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