Describing Bivariate RelationshipsMeasures of associationExample ITheoretical distribution of cells with abortion and religion independentActual and theoretical distributionExample IIStylized relationship if newspaper reading increases with incomeActual relationship between newspaper reading and incomeExample III2004 Prez. Vote vs. 2000 Pres. VoteSubtract each observation from its meanCovariance formulaCorrelation formulaPlaying with the Utts & Heckard Correlation AppletGuessing Correlations ApletWarning:Anscombe’s QuartetDescribing Bivariate Relationships17.871Spring 2006Measures of association•Discrete data (Utts & Heckard, sect. 6.4)–χ2–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 independentAbortion opinionReligion 1 2 3 4 TotalProtestant .0796 .2096 .1214 .2298 .6640Catholic .0384 .1012 .0586 .1109 .3092Jewish .0015 .0039 .0022 .0042 .0118Orthodox .0006 .0015 .0009 .0017 .0047Non-Xn/Jewish .0015 .0039 .0022 .0042 .0118Other .0004 .0010 .0006 .0011 .0031Total .1243 .3273 .1896 .3588 1.000Actual and theoretical distributionAbortion opinionReligion 1 2 3 4 TotalProtestant (101.2)118(226.4)274(154.3)160(292.0)292 844Catholic (48.8)38(128.6)133(74.5)75(141.0)141 387Jewish (1.9)0(4.9)0(2.8)0(5.4)15 15Orthodox (0.7)0(2.0)3(1.1)1(2.1)2 6Non-Xn/Non-Jewish (1.9)2(4.9)3(2.8)4(5.4)6 15Other (0.5)0(1.3)3(0.7)1(1.4)0 4Total 158 416 241 456 1271χ2=38.2Example II•What is the relationship between income and newspaper reading?Stylized relationship if newspaper reading increases with incomeIncomeReadership Low Med. HighNeverSometimesDailyTotalActual relationship between newspaper reading and incomeIncomeReadership <$65k $65K-$125K >$125K Total0-1/week 440 126 118 6842-6/week 291 124 98 513Daily 311 154 145 610Total 1042 404 361 1807χ2=25.0, Gamma = +.16Example III•What is the relationship between Bush’s vote (by county) in 2000 and in 2004?2004 Prez. Vote vs. 2000 Pres. Vote0 .2 .4 .6 .8 1bushpct20040 .2 .4 .6 .8 1bushpct2000-.6 -.4 -.2 0 .2 .4new2004-.6 -.4 -.2 0 .2 .4new2000Subtract each observation from its meanx’=x-0.588y’=y-0.609Covariance formulaC o v x yx x y yni iin( , )( ) ( ) 1Cov(BushPct00,BushPct04) =0.014858-.6 -.4 -.2 0 .2 .4new2004-.6 -.4 -.2 0 .2 .4new2000Correlation formulaC o r r x yC o v x yrx y( , )( , ) c.f. Utts & Heckard p. 166-.6 -.4 -.2 0 .2 .4new2004-.6 -.4 -.2 0 .2 .4new2000Corr(BushPct00,BushPct04) =0.96 =0 0 1 4 8 5 80 0 1 4 9 9 0 0 1 6 0 59 6.. ..Playing with the Utts & Heckard Correlation AppletGuessing Correlations Aplethttp://www.stat.uiuc.edu/~stat100/java/guess/GCApplet.htmlWarning:•Correlation only measures linear relationshipAnscombe’s QuartetI II III IVx y x y x y x y10 8.04 10 9.14 10 7.46 8 6.588 6.95 8 8.14 8 6.77 8 5.7613 7.58 13 8.74 13 12.74 8 7.719 8.81 9 8.77 9 7.11 8 8.8411 8.33 11 9.26 11 7.81 8 8.4714 9.96 14 8.1 14 8.84 8 7.046 7.24 6 6.13 6 6.08 8 5.254 4.26 4 3.1 4 5.39 19 12.512 10.84 12 9.13 12 8.15 8 5.567 4.82 7 7.26 7 6.42 8 7.915 5.68 5 4.74 5 5.73 8 6.89r =
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