1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
|
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>F Distribution Examples</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.77.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0">
<link rel="up" href="../weg.html" title="Worked Examples">
<link rel="prev" href="cs_eg/chi_sq_size.html" title="Estimating the Required Sample Sizes for a Chi-Square Test for the Standard Deviation">
<link rel="next" href="binom_eg.html" title="Binomial Distribution Examples">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td>
<td align="center"><a href="../../../../../../../index.html">Home</a></td>
<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td>
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
<td align="center"><a href="../../../../../../../more/index.htm">More</a></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="cs_eg/chi_sq_size.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../weg.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="binom_eg.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<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>
</h4></div></div></div>
<p>
Imagine that you want to compare the standard deviations of two sample
to determine if they differ in any significant way, in this situation you
use the F distribution and perform an F-test. This situation commonly occurs
when conducting a process change comparison: "is a new process more
consistent that the old one?".
</p>
<p>
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>.
The data for this case study were collected by Said Jahanmir of the NIST
Ceramics Division in 1996 in connection with a NIST/industry ceramics consortium
for strength optimization of ceramic strength.
</p>
<p>
The example program is <a href="../../../../../example/f_test.cpp" target="_top">f_test.cpp</a>,
program output has been deliberately made as similar as possible to the
DATAPLOT output in the corresponding <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda359.htm" target="_top">NIST
EngineeringStatistics Handbook example</a>.
</p>
<p>
We'll begin by defining the procedure to conduct the test:
</p>
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">f_test</span><span class="special">(</span>
<span class="keyword">double</span> <span class="identifier">sd1</span><span class="special">,</span> <span class="comment">// Sample 1 std deviation</span>
<span class="keyword">double</span> <span class="identifier">sd2</span><span class="special">,</span> <span class="comment">// Sample 2 std deviation</span>
<span class="keyword">double</span> <span class="identifier">N1</span><span class="special">,</span> <span class="comment">// Sample 1 size</span>
<span class="keyword">double</span> <span class="identifier">N2</span><span class="special">,</span> <span class="comment">// Sample 2 size</span>
<span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">)</span> <span class="comment">// Significance level</span>
<span class="special">{</span>
</pre>
<p>
The procedure begins by printing out a summary of our input data:
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</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="comment">// Print header:</span>
<span class="identifier">cout</span> <span class="special"><<</span>
<span class="string">"____________________________________\n"</span>
<span class="string">"F test for equal standard deviations\n"</span>
<span class="string">"____________________________________\n\n"</span><span class="special">;</span>
<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>
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Sample 1:\n"</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">"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>
<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>
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Sample 2:\n"</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">"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>
<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>
</pre>
<p>
The test statistic for an F-test is simply the ratio of the square of the
two standard deviations:
</p>
<p>
F = s<sub>1</sub><sup>2</sup> / s<sub>2</sub><sup>2</sup>
</p>
<p>
where s<sub>1</sub> is the standard deviation of the first sample and s<sub>2</sub>
is the standard
deviation of the second sample. Or in code:
</p>
<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>
<span class="identifier">F</span> <span class="special">*=</span> <span class="identifier">F</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">"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>
</pre>
<p>
At this point a word of caution: the F distribution is asymmetric, so we
have to be careful how we compute the tests, the following table summarises
the options available:
</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 no difference in standard deviations
(two sided test)
</p>
</td>
<td>
<p>
Reject if F <= F<sub>(1-alpha/2; N1-1, N2-1)</sub> or F >= F<sub>(alpha/2;
N1-1, N2-1)</sub>
</p>
</td>
</tr>
<tr>
<td>
<p>
The alternative hypothesis: there is a difference in means (two
sided test)
</p>
</td>
<td>
<p>
Reject if F<sub>(1-alpha/2; N1-1, N2-1)</sub> <= F <= F<sub>(alpha/2; N1-1,
N2-1)</sub>
</p>
</td>
</tr>
<tr>
<td>
<p>
The alternative hypothesis: Standard deviation of sample 1 is
greater than that of sample 2
</p>
</td>
<td>
<p>
Reject if F < F<sub>(alpha; N1-1, N2-1)</sub>
</p>
</td>
</tr>
<tr>
<td>
<p>
The alternative hypothesis: Standard deviation of sample 1 is
less than that of sample 2
</p>
</td>
<td>
<p>
Reject if F > F<sub>(1-alpha; N1-1, N2-1)</sub>
</p>
</td>
</tr>
</tbody>
</table></div>
<p>
Where F<sub>(1-alpha; N1-1, N2-1)</sub> is the lower critical value of the F distribution
with degrees of freedom N1-1 and N2-1, and F<sub>(alpha; N1-1, N2-1)</sub> is the upper
critical value of the F distribution with degrees of freedom N1-1 and N2-1.
