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25 | <div class="section"> | |
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"><</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> | |
58 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> | |
59 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></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 | <p> | |
79 | <span class="inlinemediaobject"><img src="../../../../../equations/dist_tutorial1.svg"></span> | |
80 | </p> | |
81 | <p> | |
82 | where Sp is the "pooled" standard deviation of the two samples, | |
83 | and <span class="emphasis"><em>v</em></span> is the number of degrees of freedom of the | |
84 | two combined samples. We can now write the code to calculate the t-statistic: | |
85 | </p> | |
86 | <pre class="programlisting"><span class="comment">// Degrees of freedom:</span> | |
87 | <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> | |
88 | <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> | |
89 | <span class="comment">// Pooled variance:</span> | |
90 | <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> | |
91 | <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">sp</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> | |
92 | <span class="comment">// t-statistic:</span> | |
93 | <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> | |
94 | <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> | |
95 | </pre> | |
96 | <p> | |
97 | The next step is to define our distribution object, and calculate the | |
98 | complement of the probability: | |
99 | </p> | |
100 | <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> | |
101 | <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> | |
102 | <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> | |
103 | <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> | |
104 | </pre> | |
105 | <p> | |
106 | Here we've used the absolute value of the t-statistic, because we initially | |
107 | want to know simply whether there is a difference or not (a two-sided | |
108 | test). However, we can also test whether the mean of the second sample | |
109 | is greater or is less (one-sided test) than that of the first: all the | |
110 | possible tests are summed up in the following table: | |
111 | </p> | |
112 | <div class="informaltable"><table class="table"> | |
113 | <colgroup> | |
114 | <col> | |
115 | <col> | |
116 | </colgroup> | |
117 | <thead><tr> | |
118 | <th> | |
119 | <p> | |
120 | Hypothesis | |
121 | </p> | |
122 | </th> | |
123 | <th> | |
124 | <p> | |
125 | Test | |
126 | </p> | |
127 | </th> | |
128 | </tr></thead> | |
129 | <tbody> | |
130 | <tr> | |
131 | <td> | |
132 | <p> | |
133 | The Null-hypothesis: there is <span class="bold"><strong>no difference</strong></span> | |
134 | in means | |
135 | </p> | |
136 | </td> | |
137 | <td> | |
138 | <p> | |
139 | Reject if complement of CDF for |t| < significance level | |
140 | / 2: | |
141 | </p> | |
142 | <p> | |
143 | <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> | |
144 | <span class="identifier">fabs</span><span class="special">(</span><span class="identifier">t</span><span class="special">)))</span> | |
145 | <span class="special"><</span> <span class="identifier">alpha</span> | |
146 | <span class="special">/</span> <span class="number">2</span></code> | |
147 | </p> | |
148 | </td> | |
149 | </tr> | |
150 | <tr> | |
151 | <td> | |
152 | <p> | |
153 | The Alternative-hypothesis: there is a <span class="bold"><strong>difference</strong></span> | |
154 | in means | |
155 | </p> | |
156 | </td> | |
157 | <td> | |
158 | <p> | |
159 | Reject if complement of CDF for |t| > significance level | |
160 | / 2: | |
161 | </p> | |
162 | <p> | |
163 | <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> | |
164 | <span class="identifier">fabs</span><span class="special">(</span><span class="identifier">t</span><span class="special">)))</span> | |
165 | <span class="special"><</span> <span class="identifier">alpha</span> | |
166 | <span class="special">/</span> <span class="number">2</span></code> | |
167 | </p> | |
168 | </td> | |
169 | </tr> | |
170 | <tr> | |
171 | <td> | |
172 | <p> | |
173 | The Alternative-hypothesis: Sample 1 Mean is <span class="bold"><strong>less</strong></span> | |
