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 3rd
variables, one at a time, on a
bivariate relationship.
 The elaboration technique extends
the analysis of bivariate tables
presented in Chapters 12-14.
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 yrs
resident
5+ yrs
resident
Lo
English
Profic
20
10
(80%) (40%)
30
Hi
English
Profic
5
15
(20%) (60%)
20
25
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.
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).
Res.
Relatives
Length of
Fluency
Spurious Relationships
 With Relatives
 G = 0.00
<5
5+
Low
30%
30%
High
70%
70%
Spurious Relationships
 No relatives
 G = 0.00
<5
5+
Low
65%
65%
High
35%
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%
30%
High
70%
70%
<5
5+
Low
65%
65%
High
35%
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’ fluency
depends on length
of residence.
<5
5+
Low
20%
20%
High
80%
80%
<5
5+
Low
60%
30%
High
40%
70%
Summary: Table 16.5
Partials
compared
with
bivariate
Same
Weaker
Theory
that
X  Y is
Pattern
Implication
Next Step
Direct
Disregard Z
Select
another Z
Supported
Incorporate
Z
Focus on
relationship
between Z
and Y
Not
supported
Spurious
Summary: Table 16.5
Partials
compared
with
bivariate
Weaker
Mixed
Pattern
Theory
that
X  Y is
Implication
Next Step
Intervening
Incorporate
Z
Focus on
relationship
between X,
Y, and Z
Explained
in more
detail
Interaction
Incorporate
Z
Analyze
categories of
Z
Partially
supported