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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.stat_tut.weg.f_eg"></a><a class="link" href="f_eg.html" title="F Distribution Examples">F Distribution Examples</a>
  28 </h4></div></div></div>
  29 <p>
  30           Imagine that you want to compare the standard deviations of two sample
  31           to determine if they differ in any significant way, in this situation you
  32           use the F distribution and perform an F-test. This situation commonly occurs
  33           when conducting a process change comparison: "is a new process more
  34           consistent that the old one?".
  35         </p>
  36 <p>
  37           In this example we'll be using the data for ceramic strength from <a href="http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm" target="_top">http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm</a>.
  38           The data for this case study were collected by Said Jahanmir of the NIST
  39           Ceramics Division in 1996 in connection with a NIST/industry ceramics consortium
  40           for strength optimization of ceramic strength.
  41         </p>
  42 <p>
  43           The example program is <a href="../../../../../example/f_test.cpp" target="_top">f_test.cpp</a>,
  44           program output has been deliberately made as similar as possible to the
  45           DATAPLOT output in the corresponding <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda359.htm" target="_top">NIST
  46           EngineeringStatistics Handbook example</a>.
  47         </p>
  48 <p>
  49           We'll begin by defining the procedure to conduct the test:
  50         </p>
  51 <pre class="programlisting"><span class="keyword">void</span> <span class="identifier">f_test</span><span class="special">(</span>
  52     <span class="keyword">double</span> <span class="identifier">sd1</span><span class="special">,</span>     <span class="comment">// Sample 1 std deviation</span>
  53     <span class="keyword">double</span> <span class="identifier">sd2</span><span class="special">,</span>     <span class="comment">// Sample 2 std deviation</span>
  54     <span class="keyword">double</span> <span class="identifier">N1</span><span class="special">,</span>      <span class="comment">// Sample 1 size</span>
  55     <span class="keyword">double</span> <span class="identifier">N2</span><span class="special">,</span>      <span class="comment">// Sample 2 size</span>
  56     <span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">)</span>  <span class="comment">// Significance level</span>
  57 <span class="special">{</span>
  58 </pre>
  59 <p>
  60           The procedure begins by printing out a summary of our input data:
  61         </p>
  62 <pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
  63 <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
  64
  65 <span class="comment">// Print header:</span>
  66 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
  67    <span class="string">"____________________________________\n"</span>
  68    <span class="string">"F test for equal standard deviations\n"</span>
  69    <span class="string">"____________________________________\n\n"</span><span class="special">;</span>
  70 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">);</span>
  71 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Sample 1:\n"</span><span class="special">;</span>
  72 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Number of Observations"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N1</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  73 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Sample Standard Deviation"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sd1</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
  74 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Sample 2:\n"</span><span class="special">;</span>
  75 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Number of Observations"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N2</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  76 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Sample Standard Deviation"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sd2</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
  77 </pre>
  78 <p>
  79           The test statistic for an F-test is simply the ratio of the square of the
  80           two standard deviations:
  81         </p>
  82 <p>
  83           F = s<sub>1</sub><sup>2</sup> / s<sub>2</sub><sup>2</sup>
  84         </p>
  85 <p>
  86           where s<sub>1</sub> is the standard deviation of the first sample and s<sub>2</sub>
  87 is the standard
  88           deviation of the second sample. Or in code:
  89         </p>
  90 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">F</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">sd1</span> <span class="special">/</span> <span class="identifier">sd2</span><span class="special">);</span>
  91 <span class="identifier">F</span> <span class="special">*=</span> <span class="identifier">F</span><span class="special">;</span>
  92 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Test Statistic"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">F</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
  93 </pre>
  94 <p>
  95           At this point a word of caution: the F distribution is asymmetric, so we
  96           have to be careful how we compute the tests, the following table summarises
  97           the options available:
  98         </p>
  99 <div class="informaltable"><table class="table">
 100 <colgroup>
 101 <col>
 102 <col>
 103 </colgroup>
 104 <thead><tr>
 105 <th>
 106                   <p>
 107                     Hypothesis
 108                   </p>
 109                 </th>
 110 <th>
 111                   <p>
 112                     Test
 113                   </p>
 114                 </th>
 115 </tr></thead>
 116 <tbody>
 117 <tr>
 118 <td>
 119                   <p>
 120                     The null-hypothesis: there is no difference in standard deviations