174 | than Sample 2 Mean. | |
175 | </p> | |
176 | </td> | |
177 | <td> | |
178 | <p> | |
179 | Reject if CDF of t > significance level: | |
180 | </p> | |
181 | <p> | |
182 | <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> | |
183 | <span class="identifier">t</span><span class="special">)</span> | |
184 | <span class="special">></span> <span class="identifier">alpha</span></code> | |
185 | </p> | |
186 | </td> | |
187 | </tr> | |
188 | <tr> | |
189 | <td> | |
190 | <p> | |
191 | The Alternative-hypothesis: Sample 1 Mean is <span class="bold"><strong>greater</strong></span> | |
192 | than Sample 2 Mean. | |
193 | </p> | |
194 | </td> | |
195 | <td> | |
196 | <p> | |
197 | Reject if complement of CDF of t > significance level: | |
198 | </p> | |
199 | <p> | |
200 | <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> | |
201 | <span class="identifier">t</span><span class="special">))</span> | |
202 | <span class="special">></span> <span class="identifier">alpha</span></code> | |
203 | </p> | |
204 | </td> | |
205 | </tr> | |
206 | </tbody> | |
207 | </table></div> | |
208 | <div class="note"><table border="0" summary="Note"> | |
209 | <tr> | |
210 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td> | |
211 | <th align="left">Note</th> | |
212 | </tr> | |
213 | <tr><td align="left" valign="top"><p> | |
214 | For a two-sided test we must compare against alpha / 2 and not alpha. | |
215 | </p></td></tr> | |
216 | </table></div> | |
217 | <p> | |
218 | Most of the rest of the sample program is pretty-printing, so we'll skip | |
219 | over that, and take a look at the sample output for alpha=0.05 (a 95% | |
220 | probability level). For comparison the dataplot output for the same data | |
221 | is in <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm" target="_top">section | |
222 | 1.3.5.3</a> of the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH | |
223 | e-Handbook of Statistical Methods.</a>. | |
224 | </p> | |
225 | <pre class="programlisting"> ________________________________________________ | |
226 | Student t test for two samples (equal variances) | |
227 | ________________________________________________ | |
228 | ||
229 | Number of Observations (Sample 1) = 249 | |
230 | Sample 1 Mean = 20.145 | |
231 | Sample 1 Standard Deviation = 6.4147 | |
232 | Number of Observations (Sample 2) = 79 | |
233 | Sample 2 Mean = 30.481 | |
234 | Sample 2 Standard Deviation = 6.1077 | |
235 | Degrees of Freedom = 326 | |
236 | Pooled Standard Deviation = 6.3426 | |
237 | T Statistic = -12.621 | |
238 | Probability that difference is due to chance = 5.273e-030 | |
239 | ||
240 | Results for Alternative Hypothesis and alpha = 0.0500 | |
241 | ||
242 | Alternative Hypothesis Conclusion | |
243 | Sample 1 Mean != Sample 2 Mean NOT REJECTED | |
244 | Sample 1 Mean < Sample 2 Mean NOT REJECTED | |
245 | Sample 1 Mean > Sample 2 Mean REJECTED | |
246 | </pre> | |
247 | <p> | |
248 | So with a probability that the difference is due to chance of just 5.273e-030, | |
249 | we can safely conclude that there is indeed a difference. | |
250 | </p> | |
251 | <p> | |
252 | The tests on the alternative hypothesis show that we must also reject | |
253 | the hypothesis that Sample 1 Mean is greater than that for Sample 2: | |
254 | in this case Sample 1 represents the miles per gallon for Japanese cars, | |
255 | and Sample 2 the miles per gallon for US cars, so we conclude that Japanese | |
256 | cars are on average more fuel efficient. | |
257 | </p> | |
258 | <p> | |
259 | Now that we have the simple case out of the way, let's look for a moment | |
260 | at the more complex one: that the standard deviations of the two samples | |
261 | are not equal. In this case the formula for the t-statistic becomes: | |
262 | </p> | |
263 | <p> | |
264 | <span class="inlinemediaobject"><img src="../../../../../equations/dist_tutorial2.svg"></span> | |
265 | </p> | |
266 | <p> | |
267 | 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> | |
268 | approximation: | |
269 | </p> | |
270 | <p> | |
271 | <span class="inlinemediaobject"><img src="../../../../../equations/dist_tutorial3.svg"></span> | |
272 | </p> | |
273 | <p> | |
