Chapter 14
Association Between Variables
Measured at the Ordinal Level
Chapter Outline




Introduction
Proportional Reduction in Error (PRE)
The Computation of Gamma
Determining the Direction of
Relationships
Chapter Outline
 Interpreting Association with
Bivariate Tables: What Are the
Sources of Civic Engagement in U.S.
Society?
 Spearman’s Rho (rs )
 Testing the Null Hypothesis of “No
Association” with Gamma and
Spearman’s Rho
Gamma
 Gamma is used to measure the
strength and direction of two ordinallevel variables that have been
arrayed in a bivariate table.
 Before computing and interpreting
Gamma, it will always be useful to
find and interpret the column
percentages.
An Ordinal Measure: Gamma
 To compute Gamma, two quantities
must be found:
 Ns is the number of pairs of cases
ranked in the same order on both
variables.
 Nd is the number of pairs of cases
ranked in different order on the
variables.
An Ordinal Measure: Gamma
Low
High
Low
10
12
High
17
5
 To compute Ns,
multiply each cell
frequency by all
cell frequencies
below and to the
right.
 For this table, Ns is
10 x 5 = 50.
An Ordinal Measure: Gamma
Low
auth
Low
effic
10
High
effic
17
 To compute Nd,
High
multiply each cell
author frequency by all
cell frequencies
12
below and to the
left.
5  For this table, Nd is
12 x 17 = 204.
An Ordinal Measure: Gamma
 Gamma is computed with Formula 14.1
Calculate and interpret Gamma
 Ns = 10(5)=50 Nd=12(17) = 204
 G = (Ns+Nd)/(Ns-Nd) =
 (50-204)/(50+204) = -.61
 PRE interpretation: We reduce our
errors in predicting the efficiency of a
workplace by 61% if we know the
management style
An Ordinal Measure: Gamma
 In addition to strength, gamma also
identifies the direction of the
relationship.
 This is a negative relationship: as
authoritarianism increases, efficiency
decreases.
 In a positive relationship, the
variables would change in the same
direction.
Let’s look at a more complicated
problem requiring Gamma
Let’s look at a more complicated
calculation of gamma
Low job
security
Med. Job
security
High job
security
Low job
satisf
a.
16
B
8
C
14
Medium
job satisf
D
19
E
17
F
60
High job
satisf
G
9
H
11
I
56
Calculating Gamma
Ns = 2304+1273+928+952 = 5,457
Nd= 891+814+418+238= 2361
G = (5457-2361)/(5457+2361)=.396
How do we express the PRE
interpretation?
 What is the direction of the
relationship and what does that
mean?




Spearman’s rho 2
Spearman’s rho varies between -1 and
+1
We can give it a PRE interpretation by
squaring it.
Spearman’s rho
 This measure is used with ordinal
variables that have many discrete
scores (e.g. table 14.12, p. 345)
 We could collapse the data into
high/low on each variable, but we’d
be wasting information
 Instead, we use Spearman’s rho (or
rather, we ask SPSS to do it for us)
Spearman’s rho and SPSS
 Which variables in our GSS2002 data
set might be suitable for rho?
 How do we get SPSS to calculate rho?
 Just ask for Analyze/cross tabs/ gamma
 and they’ll throw in what they call the
Spearman’s coefficient (I think that’s the
square of rho)
 Example with polyview and attend