Clinical Diagnosis by Likelihood Triangle

Differing medical conditions often exhibit similar symptoms. Rasch measures can assist clinicians by portraying the relative likelihoods of diagnoses based on observed symptom severity.

Suppose three types of injury produce similar symptoms but with different intensities. Samples of patients identified to have each type of injury are used to calibrate the symptoms on the relevant variable of injury severity. This defines three variables, gives each symptom three calibrations, and produces three raw-score-to-measure tables, one for each diagnosis.

When a new patient comes in for diagnosis, a rating of the severity of each symptom is recorded. The sum of these ratings, the patient's raw score, produces three measures, (Bj, Bk and Bl), one for each of the diagnoses, j, k, and l.

From these measures, the likelihood of the observed pattern of symptom severities is estimated for each diagnosis.

Diagj = Prod(Pji(Xi)), for i=1 to L, is the likelihood of diagnosis j. Pji(Xi) is the usual Rasch probability of the patient's observed severity level Xi, on symptom i for diagnosis j, when the patient's overall measure is Bj.

The likelihood of each diagnosis is plotted to locate the patient among the diagnoses. The plot is an equilateral triangle, in which the distance from the side opposite the diagnostic vertex is proportional to the likelihood of that diagnosis. This locates the profile of symptom severities in the plane of the three diagnoses, nearer the more likely diagnoses.

Diagnosis Triangle


The vertices of the equilateral triangle, the locations of the diagnoses, are at (0,0) for diagnosis l, (m,0) for diagnosis k, and (m/2, msqrt((3/4)) for diagnosis j, where m is a chosen to give a conveniently sized triangle. The location of the central profile point, *, is ( m(Diagk + Diagj/2)/(Diagj + Diagk + Diagl), mDiagj(sqrt((3/4))/(Diagj + Diagk + Diagl) ).

If diagnosis j is most probable, with likelihood 0.3, and diagnosis k is 0.2, diagnosis l is a least probable 0.1, the vertices are at (0,0) (1,0) and (0.5, 0.87) when m=1. The profile location is (0.58, 0.43). This shows a location nearer j than k and far from l, enabling the clinician to select at a glance an efficient screening procedure for identifying the correct diagnosis.

Empirical Triangle



Clinical diagnosis by likelihood triangle. Granger CV. … Rasch Measurement Transactions, 1994, 8:2 p.357



Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
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