Copyright © 2010 by Nelson Education Limited. Elaborating Bivariate Tables

Copyright © 2010 by Nelson Education Limited. The logic of the elaboration technique: Controlling for a third variable. The interpretation of partial tables and partial measures of association. Direct, spurious, intervening, and interactive relationships. In this presentation you will learn about:

Copyright © 2010 by Nelson Education Limited. Social science research projects are multivariate, virtually by definition. One way to conduct multivariate analysis is to observe the effect of 3 rd (control) variables, one at a time, on a bivariate relationship. –To observe how a control variable (Z ) affects the relationship between X and Y, the bivariate relationship is reconstructed for each value of the control variable. –Hence, this “elaboration” technique extends the analysis of bivariate tables presented in Chapters 11 and 12. Elaboration: Introduction

Copyright © 2010 by Nelson Education Limited. A sample of 50 immigrants, none of whom spoke English or French when they arrived to Canada, has been interviewed about their level of adjustment. –Is the pattern of adjustment, measured by level of fluency in either of the nation’s two official languages, affected by length of residence in Canada? Elaboration: An Example

Copyright © 2010 by Nelson Education Limited. The table below shows the relationship between length of residence in Canada (X ) and fluency in English or French (Y ), where the computed gamma is 0.71: English/ French Facility Length of Residence Less than Five Year (Low) More than Five Years (High)Totals Low High Totals Fluency by Length, Frequencies Elaboration: An Example (continued)

Copyright © 2010 by Nelson Education Limited. The column %’s and gamma (.71) show a strong, positive relationship: Fluency in English or French increases with length of residence in Canada. Fluency by Length, Frequencies and (Percentages) English/ French Facility Length of Residence Less than Five Year (Low) More than Five Years (High)Totals Low20 (80%)10 (40%)30 High 5 (20%)15 (60%)20 Totals25 (100%) 50 Elaboration: An Example (continued)

Copyright © 2010 by Nelson Education Limited. Will the relationship between fluency (Y ) and length of residence (X ) be affected by gender (Z )? To investigate, the bivariate relationship is reconstructed for each value of Z. The first partial table (and gamma) below shows the relationship between X and Y for men (Z 1 ) and the second partial table (and gamma) shows the relationship for women (Z 2 ). Elaboration: An Example (continued)

Copyright © 2010 by Nelson Education Limited. Fluency by Length, Controlling for Gender English/ French Facility Length of Residence Less than Five Year (Low) More than Five Years (High)Totals Low10 (83%)5 (39%)15 High 2 (17%)8 (61%)10 Totals12 (100%)13(100%)25 A. Males (Gamma = 0.78) Elaboration: An Example (continued)

Copyright © 2010 by Nelson Education Limited. Fluency by Length, Controlling for Gender English/ French Facility Length of Residence Less than Five Year (Low) More than Five Years (High)Totals Low10 (77%)5 (42%)15 High 3 (23%)7 (58%)10 Totals13 (100%)12(100%)25 B. Females (Gamma = 0.65) Elaboration: An Example (continued)

Copyright © 2010 by Nelson Education Limited. The percentage patterns and gammas for all three tables are essentially the same. –For both sexes, Y increases with X in about the same way. Sex (Z ) has little effect on the relationship between fluency (Y ) and length of residence (X ). –There seems to be a direct relationship between X and Y. Elaboration: An Example (continued) Gamma for Bivariate Table 0.71__ Gamma for Partial Tables Controlling for SexMale 0.78 Female0.65

Copyright © 2010 by Nelson Education Limited. In a direct relationship, the control variable (Z) has little effect on the relationship between X and Y. The column %’s and gammas in the partial tables are about the same as the bivariate table. This outcome supports the argument that X causes Y : X Y Direct Relationship

Copyright © 2010 by Nelson Education Limited. Spurious relationship (also called explanation): –X and Y are not related, both are caused by Z. Intervening relationship (also called interpretation): –X and Y are not directly related but are linked by Z. Interaction (also called specification): –The relationship between X and Y changes for each value of Z. We will continue to use the fluency-length of residence example to illustrate the pattern each of these outcomes would take in partial tables. Other Possible Relationships Between X, Y, and Z

Copyright © 2010 by Nelson Education Limited. X and Y are not related, both are caused by Z. –Thus, the X and Y relationship is spurious. X Z Y Spurious Relationship

