The nature of a diagnostic problem

Although for this chapter we have used a title which has often a medical connotation the problem arises in many other fields – for example, in the diagnosis of a fault in a complex industrial process, in categorising an archaeological or anthropological specimen. From an expository point of view, however, the nature of a diagnostic problem is most easily described, and the corresponding theory is best developed, within the context of a specific situation. For this purpose we have selected a medical problem concerning the differential diagnosis of three forms or types of a particular syndrome on the basis of two diagnostic tests or observable features. We have deliberately selected this three-type two-feature problem because it allows the maximum exploitation of diagrammatic means of expressing concepts and analyses. All the concepts and analyses carry over straightforwardly into higher dimensional problems. Indeed the introductory illustrative problem which we now present is a subproblem extracted from a larger real one.

Example 11.1

Differential diagnosis of Cushing's syndrome. Cushing's syndrome is a rare hypersensitive disorder associated with the over-secretion of cortisol by the adrenal cortex. For illustrative purposes we confine ourselves here to three ‘types’ of the syndrome, those types in which the cause of this over-secretion is actually within the adrenal gland itself. The types are

1. a: adenoma,

2. b: bilateral hyperplasia,

3. c: carcinoma,

and we investigate the possibilities of distinguishing the types on the basis of two observable ‘features’, the determination by paperchromatography of the urinary excretion rates (mg/24h) of two steroid metabolites, tetrahydrocortisone and pregnanetriol.

The nature of statistical prediction analysis

An essential feature of statistical prediction analysis is that it involves two experiments e and f. From the information which we gain from a performance of e, the informative experiment, we wish to make some reasoned statement concerning the performance of f, the future experiment. In order that e should provide information on f there must be some link between these two experiments. Throughout this book we shall deal with problems where this link is through the indexing parameter of the two experiments e and f, and so we make the following assumption.

A further general feature of all the problems we shall consider is contained in the following independence assumption.

By adopting this second assumption we deliberately exclude a range of prediction problems in which f is a continuation of some stochastic process of which e records a realisation to date. Techniques such as forecasting by exponential weighting, linear least squares prediction and time series analysis are thus outside the scope of this book.

To give some idea of the wide applicability of statistical prediction analysis as defined above and to motivate the development of appropriate theory we devote the remainder of this chapter to the presentation of specific prediction problems. All these problems are later analysed and extended in the sections indicated in the text.

Introduction

The myriad of possible statistical sampling inspection procedures forces us to consider in detail only a very small selection in a book of this size. We would need a separate book to do justice to the huge variety of plans. In this chapter therefore we show how some standard plans come within the framework of decisive prediction, and how the framework can readily cope with less standard problems. The application of prediction theory to this area will provide some additional justification and motivation for some of these plans. We hope that those selected will be sufficient to indicate the direction of analysis to any reader with a specific problem.

We consider both fixed size sample and sequential sampling schemes. Wetherill (1966) and Wetherill and Campling (1966) also provide a decision theory approach to sampling inspection but do not consider predictive distributions.

Fixed-size single-sample destructive testing

We consider first a fixed-size single-sample plan for deciding whether to accept or reject a batch. For a process which produces an item at each of a number of independent operations we may imagine as our basic future experiment the determination of the quality y of a single item. This quality y may be a simple counting variable taking the value 1 for an effective and 0 for a defective item, or may be more sophisticated, for example the lifetime of a component or the degree of purity of a chemical preparation. We suppose that the probabilistic mechanism which describes the production of the variable y is a density function p(y|θ) on Y where, as in previous work, θ is an indexing parameter with density function p(θ).

The nature of a treatment allocation problem

When a treatment is applied to an object or individual it is with the express purpose of altering the future of that object or individual. Thus when we choose one of a number of possible refining processes for a batch of raw material we intend that the batch will in the future attain some desirable quality. When we select a method of machining an industrial component we have in mind some future characteristic of the component. When we prescribe a particular treatment for a patient we hope that some specific aspect of his future condition will be more agreeable than his present state of disease. Because of this preoccupation with the future state of an object or individual it will not be surprising to find that statistical prediction analysis has an important role to play in the problem of treatment allocation.

