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  26 <div class="titlepage"><div><div><h5 class="title">
  27 <a name="math_toolkit.stat_tut.weg.st_eg.two_sample_students_t"></a><a class="link" href="two_sample_students_t.html" title="Comparing the means of two samples with the Students-t test">Comparing
  28           the means of two samples with the Students-t test</a>
  29 </h5></div></div></div>
  30 <p>
  31             Imagine that we have two samples, and we wish to determine whether their
  32             means are different or not. This situation often arises when determining
  33             whether a new process or treatment is better than an old one.
  34           </p>
  35 <p>
  36             In this example, we'll be using the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3531.htm" target="_top">Car
  37             Mileage sample data</a> from the <a href="http://www.itl.nist.gov" target="_top">NIST
  38             website</a>. The data compares miles per gallon of US cars with miles
  39             per gallon of Japanese cars.
  40           </p>
  41 <p>
  42             The sample code is in <a href="../../../../../../example/students_t_two_samples.cpp" target="_top">students_t_two_samples.cpp</a>.
  43           </p>
  44 <p>
  45             There are two ways in which this test can be conducted: we can assume
  46             that the true standard deviations of the two samples are equal or not.
  47             If the standard deviations are assumed to be equal, then the calculation
  48             of the t-statistic is greatly simplified, so we'll examine that case
  49             first. In real life we should verify whether this assumption is valid
  50             with a Chi-Squared test for equal variances.
  51           </p>
  52 <p>
  53             We begin by defining a procedure that will conduct our test assuming
  54             equal variances:
  55           </p>
  56 <pre class="programlisting"><span class="comment">// Needed headers:</span>
  57 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">students_t</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  58 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
  59 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
  60 <span class="comment">// Simplify usage:</span>
  61 <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>
  62 <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
  63
  64 <span class="keyword">void</span> <span class="identifier">two_samples_t_test_equal_sd</span><span class="special">(</span>
  65         <span class="keyword">double</span> <span class="identifier">Sm1</span><span class="special">,</span>       <span class="comment">// Sm1 = Sample 1 Mean.</span>
  66         <span class="keyword">double</span> <span class="identifier">Sd1</span><span class="special">,</span>       <span class="comment">// Sd1 = Sample 1 Standard Deviation.</span>
  67         <span class="keyword">unsigned</span> <span class="identifier">Sn1</span><span class="special">,</span>     <span class="comment">// Sn1 = Sample 1 Size.</span>
  68         <span class="keyword">double</span> <span class="identifier">Sm2</span><span class="special">,</span>       <span class="comment">// Sm2 = Sample 2 Mean.</span>
  69         <span class="keyword">double</span> <span class="identifier">Sd2</span><span class="special">,</span>       <span class="comment">// Sd2 = Sample 2 Standard Deviation.</span>
  70         <span class="keyword">unsigned</span> <span class="identifier">Sn2</span><span class="special">,</span>     <span class="comment">// Sn2 = Sample 2 Size.</span>
  71         <span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">)</span>     <span class="comment">// alpha = Significance Level.</span>
  72 <span class="special">{</span>
  73 </pre>
  74 <p>
  75             Our procedure will begin by calculating the t-statistic, assuming equal
  76             variances the needed formulae are:
  77           </p>
  78 <div class="blockquote"><blockquote class="blockquote"><p>
  79               <span class="inlinemediaobject"><img src="../../../../../equations/dist_tutorial1.svg"></span>
  80
  81             </p></blockquote></div>
  82 <p>
  83             where Sp is the "pooled" standard deviation of the two samples,
  84             and <span class="emphasis"><em>v</em></span> is the number of degrees of freedom of the
  85             two combined samples. We can now write the code to calculate the t-statistic:
  86           </p>
  87 <pre class="programlisting"><span class="comment">// Degrees of freedom:</span>
  88 <span class="keyword">double</span> <span class="identifier">v</span> <span class="special">=</span> <span class="identifier">Sn1</span> <span class="special">+</span> <span class="identifier">Sn2</span> <span class="special">-</span> <span class="number">2</span><span class="special">;</span>
  89 <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">"Degrees of Freedom"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">v</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  90 <span class="comment">// Pooled variance:</span>
  91 <span class="keyword">double</span> <span class="identifier">sp</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">(((</span><span class="identifier">Sn1</span><span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">Sd1</span> <span class="special">*</span> <span class="identifier">Sd1</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">Sn2</span><span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">Sd2</span> <span class="special">*</span> <span class="identifier">Sd2</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">v</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">"Pooled Standard Deviation"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sp</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  93 <span class="comment">// t-statistic:</span>
