By Wes Fulton
There is an improved goodness of fit based upon the correlation coefficient, r, called P-value. This section presents an overview of correlation coefficient P-value and its relationship to the critical correlation coefficient, CCC. The [The New Weibull Handbook] author’s (Dr. Abernethy’s) CCC value is that value of r equivalent to 10% P-value. The CCC represents a lower acceptance bound on r (or equivalently a lower bound on the square of correlation coefficient r^2) at 10% significance. Values of actual r below the CCC indicate extraordinarily poor fit and only happen 1 out of 10 times on average in good sampling. The CCC is known to increase as sample size increases and it also depends on the type of model, Weibull or lognormal, etc.
The true P-value of correlation for a particular data set is the ranking of actual correlation among all the possible correlation values for the sample size of the data set and the model type selected. P-values for probability plotting must be between 0% and 100%, the higher the better. The author suggested that P-value display should be implemented in software to facilitate model selection and approval. Wes Fulton has developed the capability in the WSW [WinSMITH Weibull] software to estimate the true P-value for the Weibull 2-parameter, Weibull 3-parameter, lognormal, normal, and extreme value (Gumbel) distributions and equivalent models. There are two P-value estimates implemented in the WSW software, the prr-value(%) and the pve%.
You can get a relatively precise estimate of the true
P-value, designated prr-value(%), after pivotal rank regression confidence has
been calculated. The prr-value(%) when available is displayed only in the WSW
software report output. The accuracy of this estimate can be improved by
increasing the quantity of
The alternative to prr-value(%) is an instantaneous estimate
of the true P-value without further simulation. It is designated pve%, and it
is displayed for both the software plot and report output when selected. It
comes from previous
We have many benefits from using the P-value as the goodness of fit. Chi Chao Liu  [download the table of contents as a 1.2 meg PDF file, or the complete dissertation as a 15.8 Meg file] concluded for regression that the P-value provides the best indication of goodness-of-fit. Both correlation P-value estimates prr-value(%) and pve% retain the same meaning independent of sample size and independent of statistical model. They also can be used directly for model comparison and selection. Distribution analysis in WSW determines the model with the best fit using the highest value of pve% as the indicator of the best selection. The pve% goodness of fit measurement for regression is now the default for the WSW software. However, there is still the capability [in WSW] to show prr-value(%) or r^2 or r^2-CCC^2 in the results.
© Fulton Findings 2004 Published with permission by Wes Fulton to Barringer & Associates, Inc. May 2, 2006
You can download this as a PDF file.