<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Tables of the power function of the χ2 test.</title>
<link rel="stylesheet" href="../../../../../../../../../../doc/src/boostbook.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.74.0">
<link rel="home" href="../../../../../index.html" title="Math Toolkit">
<link rel="up" href="../nccs_eg.html" title="Non Central Chi Squared Example">
<link rel="prev" href="../nccs_eg.html" title="Non Central Chi Squared Example">
<link rel="next" href="../error_eg.html" title="Error Handling Example">
</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="../nccs_eg.html"><img src="../../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../nccs_eg.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="../error_eg.html"><img src="../../../../../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section" lang="en">
<div class="titlepage"><div><div><h6 class="title">
<a name="math_toolkit.dist.stat_tut.weg.nccs_eg.nccs_power_eg"></a><a class="link" href="nccs_power_eg.html" title="Tables of the power function of the χ2 test.">
Tables of the power function of the χ<sup>2</sup> test.</a>
</h6></div></div></div>
<p>
</p>
<p>
This example computes a table of the power of the χ<sup>2</sup>
test at the 5%
significance level, for various degrees of freedom and non-centrality
parameters. The table is deliberately the same as Table 6 from "The
Non-Central χ<sup>2</sup> and F-Distributions and their applications.", P.
B. Patnaik, Biometrika, Vol. 36, No. 1/2 (June 1949), 202-232.
</p>
<p>
</p>
<p>
First we need some includes to access the non-central chi squared
distribution (and some basic std output of course).
</p>
<p>
</p>
<p>
</p>
<pre class="programlisting"><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">non_central_chi_squared</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="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
<span class="special">{</span></pre>
<p>
</p>
<p>
</p>
<p>
Create a table of the power of the χ<sup>2</sup> test at 5% significance level,
start with a table header:
</p>
<p>
</p>
<p>
</p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"[table\n[[[nu]]"</span><span class="special">;</span>
<span class="keyword">for</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">lam</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span> <span class="identifier">lam</span> <span class="special"><=</span> <span class="number">20</span><span class="special">;</span> <span class="identifier">lam</span> <span class="special">+=</span> <span class="number">2</span><span class="special">)</span>
<span class="special">{</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"[[lambda]="</span> <span class="special"><<</span> <span class="identifier">lam</span> <span class="special"><<</span> <span class="string">"]"</span><span class="special">;</span>
<span class="special">}</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"]\n"</span><span class="special">;</span></pre>
<p>
</p>
<p>
</p>
<p>
(Note: the enclosing [] brackets are to format as a table in Boost.Quickbook).
</p>
<p>
</p>
<p>
Enumerate the rows and columns and print the power of the test for
each table cell:
</p>
<p>
</p>
<p>
</p>
<pre class="programlisting"><span class="keyword">for</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span> <span class="identifier">n</span> <span class="special"><=</span> <span class="number">20</span><span class="special">;</span> <span class="special">++</span><span class="identifier">n</span><span class="special">)</span>
<span class="special">{</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"[["</span> <span class="special"><<</span> <span class="identifier">n</span> <span class="special"><<</span> <span class="string">"]"</span><span class="special">;</span>
<span class="keyword">for</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">lam</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span> <span class="identifier">lam</span> <span class="special"><=</span> <span class="number">20</span><span class="special">;</span> <span class="identifier">lam</span> <span class="special">+=</span> <span class="number">2</span><span class="special">)</span>
<span class="special">{</span></pre>
<p>
</p>
<p>
</p>
<p>
Calculate the χ<sup>2</sup> statistic for a 5% significance:
</p>
<p>
</p>
<p>
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">cs</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">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">chi_squared</span><span class="special">(</span><span class="identifier">n</span><span class="special">),</span> <span class="number">0.05</span><span class="special">));</span></pre>
<p>
</p>
<p>
</p>
<p>
The power of the test is given by the complement of the CDF of the
non-central χ<sup>2</sup> distribution:
</p>
<p>
</p>
<p>
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">beta</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">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">non_central_chi_squared</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span> <span class="identifier">lam</span><span class="special">),</span> <span class="identifier">cs</span><span class="special">));</span></pre>
<p>
</p>
<p>
</p>
<p>
Then output the cell value:
</p>
<p>
</p>
<p>
</p>
<pre class="programlisting"> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"["</span> <span class="special"><<</span> <span class="identifier">std</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">beta</span> <span class="special"><<</span> <span class="string">"]"</span><span class="special">;</span>
<span class="special">}</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"]"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"]"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span></pre>
<p>
</p>
<p>
</p>
<p>
The output from this program is a table in Boost.Quickbook format
as shown below.