  94 <span class="keyword">double</span> <span class="identifier">t_stat</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">Sm1</span> <span class="special">-</span> <span class="identifier">Sm2</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">sp</span> <span class="special">*</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="number">1.0</span> <span class="special">/</span> <span class="identifier">Sn1</span> <span class="special">+</span> <span class="number">1.0</span> <span class="special">/</span> <span class="identifier">Sn2</span><span class="special">));</span>
  95 <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">"T Statistic"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">t_stat</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  96 </pre>
  97 <p>
  98             The next step is to define our distribution object, and calculate the
  99             complement of the probability:
 100           </p>
 101 <pre class="programlisting"><span class="identifier">students_t</span> <span class="identifier">dist</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
 102 <span class="keyword">double</span> <span class="identifier">q</span> <span class="special">=</span> <span class="identifier">cdf</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">fabs</span><span class="special">(</span><span class="identifier">t_stat</span><span class="special">)));</span>
 103 <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">"Probability that difference is due to chance"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span>
 104    <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="number">2</span> <span class="special">*</span> <span class="identifier">q</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
 105 </pre>
 106 <p>
 107             Here we've used the absolute value of the t-statistic, because we initially
 108             want to know simply whether there is a difference or not (a two-sided
 109             test). However, we can also test whether the mean of the second sample
 110             is greater or is less (one-sided test) than that of the first: all the
 111             possible tests are summed up in the following table:
 112           </p>
 113 <div class="informaltable"><table class="table">
 114 <colgroup>
 115 <col>
 116 <col>
 117 </colgroup>
 118 <thead><tr>
 119 <th>
 120                     <p>
 121                       Hypothesis
 122                     </p>
 123                   </th>
 124 <th>
 125                     <p>
 126                       Test
 127                     </p>
 128                   </th>
 129 </tr></thead>
 130 <tbody>
 131 <tr>
 132 <td>
 133                     <p>
 134                       The Null-hypothesis: there is <span class="bold"><strong>no difference</strong></span>
 135                       in means
 136                     </p>
 137                   </td>
 138 <td>
 139                     <p>
 140                       Reject if complement of CDF for |t| &lt; significance level
 141                       / 2:
 142                     </p>
 143                     <p>
 144                       <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
 145                       <span class="identifier">fabs</span><span class="special">(</span><span class="identifier">t</span><span class="special">)))</span>
 146                       <span class="special">&lt;</span> <span class="identifier">alpha</span>
 147                       <span class="special">/</span> <span class="number">2</span></code>
 148                     </p>
 149                   </td>
 150 </tr>
 151 <tr>
 152 <td>
 153                     <p>
 154                       The Alternative-hypothesis: there is a <span class="bold"><strong>difference</strong></span>
 155                       in means
 156                     </p>
 157                   </td>
 158 <td>
 159                     <p>
 160                       Reject if complement of CDF for |t| &gt; significance level
 161                       / 2:
 162                     </p>
 163                     <p>
 164                       <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
 165                       <span class="identifier">fabs</span><span class="special">(</span><span class="identifier">t</span><span class="special">)))</span>
 166                       <span class="special">&gt;</span> <span class="identifier">alpha</span>
 167                       <span class="special">/</span> <span class="number">2</span></code>
 168                     </p>
 169                   </td>
 170 </tr>
 171 <tr>
 172 <td>
 173                     <p>
 174                       The Alternative-hypothesis: Sample 1 Mean is <span class="bold"><strong>less</strong></span>
 175                       than Sample 2 Mean.
 176                     </p>
 177                   </td>
 178 <td>
 179                     <p>
 180                       Reject if CDF of t &gt; significance level:
 181                     </p>
 182                     <p>
 183                       <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
 184                       <span class="identifier">t</span><span class="special">)</span>
 185                       <span class="special">&gt;</span> <span class="identifier">alpha</span></code>
 186                     </p>
 187                   </td>
 188 </tr>
 189 <tr>
 190 <td>
 191                     <p>
 192                       The Alternative-hypothesis: Sample 1 Mean is <span class="bold"><strong>greater</strong></span>
 193                       than Sample 2 Mean.
