# Elaborating Bivariate Tables

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Elaborating Bivariate Tables
Chapter 16 Elaborating Bivariate Tables

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

Chapter Outline Where Do Control Variables Come From?
The Limitations of Elaborating Bivariate Tables Interpreting Statistics: Analyzing Civic Engagement

In This Presentation The logic of the elaboration technique.
The construction and interpretation of partial tables. The interpretation of partial measures of association. Direct, spurious, intervening, and interactive relationships.

Introduction Social science research projects are multivariate, virtually by definition. One way to conduct multivariate analysis is to observe the effect of 3rd 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
The column %s and G show a strong, positive relationship: fluency increases with length of residence. < 5 5+ Lo 20 (80%) 10 (40%) 30 Hi 5 (20%) 15 (60%) 20 25 50

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 (Z1)and the other shows the relationship for women (Z2).

Problem 16.1: Partial Tables
Partial table for males. G = .78 < 5 5 + Lo 83% 39% Hi 17% 61%

Problem 16.1: Partial Tables
Partial table for females. G = .65 < 5 5 + Lo 77% 42% Hi 23% 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.

X 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 < 5 5+ Low 30% High 70%

Spurious Relationships
No relatives G = 0.00 < 5 5 + Low 65% High 35%

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. < 5 5+ Low 30% High 70% < 5 5 + Low 65% High 35%

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. Z1 X Y Z2 0 Z1 + X Y Z2 -

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 are less fluent regardless of residence. < 5 5+ Low 20% High 80% < 5 5 + Low 60% High 40%

Partials compared with bivariate
Summary: Table 16.5 Partials compared with bivariate Pattern Implication Next Step Theory that X  Y is Same Direct Disregard Z Select another Z Supported Weaker Spurious Incorporate Z Focus on relationship between Z and Y Not supported

Partials compared with bivariate
Summary: Table 16.5 Partials compared with bivariate Pattern Implication Next Step Theory that X  Y is Intervening Incorporate Z Focus on relationship between X, Y, and Z Partially supported Mixed Interaction Analyze categories of Z