[R] Collinearity in Moderated Multiple Regression

[David Winsemius]

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 <- rnorm(100)
  x2 <- rnorm(100)
  x3 <- x1*x2

y <- rnorm(100)
model <- lm (y ~ x1 + x2 + x1*x2)
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 <- rbind(1,1,1,2,2,2,3,3,3)
> x2 <- rbind(1,2,3,1,2,3,1,2,3)
> 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 <- x1*x2
> 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 <- rnorm(9)
> model <- lm (y ~ x1 + x2 + x1*x2)
> 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]]
>
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David Winsemius, MD
West Hartford, CT