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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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Paul

Posts: 263
Registered: 2/23/10
Re: Test constantness of normality of residuals from linear regression
Posted: Jan 10, 2013 11:20 AM
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On Jan 10, 6:50 am, "David Jones" <dajx...@ceh.ac.uk> wrote:
> "Paul" <paul.domas...@gmail.com> wrote in message
>
> news:69701577-6f8a-4caf-a535-a7969cf9e139@k6g2000yqf.googlegroups.com...
>
>
>
>
>

> > 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 fromhttp://en.wikipedia.org/wiki/Heteroscedasticity


Thanks, David! I'm off on another educational path of exploration.



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