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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.
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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.
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.
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.
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