Soc 3306a Lecture 6:
Introduction to Multivariate
Relationships
Control with Bivariate Tables
Simple Control in Regression
Control for Bivariate Tables
Occurs when we hold a third variable
constant when examining a bivariate table
 In other words, we examine the bivariate
table at each level of a third (control)
variable
 Bivariate = 0-order relationship
 1 control = 1st order, 2 = 2nd order, etc.

Elaboration
Called “elaboration” when we create
partial tables and present them in a more
detailed form by controlling for a third
variable
 Note that when the cells in the partial
tables are added together, the original
bivariate table will be reproduced

Effects of the Control Variable
Direct (replication)
 Spurious
 Intervening
 Interaction
 Suppressor

Observing Effects of Control
Look at change in the conditional
distribution of y (usually the column %)
 Examine change in Chi-square and its pvalue and in lambda for nominal
relationships
 Use gamma and Kendall’s tau-b and their
p-values for ordinal x ordinal relationships

Direct
The relationship between x and y does not
change for a third variable z
 The original bivariate relationship is “real”
 Chi-square or gamma do not change
 Can ignore z

Spurious (Figure 1 and 2)
The relationship weakens or disappears
(ie becomes non significant) when
controlling for z
 Z is the probable cause for change in y
 Need to examine relationship between z
and y further

Intervening
Z intervenes in the relationship between x
and y
 Similar to spurious, the relationship
between x and y weakens or disappears
 Need time order or theory to tell you
whether a relationship is spurious or
whether z is an intervening variable
 Further examination of relationship needs
to take z into account

Interaction (Figure 3 and 4)
Sometimes called specification
 When control for z, the relationship
between x and y changes for different
levels of z
 Relationship between x and y may also
change direction at different levels of z
 Implication: x and z interact in their effect
on y
 Categories of z need to be interpreted
separately

Suppressor Effect
The bivariate table shows no association
between x and y but when control for z,
partial relationships between x and y
appear (ie become significant)
 Need theory to tell you when z is
necessary

Issues Related to Elaboration of
Bivariate Tables
Each additional control variable reduces
cell sizes in partial tables
 Empty or small cells reduce confidence in
findings
 Need large sample sizes (not always
possible)
 Or can collapse variables to create
dichotomous variables and 2x2 tables but
will lose information

Simple Control in Bivariate Regression
by adding a “Dummy Variable”
(Figure 5 and 6)
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Can add a dichotomous (dummy) variable to the
bivariate model to achieve basic form of control
Original prediction equation: y = a + b1x1
Add Sex (x2), coded as female=1 and male=0
New prediction equation: y = a + b1x1+ b2x2
When x2 = female (1): y = a + b1x1+ b2(1)
When x2 = male (0): y = a + b1x1+ b2(0)