Week 10: Chapter 16

Controlling for a Third Variable

Multivariate Analyses

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 12-15 and 17.

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 (see Healey p. 452) will be used to illustrate these procedures.

Problem 16.1: Bivariate Table

• Sample - 50 immigrants• X = length of residence• Y = Fluency in English• G = .71

Problem 16.1: Bivariate Table

< 5 5+

Lo 80% 40%

Hi 20% 60%

100%

(N=25)

100%

(N=25)

• The column %s and G show a strong, positive relationship: fluency increases with length of residence.

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).

For both sexes, Y increases with X in about the same way.

There seems to be a direct relationship between X and Y.

A. 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: B. Spurious relationships:

X and Y are not related, both are caused by Z. C. Intervening relationships:

X and Y are not directly related but are linked by Z.

D. 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.

B. Spurious Relationships X and Y are not related, both are caused

by Z.

XZ

Y

B. Spurious Relationships Immigrants with relatives who feel at

home in the UK (Z) are more fluent (Y) and more likely to stay (X).

Length of StayRelatives

Fluency

B. Spurious Relationships With Relatives G = 0.00 < 5 5+

Low 30% 30%

High 70% 70%

B. Spurious Relationships No relatives G = 0.00 < 5 5 +

Low 65% 65%

High 35% 35%

B. 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.

C. 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.

ZX Y

Jobs

Length

Fluency

C. 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%

D. 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 YZ2 -

D. Interaction

Interaction occurs when the relationship between X and Y changes across the categories of Z.

Perhaps the relationship between fluency and residence is affected by the level of education residents bring with them.

D. Interaction Well educated

immigrants are more fluent regardless of residence.

Less educated immigrants are less fluent regardless of residence.

< 5 5+

Low 20% 20%

High 80% 80%

< 5 5 +

Low 60% 60%

High 40% 40%

Summary: Table 16.5 (see Healey, p. 441)

Partials compared

with bivariate

Pattern Implication Next Step

Theory that

X Y is

Same A. Direct Disregard ZSelect

another ZSupported

WeakerB.

SpuriousIncorporate

Z

Focus on relationship between Z

and Y

Not supported

Summary: Table 16.5 (see Healey, p. 441)

Partials compared

with bivariate

Pattern Implication Next Step

Theory that

X Y is

WeakerC.

InterveningIncorporate

Z

Focus on relationship between X,

Y, and Z

Partially supported

MixedD.

InteractionIncorporate

Z

Analyze categories of

Z

Partially supported