1. Technical Information
2. Purpose of the tables
3. References

 

CRBIND. Technical information

Version:	1.0  (2000)
Authors:	J. M. Tapia García   A. Martín Andrés
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CRBIND. Purpose

The tables provides the unconditional critical region for testing the independence in a 2´ 2 table (multinomial case).
If the cell probabilities and the observed values for the four cells in a 2´ 2 table are, respectively,  

the aim is to test:
                             Ho º OR =1 (independence or not association)
                         vs Ha º OR < 1 or Haº OR > 1 (one tail)
                     or vs Ha º OR ¹ 1 (two tails)  
with OR= p₁₁ p₂₂ / p₁₂ p₂₁ the poblational odds-ratio.  

Under Ho:
                                P( n₁₁, n₁₂, n₂₁, n₂₂ | n_.. ) =  n_..! p_.1^n.1 p_.2^n.2 p_1.^n1. p_2.^n2. /( n₁₁! n₂₂!  n₁₂! n₂₁!)  
For a target error a the critical region is a set, CR, of triplet (n₁₁, n₁₂, n₂₁), with n₁₁+ n₁₂+ n₂₁ £ n_.., so the error a is:
                                                           a (p_1., p_.1) = S_CR P( n₁₁, n₁₂, n₂₁, n₂₂ | n_.. )
and de size of the test will be:
                                                                a *= Max a (p_1., p_.1) in 0< p_1., p_.1 < 1  

This tables gives the CR for the Barnard's test following the optimal procedure. The one-sided Ha considered is that indicate by the data:

Ha º OR<1   if   (n₁₁ n₂₂ / n₁₂ n₂₁) < 1

 ---o0o---

CRBIND. References

 For further details see:
 

1. MARTÍN ANDRÉS, A. and TAPIA GARCIA, J. M. (1998). 'On determining the P-value in 2x2 multinomial trials'. Journal of Statistical Planning and Inferences 69(1), 33-49.
2. MARTÍN ANDRÉS, A. and TAPIA GARCIA, J. M. (1999). 'Optimal unconditional test in 2x2 multinomial trials'. Comput. Statis. & Data Anal. 31(3), 311-321.
3. TAPIA GARCIA, J. M. and MARTÍN ANDRÉS, A. (2000). ''Optimal unconditional critical regions for 2x2 multinomial trials'. Journal of Applied Statistics 27(6), 689-695..