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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 ordinal-level 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 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.
Low High
Low 10 12
High 17 5
An Ordinal Measure: Gamma To compute Nd,
multiply each cell frequency by all cell frequencies below and to the left.
For this table, Nd is 12 x 17 = 204.
Low auth
High author
Low effic
10 12
Higheffic
17 5
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
C14
Medium job satisf
D19
E17
F 60
High job satisf
G 9
H11
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