12. The Chi-square Test and the
Analysis of the Contingency
Tables
12.1 Contingency Table
12.2 A Words of Caution about Chi-Square
Test
12.1 Contingency Tables
General Form of a Contingency Tables Analysis: A Test for
Independence
H 0 : The two classifica tions are independen t
H a : The two classifica tions are dependent
ˆ (n )]2
[
n

E
ij
ij
Test statistic : x 2  
Eˆ (nij )
Where,
ri c j
ˆ
E (nij ) 
n
2
2
2
Rejection region: x  x , where x has (r  1)(c  1) df.
Assumption:
1. The n observed counts are a random sample from the
population of interest. We may then consider this to be a
multinomial experiment with r  c possible outcomes
2. The sample size, n, will be large enough so that, for
every cell, the expected count, E ( nij ) , will be equal to 5
or more.
12.2
A Words of Caution about ChiSquare Test
• The use of x2 probability as an approximation to the
sampling distribution should be avoided when the expected
counts are very small. An expected cell count of at least 5
means that the x2 probability distribution can be used to
determine an approximate critical value.
• If the x2 value does not exceed the established critical
value of x2, do not accept the hypothesis of independence.
(you would be risking Type II error and the probability of
committing such an error is unknown.
The usual alternative hypothesis is that the classifications
are dependent.
• If the a contingency table x2 does exceed the critical value,
we must be careful to avoid inferring that a causal
relationship exists between the classification.
The alternative hypothesis states that the two classification
are statistically dependent – and statistically dependence
does not imply causality.
Therefore, the existence of a causal relationship cannot be
established by a contingency table analysis.