</p>
<p>
</p>
<p>
We can interpret this as follows - for example if ν=10 and λ=10 then
the power of the test is 0.542 - so we have only a 54% chance of
correctly detecting that our null hypothesis is false, and a 46%
chance of incurring a type II error (failing to reject the null hypothesis
when it is in fact false):
</p>
<p>
</p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
ν
</p>
</th>
<th>
<p>
λ=2
</p>
</th>
<th>
<p>
λ=4
</p>
</th>
<th>
<p>
λ=6
</p>
</th>
<th>
<p>
λ=8
</p>
</th>
<th>
<p>
λ=10
</p>
</th>
<th>
<p>
λ=12
</p>
</th>
<th>
<p>
λ=14
</p>
</th>
<th>
<p>
λ=16
</p>
</th>
<th>
<p>
λ=18
</p>
</th>
<th>
<p>
λ=20
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
2
</p>
</td>
<td>
<p>
0.226
</p>
</td>
<td>
<p>
0.415
</p>
</td>
<td>
<p>
0.584
</p>
</td>
<td>
<p>
0.718
</p>
</td>
<td>
<p>
0.815
</p>
</td>
<td>
<p>
0.883
</p>
</td>
<td>
<p>
0.928
</p>
</td>
<td>
<p>
0.957
</p>
</td>
<td>
<p>
0.974
</p>
</td>
<td>
<p>
0.985
</p>
</td>
</tr>
<tr>
<td>
<p>
3
</p>
</td>
<td>
<p>
0.192
</p>
</td>
<td>
<p>
0.359
</p>
</td>
<td>
<p>
0.518
</p>
</td>
<td>
<p>
0.654
</p>
</td>
<td>
<p>
0.761
</p>
</td>
<td>
<p>
0.84
</p>
</td>
<td>
<p>
0.896
</p>
</td>
<td>
<p>
0.934
</p>
</td>
<td>
<p>
0.959
</p>
</td>
<td>
<p>
0.975
</p>
</td>
</tr>
<tr>
<td>
<p>
4
</p>
</td>
<td>
<p>
0.171
</p>
</td>
<td>
<p>
0.32
</p>
</td>
<td>
<p>
0.47
</p>
</td>
<td>
<p>
0.605
</p>
</td>
<td>
<p>
0.716
</p>
</td>
<td>
<p>
0.802
</p>
</td>
<td>
<p>
0.866
</p>
</td>
<td>
<p>
0.912
</p>
</td>
<td>
<p>
0.943
</p>
</td>
<td>
<p>
0.964
</p>
</td>
</tr>
<tr>
<td>
<p>
5
</p>
</td>
<td>
<p>
0.157
</p>
</td>
<td>
<p>
0.292
</p>
</td>
<td>
<p>
0.433
</p>
</td>
<td>
<p>
0.564
</p>
</td>
<td>
<p>
0.677
</p>
</td>
<td>
<p>
0.769
</p>
</td>
<td>
<p>
0.839
</p>
</td>
<td>
<p>
0.89
</p>
</td>
<td>
<p>
0.927
</p>
</td>
<td>
<p>
0.952
</p>
</td>
</tr>
<tr>
<td>
<p>
6
</p>
</td>
<td>
<p>
0.146
</p>
</td>
<td>
<p>
0.27
</p>
</td>
<td>
<p>
0.403
</p>
</td>
<td>
<p>
0.531
</p>
</td>
<td>
<p>
0.644
</p>
</td>
<td>
<p>
0.738
</p>
</td>
<td>
<p>
0.813
</p>
</td>
<td>
<p>
0.869
</p>
</td>
<td>
<p>
0.911
</p>
</td>
<td>
<p>
0.94
</p>
</td>
</tr>
<tr>
<td>
<p>
7
</p>
</td>
<td>
<p>
0.138
</p>
</td>
<td>
<p>
0.252
</p>
</td>
<td>
<p>
0.378
</p>
</td>
<td>
<p>
0.502
</p>
</td>
<td>
<p>
0.614
