```            		    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.