chapter 14 association between variables measured at the ordinal level
TRANSCRIPT
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Chapter 14
Association Between Variables Measured at the Ordinal Level
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Chapter Outline
Introduction Proportional Reduction in Error (PRE) The Computation of Gamma Determining the Direction of
Relationships
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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
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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.
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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.
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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
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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
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An Ordinal Measure: Gamma Gamma is computed with Formula 14.1
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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
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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.
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Let’s look at a more complicated problem requiring Gamma
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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
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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?
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Spearman’s rho 2
Spearman’s rho varies between -1 and +1
We can give it a PRE interpretation by squaring it.
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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)
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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