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Chapters1415PPT.pptxChapter 14
Elaborating Bivariate Tables
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
Controlling for a Third Variable
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 and relationships
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
Controlling for a Third Variable
Focus on Three Basic Patterns
1. Direct relationships
2. Spurious or Intervening relationships
3. Interaction
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
This outcome supports the argument that X causes Y
Also referred to as replication
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
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
Controlling for a Third Variable
Z1
X Y
Z2
0
Z1 +
X Y
Z2 -
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
Controlling for a Third Variable
Summary
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
Example 1
The table summarizes the relationship between Number of Memberships in Student Organizations (X, independent variable) and Satisfaction with College (Y, dependent variable)
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
Example 1
The tables introduce the control variable, GPA
Example 1
Looking first at the table for students with high GPAs, we continue to find a positive relationship (G = +0.40)
Example 1
Looking next at the table for students with high GPAs, we also find a positive relationship (G = +0.39)
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
Example 1
This conclusion is further supported by the calculation of Partial Gamma (Gp = +0.40), which is the same as the bivariate value
Example 2
The tables below introduce a new control variable, class standing
Example 2
Looking first at the table for Upperclass students, we find no relationship (G = +0.01)
Example 2
Looking next at the table for Underclass students, we also find no relationship (G = +0.01)
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
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
Example 3
The table below summarizes the relationship between Length of Residency and English Facility for a sample of 50 immigrants (problem 14.1, p. 398)
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
Example 3
We have introduced sex as a control variable in the tables below
Example 3
The Gamma for males is 0.78
The Gamma for females is 0.65
The Partial Gamma is 0.71
Example 3
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
Example 4
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
Example 4
We next control for area of residence
Example 4
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
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
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
Chapter 15
Partial Correlation and Multiple Regression and Correlation
In This Presentation
Partial correlations
Multiple regression
Using the multiple regression line to predict Y
Multiple correlation coefficient (R2)
Limitations of multiple regression and correlation
Introduction
Multiple Regression and Correlation allow us to:
Disentangle and examine the separate effects of the independent variables.
Use all of the independent variables to predict Y.
Assess the combined effects of the independent variables on Y.
Partial Correlation
Partial Correlation measures the correlation between X and Y controlling for Z
Comparing the bivariate (“zero-order”) correlation to the partial (“first-order”) correlation allows us to determine if the relationship between X and Y is direct, spurious, or intervening
Interaction cannot be determined with partial correlations
Partial Correlation
Note the subscripts in the symbol for a partial correlation coefficient:
rxy●z
which indicates that the correlation coefficient is for X and Y controlling for Z
Partial Correlation
Example
The table lists husbands’ hours of housework per week (Y), number of children (X), and husbands’ years of education (Z) for a sample of 12 dual-career households
Partial Correlation
Example
The bivariate (zero-order) correlation between husbands’ housework and number of children is +0.50
This indicates a positive relationship
Partial Correlation
Example
Calculating the partial (first-order) correlation between husbands’ housework and number of children controlling for husbands’ years of education yields +0.43
Partial Correlation
Example
Comparing the bivariate correlation (+0.50) to the partial correlation (+0.43) finds little change
The relationship between number of children and husbands’ housework controlling for husbands’ education has not changed
Therefore, we have evidence of a direct relationship
Multiple Regression
Previously, the bivariate regression equation was:
In the multivariate case, the regression equation becomes:
42
Multiple Regression
Y = a + b1X1 + b2X2
Notation
a is the Y intercept, where the regression line crosses the Y axis
b1 is the partial slope for X1 on Y
b1 indicates the change in Y for one unit change in X1, controlling for X2
b2 is the partial slope for X2 on Y
b2 indicates the change in Y for one unit change in X2, controlling for X1
Multiple Regression
The partial slopes indicate the effect of each independent variable on Y while controlling for the effect of the other independent variable(s)
The partial slopes show the effects of the X’s in their original units
These values can be used to predict scores on Y
Partial slopes must be computed before computing the Y intercept (a)
Multiple Regression
Formulas