Interactions

POL 242

Renan Levine

March 13/15, 2007

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

Focus on the relationship

X Y

X Y

When Z = α

?When Z = β

Can be positive or negative.

Can be strong, weak or have no effect.

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+PolKnowledg

e*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

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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.