 194                     </p>
 195                   </td>
 196 <td>
 197                     <p>
 198                       Reject if complement of CDF of t &gt; significance level:
 199                     </p>
 200                     <p>
 201                       <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
 202                       <span class="identifier">t</span><span class="special">))</span>
 203                       <span class="special">&gt;</span> <span class="identifier">alpha</span></code>
 204                     </p>
 205                   </td>
 206 </tr>
 207 </tbody>
 208 </table></div>
 209 <div class="note"><table border="0" summary="Note">
 210 <tr>
 211 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
 212 <th align="left">Note</th>
 213 </tr>
 214 <tr><td align="left" valign="top"><p>
 215               For a two-sided test we must compare against alpha / 2 and not alpha.
 216             </p></td></tr>
 217 </table></div>
 218 <p>
 219             Most of the rest of the sample program is pretty-printing, so we'll skip
 220             over that, and take a look at the sample output for alpha=0.05 (a 95%
 221             probability level). For comparison the dataplot output for the same data
 222             is in <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm" target="_top">section
 223             1.3.5.3</a> of the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
 224             e-Handbook of Statistical Methods.</a>.
 225           </p>
 226 <pre class="programlisting">   ________________________________________________
 227    Student t test for two samples (equal variances)
 228    ________________________________________________
 229
 230    Number of Observations (Sample 1)                      =  249
 231    Sample 1 Mean                                          =  20.145
 232    Sample 1 Standard Deviation                            =  6.4147
 233    Number of Observations (Sample 2)                      =  79
 234    Sample 2 Mean                                          =  30.481
 235    Sample 2 Standard Deviation                            =  6.1077
 236    Degrees of Freedom                                     =  326
 237    Pooled Standard Deviation                              =  6.3426
 238    T Statistic                                            =  -12.621
 239    Probability that difference is due to chance           =  5.273e-030
 240
 241    Results for Alternative Hypothesis and alpha           =  0.0500
 242
 243    Alternative Hypothesis              Conclusion
 244    Sample 1 Mean != Sample 2 Mean       NOT REJECTED
 245    Sample 1 Mean &lt;  Sample 2 Mean       NOT REJECTED
 246    Sample 1 Mean &gt;  Sample 2 Mean       REJECTED
 247 </pre>
 248 <p>
 249             So with a probability that the difference is due to chance of just 5.273e-030,
 250             we can safely conclude that there is indeed a difference.
 251           </p>
 252 <p>
 253             The tests on the alternative hypothesis show that we must also reject
 254             the hypothesis that Sample 1 Mean is greater than that for Sample 2:
 255             in this case Sample 1 represents the miles per gallon for Japanese cars,
 256             and Sample 2 the miles per gallon for US cars, so we conclude that Japanese
 257             cars are on average more fuel efficient.
 258           </p>
 259 <p>
 260             Now that we have the simple case out of the way, let's look for a moment
 261             at the more complex one: that the standard deviations of the two samples
 262             are not equal. In this case the formula for the t-statistic becomes:
 263           </p>
 264 <div class="blockquote"><blockquote class="blockquote"><p>
 265               <span class="inlinemediaobject"><img src="../../../../../equations/dist_tutorial2.svg"></span>
 266
 267             </p></blockquote></div>
 268 <p>
 269             And for the combined degrees of freedom we use the <a href="http://en.wikipedia.org/wiki/Welch-Satterthwaite_equation" target="_top">Welch-Satterthwaite</a>
 270             approximation:
 271           </p>
 272 <div class="blockquote"><blockquote class="blockquote"><p>
 273               <span class="inlinemediaobject"><img src="../../../../../equations/dist_tutorial3.svg"></span>
 274
 275             </p></blockquote></div>
 276 <p>
 277             Note that this is one of the rare situations where the degrees-of-freedom
 278             parameter to the Student's t distribution is a real number, and not an
 279             integer value.
 280           </p>
 281 <div class="note"><table border="0" summary="Note">
 282 <tr>
 283 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
 284 <th align="left">Note</th>
 285 </tr>
 286 <tr><td align="left" valign="top"><p>
 287               Some statistical packages truncate the effective degrees of freedom
 288               to an integer value: this may be necessary if you are relying on lookup
 289               tables, but since our code fully supports non-integer degrees of freedom
 290               there is no need to truncate in this case. Also note that when the
 291               degrees of freedom is small then the Welch-Satterthwaite approximation
 292               may be a significant source of error.
