Tables of the power function of the chi2 test.

Tables of the power function of the chi² test.

This example computes a table of the power of the χ² 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 χ² and F-Distributions and their applications.", P. B. Patnaik, Biometrika, Vol. 36, No. 1/2 (June 1949), 202-232.

First we need some includes to access the non-central chi squared distribution (and some basic std output of course).

#include <boost/math/distributions/non_central_chi_squared.hpp>
using boost::math::chi_squared;
using boost::math::non_central_chi_squared;

#include <iostream>
using std::cout; using std::endl;
using std::setprecision;

int main()
{

Create a table of the power of the χ² test at 5% significance level, start with a table header:

cout << "[table\n[[[nu]]";
for(int lam = 2; lam <= 20; lam += 2)
{
   cout << "[[lambda]=" << lam << "]";
}
cout << "]\n";

(Note: the enclosing [] brackets are to format as a table in Boost.Quickbook).

Enumerate the rows and columns and print the power of the test for each table cell:

for(int n = 2; n <= 20; ++n)
{
   cout << "[[" << n << "]";
   for(int lam = 2; lam <= 20; lam += 2)
   {

Calculate the χ² statistic for a 5% significance:

double cs = quantile(complement(chi_squared(n), 0.05));

The power of the test is given by the complement of the CDF of the non-central χ² distribution:

double beta = cdf(complement(non_central_chi_squared(n, lam), cs));

Then output the cell value:

         cout << "[" << setprecision(3) << beta << "]";
      }
      cout << "]" << endl;
   }
   cout << "]" << endl;
}

The output from this program is a table in Boost.Quickbook format as shown below.

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):

ν	λ=2	λ=4	λ=6	λ=8	λ=10	λ=12	λ=14	λ=16	λ=18	λ=20
2	0.226	0.415	0.584	0.718	0.815	0.883	0.928	0.957	0.974	0.985
3	0.192	0.359	0.518	0.654	0.761	0.84	0.896	0.934	0.959	0.975
4	0.171	0.32	0.47	0.605	0.716	0.802	0.866	0.912	0.943	0.964
5	0.157	0.292	0.433	0.564	0.677	0.769	0.839	0.89	0.927	0.952
6	0.146	0.27	0.403	0.531	0.644	0.738	0.813	0.869	0.911	0.94
7	0.138	0.252	0.378	0.502	0.614	0.71	0.788	0.849	0.895	0.928
8	0.131	0.238	0.357	0.477	0.588	0.685	0.765	0.829	0.879	0.915
9	0.125	0.225	0.339	0.454	0.564	0.661	0.744	0.811	0.863	0.903
10	0.121	0.215	0.323	0.435	0.542	0.64	0.723	0.793	0.848	0.891
11	0.117	0.206	0.309	0.417	0.523	0.62	0.704	0.775	0.833	0.878
12	0.113	0.198	0.297	0.402	0.505	0.601	0.686	0.759	0.818	0.866
13	0.11	0.191	0.286	0.387	0.488	0.584	0.669	0.743	0.804	0.854
14	0.108	0.185	0.276	0.374	0.473	0.567	0.653	0.728	0.791	0.842
15	0.105	0.179	0.267	0.362	0.459	0.552	0.638	0.713	0.777	0.83
16	0.103	0.174	0.259	0.351	0.446	0.538	0.623	0.699	0.764	0.819
17	0.101	0.169	0.251	0.341	0.434	0.525	0.609	0.686	0.752	0.807
18	0.0992	0.165	0.244	0.332	0.423	0.512	0.596	0.673	0.74	0.796
19	0.0976	0.161	0.238	0.323	0.412	0.5	0.584	0.66	0.728	0.786
20	0.0961	0.158	0.232	0.315	0.402	0.489	0.572	0.648	0.716	0.775

See nc_chi_sq_example.cpp for the full C++ source code.