274 | Note that this is one of the rare situations where the degrees-of-freedom | |
275 | parameter to the Student's t distribution is a real number, and not an | |
276 | integer value. | |
277 | </p> | |
278 | <div class="note"><table border="0" summary="Note"> | |
279 | <tr> | |
280 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td> | |
281 | <th align="left">Note</th> | |
282 | </tr> | |
283 | <tr><td align="left" valign="top"><p> | |
284 | Some statistical packages truncate the effective degrees of freedom | |
285 | to an integer value: this may be necessary if you are relying on lookup | |
286 | tables, but since our code fully supports non-integer degrees of freedom | |
287 | there is no need to truncate in this case. Also note that when the | |
288 | degrees of freedom is small then the Welch-Satterthwaite approximation | |
289 | may be a significant source of error. | |
290 | </p></td></tr> | |
291 | </table></div> | |
292 | <p> | |
293 | Putting these formulae into code we get: | |
294 | </p> | |
295 | <pre class="programlisting"><span class="comment">// Degrees of freedom:</span> | |
296 | <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> | |
297 | <span class="identifier">v</span> <span class="special">*=</span> <span class="identifier">v</span><span class="special">;</span> | |
298 | <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> | |
299 | <span class="identifier">t1</span> <span class="special">*=</span> <span class="identifier">t1</span><span class="special">;</span> | |
300 | <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> | |
301 | <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> | |
302 | <span class="identifier">t2</span> <span class="special">*=</span> <span class="identifier">t2</span><span class="special">;</span> | |
303 | <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> | |
304 | <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> | |
305 | <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> | |
306 | <span class="comment">// t-statistic:</span> | |
307 | <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> | |
308 | <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> | |
309 | </pre> | |
310 | <p> | |
311 | Thereafter the code and the tests are performed the same as before. Using | |
312 | are car mileage data again, here's what the output looks like: | |
313 | </p> | |
314 | <pre class="programlisting"> __________________________________________________ | |
315 | Student t test for two samples (unequal variances) | |
316 | __________________________________________________ | |
317 | ||
318 | Number of Observations (Sample 1) = 249 | |
319 | Sample 1 Mean = 20.145 | |
320 | Sample 1 Standard Deviation = 6.4147 | |
321 | Number of Observations (Sample 2) = 79 | |
322 | Sample 2 Mean = 30.481 | |
323 | Sample 2 Standard Deviation = 6.1077 | |
324 | Degrees of Freedom = 136.87 | |
325 | T Statistic = -12.946 | |
326 | Probability that difference is due to chance = 1.571e-025 | |
327 | ||
328 | Results for Alternative Hypothesis and alpha = 0.0500 | |
329 | ||
330 | Alternative Hypothesis Conclusion | |
331 | Sample 1 Mean != Sample 2 Mean NOT REJECTED | |
332 | Sample 1 Mean < Sample 2 Mean NOT REJECTED | |
333 | Sample 1 Mean > Sample 2 Mean REJECTED | |
334 | </pre> | |
335 | <p> | |
336 | This time allowing the variances in the two samples to differ has yielded | |
337 | a higher likelihood that the observed difference is down to chance alone | |
338 | (1.571e-025 compared to 5.273e-030 when equal variances were assumed). | |
339 | However, the conclusion remains the same: US cars are less fuel efficient | |
340 | than Japanese models. | |
341 | </p> | |
342 | </div> | |
343 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
344 | <td align="left"></td> | |
345 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
346 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
347 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
348 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
349 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
350 | 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>) | |
351 | </p> | |
352 | </div></td> | |
353 | </tr></table> | |
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