The Bayes Theorema. Converting pre-diagnostic odds into post-diagnostic odds. Prof. Dr. F. Vanstapel, MD PhD Laboratoriumgeneeskunde UZ KULeuven

Transcription

1 slide 1 The Bayes Theorema Converting pre-diagnostic odds into post-diagnostic odds Prof. Dr. F. Vanstapel, MD PhD Laboratoriumgeneeskunde UZ KULeuven

2 slide 2 Problem * The yearly incidence of TBC infections in the school population of inner cities is 0.5 %. We tested your child radiographically and found a suspicious lesion. The test is believed to detect 80 % of early lesions. However 5% of healthy children show lesions which turn out to be benign artifacts. Are you alarmed? Do you have all the information that you need to have an idea of the chance that the suspicion turns out to be a serious condition? O Try to calculate the odds, or explain why you cannot * The example has no bearing with the actual situation in a particular poulation

3 slide 3 Solving the problem: 1/4 Given: Yearly incidence of TBC infections in the school population of inner cities is 0.5 %. P(D+) = We tested your child radiographically and found a suspicious lesion. The test is believed to detect 80 % of early lesions. P(T+ D+) = 0.80 However 5% of healthy subjects show lesions which turn out to be benign artifacts. P(T+ D-) = 0.05 What has to be solved: The probability that your child is sick, given the positive test result. P(D+ T+) Notation: P(D+) : the probability to find the disease in the population under study P(T+ D+) : the probability that the test is positive, when disease is present P(T+ D-) : the probability that the test is positive, when disease is absent P(D+ T+) : the probability to find the disease in persons with a positive test result

4 slide 4 Solving the problem: 2/4 Given: P(D+) = P(T+ D+) = 0.80 P(T+ D-) = 0.05 Solution: From known probabilities (proportions) TruePos = P(D+) * P(T+ D+) = * 0.80 = FalsePos = P(D-) * P(T+ D-) = ( )* P(D+ T+) = TruePos / (TruePos + FalsePos) / ( ) = 7.4 % P(D+ T+) P(D+) P(T+ D+) = 1- P(D+ T+) P(D-) P(T+ D-) Prevalence Sensitivity 1 - Specificity T+ T- T+ SUM True positive False positive D D D D Post-Test Odds P(D+ T+) 4/ /53.75

5 slide 5 Solving the problem: 3/4 Given: P(D+) = P(T+ D+) = 0.80 P(T+ D-) = 0.05 T+ T- D+ D Solution: From known odds (likelihood ratio s) PreTOdds = P(D+) / P(D-) = / ( ) = 1/199 LHR = P(T+ D+) / P(T+ D-) = 0.80 / 0.05 = 16/1 PostTOdds = PreTOdds * LHR = 16/199 P(D+ T+) = PreTOdds / (1 + PostTOdds) = 16 / ( ) = 7.4 % P(D+ T+) P(D+) P(T+ D+) = 1- P(D+ T+) P(D-) P(T+ D-) T+ D Pre-Test Odds Positive Likelihood Ratio Post-Test Odds P(D+ T+) D- Odds /199 16/1 16/199 16/215

6 slide 6 Test Result S S Latin square Pre-Test Probability Condition Prevalence: D/Total D _ TN FP TN / D _ D FN TP TP / D Specificity Sensitivity Test Ability P(S D) P(D) P(S D) TN / S TP / S Post-Test Probability Predictive Value NPV PPV P(D S) P(D S) What the clinician is after What the literature has on offer

7 slide 7 Solving the problem: 4/4 Given: P(D+) = Prevalence P(T+ D+) = Sensitivity P(T+ D-) = 1- Specificity Discussion of the solution: Bayes theorema solves P(D+ T+) from P(T+ D+) & P(T+ D-) & P(D+) P(D+ T+) P(D+) P(T+ D+) = = OddsRatio 1 - P(D+ T+) P(D-) P(T+ D-) OddsRatio P(D+ T+) = = Probability 1 + OddsRatio Prevalence, Sensitivity & Specificity is needed and suffices to solve the problem. The solution can be derived from probabilities (sol. 1) or from odds (sol. 2), with exactly the same number of numerical operations (starting from a Latin square).

8 Bayes Theorema - Introduction Version slide 8 Solving the problem - Comments (1/5) Equivalence of Proportions and Odds ratio s I got 1/8 of the cake Proportions Odds ratio s p all Formula s O = p / (1 - p) p = O / (1 + O) possibilities Break-Even-Point 1/2 1/1 Range 0 1/ /7 O all alternative possibilities %-transformed 0% 12.5% 100% log-transformed

9 Bayes Theorema - Introduction Version slide 9 Solving the problem - Comments (2/5) Likelihood ratio s : what they mean An example from the physical world : Dual-phase Extraction Odds Ratio s Proportions Phase 2 Vol 1 Vol 2 x Conc 1 Conc 2 = Mass 1 Mass 2 Mass 1 Mass 1 + Mass 2 yield partition coefficient Phase 1 phase volume ratio Likelihood ratio of an individual molecule at the interface to reside in either phase

10 Bayes Theorema - Introduction Version slide 10 Solving the problem - Comments (3/5) Dichotomic Classification versus a cut-off Unit Surfaces Chance Density x prevalence Frequency Distribution Number Density D- D+ D- Note: Optimal Cut-Off is Prevalence dependent D Bayesian Evaluation of Results Sensitivity Specificity P (T + D + ) P (T - D - ) Pre-test prevalence P (D + ) P (T + D + ) P(D + ) P (D - ) P (T - D - ) P(D - ) P (D + T + ) +

11 Bayes Theorema - Introduction Version slide 11 Solving the problem - Comments (4/5) Interpreting an actual value in a continuum of possible values Unit Surfaces Chance Density x prevalence Frequency Distribution Number Density D- D+ D- D Bayesian Evaluation of Results x Pre-test Prob P (x D + ) P (D + ) P (x D + ) P(D + ) P (D + x) + P (x D - ) P (D - ) P (x D - ) P(D - )

12 Bayes Theorema - Introduction Version slide 12 Solving the problem - Comments (5/5) Calculating with Odds Ratio s Pre-test Odds P (x D + ) P (x D - ) x P (D + ) = P (x D + ) P(D + ) P (D - ) P (x D - ) P(D - ) P (D + x) Post-test Odds non-linear parameter Prevalence in actual clinical case-mix in similar clinical presentations Positive Likelihood Ratio : non-linear parameter referring to artificial 1/1 prevalence

13 Bayes Theorema - Introduction Version slide 13 Summary: The Bayes theorema describes the learning process. It explains how prior knowledge is converted by a learning experience into posterior probability (= a considered professional judgement about the likelihood of a condition in a patient and about the diagnostic ability of the test requested) (= learning that a particular test result was obtained) (= degree of belief in a given diagnosis). The value of a test depends on its position in the sequence of diagnostic procedures The posterior probability (when expressed as such) is read on a scale of 0 to 1. The resulting decision is read on an on/off scale, and is either 0 or 1. Test drive our diagnostic calculator

14 slide 14 Concepts that you should have mastered in this exercise: O Prevalence O Pre-test Probability O Sensitivity and Specificity O Positive and (by analogy) Negative Likelihood Ratio s O Post-test Probability O Equivalence of Probabilities and Odds Exercise: Give for each item in the above list an operational (what you use it for) definition, and cross-check your definition with the solution of the problem. Exercise: In this example the test was used for screening purposes, sensitivity and specificity are defined with respect to a healthy population. What should be the definition of sensitivity and specificity in a clinical setting? (hint: differential diagnosis)