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Misclassification Probabilities (In Variability Attribute Gauge Platform)

Hello Everyone, 


Could anyone point me in the right direction in terms of better understanding of the statistical details behind the "MissClassification Probabilities Report" generated in JMP's Variability Attribute Gauge Platform? 


How are these calculated on the basis of an anlysis on a continouous Y vs Categorical Xs (such as in Gauge R&R when we have Measurement vs Operator, Part and Operator*Part).


I am familiar with the concepts of alpha and beta, and their complements.  How are these related to alpha and beta?


Here's what I found in the JMP Help (Quality and Process Methods Book):  


Misclassification Probabilities
Due to measurement variation, good parts can be rejected and bad parts can be accepted. This is called misclassification. Once you select the Misclassification Probabilities option, if you have not already done so, you are prompted to select the model type and enter specification limits.
Example of the Misclassification Probabilities Report
Untitled - Misclassification Probabilities.png
Note the following:
The first and second values are conditional probabilities.
The third and fourth values are joint probabilities.
The fifth value is a marginal probability.
The first four values are probabilities of errors that decrease as the measurement variation decreases.
Thanks in advance for any insights you can offer! 
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