</p>
</td>
<td>
<p>
0.71
</p>
</td>
<td>
<p>
0.788
</p>
</td>
<td>
<p>
0.849
</p>
</td>
<td>
<p>
0.895
</p>
</td>
<td>
<p>
0.928
</p>
</td>
</tr>
<tr>
<td>
<p>
8
</p>
</td>
<td>
<p>
0.131
</p>
</td>
<td>
<p>
0.238
</p>
</td>
<td>
<p>
0.357
</p>
</td>
<td>
<p>
0.477
</p>
</td>
<td>
<p>
0.588
</p>
</td>
<td>
<p>
0.685
</p>
</td>
<td>
<p>
0.765
</p>
</td>
<td>
<p>
0.829
</p>
</td>
<td>
<p>
0.879
</p>
</td>
<td>
<p>
0.915
</p>
</td>
</tr>
<tr>
<td>
<p>
9
</p>
</td>
<td>
<p>
0.125
</p>
</td>
<td>
<p>
0.225
</p>
</td>
<td>
<p>
0.339
</p>
</td>
<td>
<p>
0.454
</p>
</td>
<td>
<p>
0.564
</p>
</td>
<td>
<p>
0.661
</p>
</td>
<td>
<p>
0.744
</p>
</td>
<td>
<p>
0.811
</p>
</td>
<td>
<p>
0.863
</p>
</td>
<td>
<p>
0.903
</p>
</td>
</tr>
<tr>
<td>
<p>
10
</p>
</td>
<td>
<p>
0.121
</p>
</td>
<td>
<p>
0.215
</p>
</td>
<td>
<p>
0.323
</p>
</td>
<td>
<p>
0.435
</p>
</td>
<td>
<p>
0.542
</p>
</td>
<td>
<p>
0.64
</p>
</td>
<td>
<p>
0.723
</p>
</td>
<td>
<p>
0.793
</p>
</td>
<td>
<p>
0.848
</p>
</td>
<td>
<p>
0.891
</p>
</td>
</tr>
<tr>
<td>
<p>
11
</p>
</td>
<td>
<p>
0.117
</p>
</td>
<td>
<p>
0.206
</p>
</td>
<td>
<p>
0.309
</p>
</td>
<td>
<p>
0.417
</p>
</td>
<td>
<p>
0.523
</p>
</td>
<td>
<p>
0.62
</p>
</td>
<td>
<p>
0.704
</p>
</td>
<td>
<p>
0.775
</p>
</td>
<td>
<p>
0.833
</p>
</td>
<td>
<p>
0.878
</p>
</td>
</tr>
<tr>
<td>
<p>
12
</p>
</td>
<td>
<p>
0.113
</p>
</td>
<td>
<p>
0.198
</p>
</td>
<td>
<p>
0.297
</p>
</td>
<td>
<p>
0.402
</p>
</td>
<td>
<p>
0.505
</p>
</td>
<td>
<p>
0.601
</p>
</td>
<td>
<p>
0.686
</p>
</td>
<td>
<p>
0.759
</p>
</td>
<td>
<p>
0.818
</p>
</td>
<td>
<p>
0.866
</p>
</td>
</tr>
<tr>
<td>
<p>
13
</p>
</td>
<td>
<p>
0.11
</p>
</td>
<td>
<p>
0.191
</p>
</td>
<td>
<p>
0.286
</p>
</td>
<td>
<p>
0.387
</p>
</td>
<td>
<p>
0.488
</p>
</td>
<td>
<p>
0.584
</p>
</td>
<td>
<p>
0.669
</p>
</td>
<td>
<p>
0.743
</p>
</td>
<td>
<p>
0.804
</p>
</td>
<td>
<p>
0.854
</p>
</td>
</tr>
<tr>
<td>
<p>
14
</p>
</td>
<td>
<p>
0.108
</p>
</td>
<td>
<p>
0.185
</p>
</td>
<td>
<p>
0.276
</p>
</td>
<td>
<p>
0.374
</p>
</td>
<td>
<p>
0.473
</p>
</td>
<td>
<p>
0.567
</p>
</td>
<td>
<p>
0.653
</p>
</td>
<td>
<p>
0.728
</p>
</td>
<td>
<p>
0.791
</p>
</td>
<td>
<p>
0.842
</p>
</td>
</tr>
<tr>
<td>
<p>
15
</p>
</td>
<td>
<p>
0.105
</p>
</td>
<td>
<p>
0.179
</p>
</td>
