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| <div class="titlepage"><div><div><h6 class="title"> |
| <a name="math_toolkit.dist.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 the means of two samples with the Students-t test</a> |
| </h6></div></div></div> |
| <p> |
| Imagine that we have two samples, and we wish to determine whether |
| their means are different or not. This situation often arises when |
| determining whether a new process or treatment is better than an old |
| one. |
| </p> |
| <p> |
| In this example, we'll be using the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3531.htm" target="_top">Car |
| Mileage sample data</a> from the <a href="http://www.itl.nist.gov" target="_top">NIST |
| website</a>. The data compares miles per gallon of US cars with |
| miles per gallon of Japanese cars. |
| </p> |
| <p> |
| The sample code is in <a href="../../../../../../../../example/students_t_two_samples.cpp" target="_top">students_t_two_samples.cpp</a>. |
| </p> |
| <p> |
| There are two ways in which this test can be conducted: we can assume |
| that the true standard deviations of the two samples are equal or not. |
| If the standard deviations are assumed to be equal, then the calculation |
| of the t-statistic is greatly simplified, so we'll examine that case |
| first. In real life we should verify whether this assumption is valid |
| with a Chi-Squared test for equal variances. |
| </p> |
| <p> |
| We begin by defining a procedure that will conduct our test assuming |
| equal variances: |
| </p> |
| <pre class="programlisting"><span class="comment">// Needed headers: |
| </span><span class="preprocessor">#include</span> <span class="special"><</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">></span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></span> |
| <span class="comment">// Simplify usage: |
| </span><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> |
| <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span> |
| |
| <span class="keyword">void</span> <span class="identifier">two_samples_t_test_equal_sd</span><span class="special">(</span> |
| <span class="keyword">double</span> <span class="identifier">Sm1</span><span class="special">,</span> <span class="comment">// Sm1 = Sample 1 Mean. |
| </span> <span class="keyword">double</span> <span class="identifier">Sd1</span><span class="special">,</span> <span class="comment">// Sd1 = Sample 1 Standard Deviation. |
| </span> <span class="keyword">unsigned</span> <span class="identifier">Sn1</span><span class="special">,</span> <span class="comment">// Sn1 = Sample 1 Size. |
| </span> <span class="keyword">double</span> <span class="identifier">Sm2</span><span class="special">,</span> <span class="comment">// Sm2 = Sample 2 Mean. |
| </span> <span class="keyword">double</span> <span class="identifier">Sd2</span><span class="special">,</span> <span class="comment">// Sd2 = Sample 2 Standard Deviation. |
| </span> <span class="keyword">unsigned</span> <span class="identifier">Sn2</span><span class="special">,</span> <span class="comment">// Sn2 = Sample 2 Size. |
| </span> <span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">)</span> <span class="comment">// alpha = Significance Level. |
| </span><span class="special">{</span> |
| </pre> |
| <p> |
| Our procedure will begin by calculating the t-statistic, assuming equal |
| variances the needed formulae are: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../../equations/dist_tutorial1.png"></span> |
| </p> |
| <p> |
| where Sp is the "pooled" standard deviation of the two samples, |
| and <span class="emphasis"><em>v</em></span> is the number of degrees of freedom of the |
| two combined samples. We can now write the code to calculate the t-statistic: |
| </p> |
| <pre class="programlisting"><span class="comment">// Degrees of freedom: |
| </span><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> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Degrees of Freedom"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">v</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> |
| <span class="comment">// Pooled variance: |
| </span><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> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Pooled Standard Deviation"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">v</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> |
| <span class="comment">// t-statistic: |
| </span><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> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"T Statistic"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">t_stat</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> |
| </pre> |
| <p> |
| The next step is to define our distribution object, and calculate the |
| complement of the probability: |
