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top="6" width="237" height="34.5" align-x="6" align-y="6" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><png item-idref="2" display-width="235.5" display-height="33" /><link href="./dataTOC.xmcd" title="Links to Table of Contents" popup="false" /><rendering item-idref="3" /></region><region region-id="930" left="156" top="44.25" width="87" height="12" align-x="156" align-y="54" 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="Titre" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"><f family="Arial" charset="0"><u><inlineAttr line-through="false"><c val="#808000"><link href="./regression.xmcd" title="Links to Section Contents" popup="false"><sp count="6" />REGRESSION</link></c></inlineAttr></u></f></p></text></region><region region-id="1321" left="36" top="90.75" width="64.5" height="13.5" align-x="36" align-y="102" 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">Quality of Fit</p></text><indexes><phrase>quality of fit</phrase><phrase>regression, quality of fit</phrase><phrase>fit, quality of</phrase></indexes></region><region region-id="2118" left="66" top="109.5" width="165" height="12.75" align-x="75" align-y="120" 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">confidence(vx,vy,F(x),vparam,conf) </p></text><indexes><phrase>confidence</phrase><phrase>functions, confidence</phrase></indexes></region><region region-id="2119" left="36" top="146.25" width="324.75" height="120" align-x="36" align-y="156" 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">Scattered throughout the previous files in the regression section we have included metrics for judging how well we have fit the data: how far away are the predicted values of y from the actual values of y. In the case of the <b><u><link href="./nonlin.xmcd" title="Links to section" popup="false">LeastSquaresFit</link></u></b> function, confidence limits are provided along with the fit parameters, and the <b><u><link href="./linreg.xmcd" title="Links to section" popup="false">stderr</link></u></b> function gives a measure of the error for linear fits. 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margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Suppose we fit some exponential parameters to the data.</p></text></region><region region-id="2138" left="66" top="382.5" width="86.25" height="45.75" align-x="91.5" align-y="408" 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="3" cols="1"><ml:id xml:space="preserve">a</ml:id><ml:id xml:space="preserve">b</ml:id><ml:id xml:space="preserve">c</ml:id></ml:matrix><ml:apply><ml:id xml:space="preserve">expfit</ml:id><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">vy</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="8" /></region><region region-id="2150" left="222" top="382.5" width="81" height="45.75" align-x="246.75" align-y="408" 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xml:space="preserve">b</ml:id><ml:id xml:space="preserve">c</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">a</ml:id><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">b</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply><ml:id xml:space="preserve">c</ml:id></ml:apply></ml:define></math><rendering item-idref="10" /></region><region region-id="2159" left="48" top="474" width="245.25" height="131.25" align-x="48" align-y="474" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="11" /><rendering item-idref="12" /></region><region region-id="2154" left="36" top="608.25" width="300.75" height="36" align-x="36" align-y="618" 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">But how close are these values to the data? How sure are we that these are the right parameters? How closely may we predict the behavior at a point that we did not measure.</p></text></region><region region-id="2206" left="0" top="654" width="6000" height="6" align-x="0" align-y="654" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2205" left="36" top="664.5" width="77.25" height="15.75" align-x="36" align-y="678" 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">Residuals and <f family="Symbol" charset="2" size="11"><b><inlineAttr font-style="normal" underline="false" line-through="false"><c val="#800000">c</c></inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false"><c val="#800000"><sup>2</sup></c></inlineAttr></f></p></text><indexes><phrase>residuals</phrase></indexes></region><region region-id="2207" left="36" top="686.25" width="318" height="36" align-x="36" align-y="696" 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">One reasonable first test of how well we did with our fit is to calculate the residuals — the difference between the calculated model values and the measured ones.