Date: Jan 9, 2013 10:38 PM
Author: Paul
Subject: Re: Test constantness of normality of residuals from linear regression

Oops, typo.  There are actually 16 data points for this problem, not 6
as typed below.

On Jan 9, 10:35 pm, Paul <> wrote:
> 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).