In the examples already mentioned there are three basic sets which must clearly play an important role. First we suppose that the present state or indicator t of the individual unit under consideration belongs to some specifiable set T of possible initial states or indicators. Secondly, there is some set A of possible treatments from which we have to select a treatment a to apply to the individual unit. Thirdly we must to some extent assess the effectiveness of treatment in terms of the future state or response y attained by the unit after application of treatment; we thus have to be in a position to envisage the set Y of possible future states or responses.

Point prediction

If we are asked to predict the outcome of a performance of a future experiment f our answer will clearly depend on how we view the consequences of being wrong. More specifically we may attempt to assess the relative consequences of being ‘close’ to the realised outcome and of being ‘badly’ wrong. If we can quantify these visualised consequences then we can present the problem as one of statistical decision theory. Since in constructing the predictive density function p(y|x) we have already carried out the information-extraction part of the problem we have a particularly simple confrontation in this decision problem. The components are as follows,

1. (i)Parameter set. The unknown outcome of the future experiment f plays the role of an unknown state of nature, so that Y, the sample space of f, is the parameter set of the statistical decision problem. Our assessment of the plausibility of a particular y at the time of making a decision is p(y|x), the predictive density at y.

2. (ii)Action set. The set A of possible actions is simply a reproduction of Y, since any element of Y is a possible prediction a.

3. (iii)Utility function. Associated with each prediction or action a and each realisable outcome y there is a utility or value U(a, y). We thus suppose defined a function U on the product domain A × Y.

Prediction by its derivation (L. praedicere, to say before) means literally the stating beforehand of what will happen at some future time. It is an occupational hazard of many professions: meteorologist, doctor, economist, market researcher, engineering designer, politician and pollster. It is indeed a precarious game because any specific prediction can eventually be compared with the actuality. Many prophets of doom predicting that the world will end at 12.30 on 7 May are left in quieter mood by 12.31. Prediction is a problem simply because of the presence of uncertainty. Seldom, if ever, is it a case of logical deduction; almost inevitably it is a matter of induction or inference. Probabilistic and statistical tools are therefore necessary components of any scientific approach to the formalisation of prediction problems.

In this book we shall be concerned with prediction not only in this narrow sense of making a reasoned statement about what is likely to happen in some future situation but with a much wider class of problems. Any inferential problem whose solution depends on our envisaging some future occurrence will be termed a problem of statistical prediction analysis. The presentation in chapter 1 of a selection of motivating examples illustrates the nature and diversity of statistical prediction analysis, and serves as an introduction to the ingredients of the problem.

A science historian, writing on the development of the concepts and practice of prediction, would probably start by pointing out how primitive man was compelled to attempt prediction, for example the forecasting of the date on which the local river would flood.

The nature of a calibration problem

Calibration is commonly regarded as the process whereby the scale of a measuring instrument is determined or adjusted on the basis of an informative or ‘calibration’ experiment. For example, if we wish to calibrate an unsealed thermometer we might note the position x1 on the liquid scale when the thermometer is immersed in boiling water at atmospheric pressure, that is, corresponding to temperature t1 (= 100°C); and the position x2 when the immersion is in ice, say corresponding to temperature t2 (= 0°C). We might then divide the scale between x1 and x2 into 100 equal divisions so that, when the thermometer is immersed into some other substance, we are able to deduce very simply from the x-scale the corresponding temperature of the substance. In this example the use of the calibration experiment yielding trial records (t1, x1), (t2, x2) is straightforward since there is, or at least we are assuming that there is, a one-to-one correspondence between the x-scale and the temperature or t-scale. But the same type of problem arises commonly in a less simple form, for usually, as in the following examples, there is no unique x corresponding to a given t.

Example 10.1

Measuring water content of soil specimens. Two methods are available for obtaining the water content in soil specimens. The first method, performed in the laboratory, is very accurate but is expensive and tedious to operate. The second method, which can be performed on site, is much quicker and cheaper, but is less accurate.