</p>
<p>
The upper and lower critical values can be computed using the quantile
function:
</p>
<p>
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>
<span class="identifier">alpha</span><span class="special">)</span></code>
</p>
<p>
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>
<span class="identifier">alpha</span><span class="special">))</span></code>
</p>
<p>
In our example program we need both upper and lower critical values for
alpha and for alpha/2:
</p>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
</pre>
<p>
The final step is to perform the comparisons given above, and print out
whether the hypothesis is rejected or not:
</p>
<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>
<span class="string">"Results for Alternative Hypothesis and alpha"</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">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>
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Alternative Hypothesis Conclusion\n"</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Standard deviations are unequal (two sided test) "</span><span class="special">;</span>
<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>
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
<span class="keyword">else</span>
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
<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>
<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>
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
<span class="keyword">else</span>
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
<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>
<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>
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
<span class="keyword">else</span>
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
<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>
</pre>
<p>
Using the ceramic strength data as an example we get the following output:
</p>
<pre class="programlisting">F test for equal standard deviations
____________________________________
Sample 1:
Number of Observations = 240
Sample Standard Deviation = 65.549
Sample 2:
Number of Observations = 240
Sample Standard Deviation = 61.854
Test Statistic = 1.123
CDF of test statistic: = 8.148e-001
Upper Critical Value at alpha: = 1.238e+000
Upper Critical Value at alpha/2: = 1.289e+000
Lower Critical Value at alpha: = 8.080e-001
Lower Critical Value at alpha/2: = 7.756e-001
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Standard deviations are unequal (two sided test) REJECTED
Standard deviation 1 is less than standard deviation 2 REJECTED
Standard deviation 1 is greater than standard deviation 2 REJECTED
</pre>
<p>
In this case we are unable to reject the null-hypothesis, and must instead
reject the alternative hypothesis.
</p>
<p>
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
data</a>:, once again from the NIST Engineering Statistics Handbook:
A new procedure to assemble a device is introduced and tested for possible
improvement in time of assembly. The question being addressed is whether
the standard deviation of the new assembly process (sample 2) is better
(i.e., smaller) than the standard deviation for the old assembly process
(sample 1).
</p>
<pre class="programlisting">____________________________________
F test for equal standard deviations
____________________________________
Sample 1:
Number of Observations = 11.00000
Sample Standard Deviation = 4.90820
Sample 2:
Number of Observations = 9.00000
Sample Standard Deviation = 2.58740
Test Statistic = 3.59847
CDF of test statistic: = 9.589e-001
Upper Critical Value at alpha: = 3.347e+000
Upper Critical Value at alpha/2: = 4.295e+000
Lower Critical Value at alpha: = 3.256e-001
Lower Critical Value at alpha/2: = 2.594e-001
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Standard deviations are unequal (two sided test) REJECTED
Standard deviation 1 is less than standard deviation 2 REJECTED
Standard deviation 1 is greater than standard deviation 2 ACCEPTED
</pre>
<p>
In this case we take our null hypothesis as "standard deviation 1
is less than or equal to standard deviation 2", since this represents
the "no change" situation. So we want to compare the upper critical
value at <span class="emphasis"><em>alpha</em></span> (a one sided test) with the test statistic,
and since 3.35 < 3.6 this hypothesis must be rejected. We therefore
conclude that there is a change for the better in our standard deviation.
</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta,
Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="cs_eg/chi_sq_size.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../weg.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="binom_eg.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>
|
