect of the other transformations, you can disable the first equation and enable another equation. </p></text></region><region region-id="1718" left="66" top="1453.5" width="48" height="24" align-x="94.5" align-y="1470" 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="stab">y</ml:id><ml:apply><ml:vectorize /><ml:apply><ml:sqrt /><ml:id xml:space="preserve">y</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="21" /></region><region region-id="1719" left="66" top="1491" width="60" height="22.5" align-x="94.5" align-y="1506" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="true"><ml:define><ml:id xml:space="preserve" subscript="stab">y</ml:id><ml:apply><ml:vectorize /><ml:apply><ml:id xml:space="preserve">ln</ml:id><ml:id xml:space="preserve">y</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="22" /></region><region region-id="1720" left="66" top="1526.25" width="48" height="33.75" align-x="94.5" align-y="1548" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="true"><ml:define><ml:id xml:space="preserve" subscript="stab">y</ml:id><ml:apply><ml:vectorize /><ml:apply><ml:div /><ml:real>1</ml:real><ml:id xml:space="preserve">y</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="23" /></region><region region-id="1878" left="36" top="1568.25" width="324" height="24" align-x="36" align-y="1578" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit"><f family="Verdana" charset="0"><inlineAttr line-through="false"><c val="#000">Below, we've plotted the new resulting residuals</c></inlineAttr></f><f family="Verdana" charset="0"><inlineAttr line-through="false"><c val="#000"> versus the predicted values, to see the effect of stabilization</c></inlineAttr></f><c val="#000">.</c></p></text></region><region region-id="1873" left="66" top="1605" width="122.25" height="16.5" align-x="100.5" align-y="1614" 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="WarnRedefinedUDScalar"><ml:id xml:space="preserve">params</ml:id><ml:apply><ml:id xml:space="preserve">regress</ml:id><ml:sequence><ml:id xml:space="preserve">X</ml:id><ml:id xml:space="preserve" subscript="stab">y</ml:id><ml:real>2</ml:real></ml:sequence></ml:apply></ml:define></math><rendering item-idref="24" /></region><region region-id="1874" left="66" top="1624.5" width="165.75" height="28.5" align-x="92.25" align-y="1644" 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:apply><ml:indexer /><ml:id xml:space="preserve">yfits</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">interp</ml:id><ml:sequence><ml:id xml:space="preserve">params</ml:id><ml:id xml:space="preserve">X</ml:id><ml:id xml:space="preserve" subscript="stab">y</ml:id><ml:apply><ml:matcol /><ml:parens><ml:apply><ml:transpose /><ml:id xml:space="preserve">X</ml:id></ml:apply></ml:parens><ml:id xml:space="preserve">i</ml:id></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="25" /></region><region region-id="1875" left="66" top="1665" width="89.25" height="16.5" align-x="96" align-y="1674" 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">resids</ml:id><ml:apply><ml:minus /><ml:id xml:space="preserve" subscript="stab">y</ml:id><ml:id xml:space="preserve">yfits</ml:id></ml:apply></ml:define></math><rendering item-idref="26" /></region><region region-id="1876" left="66" top="1687.5" width="91.5" height="46.5" align-x="97.5" align-y="1722" 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="WarnRedefinedUDScalar"><ml:id xml:space="preserve">stERR</ml:id><ml:apply><ml:sqrt /><ml:apply><ml:div /><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:id xml:space="preserve">resid</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">y</ml:id></ml:apply><ml:real>5</ml:real></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="27" /></region><region region-id="1877" left="222" top="1713" width="77.25" height="12.75" align-x="252.75" align-y="1722" 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">stERR</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.77473922422577357</ml:real></result></ml:eval></math><rendering item-idref="28" /></region><region region-id="1883" left="0" top="1746" width="6000" height="-6" align-x="0" align-y="1740" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1882" left="36" top="1752" width="276" height="143.25" align-x="36" align-y="1752" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="29" /><rendering item-idref="30" /></region><region region-id="1726" left="36" top="1904.25" width="300.75" height="60" align-x="36" align-y="1914" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">You may find that some of the transformations are too severe, actually causing a violation rather than removing it. For instance, the graph pattern could be curved rather than random, indicating that a nonlinear model might provide a better fit to the data.</p></text></region><region region-id="1728" left="36" top="1981.5" width="115.5" height="12.75" align-x="36" align-y="1992" 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">Correcting Nonnormality</p></text></region><region region-id="1729" left="36" top="2000.25" width="279" height="60" align-x="36" align-y="2010" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit"><c val="#000">Besides helping to stabilize the error variance, the same transformations may also correct any nonnormality of errors. To check for nonnormality, generate a sample from a population having error terms that are exponentially distributed.