| </p> |
| <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> |
| <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> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Probability that difference is due to chance"</span> <span class="special"><<</span> <span class="string">"= "</span> |
| <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">scientific</span> <span class="special"><<</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">q</span> <span class="special"><<</span> <span class="string">"\n\n"</span><span class="special">;</span> |
| </pre> |
| <p> |
| Here we've used the absolute value of the t-statistic, because we initially |
| want to know simply whether there is a difference or not (a two-sided |
| test). However, we can also test whether the mean of the second sample |
| is greater or is less (one-sided test) than that of the first: all |
| the possible tests are summed up in the following table: |
| </p> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Hypothesis |
| </p> |
| </th> |
| <th> |
| <p> |
| Test |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| The Null-hypothesis: there is <span class="bold"><strong>no difference</strong></span> |
| in means |
| </p> |
| </td> |
| <td> |
| <p> |
| Reject if complement of CDF for |t| < significance level |
| / 2: |
| </p> |
| <p> |
| <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> |
| <span class="identifier">fabs</span><span class="special">(</span><span class="identifier">t</span><span class="special">)))</span> |
| <span class="special"><</span> <span class="identifier">alpha</span> |
| <span class="special">/</span> <span class="number">2</span></code> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| The Alternative-hypothesis: there is a <span class="bold"><strong>difference</strong></span> |
| in means |
| </p> |
| </td> |
| <td> |
| <p> |
| Reject if complement of CDF for |t| > significance level |
| / 2: |
| </p> |
| <p> |
| <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> |
| <span class="identifier">fabs</span><span class="special">(</span><span class="identifier">t</span><span class="special">)))</span> |
| <span class="special"><</span> <span class="identifier">alpha</span> |
| <span class="special">/</span> <span class="number">2</span></code> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| The Alternative-hypothesis: Sample 1 Mean is <span class="bold"><strong>less</strong></span> |
| than Sample 2 Mean. |
| </p> |
| </td> |
| <td> |
| <p> |
| Reject if CDF of t > significance level: |
| </p> |
| <p> |
| <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> |
| <span class="identifier">t</span><span class="special">)</span> |
| <span class="special">></span> <span class="identifier">alpha</span></code> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| The Alternative-hypothesis: Sample 1 Mean is <span class="bold"><strong>greater</strong></span> |
| than Sample 2 Mean. |
| </p> |
| </td> |
| <td> |
| <p> |
| Reject if complement of CDF of t > significance level: |
| </p> |
| <p> |
| <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> |
| <span class="identifier">t</span><span class="special">))</span> |
| <span class="special">></span> <span class="identifier">alpha</span></code> |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| <div class="note"><table border="0" summary="Note"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../../../doc/src/images/note.png"></td> |
| <th align="left">Note</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| For a two-sided test we must compare against alpha / 2 and not alpha. |
| </p></td></tr> |
| </table></div> |
| <p> |
| Most of the rest of the sample program is pretty-printing, so we'll |
| skip over that, and take a look at the sample output for alpha=0.05 |
| (a 95% probability level). For comparison the dataplot output for the |
| same data is in <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm" target="_top">section |
| 1.3.5.3</a> of the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH |
| e-Handbook of Statistical Methods.</a>. |
| </p> |
| <pre class="programlisting"> ________________________________________________ |
| Student t test for two samples (equal variances) |
| ________________________________________________ |
| |
| Number of Observations (Sample 1) = 249 |
| Sample 1 Mean = 20.14458 |
| Sample 1 Standard Deviation = 6.41470 |
| Number of Observations (Sample 2) = 79 |
| Sample 2 Mean = 30.48101 |
| Sample 2 Standard Deviation = 6.10771 |
| Degrees of Freedom = 326.00000 |
| Pooled Standard Deviation = 326.00000 |
| T Statistic = -12.62059 |
| Probability that difference is due to chance = 5.273e-030 |
| |
| Results for Alternative Hypothesis and alpha = 0.0500 |
| |
| Alternative Hypothesis Conclusion |
| Sample 1 Mean != Sample 2 Mean NOT REJECTED |
