# Modernising Receiver Operating Characteristic (ROC) Curves +
## Appendix A ROC, Item Response Theory, Control Charts, Rankits and Other Measures
### [PERSON], MSA and Specific Objectivity  
A key development in modern measurement theory is the psychometric approach of [PERSON] [18]. A special feature of the [PERSON] approach [18] is being able to, with Equation (1), make separate estimates for classifier ability and classification task difficulty, which are confounded in the raw data according to [PERSON]'s principle of specific objectivity.  
The [PERSON] model, developed about twenty years after the first ROC curves, adopted a radically different approach to statistical evaluation, as summarised in [PERSON]'s [18] words: _'[PERSON] et al. (1959) expresses: \"Recourse must be had to individual statistics, treating each patient as a separate universe. Unfortunately, present day statistical methods are entirely group-centered so that there is a real need for developing individual-centered statistics.\"' Individual-centered statistical techniques require models in which each individual is characterised separately and from which, given adequate data, the individual parameters can be estimated..... Symmetrically, it ought to be possible to compare stimuli belonging to the same class--\"measuring the same thing\"--independent of which particular individuals within a class considered were instrumental for the comparison.'_  
This separation is key in metrology, and is particularly important to consider when assuring metrological quality of performance tests and other classification data.  
Although [PERSON] [25] did not use the MSA approach and terminology (described in Section 3.2), an early form of his psychometric model, in response to the contemporary demands for individual measures, was very much an MSA formulation [4]. He posited that the odds ratio of successfully performing a task is equal to the ratio of an ability, \(h\), ([PERSON] [18] used the person attribute 'inability' instead, given by \(h^{-1}\)), to a difficulty, \(k\):  
\[\frac{P_{\_}{success}}{1-P_{\_}{success}}=\frac{h}{k} \tag{1}\]  
where the test person (\"agent\") ability, \(\theta=log(h)\), and task (\"object\") difficulty, \(\delta=log(k)\) (or other item:probe pairs of attributes).  
The probability \(P_{\_}{success,j,}\) of a 'correct' response on item \(j\) by a rare \(i\) is cumulative, and the cumulative distribution function of the logistic distribution for the [PERSON] model is an operating characteristic function (OCF) [26]. The [PERSON] approach essentially compensates for the 'counted fractions' ordinality (Appendix A.1).  
The original [PERSON] [25] formulation referred to a probabilistic Poisson distribution:  
\[p(x)=\frac{e^{-\lambda}\cdot\lambda^{x}}{x!};x=0,1,\ldots\]  
This distribution is well known from quality control as a model of the number of defects or nonconformities that occur in a unit of product when classifying it, where \(x\) is the quality characteristic being classified [20]. The parameter \(\lambda\) is equal directly both to the mean and variance of the Poisson distribution. In [PERSON]'s [25] model:  
\[\lambda=h^{-1}\cdot k\]  
where \(h\) and \(k\) are defined in Equation (A1).