boost/math
2
0
Fork 0
mirror of https://github.com/boostorg/math.git synced 2026-01-25 06:22:09 +00:00
Code Issues Projects Releases Wiki Activity
Files
b78ef0ea4bf063400fc92cc512aecc609f8fdcac
math/doc/html/math_toolkit/t_test.html
jzmaddock 64154c2d4b Update history, regenerate docs. 
Add some missing files.
[CI SKIP]
2020-10-19 18:19:34 +01:00

153 lines
23 KiB
HTML
Raw Blame History

This file contains invisible Unicode characters
This file contains invisible Unicode characters that are indistinguishable to humans but may be processed differently by a computer. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>t-tests</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 2.13.0">
<link rel="up" href="../statistics.html" title="Chapter 6. Statistics">
<link rel="prev" href="anderson_darling.html" title="The Anderson-Darling Test">
<link rel="next" href="runs_test.html" title="Runs tests">
</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="anderson_darling.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../statistics.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="runs_test.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.t_test"></a><a class="link" href="t_test.html" title="t-tests"><span class="emphasis"><em>t</em></span>-tests</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.t_test.h0"></a>
<span class="phrase"><a name="math_toolkit.t_test.synopsis"></a></span><a class="link" href="t_test.html#math_toolkit.t_test.synopsis">Synopsis</a>
</h4>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">statistics</span><span class="special">/</span><span class="identifier">t_test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</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="identifier">statistics</span> <span class="special">{</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">one_sample_t_test</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">sample_mean</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">sample_variance</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">num_samples</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">assumed_mean</span><span class="special">);</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</span>
<span class="keyword">auto</span> <span class="identifier">one_sample_t_test</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">begin</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">end</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">iterator_traits</span><span class="special">&lt;</span><span class="identifier">ForwardIterator</span><span class="special">&gt;::</span><span class="identifier">value_type</span> <span class="identifier">assumed_mean</span><span class="special">);</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</span>
<span class="keyword">auto</span> <span class="identifier">one_sample_t_test</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">assumed_mean</span><span class="special">);</span>
<span class="special">}}}</span>
</pre>
<h4>
<a name="math_toolkit.t_test.h1"></a>
<span class="phrase"><a name="math_toolkit.t_test.background"></a></span><a class="link" href="t_test.html#math_toolkit.t_test.background">Background</a>
</h4>
<p>
A one-sample <span class="emphasis"><em>t</em></span>-test attempts to answer the question "given
a sample mean, is it likely that the population mean of my data is a certain
value?" The test statistic is
</p>
<p>
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/one_sample_t_test_statistic.svg" width="97" height="59"></object></span>
</p>
<p>
where µ<sub>0</sub> is the assumed mean, <span class="emphasis"><em>s</em></span><sup>2</sup> is the sample variance,
and <span class="emphasis"><em>n</em></span> is the number of samples. If the absolute value
of the test statistic is large, then we have low confidence that the population
mean is equal to µ<sub>0</sub>, and if the absolute value of the test statistic is small,
we have high confidence. We now ask the question "what constitutes large
and small in this context?"
</p>
<p>
Under reasonable assumptions, the test statistic <span class="emphasis"><em>t</em></span> can
be assumed to come from a Student's <span class="emphasis"><em>t</em></span>-distribution. Since
we wish to know if the sample mean deviates from the true mean in either direction,
the test is two-tailed. Hence the <span class="emphasis"><em>p</em></span>-value is straightforward
to calculate from the Student's <span class="emphasis"><em>t</em></span>-distribution on <span class="emphasis"><em>n</em></span>
- 1 degrees of freedom, but nonetheless it is convenient to have it computed
here.
</p>
<p>
An example usage is as follows:
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">vector</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">random</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">statistics</span><span class="special">/</span><span class="identifier">t_test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">random_device</span> <span class="identifier">rd</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">mt19937</span> <span class="identifier">gen</span><span class="special">{</span><span class="identifier">rd</span><span class="special">()};</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">normal_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">dis</span><span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">};</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">v</span><span class="special">(</span><span class="number">1024</span><span class="special">);</span>
<span class="keyword">for</span> <span class="special">(</span><span class="keyword">auto</span> <span class="special">&amp;</span> <span class="identifier">x</span> <span class="special">:</span> <span class="identifier">v</span><span class="special">)</span> <span class="special">{</span>
<span class="identifier">x</span> <span class="special">=</span> <span class="identifier">dis</span><span class="special">(</span><span class="identifier">gen</span><span class="special">);</span>
<span class="special">}</span>
<span class="keyword">auto</span> <span class="special">[</span><span class="identifier">t</span><span class="special">,</span> <span class="identifier">p</span><span class="special">]</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">statistics</span><span class="special">::</span><span class="identifier">one_sample_t_test</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="number">0.0</span><span class="special">);</span>
</pre>
<p>
The test statistic is the first element of the pair, and the <span class="emphasis"><em>p</em></span>-value
is the second element.
</p>
<h4>
<a name="math_toolkit.t_test.h2"></a>
<span class="phrase"><a name="math_toolkit.t_test.performance"></a></span><a class="link" href="t_test.html#math_toolkit.t_test.performance">Performance</a>
</h4>
<p>
There are two cases: Where the mean and sample variance have already been computed,
and the case where the mean and sample variance must be computed on the fly.
</p>
<pre class="programlisting"><span class="special">----------------------------------------------</span>
<span class="identifier">Benchmark</span> <span class="identifier">Time</span>
<span class="special">----------------------------------------------</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span> <span class="number">291</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">210.058</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16</span> <span class="number">1064</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">114.697</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32</span> <span class="number">407</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">599.213</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span> <span class="number">595</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">821.086</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">128</span> <span class="number">1475</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">662.071</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span> <span class="number">1746</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1118.85</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span> <span class="number">3303</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.15492</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span> <span class="number">6404</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.19139</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span> <span class="number">12461</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.2245</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span> <span class="number">24805</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23029</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span> <span class="number">49639</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.22956</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span> <span class="number">98685</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23698</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span> <span class="number">197434</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23656</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span> <span class="number">393929</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23952</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span> <span class="number">790967</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23466</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span> <span class="number">1582366</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23434</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span> <span class="number">3141112</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.24358</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span> <span class="number">6260407</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.24792</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2097152</span> <span class="number">12521811</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.24784</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4194304</span> <span class="number">25076257</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.24619</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8388608</span> <span class="number">50226183</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.2444</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16777216</span> <span class="number">100522789</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.24353</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span> <span class="number">5.99</span> <span class="identifier">N</span>
<span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_RMS</span> <span class="number">0</span> <span class="special">%</span>
<span class="identifier">OneSampleTTestKnownMeanAndVariance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="number">207</span> <span class="identifier">ns</span>
</pre>
</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 © 2006-2020 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher
Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Evan Miller, Jeremy Murphy,
Matthew Pulver, Johan Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson,
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="anderson_darling.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../statistics.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="runs_test.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>
Reference in New Issue View Git Blame Copy Permalink