
A twoway
tabulation is useful in understanding the nature of association between
a pair of variables. (Parasuraman, Grewal, Krishnan, 2004) 

TwoWay Table 


A
twoway table uses a simple crosstabulation. 


Categories 



Data must be
coded into fixed sets of categories. 



The
number of categories should not be large. 



Interval and/or
ratio data that can be transformed into a limited set of categories can
also be analyzed with this technique. 


Constructing the Table 



The number of
responses for each category of one variable are broken into the
categories of the second variable. 



For example ... 




The
hypothesis 



The null is a
test of no relationship or independence. 



The alternative
hypothesis asserts there is a relationship; they are not independent. 



H_{0}:
There is no relationship between height and sex. H_{a}: There
is a relationship between height and sex. 

Conducting the
test 


The
actual frequencies are compared with expected cell frequencies, created
under the assumption of the null hypothesis (e.g. there is no
relationship). 


E_{ij}
= (n_{i}n_{j})/n Where n_{i
}and n_{j }are the marginal frequencies
i = the number of sample units in category i of the row variable
j = the number of sample units in category j of the column variable 


Expected values: n = 100
(60)(40)/100, etc.

5'11" + 
< 5'11" 
Total 
Men 
24 
36 
60 
Women 
16 
24 
40 
Total 
40 
60 
100 



The
value for c^{2}
c^{2 }=
SS[(O_{ij}  E_{ij})^{2}]/E_{ij}
summed from i = 1 to r and from j = 1 to c
O =observed
r = number of rows
c = number of columns 


c^{2 }=
SS[(O_{ij}  E_{ij})^{2}]/E_{ij}
(3024)^{2}/24 + (3036)^{2}/36 +
(1016)^{2}/16 + (3024)^{2}/24 =
1.5 + 1.0 + 2.25 + 1.5 = 6.25



Degrees of Freedom d.f. = (r1)(c1)
(21)(21) = (1)(1) = 1 


The
Critical Value is taken from c^{2 }Table given a specific significance level
and the degrees of freedom. 


Our
example: a =
0.05 and d.f. = 1, the critical value of 3.84 is obtained from the
table.
Decision Rule: Reject H_{0}: if
c^{2 }> 3.84
Because, 6.25 is greater than 3.84, we reject the null
hypothesis in favor of the alternative.
Conclusion: The data strongly suggest a
relationship between height and sex. 

Most statistical
software can compute a c^{2
}test. 

Precautions in
Interpreting TwoWay Tables 


Unless the data were collected in a carefully controlled experiment,
this test is a test of association, not a causal relationship. 


Just like with small samples, small cells can generate misleading
results by giving too much weight to a few data points. The test
requires adequate numbers in each cell. 


This technique only examines pairs of data which can be problematic if
the relationship between the two variables is dependent or influenced by
one or more other variables. 




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