Chapter 16 Elaborating Bivariate Tables

Chapter Outline  Introduction  Controlling for a Third Variable  Interpreting Partial Tables  Partial Gamma (Gp )

Introduction  Social science research projects are multivariate, virtually by definition.  One way to conduct multivariate analysis is to observe the effect of 3 rd variables, one at a time, on a bivariate relationship.  The elaboration technique extends the analysis of bivariate tables presented in Chapters

Elaboration  To “elaborate”, we observe how a control variable (Z) affects the relationship between X and Y.  To control for a third variable, the bivariate relationship is reconstructed for each value of the control variable.  Problem 16.1 will be used to illustrate these procedures.

Proble m 16.1:Bivariate Table Sample - 50 immigrants X = length of residence Y = Fluency in English G =.71

Problem 16.1: Bivariate Table < 5 yrs resident 5+ yrs resident Lo English Profic 20 (80%) 10 (40%) 30 Hi English Profic 5 (20%) 15 (60%) The column %s and G show a strong, positive relationship: fluency increases with length of residence.

Problem 16.1  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.  One partial table shows the relationship between X and Y for men (Z 1 )and the other shows the relationship for women (Z 2 ).

Problem 16.1: Partial Tables  Partial table for males.  G =.78 < 55 + Lo83%39% Hi17%61%

Problem 16.1: Partial Tables  Partial table for females.  G =.65 < 55 + Lo77%42% Hi23%58%

Problem 16.1: A Direct Relationship  The percentage patterns and G’s for all three tables are essentially the same.  Sex (Z) has little effect on the relationship between fluency (Y) and length of residence (X).

Problem 16.1: A Direct Relationship  For both sexes, Y increases with X in about the same way.  There seems to be a direct relationship between X and Y.

Direct Relationships  In a direct relationship, the control variable 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

Other Possible Relationships Between X, Y, and Z:  Spurious relationships: X and Y are not related, both are caused by Z.  Intervening relationships: X and Y are not directly related but are linked by Z.

Other Possible Relationships Between X, Y, and Z:  Interaction The relationship between X and Y changes for each value of Z.  We will extend problem 16.1 beyond the text to illustrate these outcomes.

Spurious Relationships  X and Y are not related, both are caused by Z. X Z Y

Spurious Relationships  Immigrants with relatives who are Americanized (Z) are more fluent (Y) and more likely to stay (X). Length of Res. Relatives Fluency

Spurious Relationships  With Relatives  G = 0.00 < 55+ Low30% High70%

Spurious Relationships  No relatives  G = 0.00 < 55 + Low65% High35%

Spurious Relationships  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.

Intervening Relationships  X and Y and not directly related but are linked by Z.  Longer term residents may be more likely to find jobs that require English and be motivated to become fluent. Z X Y Jobs Length Fluency

Intervening Relationships  Intervening and spurious relationships look the same in the partial tables.  Intervening and spurious relationships must be distinguished on logical or theoretical grounds. < 55+ Low30% High70% < 55 + Low65% High35%

Interaction  Interaction occurs when the relationship between X and Y changes across the categories of Z.

Interaction X and Y could only be related for some categories of Z. X and Y could have a positive relationship for one category of Z and a negative one for others. Z 1 XY Z 2 0 Z 1 + X Y Z 2 -

Interaction  Perhaps the relationship between fluency and residence is affected by the level of education residents bring with them.

Interaction  Well educated immigrants are more fluent regardless of residence.  Less educated immigrants’ fluency depends on length of residence. < 55+ Low20% High80% < 55 + Low60%30% High40%70%

Summary: Table 16.5 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

Summary: Table 16.5 Partials compared with bivariate PatternImplicationNext Step Theory that X  Y is WeakerIntervening Incorporate Z Focus on relationship between X, Y, and Z Explained in more detail MixedInteraction Incorporate Z Analyze categories of Z Partially supported