</c></p></text></region><region region-id="1730" left="36" top="2066.25" width="60.75" height="12" align-x="36" align-y="2076" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">sample size:</p></text></region><region region-id="1930" left="144" top="2067" width="51.75" height="12.75" align-x="179.25" align-y="2076" 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">n_expo</ml:id><ml:real>25</ml:real></ml:define></math><rendering item-idref="31" /></region><region region-id="1929" left="66" top="2097.75" width="270.75" height="147" align-x="99.75" align-y="2112" 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">εEXPO</ml:id><ml:program><ml:localDefine><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:id xml:space="preserve">runif</ml:id><ml:sequence><ml:id xml:space="preserve">n_expo</ml:id><ml:real>0</ml:real><ml:real>1</ml:real></ml:sequence></ml:apply></ml:localDefine><ml:localDefine><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:id xml:space="preserve">runif</ml:id><ml:sequence><ml:id xml:space="preserve">n_expo</ml:id><ml:real>0</ml:real><ml:real>1</ml:real></ml:sequence></ml:apply></ml:localDefine><ml:localDefine><ml:id xml:space="preserve">ε1</ml:id><ml:apply><ml:id xml:space="preserve">rexp</ml:id><ml:sequence><ml:id xml:space="preserve">n_expo</ml:id><ml:real>10</ml:real></ml:sequence></ml:apply></ml:localDefine><ml:localDefine><ml:id xml:space="preserve">β</ml:id><ml:matrix rows="3" cols="1"><ml:real>4</ml:real><ml:real>7</ml:real><ml:real>2</ml:real></ml:matrix></ml:localDefine><ml:localDefine><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>0</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:matcol /><ml:id xml:space="preserve">DATA</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:apply><ml:id xml:space="preserve">ε1</ml:id></ml:apply></ml:localDefine><ml:id xml:space="preserve">DATA</ml:id></ml:program></ml:define></math><rendering item-idref="32" /></region><region region-id="1888" left="66" top="2259" width="62.25" height="18.75" align-x="81.75" align-y="2274" 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">ye</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">εEXPO</ml:id><ml:real>2</ml:real></ml:apply></ml:define></math><rendering item-idref="33" /></region><region region-id="1889" left="144" top="2265" width="59.25" height="12.75" align-x="152.25" align-y="2274" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define 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warning="WarnRedefinedUDScalar"><ml:apply><ml:matcol /><ml:id xml:space="preserve">Xe</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:matcol /><ml:id xml:space="preserve">εEXPO</ml:id><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="36" /></region><region region-id="1891" left="144" top="2289" width="75.75" height="18.75" align-x="173.25" align-y="2304" 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="WarnRedefinedUDScalar"><ml:apply><ml:matcol /><ml:id xml:space="preserve">Xe</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:matcol /><ml:id xml:space="preserve">εEXPO</ml:id><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="37" /></region><region region-id="1734" left="0" top="2322" width="6000" height="6" align-x="0" align-y="2322" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1735" left="36" top="2336.25" width="295.5" height="36" align-x="36" align-y="2346" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Nonnormality can be detected in a <b><u><link href="./QQplot.xmcd" title="Links to section" popup="false">normal plot</link></u></b> of the standardized residuals. If the plot resembles a straight line, then most likely the errors are from a normal distribution. </p></text></region><region region-id="1917" left="66" top="2391.75" width="134.25" height="24" align-x="100.5" align-y="2412" 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="WarnRedefinedUDScalar"><ml:id xml:space="preserve">params</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:transpose /><ml:id xml:space="preserve">Xe</ml:id></ml:apply><ml:id xml:space="preserve">Xe</ml:id></ml:apply></ml:parens><ml:real>-1</ml:real></ml:apply><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:transpose /><ml:id xml:space="preserve">Xe</ml:id></ml:apply><ml:id xml:space="preserve">ye</ml:id></ml:apply></ml:parens></ml:apply></ml:define></math><rendering item-idref="38" /></region><region 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cols="1"><ml:real>4.1595424932013252</ml:real><ml:real>6.9145374700881668</ml:real><ml:real>1.986304369855155</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="40" /></region><region region-id="1899" left="66" top="2457" width="72" height="12.75" align-x="96" align-y="2466" 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">reside</ml:id><ml:apply><ml:minus /><ml:id xml:space="preserve">ye</ml:id><ml:id xml:space="preserve">yfit</ml:id></ml:apply></ml:define></math><rendering item-idref="41" /></region><region region-id="1900" left="66" top="2473.5" width="127.5" height="46.5" align-x="97.5" align-y="2508" 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="WarnRedefinedUDScalar"><ml:id xml:space="preserve">stERR</ml:id><ml:apply><ml:sqrt /><ml:apply><ml:div /><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:id xml:space="preserve">reside</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">ye</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">cols</ml:id><ml:id xml:space="preserve">Xe</ml:id></ml:apply></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="42" /></region><region region-id="1901" left="222" top="2499" width="77.25" height="12.75" align-x="252.75" align-y="2508" 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">stERR</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.1115870570001352</ml:real></result></ml:eval></math><rendering item-idref="43" /></region><region 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align-y="2772" 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">As expected, the residuals are not normally distributed, as shown on this nonlinear scatter plot.