Probability diagnostics. Application of probabilistic logic in diagnostics
Diagnostics is a process of stage-by-stage processing of the information in the system of "patient - doctor- patient", which purpose is creation the most adequate model of a state of the patient organism.
The diagnostical process it is possible to divide into three logically connected stages:
1) Collection of information of the patient state (detection of signs, carrying out of analyses, etc.);
2) Selection of the most important tags (signs), their comparison with a norm range with taking into account a sex, age, genetic tags, a mode of life, etc.; systematization of sick person tags in defined a symptom-complex (syndromes) is a stage of processing of the information;
3) Comparison to signs of known illnesses - the diagnosis setting.
The process of diagnosis setting is carried by the defined rules described as an algorithm.
The diagnostic algorithm is a logical sequence of rules in which the information on tags of a state of the patient is compared to the complex of tags which characterize typical illnesses.
On the basis of outcomes comparison is made a decision on the probable diagnosis. The diagnosis setting is easy only at typical clinical course, when set of given patient signs completely coincides with a symptom-complex of the defined illness. In practice it happens farly from being always. More often solution starts as choice from several probable diagnoses. Any diagnostic algorithm it is possible to automize and organize machine diagnostics.
Type of diagnostic algorithm will depend on accepted medical logic. In modern medical diagnostic intellection is accepted to distinguish three types of medical logic: 1) deterministic logic; 2) phase interval logic; 3) probabilistic logic.
Nowadays most spread method is grounded on probabilistic logic.
The probabilistic logic is a diagnostic method, whether the probability settles up with the help that one or other diagnosis for defined a set of signs. For this purpose it is necessary to know probability of each sign for different illnesses. This probability (frequency of sign at different illnesses) is received after handling a great quantity of case histories with precisely established diagnoses, as a rule.
Bases of probabilistic diagnostics theory
Probability of observation of this or that medicobiological event (presence or absence of tag, a degree of expressiveness in this case the question is any value of the parameter, deviation, performances of physiological processes, etc.) similar to theoretical concept of probability.
We shall designate a total number of examined patients as N. An amount of patients with different deviations – accordingly as S1, S2, S3, S4.
Frequency of event is the ratio of quantity of event (amount of real cases which are observed) to total of observations P=S/N.
Correspondingly P1=S1/N, P2= S2/N, P3=S3/N, P4=S4/N.
Any of P1, P2, P3, P4 values is named as frequency of event (or sign which is observed).
The probability of observation of event (whether tag which is observed) is the frequency of event if total amount of observations aspire to infinity.
The probability can be expressed in parts of unit or in percents (after multiplication by 100%).
For example, after examination of great group of patients with diabetes follows data were obtained: among such patients obesity happens in 30% cases (POb=0,3), retinopathies happens in 25% cases (PR=0,25), arterial hypertension happens in 10% cases (PAH=0,1). Reliability of these data depends on amount of studied group: the amount is more the reliability is more. Level of significance can be calculated by statistical methods.
In bases of probabilistic diagnostics such data obtained during examinations of many patients with different diagnoses lay. Then for every sing conditional probabilities at every disease are calculated. Conditional probability is probability of any event on the background (at a presence) of other event.
Conditional probabilities of presence of signs Si calculated at disease Dj in notated as P(Si/Dj) (it is read: “Probability of event Si at event Dj”).
Conditional probability P(Si/Dj) means, that if it is defined for the patient illness with diagnosis Dj then signs Si of this illness have probability P(Si/Dj).
Systems of probability diagnostics and Bayes' formula
1) It is necessary to obtain all probabilities of all signs for the expected diseases. If it is three diseases (D1, D2, D3) with four sings (S2, S7, S9, S11) then three groups of numbers should appear:
P(S2/D1) P(S2/D2) P(S2/D3)
P(S7/D1) P(S7/D2) P(S7/D3)
P(S9/D1) P(S9/D2) P(S9/D3)
P(S11/D1) P(S11/D2) P(S11/D3)
If it is a lot of signs and many probable diagnoses (it happens in practice) then given method is possible only with usage of computer technologies.
2) Conditional probability of symptom-complex Р (Ssc/Dj). According to probability theory, the conditional probability of a symptom-complex (symptom group) is a product of probabilities of all signs of symptom-complex at the given diagnosis. For example, for a symptom-complex with 4 signs for defined diagnosis J:
P (Ssc/Dj) = P(S1/Dj)*P(S2/Dj) * P(S3/Dj) *P(Sn/Dj),
In case of n signs:
P (Ssc/Dj) = P(S1/Dj)*P(S2/Dj) * … *P(Sn/Dj)
Such conditional probabilities obtained for every diagnosis in studied set.
3) Definition of a priori probability of disease.
A priori probability of the certain diagnosis (Dj) is empirical frequency of observation of the given disease in the certain concrete conditions. The priori probability designates P(Dj) and is evaluated for every studied disease.
“A priori” means knowledge foregoing to experiment with new observation (new patient). A priori probabilities of diseases depend on season, genetic, geographic, social, and other factors. They must be calculated in every research conditions, checked and corrected regularly.
4) An evaluation of the normalized coefficient (Pnc).
The normalized coefficient is a composite probability of available symptom-complex for all diseases. This value is the complete sum of paired products conditional probabilities of every symptom-complex for the given diagnosis on a priori probability of this diagnosis:
Pnc = P(Ssc/D1) * P(D1) + P(Ssc/D2) *P(D2) +... +P(Ssc/Dn) * P(Dn)
Here n – to number of diagnoses which are considered in this system.
5) An evaluation of probabilities of diagnoses for given a symptom-complex Р(Dj/Ssc)
Bayes theorem (the formula of probability of hypotheses):
This stage penultimate in the circuit{scheme} of operation of the system also is grounded on usage of Bayes theorem (the formula of probability of hypotheses):
P (Dj/Ssc) = [P(Ssc/Dj) *P(Dj)] / [Psc]
Such probability is calculated for every diagnosis; them amount is equaled to number of diagnoses of the system.
6) Identification of the diagnosis
The greatest probability value of P(Dj/Ssc) set corresponds to the most probable diagnosis. If probabilities of some diagnoses peer, it is necessary to declare that the diagnostic table which is used in the system, insufficiently perfect "to distinguish" these diagnoses.
Main sources.
Handbook of Medical Informatics. Editors: J.H. van Bemmel, M.A. Musen. – http://www.mieur.nl/mihandbook; http://www.mihandbook.stanford.edu
Kimble G. How correctly to use statistics.
Neumann J. Introductory course of probability theory and mathematical statistics.
Additional textbook:
John Truss, “Discrete Mathematics for Computer Scientists”, Addison-Wesley, Second Edition, 1999
Nimal Nissanke, “Introductory Logic and Sets for Computer Scientists”, Addison-Wesley, 1999
James A. Anderson, “Discrete Mathematics with Combinatorics (2nd edition)”, Prentice-Hall, 2004
Rod Haggarty “Discrete mathematics for computing” Addison Wesley
Chetwynd and P. Diggle, “Discrete Mathematics” Butterworth-Heinemann
Geoffrey Finch, “How to study linguistics”, Macmillan, 1998
J. N. Crossley and others, “What is mathematical logic?”, Oxford University Press, 1972
Lecrure have prepared by lecturer Korovina L.D.