Copyright © 2010 by Nelson Education Limited. Elaborating Bivariate Tables

Copyright © 2010 by Nelson Education Limited.

Elaborating Bivariate Tables


• The logic of the elaboration technique: Controlling for a third variable.

• The interpretation of partial tables and partial measures of association.

• Direct, spurious, intervening, and interactive relationships.

In this presentation you will learn about:


• Social science research projects are multivariate, virtually by definition.

• One way to conduct multivariate analysis is to observe the effect of 3rd (control) variables, one at a time, on a bivariate relationship. – To observe how a control variable (Z ) affects the relationship

between X and Y, the bivariate relationship is reconstructed for each value of the control variable.

– Hence, this “elaboration” technique extends the analysis of bivariate tables presented in Chapters 11 and 12.

Elaboration: Introduction


• A sample of 50 immigrants, none of whom spoke English or French when they arrived to Canada, has been interviewed about their level of adjustment.– Is the pattern of adjustment, measured by level of

fluency in either of the nation’s two official languages, affected by length of residence in Canada?

Elaboration: An Example


• The table below shows the relationship between length of residence in Canada (X ) and fluency in English or French (Y ), where the computed gamma is 0.71:

English/French Facility

Length of Residence

Less than Five Year (Low)

More than Five Years (High) Totals

Low 20 10 30 High 5 15 20

Totals 25 25 50

Fluency by Length, Frequencies

Elaboration: An Example (continued)


• The column %’s and gamma (.71) show a strong, positive relationship: Fluency in English or French increases with length of residence in Canada.

Fluency by Length, Frequencies and (Percentages)


Length of Residence



Low 20 (80%) 10 (40%) 30

High 5 (20%) 15 (60%) 20

Totals 25 (100%) 25 (100%) 50



• 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.

• The first partial table (and gamma) below shows the relationship between X and Y for men (Z1) and the second partial table (and gamma) shows the relationship for women (Z2).



Fluency by Length, Controlling for Gender


Length of Residence



Low 10 (83%) 5 (39%) 15

High 2 (17%) 8 (61%) 10

Totals 12 (100%) 13(100%) 25

A. Males (Gamma = 0.78)



Fluency by Length, Controlling for Gender


Length of Residence



Low 10 (77%) 5 (42%) 15

High 3 (23%) 7 (58%) 10

Totals 13 (100%) 12(100%) 25

B. Females (Gamma = 0.65)



• The percentage patterns and gammas for all three tables are essentially the same. – For both sexes, Y increases with X in about the same way.

• Sex (Z ) has little effect on the relationship between fluency (Y ) and length of residence (X ).– There seems to be a direct relationship between X and Y.


Gamma for Bivariate Table 0.71__ Gamma for Partial Tables Controlling for Sex Male 0.78 Female 0.65


• In a direct relationship, the control variable (Z) 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

Direct Relationship


• Spurious relationship (also called explanation): – X and Y are not related, both are caused by Z.

• Intervening relationship (also called interpretation):– X and Y are not directly related but are linked by Z.

• Interaction (also called specification):– The relationship between X and Y changes for each value

of Z.

• We will continue to use the fluency-length of residence example to illustrate the pattern each of these outcomes would take in partial tables.

Other Possible Relationships Between X, Y, and Z


• X and Y are not related, both are caused by Z.– Thus, the X and Y relationship is spurious.

XZ Y

Spurious Relationship


• Immigrants with Canadian relatives (Z ) are more fluent in English or French (Y ) and are more likely to stay in Canada (X ):

Length of Res.Relatives

Fluency

Spurious Relationship (continued)


Fluency by Length, Controlling for Relatives(only Percentages shown)


Length of ResidenceLess than Five Year (Low)

More than Five Years (High)

Low 30% 30%

High 70% 70%

Totals 100% 100%

A. With Relatives (Gamma= 0.00)



Fluency by Length, Controlling for Relatives(only Percentages shown)




Low 65% 65%

High 35% 35%

Totals 100% 100%

B. Without Relatives (Gamma= 0.00)



• 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.


Gamma for Bivariate Table 0.71__

Gamma for Partial Tables ______ Controlling for Relatives With Relatives 0.00 Without Relatives 0.00


• X andY are not directly related but are linked by Z.

• Longer term residents may be more likely to find jobs that require English or French and be motivated to become fluent.

ZX Y

Jobs

Length Fluency

Intervening Relationship


• In an intervening relationship, the gammas in the partial tables are dramatically lower than the gamma in the bivariate table, perhaps even falling to zero.

Intervening Relationship (continued)


–The partial tables look the same for in intervening and spurious relationships, although for different causal reasons. So:• Intervening and spurious relationships cannot be

differentiate on statistical grounds (inspecting the partial tables).

• They must be distinguished on causal (temporal) grounds: Z may be theorized as antecedent –prior- to both X and Y (spurious) or Z may be theorized as intervening –between- X and Y (intervening).

Intervening Relationship (continued)


• 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.

Z1

X Y

Z2

0

Interaction (continued)


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



• Fluency by Length, Controlling for Education (Percentages)




Low 80% 5%

High 20% 95%

Totals 100% 100%

A. Higher Education (Gamma= 0.90)



• Fluency by Length, Controlling for Education (Percentages)


Length of Residence



Low 80% 80%

High 20% 20%

Totals 100% 100%

B. Lower Education (Gamma= 0.00)



• The relationship between length of residence and fluency changes markedly for the categories of education.– For higher educated immigrants the relationship between length of

residence and fluency is positive and stronger than in the bivariate table, while the relationship between length of residence and fluency drops to zero for lower educated immigrants.

Gamma for Bivariate Table 0.71

Gamma for Partial Tables __________ Controlling for Education Higher 0.90 Lower 0.00



• Interaction can be manifested in various ways in the partial tables.– In the example above the strength of the relationship

between X andY varied in the partial tables.– Another form interaction can take is in the direction of

the relationship between X andY in the partial tables. For instance, X and Y could have a positive relationship for one category of Z (Z1) and a negative relationship for the other category of Z (Z2):

Z1 +

X YZ2 -



Partials compared

with bivariate

Pattern Implication Next StepTheory that

X Y is

Same Direct Disregard Z Select another Z Supported

Weaker Spurious Incorporate Z

Focus on relationship between Z

and Y

Not supported

Weaker Intervening Incorporate Z

Focus on relationship between X,

Y, and Z

Partially supported

Mixed Interaction Incorporate Z

Analyze categories of

ZPartially supported

Summary

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Copyright © 2010 by Nelson Education Limited. Elaborating Bivariate Tables