Copyright © 2010 by Nelson Education Limited. Immigrants with Canadian relatives (Z ) are more fluent in English or French (Y ) and are more likely to stay in Canada (X ): Length of Res. Relatives Fluency Spurious Relationship (continued)

Copyright © 2010 by Nelson Education Limited. Fluency by Length, Controlling for Relatives (only Percentages shown) English/ French Facility Length of Residence Less than Five Year (Low) More than Five Years (High) Low30% High70% Totals100% A. With Relatives (Gamma= 0.00) Spurious Relationship (continued)

Copyright © 2010 by Nelson Education Limited. Fluency by Length, Controlling for Relatives (only Percentages shown) English/ French Facility Length of Residence Less than Five Year (Low) More than Five Years (High) Low65% High35% Totals100% B. Without Relatives (Gamma= 0.00) Spurious Relationship (continued)

Copyright © 2010 by Nelson Education Limited. In a spurious relationship, the gammas in the partial tables are dramatically lower than the gamma in the bivariate table, perhaps even falling to zero. Spurious Relationship (continued) Gamma for Bivariate Table 0.71__ Gamma for Partial Tables ______ Controlling for RelativesWith Relatives 0.00 Without Relatives 0.00

Copyright © 2010 by Nelson Education Limited. X andY are not directly related but are linked by Z. Longer term residents may be more likely to find jobs that require English or French and be motivated to become fluent. Z X Y Jobs Length Fluency Intervening Relationship

Copyright © 2010 by Nelson Education Limited. In an intervening relationship, the gammas in the partial tables are dramatically lower than the gamma in the bivariate table, perhaps even falling to zero. Intervening Relationship (continued)

Copyright © 2010 by Nelson Education Limited. –The partial tables look the same for in intervening and spurious relationships, although for different causal reasons. So: Intervening and spurious relationships cannot be differentiate on statistical grounds (inspecting the partial tables). They must be distinguished on causal (temporal) grounds: Z may be theorized as antecedent –prior- to both X and Y (spurious) or Z may be theorized as intervening –between- X and Y (intervening). Intervening Relationship (continued)

Copyright © 2010 by Nelson Education Limited. Interaction occurs when the relationship between X and Y changes across the categories of Z. Interaction

Copyright © 2010 by Nelson Education Limited. X and Y could only be related for some categories of Z. Z 1 XY Z 2 0 Interaction (continued)

Copyright © 2010 by Nelson Education Limited. Perhaps the relationship between fluency and residence is affected by the level of education immigrants bring with them. Interaction (continued)

Copyright © 2010 by Nelson Education Limited. Fluency by Length, Controlling for Education (Percentages) English/ French Facility Length of Residence Less than Five Year (Low) More than Five Years (High) Low80% 5% High20%95% Totals100% A. Higher Education (Gamma= 0.90) Interaction (continued)

Copyright © 2010 by Nelson Education Limited. Fluency by Length, Controlling for Education (Percentages) English/ French Facility Length of Residence Less than Five Year (Low) More than Five Years (High) Low80% High20% Totals100% B. Lower Education (Gamma= 0.00) Interaction (continued)

Copyright © 2010 by Nelson Education Limited. The relationship between length of residence and fluency changes markedly for the categories of education. –For higher educated immigrants the relationship between length of residence and fluency is positive and stronger than in the bivariate table, while the relationship between length of residence and fluency drops to zero for lower educated immigrants. Gamma for Bivariate Table 0.71 Gamma for Partial Tables __________ Controlling for EducationHigher0.90 Lower0.00 Interaction (continued)

Copyright © 2010 by Nelson Education Limited. Interaction can be manifested in various ways in the partial tables. –In the example above the strength of the relationship between X andY varied in the partial tables. –Another form interaction can take is in the direction of the relationship between X andY in the partial tables. For instance, X and Y could have a positive relationship for one category of Z (Z 1 ) and a negative relationship for the other category of Z (Z 2 ): Z 1 + X Y Z 2 - Interaction (continued)

Copyright © 2010 by Nelson Education Limited. Partials compared with bivariate PatternImplicationNext Step Theory that X  Y is SameDirectDisregard Z Select another Z Supported WeakerSpurious Incorporate Z Focus on relationship between Z and Y Not supported WeakerIntervening Incorporate Z Focus on relationship between X, Y, and Z Partially supported MixedInteraction Incorporate Z Analyze categories of Z Partially supported Summary