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The
following are a few of the many measures of association
used with chi-square and other contingency table analyses. When using the chi-square
statistic, these coefficients can be helpful in interpreting the relationship between two
variables once statistical significance has been established. The logic for using measures
of association is as follows:
Even
though a chi-square test may show statistical significance between two variables,
the relationship between those variables may not be substantively important. These and
many other measures of association are available to help evaluate the relative strength of
a statistically significant relationship. In
most cases, they are not used in interpreting the data unless the chi-square statistic
first shows there is statistical significance (i.e., it doesn't make sense to say there is
a strong relationship between two variables when your statistical test shows this
relationship is not statistically significant).
Nominal
and Ordinal Variables
Phi
Only
used on 2x2 contingency tables. Interpreted as a measure of the relative (strength) of an
association
between two variables
ranging from 0 to 1.
Pearson's
Contingency Coefficient
(C)
It
is interpreted as a measure of the relative (strength) of an association
between two variables.
The coefficient will always be less than 1 and varies according to the number of rows and
columns.
Cramer's
V Coefficient
(V)
Useful
for comparing multiple X2 test statistics and is generalizable across
contingency tables of varying sizes. It is not affected by sample size and therefore is
very useful in situations where you suspect a statistically significant chi-square was the
result of large sample size instead of any substantive relationship between the variables.
It is interpreted as a measure of the relative (strength) of an association
between two variables. The coefficient ranges from 0 to 1 (perfect association). In
practice, you may find that a Cramer's V of .10 provides a good minimum threshold for
suggesting there is a substantive relationship between two variables.
where q
= smaller # of rows or columns
Describing
Strength of Association
Characterizations
>.5
high association
.3
to .5
moderate association
.1
to .3
low association
0
to .1
little if any association
Proportional
Reduction of Error (PRE)
Lambda
This
is a proportional reduction in error (PRE) measure that ranges from 0 to 1. Lambda indicates the extent to which the
independent variable reduces the error associated with predicting the value of a dependent
variable. Multiplied by 100, it represents the percent reduction in error.
Ordinal
Variables Only
Gamma
Another
PRE measure ranging from -1 to 1 that estimates the extent errors are reduced in
predicting the order of paired cases. Gamma ignores ties.
Kendall’s
Tau b
Similar
to Gamma but includes ties. Ranges from -1 to
1 but since standardization is different from Gamma, it provides no clear explanation of
PRE.
Inter-rater
Agreement
Cohen’s
Kappa
Measures
agreement beyond chance. Although a negative value is possible, it commonly ranges from 0
to 1 (perfect agreement). This measure requires a balanced table where the number of rows
is the same as the number of columns. The
diagonal cells represent agreement.
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