</p></text></region><region region-id="2216" left="66" top="729" width="119.25" height="12.75" align-x="103.5" align-y="738" 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">residual</ml:id><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">fit</ml:id><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">a</ml:id><ml:id xml:space="preserve">b</ml:id><ml:id xml:space="preserve">c</ml:id></ml:sequence></ml:apply><ml:id xml:space="preserve">vy</ml:id></ml:apply></ml:define></math><rendering item-idref="13" /></region><region region-id="2211" left="48" top="744" width="219" height="131.25" align-x="48" align-y="744" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="14" /><rendering item-idref="15" /></region><region region-id="2213" left="36" top="884.25" width="303" height="60" align-x="36" align-y="894" 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 sum of the residuals squared is the error in the calculation; that is, it's the quantity we minimize when calculating a least-squares solution. The error is also a measure of how well we did, since it shows how much deviation is unexplained by the regression.</p></text></region><region region-id="2221" left="66" top="957" width="73.5" height="30" align-x="81.75" 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:define><ml:id xml:space="preserve">χ2</ml:id><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:id xml:space="preserve">residual</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="16" /></region><region region-id="2222" left="222" top="962.25" width="57" height="13.5" align-x="237" 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:id xml:space="preserve">χ2</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>6.998292386055371</ml:real></result></ml:eval></math><rendering item-idref="17" /></region><region region-id="2227" left="36" top="989.25" width="307.5" height="27" align-x="44.25" align-y="1002" 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">This value is also known as <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><f family="Verdana" charset="0"><inlineAttr line-through="false"><c val="#000"><sup>2</sup></c></inlineAttr></f>, and is a general metric for the goodness of fit.</p></text></region><region region-id="2256" left="36" top="1033.5" width="118.5" height="12.75" align-x="53.25" align-y="1044" 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">Analysis of variance table</p></text><indexes><phrase>analysis of variance table</phrase><phrase>variance table, analysis of</phrase></indexes></region><region region-id="2286" left="36" top="1058.25" width="289.5" height="60" align-x="36" align-y="1068" 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 amount of variation <b><u><link href="./linreg.xmcd#goodfit" title="Links to section" popup="false">explained by a linear model</link></u></b> is described in terms of deviations from the mean of y. In a nonlinear model, however, we'll define the sum of squares due to the regression (<f family="Verdana" charset="0"><inlineAttr font-style="normal" line-through="false"><c val="#000">SSR</c></inlineAttr></f>) with respect to the total sum of squares.</p></text></region><region region-id="2287" left="66" top="1125" width="60" height="30" align-x="89.25" align-y="1140" 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">SSY</ml:id><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:id xml:space="preserve">vy</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="18" /></region><region region-id="2289" left="222" top="1130.25" width="39" height="13.5" align-x="243.75" align-y="1140" 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">SSE</ml:id><ml:id xml:space="preserve">χ2</ml:id></ml:define></math><rendering item-idref="19" /></region><region region-id="2290" left="66" top="1155" width="75" height="12.75" align-x="89.25" align-y="1164" 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">SSR</ml:id><ml:apply><ml:minus /><ml:id xml:space="preserve">SSY</ml:id><ml:id xml:space="preserve">SSE</ml:id></ml:apply></ml:define></math><rendering item-idref="20" /></region><region region-id="2369" left="36" top="1184.25" width="299.25" height="24" align-x="36" align-y="1194" 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">Degrees of freedom are the length of the data less the number of fit parameters.</p></text></region><region region-id="2262" left="66" top="1221" width="90" height="16.5" align-x="106.5" 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" subscript="unadj">DF</ml:id><ml:apply><ml:id xml:space="preserve">length</ml:id><ml:id xml:space="preserve">vx</ml:id></ml:apply></ml:define></math><rendering item-idref="21" /></region><region region-id="2265" left="66" top="1245" width="54" height="16.5" align-x="108" 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="param">DF</ml:id><ml:real>3</ml:real></ml:define></math><rendering item-idref="22" /></region><region region-id="2266" left="144" top="1244.25" width="129" height="12" align-x="144" align-y="1254" 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 number of parameters</p></text></region><region region-id="2268" left="66" top="1269" width="126.75" height="16.5" align-x="103.5" align-y="1278" 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="resid">DF</ml:id><ml:apply><ml:minus /><ml:id xml:space="preserve" subscript="unadj">DF</ml:id><ml:id xml:space="preserve" subscript="param">DF</ml:id></ml:apply></ml:define></math><rendering item-idref="23" /></region><region region-id="2321" left="0" top="1296" width="6000" height="6" align-x="0" align-y="1296" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2273" left="36" top="1310.25" width="329.25" height="24" align-x="47.25" align-y="1320" 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">Mean square error and regression mean square are given by the SSE divided by the appropriate degrees of freedom.