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. LowHigh Low1012 High175

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 1012 High effic 175

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 = = 5,457  Nd= = 2361  G = ( )/( )=.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