Measures of association

• There are many summary measures of association in tables.
• In the simplest case of a $2\times2$ table, a popular measure is $\phi$ (phi):
  • Given this table:
    

    	1	2	Total
    1	a	b	a+b
    2	c	d	c+d
    Total	a+c	b+d	a+b+c+d

    
    
  • Phi is defined as:
    
    $$\phi = \frac{ad - bc}%% {\sqrt{(a + b)(c + d)(b + d)(a + c)}}$$
    
    

  • Where there are higher than expected numbers in cells a and/or d, $\phi$ will tend to 1, and where the opposite is true, to $-1$.

• For tables of variables with more than two categories an analogous measure is Cramér's $V$.
  • For $2\times2$ tables it is equivalent.
  • Same range $-1<V<+1$.
  • Based on Pearson's $X^2$, a more general measure (Cramér's $V$ is a version of $X^2$ scaled to be independent of table size).

• Pearson's $X^2$ is based on the deviation between the observed values ($O$) and the expected ($E$) values under the assumption of independence:

  
  

  $$X^2 = \Sigma \frac{(O-E)^2}{E}$$
  
  

  That is, for each cell it calculates a measure of the observed-expected difference, and adds them up.

• Pearson's $X^2$ has a $\chi ^2$ (chi-squared) distribution which allows us to make inferences about association:
  • Sampling from a population where two variables are truly independent will result in tables which do not exactly match the independence table.
  • The probability distribution of the calculated $X^2$ value under these circumstances follows a $\chi ^2$ distribution, with degrees of freedom equal to $(r-1)(c-1)$.
  • By comparing the $X^2$ for a real table with the cumulative $\chi ^2$ distribution we can test the null hypothesis that there is no association.
  • e.g., if the $X^2$ is at least as big as the value you could theoretically get from truly independent variables no more than, say, 1% of the time, then you can be 99% confident that there is really association.

• Many measures relevant to loglinear models approximate a $\chi ^2$ distribution.