</p></text></region><region region-id="2278" left="66" top="1345.5" width="67.5" height="32.25" align-x="91.5" align-y="1362" 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">MSE</ml:id><ml:apply><ml:div /><ml:id xml:space="preserve">χ2</ml:id><ml:id xml:space="preserve" subscript="resid">DF</ml:id></ml:apply></ml:define></math><rendering item-idref="24" /></region><region region-id="2279" left="222" top="1345.5" width="73.5" height="32.25" align-x="249" align-y="1362" 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">MSR</ml:id><ml:apply><ml:div /><ml:id xml:space="preserve">χ2</ml:id><ml:id xml:space="preserve" subscript="param">DF</ml:id></ml:apply></ml:define></math><rendering item-idref="25" /></region><region region-id="2283" left="36" top="1388.25" width="312.75" height="24" align-x="36" align-y="1398" 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">This set of statistics will characterize the fit, and forms an analysis of variance table.</p></text></region><region region-id="2509" left="66" top="1430.25" width="14.25" height="12" align-x="66" align-y="1440" 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"><b>DF</b></p></text></region><region region-id="2510" left="144" top="1430.25" width="13.5" height="12" align-x="144" align-y="1440" 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"><b>SS</b></p></text></region><region region-id="2512" left="240" top="1430.25" width="15.75" height="12" align-x="240" align-y="1440" 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"><b>MS</b></p></text></region><region region-id="2307" left="66" top="1449" width="58.5" height="16.5" align-x="107.25" align-y="1458" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" 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xmlns="http://schemas.mathsoft.com/math20"><ml:real>6.998292386055371</ml:real></result></ml:eval></math><rendering item-idref="30" /></region><region region-id="2372" left="240" top="1467" width="66.75" height="12.75" align-x="264.75" align-y="1476" 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">MSE</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>1.3996584772110743</ml:real></result></ml:eval></math><rendering item-idref="31" /></region><region region-id="2375" left="66" top="1485" width="57" height="16.5" align-x="105.75" align-y="1494" 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="unadj">DF</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>8</ml:real></result></ml:eval></math><rendering item-idref="32" /></region><region region-id="2376" left="144" top="1479" width="80.25" height="18.75" align-x="166.5" align-y="1494" 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">SSY</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>14730</ml:real></result></ml:eval></math><rendering item-idref="33" /></region><region region-id="2325" left="36" top="1514.25" width="285.75" height="24" align-x="36" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">For a rough estimate of how well the model fits the data, use the value </p></text></region><region region-id="2326" left="66" top="1556.25" width="129.75" height="45" align-x="82.5" align-y="1572" 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">Fit</ml:id><ml:apply><ml:minus /><ml:real>1</ml:real><ml:apply><ml:div /><ml:id xml:space="preserve">SSE</ml:id><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:id xml:space="preserve">vy</ml:id><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:id xml:space="preserve">vy</ml:id></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="34" /></region><region region-id="2332" left="222" top="1563" width="57.75" height="12.75" align-x="237.75" align-y="1572" 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">Fit</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.99886197375623131</ml:real></result></ml:eval></math><rendering item-idref="35" /></region><region region-id="2471" left="36" top="1615.5" width="95.25" height="12.75" align-x="36" align-y="1626" 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">Confidence intervals</p></text><indexes><phrase>confidence intervals</phrase><phrase>intervals, confidence</phrase></indexes></region><region region-id="2379" left="36" top="1634.25" width="320.25" height="84" align-x="36" align-y="1644" 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">Because our parameter estimates are only approximations, we'll need to rely on asymptotic rather than on exact properties to form confidence intervals for population parameters. 