Describing Bivariate Relationships17.871Spring 20022/26/02Measures of association• Discrete data– ?2– Gamma, Beta, etc.• Continuous or discrete data– (Pearson) correlation coefficient– (Spearman) rank-order correlation coefficientExample I• What is the relationship between religion and abortion sentiments?• The abortion scale:1. BY LAW, ABORTION SHOULD NEVER BE PERMITTED.2. THE LAW SHOULD PERMIT ABORTION ONLY IN CASE OF RAPE, INCEST, OR WHEN THE WOMAN'S LIFE IS IN DANGER.3. THE LAW SHOULD PERMIT ABORTION FOR REASONS OTHER THAN RAPE, INCEST, OR DANGER TO THE WOMAN'S LIFE, BUT ONLY AFTER THE NEED FOR THE ABORTION HAS BEEN CLEARLY ESTABLISHED.4. BY LAW, A WOMAN SHOULD ALWAYS BE ABLE TO OBTAIN AN ABORTION AS A MATTER OF PERSONAL CHOICE.Theoretical distribution of cells with abortion and religion independent1.000.3588.1896.3273.1243Total.0031.0011.0006.0010.0004Other.0118.0042.0022.0039.0015Non-Xn/Jewish.0047.0017.0009.0015.0006Orthodox.0118.0042.0022.0039.0015Jewish.3092.1109.0586.1012.0384Catholic.6640.2298.1214.2096.0796ProtestantTotal4321ReligionAbortion opinionActual and theoretical distribution1271456241416158Total4(1.4)0(0.7)1(1.3)3(0.5)0Other15(5.4)6(2.8)4(4.9)3(1.9)2Non-Xn/Non-Jewish6(2.1)2(1.1)1(2.0)3(0.7)0Orthodox15(5.4)15(2.8)0(4.9)0(1.9)0Jewish387(141.0)141(74.5)75(128.6)133(48.8)38Catholic844(292.0)292(154.3)160(226.4)274(101.2)118ProtestantTotal4321ReligionAbortion opinion?2=38.2Example II• What is the relationship between income and newspaper reading?Stylized relationship if newspaper reading increases with incomeTotalDailySometimesNeverHighMed.LowReadershipIncomeActual relationship between newspaper reading and income18073614041042Total610145154311Daily513981242912-6/week6841181264400-1/weekTotal>$125K$65K-$125K<$65kReadershipIncome?2=25.0, Gamma = +.16Example III• What is the relationship between voter turnout in the 1998 general election and turnout in the 1996 general election?1998 turnout vs. 1996 turnout1998 turnout1996 turnout0 .5 10.511998 turnout1996 turnout-.5 0 .5-.50.5Subtract each observation from its meanx’=x-0.562y’=y-0.462Covariance formula1998 turnout1996 turnout-.5 0 .5-.50.5Cov x yx x y yni iin( , )( )( )=− −=∑1Cov(Turnout96,Turnout98) =0.0088Correlation formulaCorr x yCov x yrx y( , )( , )= =σ σ1998 turnout1996 turnout-.5 0 .5-.50.5Corr(Turnout96,Turnout98) =.79Warning:• Correlation only measures linear relationshipAnscombe’s Quartet6.8985.7354.7455.6857.9186.4277.2674.8275.5688.15129.131210.841212.5195.3943.144.2645.2586.0866.1367.2467.0488.84148.1149.96148.4787.81119.26118.33118.8487.1198.7798.8197.71812.74138.74137.58135.7686.7788.1486.9586.5887.46109.14108.0410yxyxyxyxIVIIIIIIr =
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