Open and Distance Learning

Independence

In an r x c table, two cross-classified variables are independent if all the joint probabilities equal the product of the corresponding marginal probabilities, that is,

pij = pi.p.j  for i = 1,2,...,r and j = 1,2,...,c .

(2.6)

Under the assumption of independence of the two variables, the ML estimate of p_ij, denoted p^_ij is

and, under the same assumption,

^ m   ij  = N ^ p   ij  = N ^ p   i.  ^ p   .j  = nij = ni. n.jN

(2.7)

Consequence. Independence is true when we demonstrate nij =

^ mij

, where

^ mij = ni .   n. j N .

(2.8)

In order to test the hypothesis of independence, we introduce two statistics: The Chi-square statistic [Pearson 1900]

and the Likelihood ratio statistic (e.g., L76)

When independence is true, both c² and G² have asymptotic (as N ® ¥) Chi-square distribution with degrees of freedom (r-1)(c-1). For either statistics, larger values provide more evidence against the hypothesis (null hypothesis) of independence (For more information about the likelihood statistic consult C90 and BFH75).

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