```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 6as typed below.On Jan 9, 10:35 pm, Paul <paul.domas...@gmail.com> 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).
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