Statistical Evaluation of Diagnostic Tests – Part 2 [Pre-Test and Post-Test Probability and Odds, Likelihood Ratios, Receiver
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86 Journal of The Association of Physicians of India ■ Vol. 65 ■ July 2017 STATISTICS FOR RESEARCHERS Statistical Evaluation of Diagnostic Tests – Part 2 [Pre-test and post-test probability and odds, Likelihood ratios, Receiver Operating Characteristic Curve, Youden’s Index and Diagnostic test biases] NJ Gogtay, UM Thatte Introduction Understanding Probability From the example, it follows that and Odds and the Odds = p/1-p, where p is the n the previous article on probability of the event occurring. the statistical evaluation Relationship between the I Probability, on the other hand, of diagnostic tests –Part 1, we Two is given by the formula understood the measures of sensitivity, specificity, positive Let us understand probability p = Odds/1+Odds and negative predictive values. and odds with the example of The use of these metrices stems a drug producing bleeding in Bayesian Statistics, Pre- from the fact that no diagnostic 10/100 patients treated with it. Test Probability and Pre- test is ever perfect and every time The probability of bleeding will Test Odds we carry out a test, it will yield be 10/100 [10%], while the odds of one of four possible outcomes– bleeding will be 10/90 [11%]. This A clinician often suspects that true positive, false positive, true is because odds is defined as the a patient has the disease even negative or false negative. The 2 x probability of the event occurring before he orders a test [screening 2 table [Table 1] gives each of these divided by the probability of or diagnostic] on the patient. For four possibilities along with their the event not occurring.2 Thus, example, when a patient who is mathematical calculations when a every odds can be expressed as a chronic smoker and presents new test is compared with a gold probability and every probability with cough and weight loss of a standard test.1 as odds as these are two ways of six-month duration, the suspicion In this article, the second in explaining the same concept. of lung cancer has already entered the diagnostic test series, we will Table 1: A 2 x 2 table of depicting the results of a new test vis à vis a gold standard discuss single summary statistics test that help us understand and use Posi�ve predic�ve value = a/a +b these tests appropriately both in the clinical context and when Disease these summary statistics appear in literature. Before we discuss Test Present Absent these, we need to recapitulate a Sensi�vity = a/a +c Specificity = d/b+d few concepts presented in earlier Posi�ve True Posi�ve [TP] a False posi�ve [FP] b a+b +c articles [odds and probability] and also some novel concepts [Bayesian Nega�ve False Nega�ve [FN] c True Nega�ve [TN] d c+d statistics, pre-test and post-test probabilities and odds]. Nega�ve predic�ve value = d/d +c Department of Clinical Pharmacology, Seth GS Medical College & KEM Hospital, Mumbai, Maharashtra Received: 06.05.2017; Accepted: 10.05.2017 Journal of The Association of Physicians of India ■ Vol. 65 ■ July 2017 87 the physician’s mind. Thus, the Mathematically, this is calculated Likelihood ratio Sensitivity [TP] = clinician has already, mentally, as [positive] LR+ 1 -Specificity [FP] identified some “pre-test” probability Pre-test probability = while a negative Likelihood ratio of the patient having the disease; Number of patients with is given by lung cancer in this case. complaints actually diagnosed to Likelihood ratio 1- Sensitivity [FN] Clinical decision-making, by = have the disease [negative] LR - Specificity [TN] and large, requires a combination Total number of patients who of clinical acumen along with a Let us understand this with present with the same complaints correctly performed and interpreted an example. When physical screening or diagnostic test. When [In this case, it would be 60/100 examination is carried out in patients the physician allocates a “pre-test or 60%]. with suspected acute appendicitis, probability”, what he is applying is Pretest odds, however, would there-is-rebound tenderness at or a field of statistics called Bayesian be 0.6/0.4 or 1.5 (the probability about the McBurney’s point, pain on statistics. Herein, the knowledge of of the event occurring divided by percussion, rigidity, and guarding. prior beliefs is used and quantified the probability of the event not The positive likelihood ratio for the as a numerical value ranging from occurring). diagnosis of appendicitis would be 3 the ratio of those with appendicitis 0 -100%. This value is then used for The clinician next orders a test, who have tenderness at McBurney’s subsequent calculations. Bayesian which he hopes, will confirm [or point [sensitivity] by those without statistics allows us to interpret refute] his diagnosis. The test appendicitis who have tenderness