The Role of Likelihood Ratio in Clinical Diagnosis: Applicability in the Setting of Spontaneous Bacterial Peritonitis

Transcription

1 CLINICAL GASTROENTEROLOGY AND HEPATOLOGY 2005;3:85 89 EVIDENCE-BASED MEDICINE The Role of Likelihood Ratio in Clinical Diagnosis: Applicability in the Setting of Spontaneous Bacterial Peritonitis FERNANDO SIERRA,* DIANA TORRES, and ANDRÉS CÁRDENAS *Division of Gastroenterology and Hepatology, Fundación Santa Fe de Bogotá, Colombia; Social Security Institute, Bogotá, Colombia; and Division of Gastroenterology and Hepatology, Beth Israel Deaconess Medical Center and Harvard Medical School, Boston, Massachusetts Despite imperfect clinical information and uncertainty about clinical course and outcome, the clinician s main task is to make reasonable decisions about patient care. The clinical history and physical examination typically provide information that is useful for making a diagnosis; however, we still rely on laboratory and radiologic tests to confirm a diagnosis in most cases. Understanding the operative characteristic of a test is of key importance because it can change the probability that a patient has a disease before the result of a test is known. This operative characteristic, better known as the likelihood ratio (LR), is a global assessment of the information provided by a test. The LR allows calculating the odds that a patient has a disease after a test is performed. In this article, we explain the meaning of the LR, how it works, and the applicability of this tool in the setting of a challenging scenario in clinical practice, spontaneous bacterial peritonitis. One of the most daunting challenges facing clinicians on a daily basis is establishing an accurate diagnosis in a reliable and cost-effective manner. A physician uses inductive reasoning within an ordinary clinical scenario to make an accurate diagnosis; this type of reasoning is derived from data such as clinical history, physical findings, and test results to arrive at a final diagnosis. Within this context, the information given by a test is key for establishing the diagnosis. When a physician is in the process of making a diagnosis, he needs to ask: which is the most reliable test for making a diagnosis in a patient s specific setting? To make the best selection, a physician needs to know what measurement of test performance gives the most accurate information when making a specific diagnosis. Sensitivity and specificity are operative characteristics of a test; they are measurements of test performance used for judging the quality of a diagnostic test for a particular disease. Sensitivity is the likelihood that a patient who has a disease has a positive test, and specificity is the likelihood that a healthy patient has a negative test. However, the real value of a test in an ordinary clinical scenario relies not only on its sensitivity and specificity, but also on the ability of the information provided by the results of such a test to bring about a substantial change in the pretest probability regarding a particular disease (ie, clinical impression) and in the posttest probability (ie, accurate diagnosis). To accomplish this task, the test s sensitivity and specificity must be combined by creating a likelihood ratio (LR) so that a diagnosis can be made accurately. The LR is a global assessment of the information provided by the test. It also answers the question, Given a positive or negative test, what is the likelihood or odds of having the disease? 2 The Likelihood Ratio LRs represent the odds of having a disease if the result of the laboratory test used is positive (LR ) orthe odds of not having a disease if the result of the laboratory test used is negative (LR ). 2,3,5,6 To calculate the LR, the sensitivity and specificity of the test are combined into an equation: LR Probability of the result in diseased people (true positive rate) Probability of the result in non-diseased people (false positive rate) Every test has 2 likelihood ratios, one corresponding to a positive test result (LR ) and one corresponding to a negative test result (LR ). Abbreviations used in this paper: LR, likelihood ratio; PMN, polymorphonuclear; SBP, spontaneous bacterial peritonitis by the American Gastroenterological Association /05/$30.00 PII: /S (04)

