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Chapter 15 – Elaborating Bivariate Tables

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1 Chapter 15 – Elaborating Bivariate Tables
Week 13 Chapter 15 – Elaborating Bivariate Tables

2 Elaborating Bivariate Tables
Chapter 15 Elaborating Bivariate Tables

3 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. Partial Gamma Limitations of elaboration Key role of theory

4 Controlling for a Third Variable
Three criteria must be met to determine causation: Time order Correlation Non-Spuriousness This is determined by controlling for a third variable (Z)

5 Controlling for a Third Variable
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 Tables that display the relationship between X and Y for each value of Z (a third variable) are called partial tables (Stacked Tables)

6 Controlling for a Third Variable
Focus on Three Basic Patterns Direct relationships Spurious or Intervening relationships Interaction

7 Controlling for a Third Variable
Direct Relationships In a direct relationship, the control variable has little effect on the relationship between X and Y The column percentages and Gammas in the partial tables are about the same as the bivariate table (Original table) This outcome supports the argument that X causes Y Also referred to as replication

8 Controlling for a Third Variable
Spurious Relationships In a spurious relationship X and Y are not related, both are caused by Z 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 Also referred to as explanation X Z Y

9 Controlling for a Third Variable
Intervening Relationships In an intervening relationship X and Y are not directly related to each other but are linked by Z, which “intervenes” between the two Also referred to as interpretation Z X Y

10 Controlling for a Third Variable
Interaction Interaction occurs when the relationship between X and Y changes across the categories of Z 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 - Z1 X Y Z2

11 Controlling for a Third Variable
Summary

12 Partial Gamma Partial Gamma indicates the overall strength of association between X and Y after the effects of the control variable, Z, have been removed Compare Partial Gamma to the Gamma for the bivariate table to see if the relationship has changed

13 Example 1 The table below summarizes the relationship between Number of Memberships in Student Organizations (X, independent variable) and Satisfaction with College (Y, dependent variable)

14 Example 1 Comparing the conditional distributions of Y (the column percentages), we find a positive relationship; this direction is confirmed by the sign of Gamma (G = +0.40), which is positive as well College students with at least one membership in a student organization are more likely than students with no memberships to rate their satisfaction as high

15 Example 1 The tables below introduce the control variable, GPA

16 Example 1 Looking first at the table for students with high GPAs, we continue to find a positive relationship (G = +0.40)

17 Example 1 Looking next at the table for students with high GPAs, we also find a positive relationship (G = +0.39)

18 Example 1 The relationship between integration and satisfaction is the same in the partial tables as it was in the bivariate table Therefore, we have evidence of a direct relationship

19 Example 1 This conclusion is further supported by the calculation of Partial Gamma (Gp = +0.40), which is the same as the bivariate value

20 Example 2 The tables below introduce a new control variable, class standing

21 Example 2 Looking first at the table for Upperclass students, we find no relationship (G = +0.01)

22 Example 2 Looking next at the table for Underclass students, we also find no relationship (G = +0.01)

23 Example 2 The original bivariate relationship between memberships and satisfaction disappears in the partial tables When the relationship disappears, we have either a spurious or an intervening relationship

24 Example 2 We base the decision on which (spurious or intervening) on temporal (timing) or theoretical grounds In this case, a spurious relationship makes more sense because class standing (being an Upper- or Underclass student) likely predicts the number of memberships, and not the other way around The Partial Gamma also supports our conclusion; it too was reduced to zero

25 Example 3 The tables below introduce a third control variable, race

26 Example 3 Looking first at the table for White students, we find a strong, positive relationship (G = +0.67)

27 Example 3 Looking next at the table for Black students, we find a moderate, negative relationship (G = -0.31)

28 Example 3 The relationship between memberships and satisfaction differs across categories of the control variable For one partial table, the relationship is positive For the other partial table, the relationship is negative Therefore, we have evidence of interaction

29 Example 4 The table below summarizes the relationship between Length of Residency and English Facility for a sample of 50 immigrants (problem 15.1, p. 425) The independent variable (X) is length of residence, and the dependent variable (Y) is facility with English Gamma = +0.67, which indicates a strong, positive relationship between the variables; as length of residence increases, facility with English also increases

30 Example 4 We have introduced sex as a control variable in the tables below

31 Example 4 The Gamma for males is 0.78 The Gamma for females is 0.65
The Partial Gamma is 0.71

32 Example 4 While the two Gammas for the partial tables (0.78 and 0.65) differ slightly, they both indicate a strong, positive relationship between length of residence and English facility Comparing Partial Gamma (0.71) to the original Gamma (0.67), we find little difference Therefore, we have evidence of a direct relationship Controlling for sex does not affect the relationship between length of residence and English facility for immigrants

33 Example 5 The table below summarizes the relationship between Academic Record (X) and Delinquency (Y) for a sample of 78 juvenile males Comparing the column percentages and examining the sign of Gamma (-0.69), we conclude that juvenile males with better academic records have lower delinquency

34 Example 5 We next control for area of residence

35 Example 5 The Gamma for the “Urban Areas” table is -0.05, indicating no relationship between academic record and delinquency The Gamma for the “Nonurban Areas” table is -0.89, indicating a strong, negative relationship between academic record and delinquency The relationships between X and Y differ across our partial tables Therefore, we have interaction

36 Where Do Control Variables Come From?
Understanding whether elaboration results in a spurious relationship (explanation) or an intervening relationship (interpretation) cannot be based on statistical grounds or inspecting the partial tables Control variables are based on theory Research projects are anchored in theory so control variables come mainly from theory “Control variables that might be appropriate to incorporate in the data-analysis phase will be suggested or implied in the theoretical backdrop of the research project, along with the researcher’s imagination and sensitivity to the problem being addressed.” (p. 420)

37 Limitations of Elaboration
Basic limitation: Sample size Greater the number of partial tables, the more likely to run out of cells or have small cells Potential solutions Reduce number of cells by collapsing categories (recoding) Use very large samples Use techniques appropriate for interval-ratio level (see Chapter 16)


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