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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"><<</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"><<</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"><<</span> <span class="string">"Sample 1:\n"</span><span class="special">;</span> | |
72 | <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">"Number of Observations"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">N1</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> | |
73 | <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">"Sample Standard Deviation"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">sd1</span> <span class="special"><<</span> <span class="string">"\n\n"</span><span class="special">;</span> | |
74 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Sample 2:\n"</span><span class="special">;</span> | |
75 | <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">"Number of Observations"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">N2</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> | |
76 | <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">"Sample Standard Deviation"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">sd2</span> <span class="special"><<</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"><<</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">"Test Statistic"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">F</span> <span class="special"><<</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 <= F<sub>(1-alpha/2; N1-1, N2-1)</sub> or F >= 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> <= F <= 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 < 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 > 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"><<</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">"Upper Critical Value at alpha: "</span> <span class="special"><<</span> <span class="string">"= "</span> | |
199 | <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="identifier">ucv</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> | |
200 | <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">"Upper Critical Value at alpha/2: "</span> <span class="special"><<</span> <span class="string">"= "</span> | |
201 | <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="identifier">ucv2</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> | |
202 | <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">"Lower Critical Value at alpha: "</span> <span class="special"><<</span> <span class="string">"= "</span> | |
203 | <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="identifier">lcv</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> | |
204 | <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">"Lower Critical Value at alpha/2: "</span> <span class="special"><<</span> <span class="string">"= "</span> | |
205 | <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="identifier">lcv2</span> <span class="special"><<</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"><<</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> | |
212 | <span class="string">"Results for Alternative Hypothesis and alpha"</span> <span class="special"><<</span> <span class="string">"= "</span> | |
213 | <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">4</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">alpha</span> <span class="special"><<</span> <span class="string">"\n\n"</span><span class="special">;</span> | |
214 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Alternative Hypothesis Conclusion\n"</span><span class="special">;</span> | |
215 | ||
216 | <span class="identifier">cout</span> <span class="special"><<</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"><</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">></span> <span class="identifier">F</span><span class="special">))</span> | |
218 | <span class="identifier">cout</span> <span class="special"><<</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"><<</span> <span class="string">"REJECTED\n"</span><span class="special">;</span> | |
221 | ||
222 | <span class="identifier">cout</span> <span class="special"><<</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">></span> <span class="identifier">F</span><span class="special">)</span> | |
224 | <span class="identifier">cout</span> <span class="special"><<</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"><<</span> <span class="string">"REJECTED\n"</span><span class="special">;</span> | |
227 | ||
228 | <span class="identifier">cout</span> <span class="special"><<</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"><</span> <span class="identifier">F</span><span class="special">)</span> | |
230 | <span class="identifier">cout</span> <span class="special"><<</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"><<</span> <span class="string">"REJECTED\n"</span><span class="special">;</span> | |
233 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span> <span class="special"><<</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 < 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> | |
313 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
314 | <td align="left"></td> | |
315 | <td align="right"><div class="copyright-footer">Copyright © 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 Råde, 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 | |
320 | 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>) | |
321 | </p> | |
322 | </div></td> | |
323 | </tr></table> | |
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