</p></text></region><region region-id="1941" left="36" top="2803.5" width="127.5" height="12.75" align-x="36" align-y="2814" 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">Transforming for normality</p></text></region><region region-id="1741" left="36" top="2822.25" width="283.5" height="48" align-x="36" align-y="2832" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Again, we've defined the transformations in order of increasing severity. The first equation is enabled. To see the effect of the other transformations, you can disable the first equation, and enable another equation. </p></text></region><region region-id="1945" left="66" top="2881.5" width="55.5" height="18" align-x="97.5" align-y="2892" 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="norm">y</ml:id><ml:apply><ml:sqrt /><ml:id xml:space="preserve">ye</ml:id></ml:apply></ml:define></math><rendering item-idref="47" /></region><region region-id="1946" left="66" top="2913" width="67.5" height="16.5" align-x="97.5" align-y="2922" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="true"><ml:define><ml:id xml:space="preserve" subscript="norm">y</ml:id><ml:apply><ml:id xml:space="preserve">ln</ml:id><ml:id 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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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Inspecting the new standardized residual normal plot.</p></text></region><region region-id="1956" left="66" top="3015.75" width="154.5" height="27.75" align-x="100.5" align-y="3036" 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="WarnRedefinedUDScalar"><ml:id xml:space="preserve">params</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:transpose /><ml:id xml:space="preserve">Xe</ml:id></ml:apply><ml:id xml:space="preserve">Xe</ml:id></ml:apply></ml:parens><ml:real>-1</ml:real></ml:apply><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:transpose /><ml:id xml:space="preserve">Xe</ml:id></ml:apply><ml:id xml:space="preserve" subscript="norm">y</ml:id></ml:apply></ml:parens></ml:apply></ml:define></math><rendering item-idref="50" /></region><region region-id="1957" left="66" top="3057" width="70.5" height="12.75" align-x="85.5" align-y="3066" 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="WarnRedefinedUDScalar"><ml:id xml:space="preserve">yfit</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">Xe</ml:id><ml:id xml:space="preserve">params</ml:id></ml:apply></ml:define></math><rendering item-idref="51" /></region><region region-id="1958" left="222" top="3040.5" width="94.5" height="45.75" align-x="255.75" align-y="3066" 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">params</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="3" cols="1"><ml:real>2.1462438304143081</ml:real><ml:real>1.1871653510802784</ml:real><ml:real>0.35004234802897471</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="52" /></region><region region-id="1959" left="66" top="3081" width="87.75" height="16.5" align-x="96" align-y="3090" 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="WarnRedefinedUDScalar"><ml:id xml:space="preserve">reside</ml:id><ml:apply><ml:minus /><ml:id xml:space="preserve" subscript="norm">y</ml:id><ml:id xml:space="preserve">yfit</ml:id></ml:apply></ml:define></math><rendering item-idref="53" /></region><region region-id="1960" left="66" top="3103.5" width="143.25" height="50.25" align-x="97.5" align-y="3138" 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="WarnRedefinedUDScalar"><ml:id xml:space="preserve">stERR</ml:id><ml:apply><ml:sqrt /><ml:apply><ml:div /><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:id xml:space="preserve">reside</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve" subscript="norm">y</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">cols</ml:id><ml:id xml:space="preserve">Xe</ml:id></ml:apply></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="54" /></region><region region-id="1961" left="222" top="3129" width="77.25" height="12.75" align-x="252.75" align-y="3138" 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">stERR</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.029613268193610629</ml:real></result></ml:eval></math><rendering item-idref="55" /></region><region region-id="1962" left="66" top="3164.25" width="141" height="27.75" align-x="90.75" align-y="3180" 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="WarnRedefinedUDScalar"><ml:id xml:space="preserve">rplot</ml:id><ml:apply><ml:id xml:space="preserve">qqplot</ml:id><ml:sequence><ml:apply><ml:div /><ml:id xml:space="preserve">reside</ml:id><ml:id xml:space="preserve">stERR</ml:id></ml:apply><ml:str xml:space="preserve">normal</ml:str></ml:sequence></ml:apply></ml:define></math><rendering item-idref="56" /></region><region region-id="1965" left="48" top="3210" width="252.75" height="177" align-x="48" align-y="3210" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="57" /><rendering item-idref="58" /></region><region region-id="1971" left="36" top="3398.25" width="294.75" height="60" align-x="36" align-y="3408" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">You can investigate the relationship between the transformation and the nonnormality of error terms through enabling and disabling different equations. 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