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xml:space="preserve">A</ml:id><ml:apply><ml:id xml:space="preserve">augment</ml:id><ml:sequence><ml:id xml:space="preserve">DFA</ml:id><ml:id xml:space="preserve">DFB</ml:id><ml:id xml:space="preserve">DFC</ml:id></ml:sequence></ml:apply></ml:define></ml:provenance></math><rendering item-idref="40"><element-image-map><box left="1.5" top="0.75" width="126" height="11.25" expr-idref="1" /></element-image-map></rendering></region><region region-id="2661" left="66" top="1809.75" width="80.25" height="24" align-x="96" align-y="1830" 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">Covar</ml:id><ml:apply><ml:pow /><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:transpose /><ml:id xml:space="preserve">A</ml:id></ml:apply><ml:id xml:space="preserve">A</ml:id></ml:apply></ml:parens><ml:real>-1</ml:real></ml:apply></ml:define></math><rendering item-idref="41" /></region><region region-id="2662" left="198" top="1820.25" width="71.25" height="13.5" align-x="215.25" align-y="1830" 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">Vβ</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">MSE</ml:id><ml:id xml:space="preserve">Covar</ml:id></ml:apply></ml:define></math><rendering item-idref="42" /></region><region region-id="2602" left="36" top="1844.25" width="297.75" height="24" align-x="36" align-y="1854" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Taking the square root of the diagonal elements produces asymptotic standard errors for the parameter estimates. </p></text></region><region region-id="2593" left="66" top="1881" width="75" height="12.75" align-x="74.25" align-y="1890" 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">i</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">cols</ml:id><ml:id xml:space="preserve">A</ml:id></ml:apply><ml:real>1</ml:real></ml:apply></ml:range></ml:define></math><rendering item-idref="43" /></region><region region-id="2629" left="198" top="1878.75" width="66" height="20.25" align-x="229.5" align-y="1890" 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="WarnRedefinedBIFunction"><ml:apply><ml:indexer /><ml:id xml:space="preserve">stderr</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:apply><ml:sqrt /><ml:apply><ml:indexer /><ml:id xml:space="preserve">Vβ</ml:id><ml:sequence><ml:id xml:space="preserve">i</ml:id><ml:id xml:space="preserve">i</ml:id></ml:sequence></ml:apply></ml:apply></ml:define></math><rendering item-idref="44" /></region><region region-id="2631" left="36" top="1904.25" width="309.75" height="36" 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">We can form a confidence interval from the errors using the Student's t-Distribution, much as we did in the case of a <b><u><link href="./linreg.xmcd" title="Links to section" popup="false">linear fit</link></u></b>. </p></text></region><region region-id="2635" left="0" top="1950" width="6000" height="6" align-x="0" align-y="1950" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2645" left="36" top="1964.25" width="7.5" height="12" align-x="36" align-y="1974" 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">If</p></text></region><region region-id="2646" left="66" top="1964.25" width="36" height="13.5" align-x="78.75" align-y="1974" 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">α</ml:id><ml:real>0.01</ml:real></ml:define></math><rendering item-idref="45" /></region><region region-id="2650" left="144" top="1957.5" width="98.25" height="28.5" align-x="153" align-y="1974" 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">t</ml:id><ml:apply><ml:id xml:space="preserve">qt</ml:id><ml:sequence><ml:apply><ml:minus /><ml:real>1</ml:real><ml:apply><ml:div /><ml:id xml:space="preserve">α</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:id xml:space="preserve" subscript="resid">DF</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="46" /></region><region region-id="2651" left="264" top="1965" width="50.25" height="12.75" align-x="272.25" align-y="1974" 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">t</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>4.0321429835552305</ml:real></result></ml:eval></math><rendering item-idref="47" /></region><region region-id="2674" left="36" top="1994.25" width="240.75" height="12" align-x="36" 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="true" lock-width="false"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">giving confidence intervals on the parameters of</p></text></region><region region-id="2672" left="66" top="2014.5" width="90.75" height="45.75" align-x="100.5" 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:mult style="default" /><ml:id xml:space="preserve">t</ml:id><ml:id xml:space="preserve">stderr</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="3" cols="1"><ml:real>3.0426097305830493</ml:real><ml:real>0.076979187966543891</ml:real><ml:real>5.8725038525492144</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="48" /></region><region region-id="2667" left="36" top="2066.25" width="296.25" height="12" align-x="44.25" 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="true" lock-width="false"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This functionality is wrapped up in the confidence function.