<td>
<p>
0.267
</p>
</td>
<td>
<p>
0.362
</p>
</td>
<td>
<p>
0.459
</p>
</td>
<td>
<p>
0.552
</p>
</td>
<td>
<p>
0.638
</p>
</td>
<td>
<p>
0.713
</p>
</td>
<td>
<p>
0.777
</p>
</td>
<td>
<p>
0.83
</p>
</td>
</tr>
<tr>
<td>
<p>
16
</p>
</td>
<td>
<p>
0.103
</p>
</td>
<td>
<p>
0.174
</p>
</td>
<td>
<p>
0.259
</p>
</td>
<td>
<p>
0.351
</p>
</td>
<td>
<p>
0.446
</p>
</td>
<td>
<p>
0.538
</p>
</td>
<td>
<p>
0.623
</p>
</td>
<td>
<p>
0.699
</p>
</td>
<td>
<p>
0.764
</p>
</td>
<td>
<p>
0.819
</p>
</td>
</tr>
<tr>
<td>
<p>
17
</p>
</td>
<td>
<p>
0.101
</p>
</td>
<td>
<p>
0.169
</p>
</td>
<td>
<p>
0.251
</p>
</td>
<td>
<p>
0.341
</p>
</td>
<td>
<p>
0.434
</p>
</td>
<td>
<p>
0.525
</p>
</td>
<td>
<p>
0.609
</p>
</td>
<td>
<p>
0.686
</p>
</td>
<td>
<p>
0.752
</p>
</td>
<td>
<p>
0.807
</p>
</td>
</tr>
<tr>
<td>
<p>
18
</p>
</td>
<td>
<p>
0.0992
</p>
</td>
<td>
<p>
0.165
</p>
</td>
<td>
<p>
0.244
</p>
</td>
<td>
<p>
0.332
</p>
</td>
<td>
<p>
0.423
</p>
</td>
<td>
<p>
0.512
</p>
</td>
<td>
<p>
0.596
</p>
</td>
<td>
<p>
0.673
</p>
</td>
<td>
<p>
0.74
</p>
</td>
<td>
<p>
0.796
</p>
</td>
</tr>
<tr>
<td>
<p>
19
</p>
</td>
<td>
<p>
0.0976
</p>
</td>
<td>
<p>
0.161
</p>
</td>
<td>
<p>
0.238
</p>
</td>
<td>
<p>
0.323
</p>
</td>
<td>
<p>
0.412
</p>
</td>
<td>
<p>
0.5
</p>
</td>
<td>
<p>
0.584
</p>
</td>
<td>
<p>
0.66
</p>
</td>
<td>
<p>
0.728
</p>
</td>
<td>
<p>
0.786
</p>
</td>
</tr>
<tr>
<td>
<p>
20
</p>
</td>
<td>
<p>
0.0961
</p>
</td>
<td>
<p>
0.158
</p>
</td>
<td>
<p>
0.232
</p>
</td>
<td>
<p>
0.315
</p>
</td>
<td>
<p>
0.402
</p>
</td>
<td>
<p>
0.489
</p>
</td>
<td>
<p>
0.572
</p>
</td>
<td>
<p>
0.648
</p>
</td>
<td>
<p>
0.716
</p>
</td>
<td>
<p>
0.775
</p>
</td>
</tr>
</tbody>
</table></div>
<p>
</p>
<p>
See <a href="../../../../../../../../example/nc_chi_sq_example.cpp" target="_top">nc_chi_sq_example.cpp</a>
for the full C++ source code.
</p>
<p>
</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 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>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="../nccs_eg.html"><img src="../../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../nccs_eg.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="../error_eg.html"><img src="../../../../../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>
