In reply to Mike's question Allan makes the important point:
>
> There is absolutely no requirement that the predictors (or
> independent variables) should have a normal distribution, in fact
> the opposite. Ideally, the predictors should be from a designed
> experiment and hence will not even be random. Most of them
> should be towards the outer bounbaries of the predictor space.
> I guess that most designed experiments would give negative
> kurtoses if you go through the mechanics of calculating
> coefficients of kurtosis. If the predictors are from a multivariate
> normal distribution, then there is far too much clustering in the
> centre of the design space. Quite often, some of the predictors
> are discrete - perhaps (0,1) variables, and hence cannot have
> normal distributions.
Few researchers using multiple regression realize that normal
distributions for their predictors are, in terms of statistical
power, about the worst-case scenario. And the power problems
exacerbate when testing interactions and polynomial terms.
In McClelland & Judd (1993, Psych Bulletin, 114:376-390) we
show that to achieve equivalent statistical power for
detecting the linear-by-linear interaction, a study with
normally distributed predictors requires over 16 times as
many observations as a designed experiment. And contrary
to most researchers' expectations, skewnews helps and even
correlation between predictors helps because it avoids the
"too much clustering in the centre of the design space" that
Allan refers to.
gary
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