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Tables of the power function of the χ2 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.


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