

Generalized Linear Models on a P/C Spreadsheet



Regression and GLM 

Ordinary least squares linear
regression assumes that the model response varies continuously and is unbounded, and so is
inappropriate for binary data for which the observed outcome is bounded and discrete,
having only 0 or 1 as possible values. The resulting error is decidedly nonnormal and so
produces unreliable parameter estimates even when the model is restricted to realistic
values (0 < y < 1). Generalized Linear Models (GLM) overcome this difficulty by
"linking" the binary response to the explanatory covariates through the
probability of either outcome, which does vary continuously from 0 to 1. The transformed
probability is then modeled with an ordinary polynomial function, linear in the
explanatory variables, so is a generalized linear model. Nondestructive evaluation (NDE) to detect cracking in a structure provides an example. A perfect inspection would be a step function with POD = 1 for a > a_{crit} and POD = 0 when a < a_{crit}. (Notice that an inspection that finds everything cannot discriminate between a pernicious crack and a benign microstructural artifact, and is therefore useless.) A crack is either detected or it is not (a binary response) but the probability of detection (POD) usually varies continuously from nearly zero for small cracks, to nearly one for large ones. 

aside ...  
This is an
oversimplification, since cracks of similar size exhibit large differences in
detectability due to fixed effects like surface preparation and part geometry, random
effects like crack orientation, inspectortoinspector differences and residual stresses,
and other factors. While a thorough discussion of NDE isn't intended here, modeling POD
provides a good example of implementing GLM on a P/C. An excellent survey of
applications of statistics to NDE can be found in Olin and Meeker (1996).


Reference:  
Olin, B.D. and Meeker,
W.Q., (1996), Applications of Statistics in Nondestructive Evaluation. (with discussion),
Technometrics 38, 95112.

Mail to Charles.Annis@StatisticalEngineering.com 