| Sample 1 Mean < Sample 2 Mean NOT REJECTED |
| Sample 1 Mean > Sample 2 Mean REJECTED |
| </pre> |
| <p> |
| So with a probability that the difference is due to chance of just |
| 5.273e-030, we can safely conclude that there is indeed a difference. |
| </p> |
| <p> |
| The tests on the alternative hypothesis show that we must also reject |
| the hypothesis that Sample 1 Mean is greater than that for Sample 2: |
| in this case Sample 1 represents the miles per gallon for Japanese |
| cars, and Sample 2 the miles per gallon for US cars, so we conclude |
| that Japanese cars are on average more fuel efficient. |
| </p> |
| <p> |
| Now that we have the simple case out of the way, let's look for a moment |
| at the more complex one: that the standard deviations of the two samples |
| are not equal. In this case the formula for the t-statistic becomes: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../../equations/dist_tutorial2.png"></span> |
| </p> |
| <p> |
| 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> |
| approximation: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../../equations/dist_tutorial3.png"></span> |
| </p> |
| <p> |
| Note that this is one of the rare situations where the degrees-of-freedom |
| parameter to the Student's t distribution is a real number, and not |
| an integer value. |
| </p> |
| <div class="note"><table border="0" summary="Note"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../../../doc/src/images/note.png"></td> |
| <th align="left">Note</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| Some statistical packages truncate the effective degrees of freedom |
| to an integer value: this may be necessary if you are relying on |
| lookup tables, but since our code fully supports non-integer degrees |
| of freedom there is no need to truncate in this case. Also note that |
| when the degrees of freedom is small then the Welch-Satterthwaite |
| approximation may be a significant source of error. |
| </p></td></tr> |
| </table></div> |
| <p> |
| Putting these formulae into code we get: |
| </p> |
| <pre class="programlisting"><span class="comment">// Degrees of freedom: |
| </span><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> |
| <span class="identifier">v</span> <span class="special">*=</span> <span class="identifier">v</span><span class="special">;</span> |
| <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> |
| <span class="identifier">t1</span> <span class="special">*=</span> <span class="identifier">t1</span><span class="special">;</span> |
| <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> |
| <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> |
| <span class="identifier">t2</span> <span class="special">*=</span> <span class="identifier">t2</span><span class="special">;</span> |
| <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> |
| <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> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Degrees of Freedom"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">v</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> |
| <span class="comment">// t-statistic: |
| </span><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> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"T Statistic"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">t_stat</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> |
| </pre> |
| <p> |
| Thereafter the code and the tests are performed the same as before. |
| Using are car mileage data again, here's what the output looks like: |
| </p> |
| <pre class="programlisting"> __________________________________________________ |
| Student t test for two samples (unequal variances) |
| __________________________________________________ |
| |
| Number of Observations (Sample 1) = 249 |
| Sample 1 Mean = 20.145 |
| Sample 1 Standard Deviation = 6.4147 |
| Number of Observations (Sample 2) = 79 |
| Sample 2 Mean = 30.481 |
| Sample 2 Standard Deviation = 6.1077 |
| Degrees of Freedom = 136.87 |
| T Statistic = -12.946 |
| Probability that difference is due to chance = 1.571e-025 |
| |
| Results for Alternative Hypothesis and alpha = 0.0500 |
| |
| Alternative Hypothesis Conclusion |
| Sample 1 Mean != Sample 2 Mean NOT REJECTED |
| Sample 1 Mean < Sample 2 Mean NOT REJECTED |
| Sample 1 Mean > Sample 2 Mean REJECTED |
| </pre> |
| <p> |
| This time allowing the variances in the two samples to differ has yielded |
| a higher likelihood that the observed difference is down to chance |
| alone (1.571e-025 compared to 5.273e-030 when equal variances were |
| assumed). However, the conclusion remains the same: US cars are less |
| fuel efficient than Japanese models. |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
| </p> |
| </div></td> |
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