Paul
Posts:
517
Registered:
2/23/10


Test constantness of normality of residuals from linear regression
Posted:
Jan 9, 2013 10:35 PM


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).