</p></text></region><region region-id="2670" left="66" top="2086.5" width="162" height="45.75" align-x="81" 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">CI</ml:id><ml:apply><ml:id xml:space="preserve">confidence</ml:id><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">vy</ml:id><ml:id xml:space="preserve">fit</ml:id><ml:matrix rows="3" cols="1"><ml:id xml:space="preserve">a</ml:id><ml:id xml:space="preserve">b</ml:id><ml:id xml:space="preserve">c</ml:id></ml:matrix><ml:apply><ml:minus /><ml:real>1</ml:real><ml:id xml:space="preserve">α</ml:id></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="49" /></region><region region-id="2676" left="66" top="2146.5" width="106.5" height="45.75" align-x="80.25" align-y="2172" 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">CI</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="3" cols="2"><ml:real>3.0426097305821869</ml:real><ml:real>0.076979187966522158</ml:real><ml:real>5.8725038525477862</ml:real><ml:real>4.0321429835552305</ml:real><ml:real>0</ml:real><ml:real>0</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="50" /></region><region region-id="2677" left="192" top="2146.5" width="81" height="45.75" align-x="216.75" align-y="2172" 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="3" cols="1"><ml:id xml:space="preserve">a</ml:id><ml:id xml:space="preserve">b</ml:id><ml:id xml:space="preserve">c</ml:id></ml:matrix><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="3" cols="1"><ml:real>5.2899511305537219</ml:real><ml:real>0.40489190001799158</ml:real><ml:real>0.27781874466875323</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="51" /></region><region region-id="2443" left="36" top="2210.25" width="327" height="62.25" align-x="42.75" align-y="2220" 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 first column of the result contains the distance of the interval borders from the current parameter value. The second column contains the value for the percent point function value of the t-distribution. With a probability of <region region-id="2678" left="232.5" top="2246.25" width="427.5" height="13.5" align-x="260.25" align-y="2256" 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:minus /><ml:real>1</ml:real><ml:id xml:space="preserve">α</ml:id></ml:apply><ml:unitOverride><ml:id xml:space="preserve">%</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>99</ml:real></result></ml:eval></math><rendering item-idref="52" /></region>, values for the estimated parameters are contained in intervals </p></text></region><region region-id="2694" left="66" top="2284.5" width="113.25" height="45.75" align-x="123" align-y="2310" 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:plus /><ml:matrix rows="3" cols="1"><ml:id xml:space="preserve">a</ml:id><ml:id xml:space="preserve">b</ml:id><ml:id xml:space="preserve">c</ml:id></ml:matrix><ml:apply><ml:matcol /><ml:id xml:space="preserve">CI</ml:id><ml:real>0</ml:real></ml:apply></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="3" cols="1"><ml:real>8.3325608611359083</ml:real><ml:real>0.48187108798451372</ml:real><ml:real>6.1503225972165394</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="53" /></region><region region-id="2695" left="222" top="2284.5" width="117.75" height="45.75" align-x="278.25" align-y="2310" 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:minus /><ml:matrix rows="3" cols="1"><ml:id xml:space="preserve">a</ml:id><ml:id xml:space="preserve">b</ml:id><ml:id xml:space="preserve">c</ml:id></ml:matrix><ml:apply><ml:matcol /><ml:id xml:space="preserve">CI</ml:id><ml:real>0</ml:real></ml:apply></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="3" cols="1"><ml:real>2.247341399971535</ml:real><ml:real>0.32791271205146943</ml:real><ml:real>-5.5946851078790329</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="54" /></region><region region-id="2696" left="36" top="2354.25" width="326.25" height="49.5" align-x="36" align-y="2364" 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">That is, we are <region region-id="2004" left="115.5" top="2354.25" width="60.75" height="13.5" align-x="150.75" align-y="2364" 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:parens><ml:apply><ml:minus /><ml:real>1</ml:real><ml:id xml:space="preserve">α</ml:id></ml:apply></ml:parens><ml:unitOverride><ml:id xml:space="preserve">%</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>99</ml:real></result></ml:eval></math><rendering item-idref="55" /></region><f family="Verdana" charset="0" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000"><sp /></c></inlineAttr></f>confident that the correct value of the parameter a, for this data, lies on the interval between <region region-id="1971" left="36" top="2380.5" width="95.25" height="22.5" align-x="89.25" align-y="2394.75" 