[R] Collinearity in Moderated Multiple Regression

Bert Gunter gunter.berton at gene.com
Tue Aug 3 22:37:01 CEST 2010 

Absolutely right.

But I think it's also worth adding that when the predictors _are_
correlated, the estimates of their coefficients depend on which are
included in the model. This means that one should generally not try to
interpret the individual coefficients, e.g. as a way to assess their
relative importance. Rather, they should just be viewed as the
machinery that produces the prediction surface, and that is what one
needs to consider to understand the model.

In my experience, this elementary fact is not understood by many
(most?) nonstatistical practicioners using multiple regression -- and
this ignorance gets them into a world of trouble.

-- Bert

Bert Gunter
Genentech Nonclinical Biostatistics


On Tue, Aug 3, 2010 at 12:57 PM, Dennis Murphy <djmuser at gmail.com> wrote:
>
> Hi:
>
> On Tue, Aug 3, 2010 at 6:51 AM, <haenlein at gmail.com> wrote:
>
> > I'm sorry -- I think I chose a bad example. Let me start over again:
> >
> > I want to estimate a moderated regression model of the following form:
> > y = a*x1 + b*x2 + c*x1*x2 + e
> >
>
> No intercept? What's your null model, then?
>
>
> >
> > Based on my understanding, including an interaction term (x1*x2) into the
> > regression in addition to x1 and x2 leads to issues of multicollinearity,
> > as x1*x2 is likely to covary to some degree with x1 (and x2).
>
>
> Is it possible you're confusing interaction with multicollinearity? You've
> stated that x1 and x2 are weakly correlated;  the product term is going to
> be correlated with each of its constituent covariates, but unless that
> correlation is above 0.9 (some say 0.95) in magnitude, multicollinearity is
> not really a substantive issue. As others have suggested, if you're
> concerned about multicollinearity, then fit the interaction model and use
> the vif() function from package car or elsewhere to check for it.
> Multicollinearity has to do with ill-conditioning in the model matrix;
> interaction means that the response y is influenced by the product of x1 and
> x2 covariates as well as the individual covariates. They are not the same
> thing. Perhaps an example will help.
>
> Here's your x1 and x2 with a manufactured response:
>
> df <- data.frame(x1 = rep(1:3, each = 3),
>                  x2 = rep(1:3, 3))
> df$y <- 0.5 + df$x1 + 1.2 * df$x2 + 2.5 * df$x1 * df$x2 + rnorm(9)
> # Response is generated to produce a significant interaction
> df
>  x1 x2         y
> 1  1  1  5.968255
> 2  1  2  7.566212
> 3  1  3 13.420006
> 4  2  1  9.025791
> 5  2  2 16.382381
> 6  2  3 20.923113
> 7  3  1 11.669916
> 8  3  2 20.714224
> 9  3  3 31.757423
>
> m1 <- lm(y ~ x1 * x2, data = df)
> > summary(m1)
>     <snip>
>
> Coefficients:
>            Estimate Std. Error t value Pr(>|t|)
> (Intercept)   2.3642     2.6214   0.902  0.40846
> x1           -0.1200     1.2135  -0.099  0.92505
> x2            0.2549     1.2135   0.210  0.84193
> x1:x2         3.1589     0.5617   5.624  0.00246 **
> ---
> Residual standard error: 1.123 on 5 degrees of freedom
> Multiple R-squared: 0.9882,     Adjusted R-squared: 0.9812
> F-statistic: 139.9 on 3 and 5 DF,  p-value: 3.053e-05
>
> # So the model has insignificant marginal covariate effects but a strong
> interaction effect.
>
> library(car)
> vif(m1)
>   x1    x2 x1:x2
>    7     7    13
>
> # None of these is big enough to raise a red flag
> # re multicollinearity. Let's look at the correlation
> # matrix of the two covariates and their interaction.
>
> with(df, cor(cbind(x1, x2, x1 * x2)))
>          x1        x2
> x1 1.0000000 0.0000000 0.6793662
> x2 0.0000000 1.0000000 0.6793662
>   0.6793662 0.6793662 1.0000000
>
> The correlation of the interaction with the other two covariates is 0.68,