screening and diagnostic tests in result and the pre-test probability at McBurney’s point [falsely their clinical context. together will now be used to positive or 1- specificity] Logically, the next question calculate the post-test probability would be - what are the ways in as described below. OR which these pre-test probabilities Likelihood ratio [positive] LR+ can be allocated? These are listed Post-test Probability and The number of patients with below Post-Test Odds appendicitis who have localized • Subjectively based on informed tenderness at the McBurney’s point Since the result of a diagnostic opinion, consensus guidelines test can be either positive or The number of patients without or experience in treating the negative, post-test probabilities appendicitis who have localized disease in question are either positive or negative. tenderness at the McBurney’s point • An understanding of the Mathematically, The negative likelihood ratio evolution of the disease and • Post-test probability = Pre-test LR- would be matching it with how the probability x Likelihood ratio The number of patients with disease has actually evolved (see below for explanation), appendicitis who don’t have localized in the patient while tenderness at the McBurney’s point • Objectively based on available • Post-test odds = Post-test The number of patients evidence [prevalence data for probability/1 – post-test without appendicitis who don’t example] probability have localized tenderness at the In the example presented, the McBurney’s point treating physician may assign a The Likelihood Ratio If we were to express both these pretest probability of 60% or even [A Summary Statistic] mathematically, based on the 2 higher based on his clinical acumen x2 table, these would be as given and what he sees in practice. Likelihood ratios [LR] combine below How is this calculated? Let us say both sensitivity and specificity Likelihood ratio positive or LR + that the clinician is a lung cancer into a single measure and are an specialist and he sees 100 patients alternate way of evaluating and The probability of obtaining a in three months who are chronic interpreting diagnostic tests.4 positive test result in patients with smokers with persistent cough They help in making a choice of a disease [TP] and weight loss. Sixty of them diagnostic test or sequence of tests. The probability of obtaining eventually return a diagnosis of LR essentially tell us how many a positive test result in patients lung cancer based on one more times more [or less] a test result is without the disease [FP] tests. The pretest probability for a to be found in diseased compared On the other hand, a negative new patient with a similar history to non-diseased people. LRs are of likelihood ratio or LR- would be and complaints who presents to two types – positive and negative. him in the fourth month would A positive Likelihood ratio is given thus be 60%. by 88 Journal of The Association of Physicians of India ■ Vol. 65 ■ July 2017 The probability of obtaining a of 5% that we use routinely use to of lung cancer. In other words, negative test result in patients with check for statistical significance of the test is “positive”. Literature disease [FN] a LR that is calculated. tells us7 that low dose CT has an approximate sensitivity of 80% The probability of obtaining Clinical Application a negative test result in patients and a specificity of 90%. Thus, the without the disease [TN] – putting Together positive likelihood ratio would be Since different tests for the same Probability, Odds and the • LR + = Sensitivity [.8]/ 1- disease have different sensitivities Likelihood ratio specificity [1-0.9] = 8 [this LR and specificities, each test would + indicates that the test result yield a different likelihood ratio Having understood the concepts is more likely in someone with for the same disease. Let us of probability and odds, pre-test lung cancer than someone understand this with an example. and post-test probabilities and the without] The diagnosis of prostate cancer likelihood ratios we need to put all We now calculate the post-test can be made by both digital rectal of them together to see how they odds as pre-test odds x likelihood examination [DRE] and Trans rectal actually help in clinical decision ratio ultrasonography [TRUS]. Manyahi making; the sequence for which is • Thus, post-test odds = 1.5 x 8 = 5 JP and colleagues in their study given below 12 found the sensitivity of DRE to • Calculate Pre – test probability Finally, we want to convert be 66.7%, and the specificity to be (p) the post-test odds into post-test 88.6%. The values for TRUS were • Derive Pre- test odds as p/1-p probability 58.3% and 85.7% respectively. The LR + for DRE thus would be 5.8 • Conduct the test [screening • i.e., 12/1 + 12 = 12/13 0r 0.92 [.667/1-.886], while that for TRUS or diagnostic] with an or 92% [indicating a high would be 4.1[.583/1-.857].