MEASURES OF ASSOCIATION

  I.  Introduction.
	A.  Chi-square only tells you if a relationship exists.  It 
	    tells you nothing about the strength or direction (for
	    ordinal categorical variables only) of the relationship.
	B.  However, there are several measures of association that 
	    are frequently calculated with chi-square that do provide
	    this information.
			
 II.  Measures of association for nominal variables.
	A.  Chi-square based measures.
	      1.  Phi is a chi-square based measure of association 
	  	  appropriate for 2x2 tables.
	      2.  Calculating Phi.
	      3.  Cramer's V is a chi-square based measure of 
	  	  association appropriate for all other tables.
	      4.  Calculating Cramer's V.
	      5.  Both phi and Cramer's V range from 0 to 1 with zero 
	  	  meaning no association and 1 meaning a perfect
	  	  association.
	      6.  Although easy to calculate, phi and Cramer's V are
	          difficult to interpret and not terribly sensitive
	          measures.
	B.  Preportional Reduction in Error Measures.
	      1.  Proportional Reduction in Error (PRE) measures are 
	  	  more meaningful and sensitive.
	      2.  PRE measures indicate how much more accurately we 
	  	  can predict scores on the dependent variable when
	  	  we know scores on the independent variable compared
	  	  to when we do not know scores on the independent
	  	  variable.
	      3.  Lamda.
	  	    a.  Calculating lamda
		    b.  Interpreting lamda.
			  i.  Ranges from 0 to 1 with 0 meaning no 
		    	      association and 1 meaning perfect
			      association.
			 ii.  It represents proportionally how much 
			      more accurately we can predict the
			      dependent variable when we know the
			      independent variable.
			iii.  Lamda is asymmetrical meaning its value 
			      differs depending on which variable is
			      taken as the dependent variable.
	      4.  Tau.
	       	    a.  Calculating tau.
		    b.  Interpreting tau.
					
III.  Measures of association for ordinal variables.
	A.  Ordinal measures of association not only indicate how 
 	    strong an association exists, but also whether the
	    association is negative or positive.
	B.  Ordinal measures of association can only be used if both   
	    variables are at least ordinal.
	C.  Gamma is the simplest PRE measure of association for 
	    ordinal variables.  
	      1.  Calculating gamma.
	      2.  Interpreting gamma.
		    a.  Gamma ranges from -1 to +1 with -1 indicating 
			a perfect negative association, +1 a perfect
			positive association, and 0 indicating no
			association.  
		    b.  It represents the increase in our ability to
		        predict whether a case will be higher or
		        lower on one variable if we know their rank
		        on the other variable.  
		    c.  Gamma is symmetrical which means its value is 
			the same regardless of which variable is the
			dependent variable.
	D.  Other ordinal measures of association.
	      1.  Somer's d is another PRE measure of association for 
	  	  ordinal variables.  It is similar to gamma but
	  	  differs in that it takes into account cases whose
	  	  ranks tie one the dependent variable. 
	      2.  Like gamma it ranges from -1 to +1, but unlike 
	  	  gamma, it is asymmetrical.
	      3.  Kendall's tau-b is also a PRE measure of 
	  	  association for ordinal variables.  It is similar
	  	  to gamma and Somer's d but differs in that it takes
	  	  into account cases who are tied on both variables.  
	      4.  Like gamma and tau-b, it ranges from -1 to +1.  
	  	  Tau-b is symmetrical like gamma.

 IV.  Testing hypotheses about association.  
	A.  Some measures of association (like gamma) can be used
	    without chi-square to test hypotheses about the
	    relationship between two variables.
	B.  This is because the sampling distribution of these
	    measures is basically normal allowing a z or t-test to be
	    used.