 121                     (two sided test)
 122                   </p>
 123                 </td>
 124 <td>
 125                   <p>
 126                     Reject if F &lt;= F<sub>(1-alpha/2; N1-1, N2-1)</sub> or F &gt;= F<sub>(alpha/2;
 127                     N1-1, N2-1)</sub>
 128                   </p>
 129                 </td>
 130 </tr>
 131 <tr>
 132 <td>
 133                   <p>
 134                     The alternative hypothesis: there is a difference in means (two
 135                     sided test)
 136                   </p>
 137                 </td>
 138 <td>
 139                   <p>
 140                     Reject if F<sub>(1-alpha/2; N1-1, N2-1)</sub> &lt;= F &lt;= F<sub>(alpha/2; N1-1,
 141                     N2-1)</sub>
 142                   </p>
 143                 </td>
 144 </tr>
 145 <tr>
 146 <td>
 147                   <p>
 148                     The alternative hypothesis: Standard deviation of sample 1 is
 149                     greater than that of sample 2
 150                   </p>
 151                 </td>
 152 <td>
 153                   <p>
 154                     Reject if F &lt; F<sub>(alpha; N1-1, N2-1)</sub>
 155                   </p>
 156                 </td>
 157 </tr>
 158 <tr>
 159 <td>
 160                   <p>
 161                     The alternative hypothesis: Standard deviation of sample 1 is
 162                     less than that of sample 2
 163                   </p>
 164                 </td>
 165 <td>
 166                   <p>
 167                     Reject if F &gt; F<sub>(1-alpha; N1-1, N2-1)</sub>
 168                   </p>
 169                 </td>
 170 </tr>
 171 </tbody>
 172 </table></div>
 173 <p>
 174           Where F<sub>(1-alpha; N1-1, N2-1)</sub> is the lower critical value of the F distribution
 175           with degrees of freedom N1-1 and N2-1, and F<sub>(alpha; N1-1, N2-1)</sub> is the upper
 176           critical value of the F distribution with degrees of freedom N1-1 and N2-1.
 177         </p>
 178 <p>
 179           The upper and lower critical values can be computed using the quantile
 180           function:
 181         </p>
 182 <p>
 183           F<sub>(1-alpha; N1-1, N2-1)</sub> = <code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">fisher_f</span><span class="special">(</span><span class="identifier">N1</span><span class="special">-</span><span class="number">1</span><span class="special">,</span> <span class="identifier">N2</span><span class="special">-</span><span class="number">1</span><span class="special">),</span>
 184           <span class="identifier">alpha</span><span class="special">)</span></code>
 185         </p>
 186 <p>
 187           F<sub>(alpha; N1-1, N2-1)</sub> = <code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">fisher_f</span><span class="special">(</span><span class="identifier">N1</span><span class="special">-</span><span class="number">1</span><span class="special">,</span> <span class="identifier">N2</span><span class="special">-</span><span class="number">1</span><span class="special">),</span>
 188           <span class="identifier">alpha</span><span class="special">))</span></code>
 189         </p>
 190 <p>
 191           In our example program we need both upper and lower critical values for
 192           alpha and for alpha/2:
 193         </p>
 194 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">ucv</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">));</span>
 195 <span class="keyword">double</span> <span class="identifier">ucv2</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span> <span class="special">/</span> <span class="number">2</span><span class="special">));</span>
 196 <span class="keyword">double</span> <span class="identifier">lcv</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">);</span>
 197 <span class="keyword">double</span> <span class="identifier">lcv2</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span> <span class="special">/</span> <span class="number">2</span><span class="special">);</span>
 198 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Upper Critical Value at alpha: "</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span>
 199    <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">ucv</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
 200 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Upper Critical Value at alpha/2: "</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span>
 201    <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">ucv2</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
 202 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Lower Critical Value at alpha: "</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span>
 203    <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">lcv</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
 204 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Lower Critical Value at alpha/2: "</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span>
 205    <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">lcv2</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
 206 </pre>
 207 <p>
 208           The final step is to perform the comparisons given above, and print out
 209           whether the hypothesis is rejected or not:
 210         </p>
 211 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span>
 212    <span class="string">"Results for Alternative Hypothesis and alpha"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span>
 213    <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">4</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">alpha</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
 214 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Alternative Hypothesis                                    Conclusion\n"</span><span class="special">;</span>
 215
 216 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard deviations are unequal (two sided test)          "</span><span class="special">;</span>