> which is nowhere close to the 0.9 or above correlations that signal
> potential multicollinearity.
>
> HTH,
> Dennis
>
>
> One
> > recommendation I have seen in this context is to use mean centering, but
> > apparently this does not solve the problem (see: Echambadi, Raj and James
> > D. Hess (2007), "Mean-centering does not alleviate collinearity problems in
> > moderated multiple regression models," Marketing science, 26 (3), 438 -
> > 45). So my question is: Which R function can I use to estimate this type of
> > model.
> >
>
> > Sorry for the confusion caused due to my previous message,
> >
> > Michael
> >
> >
> >
> >
> >
> >
> > On Aug 3, 2010 3:42pm, David Winsemius <dwinsemius at comcast.net> wrote:
> > > I think you are attributing to "collinearity" a problem that is due to
> > > your small sample size. You are predicting 9 points with 3 predictor
> > > terms, and incorrectly concluding that there is some "inconsistency"
> > > because you get an R^2 that is above some number you deem surprising. (I
> > > got values between 0.2 and 0.4 on several runs.
> >
> >
> >
> > > Try:
> >
> > > x1
> > > x2
> > > x3
> >
> >
> > > y
> > > model
> > > summary(model)
> >
> >
> >
> > > # Multiple R-squared: 0.04269
> >
> >
> >
> > > --
> >
> > > David.
> >
> >
> >
> > > On Aug 3, 2010, at 9:10 AM, Michael Haenlein wrote:
> >
> >
> >
> >
> > > Dear all,
> >
> >
> >
> > > I have one dependent variable y and two independent variables x1 and x2
> >
> > > which I would like to use to explain y. x1 and x2 are design factors in
> > an
> >
> > > experiment and are not correlated with each other. For example assume
> > > that:
> >
> >
> >
> > > x1
> > > x2
> > > cor(x1,x2)
> >
> >
> >
> > > The problem is that I do not only want to analyze the effect of x1 and x2
> > > on
> >
> > > y but also of their interaction x1*x2. Evidently this interaction term
> > > has a
> >
> > > substantial correlation with both x1 and x2:
> >
> >
> >
> > > x3
> > > cor(x1,x3)
> >
> > > cor(x2,x3)
> >
> >
> >
> > > I therefore expect that a simple regression of y on x1, x2 and x1*x2 will
> >
> > > lead to biased results due to multicollinearity. For example, even when y
> > > is
> >
> > > completely random and unrelated to x1 and x2, I obtain a substantial R2
> > > for
> >
> > > a simple linear model which includes all three variables. This evidently
> >
> > > does not make sense:
> >
> >
> >
> > > y
> > > model
> > > summary(model)
> >
> >
> >
> > > Is there some function within R or in some separate library that allows
> > me
> >
> > > to estimate such a regression without obtaining inconsistent results?
> >
> >
> >
> > > Thanks for your help in advance,
> >
> >
> >
> > > Michael
> >
> >
> >
> >
> >
> > > Michael Haenlein
> >
> > > Associate Professor of Marketing
> >
> > > ESCP Europe
> >
> > > Paris, France
> >
> >
> >
> > > [[alternative HTML version deleted]]
> >
> >
> >
> > > ______________________________________________
> >
> > > R-help at r-project.org mailing list
> >
> > > https://stat.ethz.ch/mailman/listinfo/r-help
> >
> > > PLEASE do read the posting guide
> > > http://www.R-project.org/posting-guide.html
> >
> > > and provide commented, minimal, self-contained, reproducible code.
> >
> >
> >
> >
> > > David Winsemius, MD
> >
> > > West Hartford, CT
> >
> >
> >
> >
> >        [[alternative HTML version deleted]]
> >
> > ______________________________________________
> > R-help at r-project.org mailing list
> > https://stat.ethz.ch/mailman/listinfo/r-help
> > PLEASE do read the posting guide
> > http://www.R-project.org/posting-guide.html
> > and provide commented, minimal, self-contained, reproducible code.
> >
>
>        [[alternative HTML version deleted]]
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.

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