%0 Journal Article
%T 1/n - Proportional Correction to Tests of Independence
%A Jan Vrbik
%J Applied Mathematics
%P 262-274
%@ 2152-7393
%D 2025
%I Scientific Research Publishing
%R 10.4236/am.2025.163013
%X It is known that both the Pearson and G-test of independence have the same asymptotic distribution, namely <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msup>
   <mi>&#x03C7;</mi>
   <mn>2</mn>
  </msup>
  </mrow>
</math>

 with <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mrow><mo>(</mo>
   <mrow>
    <mi>K</mi><mo>&#x2212;</mo><mn>1</mn></mrow>
  <mo>)</mo></mrow><mrow><mo>(</mo>
   <mrow>
    <mi>M</mi><mo>&#x2212;</mo><mn>1</mn></mrow>
  <mo>)</mo></mrow></mrow>
</math>

 degrees of freedom, where <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>K</mi>
</math>

 and <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>M</mi>
</math>

 are the number of levels, respectively, of two attributes. However, when extending the accuracy ot this approximation by <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mfrac>
   <mn>1</mn>
   <mi>n</mi>
  </mfrac>
  </mrow>
</math>

-proportional terms, the resulting corrections differ quite dramatically. The purpose of this article is to derive each of these corrections and demonstrate their use in practical application.
%K Test of Independence
%K Asymptotic Distribution
%K Chi-Square Test
%K Contingency Tables
%K G-Test
%K Kronecker Product
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=141618