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:minus /><ml:id xml:space="preserve">a</ml:id><ml:apply><ml:indexer /><ml:parens><ml:apply><ml:matcol /><ml:id xml:space="preserve">CI</ml:id><ml:real>0</ml:real></ml:apply></ml:parens><ml:real>0</ml:real></ml:apply></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>2.247341399971535</ml:real></result></ml:eval></math><rendering item-idref="56" /></region> and <region region-id="1972" left="156.75" top="2380.5" width="427.5" height="22.5" align-x="210.75" align-y="2394.75" 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:plus /><ml:id xml:space="preserve">a</ml:id><ml:apply><ml:indexer /><ml:parens><ml:apply><ml:matcol /><ml:id xml:space="preserve">CI</ml:id><ml:real>0</ml:real></ml:apply></ml:parens><ml:real>0</ml:real></ml:apply></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>8.3325608611359083</ml:real></result></ml:eval></math><rendering item-idref="57" /></region><f family="Verdana" charset="0" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">, and so on</c></inlineAttr></f>.</p></text></region><region region-id="2697" left="36" top="2414.25" width="314.25" height="24" align-x="52.5" align-y="2424" 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 standard deviation (standard error) for each parameter can be computed as follows:</p></text></region><region region-id="2698" left="66" top="2457" width="71.25" height="42.75" align-x="94.5" align-y="2478" 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">StdFx</ml:id><ml:apply><ml:div /><ml:apply><ml:matcol /><ml:id xml:space="preserve">CI</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:indexer /><ml:parens><ml:apply><ml:matcol /><ml:id xml:space="preserve">CI</ml:id><ml:real>1</ml:real></ml:apply></ml:parens><ml:real>0</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="58" /></region><region region-id="2708" left="168" top="2452.5" width="84" height="45.75" align-x="195.75" align-y="2478" 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">StdFx</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="3" cols="1"><ml:real>0.75458874920637109</ml:real><ml:real>0.019091383485277075</ml:real><ml:real>1.4564225218446665</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="59" /></region><region region-id="2709" left="264" top="2452.5" width="84.75" height="45.75" align-x="292.5" align-y="2478" 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">stderr</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="3" cols="1"><ml:real>0.754588749206585</ml:real><ml:real>0.019091383485282463</ml:real><ml:real>1.4564225218450206</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="60" /></region><region region-id="2705" left="42" top="2510.25" width="299.25" height="12" align-x="48" align-y="2520" 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 correlation matrix of parameter estimates is given by</p></text></region><region region-id="2721" left="66" top="2541" width="75.75" height="12.75" align-x="75" align-y="2550" 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">j</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">cols</ml:id><ml:id xml:space="preserve">A</ml:id></ml:apply><ml:real>1</ml:real></ml:apply></ml:range></ml:define></math><rendering item-idref="61" /></region><region region-id="2717" left="180" top="2531.25" width="92.25" height="36" align-x="207" align-y="2550" show-border="false" show-highlight="false" 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/></element-image-map></rendering></region><region region-id="2738" left="36" top="2648.25" width="324.75" height="36" align-x="42" align-y="2658" 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">From the confidence intervals and the correlation matrix for the parameter estimates it is pretty clear that the model is overparameterized and that parameter c can be dropped.<sp count="4" /></p></text></region><region region-id="2732" left="36" top="2701.5" width="267" height="25.5" align-x="36" align-y="2712" show-border="false" show-highlight="false" is-protected="false" 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="left" list-style-type="inherit" tabs="inherit">Approximate confidence intervals on calculated and measured y</p></text><indexes><phrase>confidence intervals</phrase><phrase>intervals, confidence</phrase></indexes></region><region region-id="2743" left="36" top="2738.25" width="318" height="24" align-x="36" align-y="2748" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="false"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Similarly, we can calculate confidence intervals on any y values predicted by the model at any point along the range of vx.