 217 <span class="keyword">if</span><span class="special">((</span><span class="identifier">ucv2</span> <span class="special">&lt;</span> <span class="identifier">F</span><span class="special">)</span> <span class="special">||</span> <span class="special">(</span><span class="identifier">lcv2</span> <span class="special">&gt;</span> <span class="identifier">F</span><span class="special">))</span>
 218    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
 219 <span class="keyword">else</span>
 220    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
 221
 222 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard deviation 1 is less than standard deviation 2    "</span><span class="special">;</span>
 223 <span class="keyword">if</span><span class="special">(</span><span class="identifier">lcv</span> <span class="special">&gt;</span> <span class="identifier">F</span><span class="special">)</span>
 224    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
 225 <span class="keyword">else</span>
 226    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
 227
 228 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard deviation 1 is greater than standard deviation 2 "</span><span class="special">;</span>
 229 <span class="keyword">if</span><span class="special">(</span><span class="identifier">ucv</span> <span class="special">&lt;</span> <span class="identifier">F</span><span class="special">)</span>
 230    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
 231 <span class="keyword">else</span>
 232    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
 233 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 234 </pre>
 235 <p>
 236           Using the ceramic strength data as an example we get the following output:
 237         </p>
 238 <pre class="programlisting">F test for equal standard deviations
 239 ____________________________________
 240
 241 Sample 1:
 242 Number of Observations                                 =  240
 243 Sample Standard Deviation                              =  65.549
 244
 245 Sample 2:
 246 Number of Observations                                 =  240
 247 Sample Standard Deviation                              =  61.854
 248
 249 Test Statistic                                         =  1.123
 250
 251 CDF of test statistic:                                 =  8.148e-001
 252 Upper Critical Value at alpha:                         =  1.238e+000
 253 Upper Critical Value at alpha/2:                       =  1.289e+000
 254 Lower Critical Value at alpha:                         =  8.080e-001
 255 Lower Critical Value at alpha/2:                       =  7.756e-001
 256
 257 Results for Alternative Hypothesis and alpha           =  0.0500
 258
 259 Alternative Hypothesis                                    Conclusion
 260 Standard deviations are unequal (two sided test)          REJECTED
 261 Standard deviation 1 is less than standard deviation 2    REJECTED
 262 Standard deviation 1 is greater than standard deviation 2 REJECTED
 263 </pre>
 264 <p>
 265           In this case we are unable to reject the null-hypothesis, and must instead
 266           reject the alternative hypothesis.
 267         </p>
 268 <p>
 269           By contrast let's see what happens when we use some different <a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc32.htm" target="_top">sample
 270           data</a>:, once again from the NIST Engineering Statistics Handbook:
 271           A new procedure to assemble a device is introduced and tested for possible
 272           improvement in time of assembly. The question being addressed is whether
 273           the standard deviation of the new assembly process (sample 2) is better
 274           (i.e., smaller) than the standard deviation for the old assembly process
 275           (sample 1).
 276         </p>
 277 <pre class="programlisting">____________________________________
 278 F test for equal standard deviations
 279 ____________________________________
 280
 281 Sample 1:
 282 Number of Observations                                 =  11.00000
 283 Sample Standard Deviation                              =  4.90820
 284
 285 Sample 2:
 286 Number of Observations                                 =  9.00000
 287 Sample Standard Deviation                              =  2.58740
 288
 289 Test Statistic                                         =  3.59847
 290
 291 CDF of test statistic:                                 =  9.589e-001
 292 Upper Critical Value at alpha:                         =  3.347e+000
 293 Upper Critical Value at alpha/2:                       =  4.295e+000
 294 Lower Critical Value at alpha:                         =  3.256e-001
 295 Lower Critical Value at alpha/2:                       =  2.594e-001
 296
 297 Results for Alternative Hypothesis and alpha           =  0.0500
 298
 299 Alternative Hypothesis                                    Conclusion
 300 Standard deviations are unequal (two sided test)          REJECTED
 301 Standard deviation 1 is less than standard deviation 2    REJECTED
 302 Standard deviation 1 is greater than standard deviation 2 ACCEPTED
 303 </pre>
 304 <p>
 305           In this case we take our null hypothesis as "standard deviation 1
 306           is less than or equal to standard deviation 2", since this represents
 307           the "no change" situation. So we want to compare the upper critical
 308           value at <span class="emphasis"><em>alpha</em></span> (a one sided test) with the test statistic,
 309           and since 3.35 &lt; 3.6 this hypothesis must be rejected. We therefore
 310           conclude that there is a change for the better in our standard deviation.
 311         </p>
 312 </div>
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 315 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 316       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 317       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 318       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 319         Distributed under the Boost Software License, Version 1.0. (See accompanying
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