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Topic: Test constantness of normality of residuals from linear regression
Replies: 8   Last Post: Jan 10, 2013 6:09 PM

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 David Jones Posts: 324 Registered: 2/28/07
Re: Test constantness of normality of residuals from linear regression
Posted: Jan 10, 2013 6:50 AM

> After much browsing of Wikipedia and the web, I used both normal
> probability plot and Anderson-Darling to test the normality of
> residuals from a simple linear regression (SLR) of 6 data points.
> Results were very good. However, SLR doesn't just assume that the
> residuals are normal. It assumes that the standard deviation of the
> PDF that gives rise to the residuals is constant along the horizontal
> axis. Is there a way to test for this if none of the data points have
> the same value for the independent variable? I want to be able to
> show that there is no gross curves or spreading/focusing of the
> scatter.
>
> In electrical engineering signal theory, the horizontal axis is time.
> Using Fourier Transform (FT), time-frequency domains can show trends.
> Intuitively, I would set up the data as a scatter graph of residuals
> plotted against the independent variable (which would be treated as
> time). Gross curves show up as low-frequency content. There should
> be none if residuals are truly iid. The spectrum should look like
> white noise. The usual way to get the power spectrum is the FT of the
> autocorrelation function, which itself should resemble an impulse at
> zero. This just shows indepedence of samples, not constant iid normal
> along the horizontal axis.
>
> As for spreading or narrowing of the scatter, I guess that can be
> modelled in time as a multiplication of a truly random signal by a
> linear (or exponential) attenuation function. The latter acts like a
> modulation envelope. Their power spectrums will then convolve in some
> weird way. I'm not sure if this is a fruitful direction for
> identifying trends in the residuals. It starts to get convoluted
> pretty quickly.
>
> Surely there must be a less klugy way from the world of statistics? I
> realize that my sample size will probably be too small for many
> conceptual approaches. For example, if I had a wealth of data points,
> I could segment the horizontal axis, then do a normality test on each
> segment. This would generate mu's and sigma's as well, which could
> then be compared across segments. So for the sake of conceptual
> gratification, I'm hoping for a more elegant test for the ideal case
> of many data points. If there is also a test for small sample sizes,
> so much the better (though I don't hold my breath).

Wikipedia outlines a number of tests, which you can find by starting from
http://en.wikipedia.org/wiki/Heteroscedasticity

Date Subject Author
1/9/13 Paul
1/9/13 Paul
1/10/13 Ray Koopman
1/10/13 Paul
1/10/13 Ray Koopman
1/10/13 Michael Press
1/10/13 Paul
1/10/13 David Jones
1/10/13 Paul