</p></text></region><region region-id="2739" left="66" top="2769.75" width="62.25" height="18" align-x="83.25" 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">Vy</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">A</ml:id><ml:id xml:space="preserve">Vβ</ml:id></ml:apply><ml:apply><ml:transpose /><ml:id xml:space="preserve">A</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="64" /></region><region region-id="2757" left="144" top="2775" width="60.75" height="12.75" align-x="154.5" align-y="2784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">k</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">vx</ml:id></ml:apply></ml:range></ml:define></math><rendering item-idref="65" /></region><region 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/><ml:apply><ml:id xml:space="preserve">fit</ml:id><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">a</ml:id><ml:id xml:space="preserve">b</ml:id><ml:id xml:space="preserve">c</ml:id></ml:sequence></ml:apply><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">t</ml:id><ml:id xml:space="preserve">S</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="67" /></region><region region-id="2759" left="270" top="2805" width="108" height="12.75" align-x="295.5" align-y="2814" show-border="false" show-highlight="false" is-protected="false" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">dlow</ml:id><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">fit</ml:id><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">a</ml:id><ml:id xml:space="preserve">b</ml:id><ml:id 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bars on each data point:</p></text></region><region region-id="2750" left="66" top="3012" width="239.25" height="180" align-x="66" align-y="3012" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="71" /><rendering item-idref="72" /></region><region region-id="2506" left="0" top="3204" width="6000" height="6" align-x="0" align-y="3204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2785" left="36" top="3218.25" width="347.25" height="24" align-x="36" align-y="3228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">And finally, for the measured values of y, we can calculate confidence intervals at any measured point in vx.</p></text></region><region region-id="2791" left="66" top="3253.5" width="106.5" height="19.5" align-x="102" align-y="3264" 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">Smeas</ml:id><ml:id xml:space="preserve">k</ml:id></ml:apply><ml:apply><ml:sqrt /><ml:apply><ml:plus /><ml:id xml:space="preserve">MSE</ml:id><ml:apply><ml:indexer /><ml:id xml:space="preserve">Vy</ml:id><ml:sequence><ml:id xml:space="preserve">k</ml:id><ml:id xml:space="preserve">k</ml:id></ml:sequence></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="73" /></region><region region-id="2798" left="66" top="3285" width="132.75" height="12.75" align-x="95.25" align-y="3294" show-border="false" show-highlight="false" 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xml:space="preserve">dlow</ml:id><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">fit</ml:id><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">a</ml:id><ml:id xml:space="preserve">b</ml:id><ml:id xml:space="preserve">c</ml:id></ml:sequence></ml:apply><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">t</ml:id><ml:id xml:space="preserve">Smeas</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="75" /></region><region region-id="2790" left="66" top="3312" width="233.25" height="156" align-x="66" align-y="3312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="76" /><rendering item-idref="77" /></region><region region-id="2493" left="36" top="3482.25" width="80.25" height="12" align-x="36" align-y="3492" 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">Note that either</p></text></region><region region-id="2498" left="66" top="3501" width="101.25" height="18.75" align-x="119.25" align-y="3516" 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="WarnRedefinedUDFunction"><ml:function><ml:id xml:space="preserve">fit</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">a</ml:id><ml:id xml:space="preserve">b</ml:id><ml:id xml:space="preserve">c</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">a</ml:id><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">b</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply><ml:id xml:space="preserve">c</ml:id></ml:apply></ml:define></math><rendering item-idref="78" /></region><region region-id="2507" left="222" top="3499.5" width="100.5" height="22.5" align-x="258" align-y="3516" 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="WarnRedefinedUDFunction"><ml:function><ml:id xml:space="preserve">fit</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">β</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><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:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="79" /></region><region region-id="2502" left="36" top="3530.25" width="328.5" height="24" align-x="36" align-y="3540" 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">are acceptable ways to specify the fit function for the <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">confidence</c></inlineAttr></b></f> function.</p></text></region><region region-id="2508" left="6" top="3570" width="427.5" height="5.25" 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