 293             </p></td></tr>
 294 </table></div>
 295 <p>
 296             Putting these formulae into code we get:
 297           </p>
 298 <pre class="programlisting"><span class="comment">// Degrees of freedom:</span>
 299 <span class="keyword">double</span> <span class="identifier">v</span> <span class="special">=</span> <span class="identifier">Sd1</span> <span class="special">*</span> <span class="identifier">Sd1</span> <span class="special">/</span> <span class="identifier">Sn1</span> <span class="special">+</span> <span class="identifier">Sd2</span> <span class="special">*</span> <span class="identifier">Sd2</span> <span class="special">/</span> <span class="identifier">Sn2</span><span class="special">;</span>
 300 <span class="identifier">v</span> <span class="special">*=</span> <span class="identifier">v</span><span class="special">;</span>
 301 <span class="keyword">double</span> <span class="identifier">t1</span> <span class="special">=</span> <span class="identifier">Sd1</span> <span class="special">*</span> <span class="identifier">Sd1</span> <span class="special">/</span> <span class="identifier">Sn1</span><span class="special">;</span>
 302 <span class="identifier">t1</span> <span class="special">*=</span> <span class="identifier">t1</span><span class="special">;</span>
 303 <span class="identifier">t1</span> <span class="special">/=</span>  <span class="special">(</span><span class="identifier">Sn1</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span>
 304 <span class="keyword">double</span> <span class="identifier">t2</span> <span class="special">=</span> <span class="identifier">Sd2</span> <span class="special">*</span> <span class="identifier">Sd2</span> <span class="special">/</span> <span class="identifier">Sn2</span><span class="special">;</span>
 305 <span class="identifier">t2</span> <span class="special">*=</span> <span class="identifier">t2</span><span class="special">;</span>
 306 <span class="identifier">t2</span> <span class="special">/=</span> <span class="special">(</span><span class="identifier">Sn2</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span>
 307 <span class="identifier">v</span> <span class="special">/=</span> <span class="special">(</span><span class="identifier">t1</span> <span class="special">+</span> <span class="identifier">t2</span><span class="special">);</span>
 308 <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">"Degrees of Freedom"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">v</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
 309 <span class="comment">// t-statistic:</span>
 310 <span class="keyword">double</span> <span class="identifier">t_stat</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">Sm1</span> <span class="special">-</span> <span class="identifier">Sm2</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">Sd1</span> <span class="special">*</span> <span class="identifier">Sd1</span> <span class="special">/</span> <span class="identifier">Sn1</span> <span class="special">+</span> <span class="identifier">Sd2</span> <span class="special">*</span> <span class="identifier">Sd2</span> <span class="special">/</span> <span class="identifier">Sn2</span><span class="special">);</span>
 311 <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">"T Statistic"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">t_stat</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
 312 </pre>
 313 <p>
 314             Thereafter the code and the tests are performed the same as before. Using
 315             are car mileage data again, here's what the output looks like:
 316           </p>
 317 <pre class="programlisting">   __________________________________________________
 318    Student t test for two samples (unequal variances)
 319    __________________________________________________
 320
 321    Number of Observations (Sample 1)                      =  249
 322    Sample 1 Mean                                          =  20.145
 323    Sample 1 Standard Deviation                            =  6.4147
 324    Number of Observations (Sample 2)                      =  79
 325    Sample 2 Mean                                          =  30.481
 326    Sample 2 Standard Deviation                            =  6.1077
 327    Degrees of Freedom                                     =  136.87
 328    T Statistic                                            =  -12.946
 329    Probability that difference is due to chance           =  1.571e-025
 330
 331    Results for Alternative Hypothesis and alpha           =  0.0500
 332
 333    Alternative Hypothesis              Conclusion
 334    Sample 1 Mean != Sample 2 Mean       NOT REJECTED
 335    Sample 1 Mean &lt;  Sample 2 Mean       NOT REJECTED
 336    Sample 1 Mean &gt;  Sample 2 Mean       REJECTED
 337 </pre>
 338 <p>
 339             This time allowing the variances in the two samples to differ has yielded
 340             a higher likelihood that the observed difference is down to chance alone
 341             (1.571e-025 compared to 5.273e-030 when equal variances were assumed).
 342             However, the conclusion remains the same: US cars are less fuel efficient
 343             than Japanese models.
 344           </p>
 345 </div>
 346 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 347 <td align="left"></td>
 348 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 349       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 350       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 351       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 352       Daryle Walker and Xiaogang Zhang<p>
 353         Distributed under the Boost Software License, Version 1.0. (See accompanying
 354         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
 355       </p>
 356 </div></td>
 357 </tr></table>
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