Paul
Posts:
513
Registered:
2/23/10


Re: Test constantness of normality of residuals from linear regression
Posted:
Jan 9, 2013 10:38 PM


Oops, typo. There are actually 16 data points for this problem, not 6 as 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 AndersonDarling 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), timefrequency 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 lowfrequency 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).

