Week 13

Chapter 15 – Elaborating Bivariate Tables

Chapter 15

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 Three criteria must be met to determine

causation: Time order Correlation Non-Spuriousness

This is determined by controlling for a third variable (Z)

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)

Controlling for a Third Variable

Focus on Three Basic Patterns Direct relationships Spurious or Intervening relationships 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 (Original 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

ZX Y

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

XY

Z2

0

Z1 +

X Y

Z2 -

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 below summarizes the relationship between Number

of Memberships in Student Organizations (X, independent variable) and Satisfaction with College (Y, dependent variable)

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

Example 1 The tables below 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 tables below introduce a third control variable, race

Example 3

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

Example 3 Looking next at the table for Black students, we find a

moderate, negative relationship (G = -0.31)

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

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

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

below

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

71.0)1510()7080(

)1510()7080(

NN

NNG

65.01570

1570

NN

NNG

78.01080

1080

NN

NNG

ds

dspartial

ds

dsfemales

ds

dsmales

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

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

Example 5 We next control for area of residence

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

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)

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

1. Reduce number of cells by collapsing categories (recoding)

2. Use very large samples

3. Use techniques appropriate for interval-ratio level (see Chapter 16)