Linear Regression: Interaction Effects and Collinearity Issues

[Tanja Gabriele Baudson]

Hi all,

I am trying to predict teacher expectations from a number of variables
and would like to include interaction effects as predictors. Research
has shown, for instance, that child gender and class size havea direct
effect. However, I would also like to know whether, besides these
direct effects, the interaction between the two plays a role (such
that, e.g., girls in smaller classes are less likely to be
underestimated and boys in bigger classes). As far as I know,
interaction effects can be modeled by multiplying the variables in
question; however, if I do so, it is no longer possible to include the
two direct effects, due to collinearity of the predictors. If I use
the residuals only (here, by regressing the interaction term on both
child gender and class size), the effects are minuscule (I tried
several interactions, which all make sense from a theoretical point of
view, and this was the case for all of them).

To solve the problem:
* Do I have to make a choice (direct effects or interaction effects,
but not the two)?
* How would I proceed? My idea would be to have a look at the
correlations with the criterion (if they are higher for the
interaction term than for the two individual terms, would I choose the
interaction term?)

I hope the question is not absolutely silly; if it is, please let me
know so I know at least the answer is closer than my blocked brain is
able to recognize right now ;)

Thanks very much in advance!
Tanya

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[Bruce Weaver]

Hi Tanja.  Two things.

1. You have explanatory variables from at least 2 levels here.  Characteristics of individual students (e.g., sex, IQ) are Level 1 variables; and characteristics of the classes (e.g., class size) are level 2 variables.  So IMO, you should be using MIXED, not REGRESSION.

2. Try centering your variables on some meaningful in-range value*.  This will not only reduce the (appearance of) multicollinearity, but will also yield a constant that is interpretable.  

* Authors often describe mean-centering.  But you do not have to center on the mean.  I like to choose a nice round figure somewhere near the bottom end of the range for a variable, for example.

HTH.

Tanja Gabriele Baudson wrote

Hi all,

I am trying to predict teacher expectations from a number of variables
and would like to include interaction effects as predictors. Research
has shown, for instance, that child gender and class size havea direct
effect. However, I would also like to know whether, besides these
direct effects, the interaction between the two plays a role (such
that, e.g., girls in smaller classes are less likely to be
underestimated and boys in bigger classes). As far as I know,
interaction effects can be modeled by multiplying the variables in
question; however, if I do so, it is no longer possible to include the
two direct effects, due to collinearity of the predictors. If I use
the residuals only (here, by regressing the interaction term on both
child gender and class size), the effects are minuscule (I tried
several interactions, which all make sense from a theoretical point of
view, and this was the case for all of them).

To solve the problem:
* Do I have to make a choice (direct effects or interaction effects,
but not the two)?
* How would I proceed? My idea would be to have a look at the
correlations with the criterion (if they are higher for the
interaction term than for the two individual terms, would I choose the
interaction term?)

I hope the question is not absolutely silly; if it is, please let me
know so I know at least the answer is closer than my blocked brain is
able to recognize right now ;)

Thanks very much in advance!
Tanya

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command. To leave the list, send the command
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INFO REFCARD

[jmdpulido]

Hi Tanya,

Bruce is right (like always). You need to fit a multilevel (a.k.a. random effects, a.k.a. mixed model) in order to be able to include at the same time independent variables related to the teachers (the individuals) and independent variables related to the schools (groups).

As Bruce suggest, centering in multilevell is very useful indeed.

If you want to read about multilevell I recommend you Hox (2010): Multilevel analysis. It is an excelent text-book, it is clear and it has many examples with datasets from education data.

The only alternative to to multilevel here will be to fit a 2-stages fixed effects model, which I think would not be very adequate in your case.

Kind Regards