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Recap

- Learned how to do bivariate analyses
- Cross-tabs, measures of association, correlations.
- Added variables.
- Learned to do multivariate regression analyses.
- Learned to interpret coefficients when controlling for all other variables.
- Today: What if relationship between one IV (X) and the DV (Y) is different at different levels of another variable?

Start with a basic bivariate relationship

X

Y

Question:

Will this relationship be the same at all levels of Z???

X

Y

Y

Focus on the relationshipWhen Z = α

Can be positive or negative.

Can be strong, weak or have no effect.

?

When Z = β

NOT what is the effect of Z on Y.

See Pollock p. 86 for a complete set of possible interactions.

Step 1

- Go back to contingency tables or correlations.
- Recode variables if necessary (reduce number of categories).
- Run a different cross-tab (or correlation) for each value of Z
- Look to see if relationship changes.
- Are the measures of association different?

Focus is on X & Y

- The question is:
- Did the relationship between X and Y change at different levels of Z?
- Did the relationship get weaker? stronger?
- Did the sign change or stay the same?
- Focus on the relationship between X & Y
- Not on how Z affects Y until Step 2…

Step 2

- Run a cross-tab or a correlation between new variable and the independent variable.
- Is there a relationship?

Evaluate

- Is new variable affecting the IV, the DV, and/or the relationship between the DV and the IV.
- Spurious?
- Specification?
- Antecedent?
- Reference your qualitative research!

Possible Outcome - I

- Relationship between independent and dependent variables remains unchanged &
- New variable is not related to dependent variable.
- What to do: Eliminate new variable from further analysis UNLESS you anticipate that people will expect this variable to be included and you need to demonstrate it has no effect.
- You can have IVs that are control variables and have no hypothesized effect on the DV

Possible Outcome - IIA

- Relationship between independent and dependent variables remains unchanged BUT
- New variable is related to dependent variable.
- What to do: Consider adding new variable to regression.

Possible Outcomes IIB

- Relationship between independent variable and dependent variable is slightly changed and remains consistent across categories of control.
- Both IV and the 3rd variable are related to DV.
- What to do: Consider including IV and 3rd variable in future analyses.
- Might consider running separate regressions for each category of 3rd variable if you are very interested in that relationship.
- Probably no reason to do anything special.

Possible Outcomes - III

- When you add a third variable…
- Relationship between independent and dependent variables virtually disappears.
- Independent variable is not related to dependent variable OR
- There is a sequence: independent variable affects third variable which affects DV.
- Recall example: Race, income and the vote in the US
- New variable replaces IV in the regression.

Possible Outcomes IV

- Relationship between independent and dependent variables changes (Specification) BUT
- New variable is not related to dependent variable.
- What to do:
- Run separate regressions for each level of new variable (only works when new variable has few categories – like Francophobes/Anglophones).
- Add new variables to regression and create interaction term between new variable and IV.

Specification

- Z specifies relationship of x and y.
- Example: when z=1, x has a strong, positive relationship with y, but when z=0, x has a weak, negative relationship with y.’

Interaction

- Interaction term = Z * X
- Example, if X = Education, Z = Female (1)
- IVs:
- X (weak / insignificant)
- Z (insignificant)
- Z * X (strong, significant)

Possible Outcomes V

- Relationship between independent and dependent variables changes “markedly” like when relationship between IV and DV changes sign across categories of control variable.
- The relationship is interactive; the control variable specifies the relationship between DV and IV.
- What to do:
- Include IV & new variable in all future analyses.
- Add variable and interaction term

Interaction

- Treat Z as another independent variable, X2.
- X1 and X2 do not have an additive effect on Y. Form is not Y=a+bX1+bX2
- Relationship is interactive. Y=a+bX1+bX2+b(X1*X2)

Interaction Terms

- Example:
- X1= Attitude towards abortion
- Y= Opinion towards feminists
- X2= Political Knowledge
- In the U.S., those with high levels of knowledge equate feminism and feminists with pro-choice stances. Relationship is much weaker at low levels of political knowledge.
- So, we need to interact political knowledge with attitudes towards abortion to best explain attitudes towards
- OpinionFeminists=AttitudeAbortion+PolKnowledge+PolKnowledge*AttitudeAbortion
- Note: you always include the “direct” effect of both interaction terms in equation too!

Problems and Opportunities

- You can interact more than two variables.
- Interaction can be Interval/Ordinal*Interval/Ordinal OR Interval/Ordinal*Dummy OR Dummy*Dummy
- But every time you run an interaction, you risk multicollinearity since the interaction term is necessarily related to direct effects of the variables that are interacting.

Tricky interpretation

- “Direct” effect = effect of X1 is when X2 is zero and vice versa.

Example – Gender & Language

- Three dummy variables:
- Gender (1=Women, 0=Men)
- Language (1=French, 0=English)
- Gender*Language (Interaction)
- Interpret direct effect of Gender as effect of English speaking women compared to English speaking men.
- Since 0=English and 0=Men, reference category is always English speaking men.
- Interpret direct effect of Language as effect of French speaking men compared to English speaking men.
- Interaction is understood as effect of French speaking women compared to English speaking men.

Example – Age & Religiosity

- Three variables:
- Age (ordinal, young->old recoded into cohort groups)
- Religiosity (ordinal, high=regular church-goer)
- Gender*Language (Interaction)
- Interpret direct effect of Age as effect of increasing age for non-religious people.
- Reference category is always non-religious young.
- Interpret direct effect of Religiosity as effect of religion on youngest group.
- Interaction is understood as effect of increasing both Age and Religiosity, in other words, what is effect of older, religious people compared to non-religious young.

Another possible option

- When one variable is dichotomous it is often easier to just run separate regressions for each category of the control variable.
- So, one regression for francophones, and one for anglophones. Or one for men, and one for women…

To – Do:

- Lab 7 – but can also be done with correlations (for interval level data or ordinal data with many categories)
- Foundation for worksheet
- Lab 9B – Interactions
- Put an interaction variable in the equation OR
- Run multiple regressions on different parts of the data

Announcements

- Turnitin.com; 2653464 Pwd = Tables
- Thursday: 2653473, Pwd = spring
- Quiz results
- Next week may be a little different than what is on the syllabus
- Encouraged to speak to me and the TAs about papers OR whether you are best off taking the test.

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