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 = 1 st order, 2 = 2 nd 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 p- value 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) Can add a dichotomous (dummy) variable to the bivariate model to achieve basic form of control Original prediction equation: y = a + b 1 x 1 Add Sex (x 2 ), coded as female=1 and male=0 New prediction equation: y = a + b 1 x 1 + b 2 x 2 When x 2 = female (1): y = a + b 1 x 1 + b 2 (1) When x 2 = male (0): y = a + b 1 x 1 + b 2 (0)