Question

2x2 contingency table formula derivation

Answer 1

2 x 2 

As the name it has two columns and two rows. It is very simple to use as tables with more numbers of variables are quite confusing. Here we make use of chi - square test, phi coefficient, or fisher exact probability test in order to make comparisons. In general we have the following 2 x 2 contingency table.


Variable Data 1 Data 2 Total  Category 1 m n m+n Category 2 p r p+r Total m+p n+r N


We make use of following formula in a 2 x 2 contingency table to determine the chi square statistic:


Z2Z2 = [(mr–np)2(m+n+p+r)][(m+n)(p+r)(m+p)(n+r)][(mr–np)2(m+n+p+r)][(m+n)(p+r)(m+p)(n+r)]


In other words we can do it as 


Z2Z2 = ∑∑ [(observedvalue–expectedvalue)2expectedvalue(observedvalue–expectedvalue)2expectedvalue]


We perform the following steps:


1) Obtain column of difference in observed and expected.


2) Square this difference


3) Find the quotient of this square by the expected value.


4) Sum all these values obtained in 3.


5) The result is chi square statistic.


Examples

Example on Contingency table is given below:

Example: Find the chi square statistic:



 A B Total A  10 40 50B 30 20 50 Total 40 60 100

Solution:


Here m = 10, n = 40, p = 30 and r = 20


We make use of following


Z2Z2 = (mr−np)2(m+n+p+r)(m+n)(p+r)(m+p)(n+r)(mr−np)2(m+n+p+r)(m+n)(p+r)(m+p)(n+r)


On substituting the given values in it we get


Z2Z2 = (200−1200)2(100)(50∗50∗40∗60)(200−1200)2(100)(50∗50∗40∗60)


→→ Z2Z2 = 1000∗1000∗1003000∗20001000∗1000∗1003000∗2000


→→ Z2Z2 = 10061006


→→ Z2Z2 = 16.67


So here the association between the rows and the columns is statistically extremely significant.