示例：ANOVA 和分块

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示例：ANOVA 和分块

使用 block 和 anova 函数将设计矩阵分为两块并检验分块是否影响结果。

1. 调用函数 fullfact 创建完全析因设计矩阵。

2. 调用函数 block 将设计矩阵 X 分为两块。

前八个游程位于 Block 1 中而其余游程位于 Block 2 中。

3. 执行实验前调用函数 randomize。对每个块都会分别执行该随机函数。

4. 将实验结果记录在矩阵 Y 中，在分块后的设计矩阵 B 中，每个游程为一行，每个副本为一列。

5. 调用函数 quickscreen 计算各因子、二阶交互作用和分块的效果。

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    <define xmlns="http://schemas.mathsoft.com/math50">
      <id labels="VARIABLE" xml:space="preserve">Q</id>
      <eval>
        <apply>
          <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">quickscreen</id>
          <sequence>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">B</id>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Y</id>
            <real>2</real>
          </sequence>
        </apply>
        <unitOverride>
          <placeholder />
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    <resultFormat>
      <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
      <matrix size="17,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
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6. 使用函数 augment 和 submatrix 从 Q 中提取因子及其效果，并移除页眉。

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      <id labels="VARIABLE" xml:space="preserve">data</id>
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        <sequence>
          <apply>
            <matcol />
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q</id>
            <real>0</real>
          </apply>
          <apply>
            <matcol />
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q</id>
            <real>3</real>
          </apply>
        </sequence>
      </apply>
    </define>
    <resultFormat>
      <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
      <matrix size="17,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
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    <define xmlns="http://schemas.mathsoft.com/math50">
      <id labels="VARIABLE" xml:space="preserve">data</id>
      <eval>
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          <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">submatrix</id>
          <sequence>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">data</id>
            <real>1</real>
            <apply>
              <minus />
              <apply>
                <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">rows</id>
                <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">data</id>
              </apply>
              <real>1</real>
            </apply>
            <real>0</real>
            <real>1</real>
          </sequence>
        </apply>
        <unitOverride>
          <placeholder />
        </unitOverride>
      </eval>
    </define>
  </math>
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7. 用半数效应的绝对值替换该效果。

8. 调用函数 pareto 并创建柏拉图。

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          <traceStyle color="#FF008000" symbol="filledRectangle" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
      </traces>
      <graph-size width="422.015384615385" height="283.2" />
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="0" end="10">
          <axisLine position="origin" positionticmark="0" legendWidth="73.44" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
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                  <real>0</real>
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          <axisLine position="left" positionticmark="-1" legendWidth="123.046666666667" />
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            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
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            <tickMarks display="true" />
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          <axisLabel />
          <markers />
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                <apply xmlns="http://schemas.mathsoft.com/math50">
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                  <real>1</real>
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                    <summation />
                    <lambda>
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                      <apply>
                        <matcol />
                        <id xml:space="preserve">P</id>
                        <real>1</real>
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                    <upperBound>
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                  <real>100</real>
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                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <matcol />
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因子 A、B、D、交互作用 AD 和 BD，以及 Blocks 看起来影响显著。

9. 调用函数 anova 执行方差分析。计算因子、交互作用和分块的临界 F-value。比较其 F-value 与临界 F-value。

10. 使用函数 qF 计算因子、交互作用和分块的临界 F-value。比较其 F-value 与临界 F-value。

将级设为 5%：	<region id="ID0E3DGW" actualWidth="62.706666666666671" actualHeight="25.6" top="3878.4000000000005" left="326.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="80"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">α</id> <real>0.05</real> </define> <resultFormat> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" /> <matrix size="17,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
设置最低自由度 DF：	<region id="ID0E5EGW" actualWidth="56.150000000000006" actualHeight="27.791200000000003" top="3916.8000000000006" left="326.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="82"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">df<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">A</Subscript></Span> </id> <real>1</real> </define> <resultFormat> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" /> <matrix size="17,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
设置最高自由度 DF：	<region id="ID0EAGGW" actualWidth="64.233333333333348" actualHeight="27.791200000000003" top="3955.2000000000007" left="326.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="84"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">df<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">E</Subscript></Span> </id> <real>36</real> </define> <resultFormat> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" /> <matrix size="17,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EUGGW" actualWidth="241.32000000000002" actualHeight="27.791200000000003" top="3993.6000000000004" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="86"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="*" xml:space="preserve"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">F<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">crit</Subscript></Span> </id> <eval> <apply> <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">qF</id> <sequence> <apply> <minus /> <real>1</real> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">α</id> </apply> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">df<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">A</Subscript></Span> </id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true"> <Span xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation">df<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">E</Subscript></Span> </id> </sequence> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </define> <resultFormat> <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" /> <matrix size="17,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>

级为 5% 时，因子 A、B、D、交互作用 AD 和 BD，以及 Blocks 影响显著，因为其 F 值大于 Fcrit。此方差分析可强化由柏拉图得出的主观结论。

参考文献

Montgomery, D.C., Design and Analysis of Experiments, 5th ed., John Wiley & Sons, New York, 2001, pp. 295.