2 86 SIERRA ET AL CLINICAL GASTROENTEROLOGY AND HEPATOLOGY Vol. 3, No. 1 LR LR Probability that test is positive in diseased people Probability that test is positive in non-diseased people Sensitivity 1 Specificity Probability that test is negative in diseased people Probability that test is negative in non-diseased people 1 Sensitivity Specificity When calculating LR, the odds are constituted in the numerator by the test s sensitivity, and in the denominator by the rate of false-positive results or 1 specificity. LR answers the question: In those people who have a disease as opposed to those who do not have it, what is the probability that the laboratory test result could be positive? Therefore, if a laboratory test has 7 LR, then the probability of the test being positive is 7 times more frequent in patients having the disease than in people who do not have the disease. With LR, the odds are constituted in the numerator by the rate of false-negative results or 1 sensitivity (which is the same as the false-negative rate), and in the denominator by the specificity of the laboratory test. LR answers the question, In those people who have a disease as opposed to those who do not have it, what is the probability of the result of a laboratory test being negative? Therefore, if a laboratory test has.2 LR, then the probability of the test being negative in patients having the disease is.2 times less than in those patients who do not have the disease. This is more understandable if the inverse is calculated (1/.2 5). Patients who do not have a disease are 5 times more likely to have a negative test result than those patients who have a disease. What is the value of LRs? The value of LRs depends on how they are combined with the initial diagnostic impression to increase or decrease the probability that a patient has a disease. LRs represent the best way clinicians have of talking about data obtained by laboratory tests 3,6 as they modify the first impression concerning a patient s diagnosis (pretest probability of having the disease) by translating it into a posttest probability of having the disease. To accomplish this task, clinicians have to calculate the prior probability of having the disease and subsequently translate it into Table 1. Recommended Approach to Determine the Pretest Probability of a Disease 1% Pretest probability of some risk factors for a common disease but with no symptoms 10% Pretest probability when a disease is improbable but clinically possible and the doctor wishes to exclude it 50% Pretest probability when there is considerable uncertainty but clinical presentation is compatible with a disease and the doctor wishes to establish or discard such disease 80% Pretest probability when a disease clinically is very possible but the doctor wishes to establish it fully prior or pretest odds (odds probability/1 probability). The pretest odds then are multiplied by the LR or LR from the specific diagnostic test. This leads to posttest odds that must be converted into a new probability for them to become intuitive and practical (probability odds/1 odds), thus indicating the possibility (on a numeric basis) that a patient could have a disease. The difference between odds and probability can be compared to rolling a die. The probability of getting number 3 on rolling a die is 1 of 6, whereas the odds are 1 against 5 (ie, of not rolling that number). The usefulness of the LR may be applied on a daily basis in clinical practice. There are 2 ways of calculating the LR. One is through conversion of probability to odds and then back to probability, and the other is through calculation by a nomogram. Quantifying the pretest probability of a disease is a challenging exercise involving knowledge, clinical experience, prevalence statistics, studies specifically focused on determining pretest probabilities, and expertise regarding a disease state. Although there are no guidelines allowing one to exactly quantify the probability of a disease before doing a test, Katz 4 has suggested some helpful parameters (Table 1). For example, a patient with acute abdominal pain and a positive Murphy sign has a pretest probability of having acute cholecystitis around 60% and, therefore, a liver ultrasound is ordered to confirm the diagnosis. The ultrasound turns out to be positive for acute cholecystitis. Given that the LR of this test for acute cholecystitis is 5, how is it that LR worksinthis setting? If the pretest probability of the disease is 60%, then (the first method is described here): Pretest odds Probability 1 Probability

3 January 2005 LR IN THE SBP SETTING 87 Odds : The posttest odds are obtained by multiplying the pretest odds by the LR value, and then it can be converted into a posttest probability. In the hypothetical case, the posttest odds are This needs to be converted into a posttest probability to make it understandable. Thus, it is calculated as: Posttest odds Posttest probability 1 Posttest odds In this hypothetical case, 7.5/ ; the patient has an 88% probability of having acute cholecystitis. The other method, which is more simple for calculating the posttest probability from the pretest probability and LR for using LR at the bedside, was designed by Fagan. 5 This method uses a nomogram (Figure 1) in which the physician places a straight line through the points representing the pretest probability and the LR and then notes where the straight edge crosses the pretest probabilities line. 5 In the case of acute cholecystitis, if a line is drawn from the pretest probability of 60% to the second line of a LR of 5, then in the third line, the patient shows an 88% probability of having this disease. A diagnostic test s LR indicates that a laboratory test result can increase or decrease the probability that a patient has or does not have a particular disease. 11 An LR of 1 means that the probability of having a disease after a laboratory test is equal to the probability of having a disease after a laboratory test has been performed. An LR 1 increases the probability that a disease being investigated is present. An LR 1 works in the opposite direction; it diminishes the possibility of a patient having a particular disease. It is believed that an LR 10 or.1 generates an important (almost definitive) change from pre-examination probability to postexamination probability of having a particular disease. LRs from 5 10 and.1.2 generate moderate changes from pretest probability to posttest probability. LRs from 2 5 and.5.2 generate small changes in diagnostic probability. LRs from 1 2 and.5 1 do not significantly alter diagnostic probability before or after the clinical history and examination. 11 Figure 1. The Fagan nomogram. The use of this nomogram allows calculating the posttest probability of a disease from pretest probability and the LR. Reprinted with permission. 5 Copyright 1975 Massachusetts Medical Society. All rights reserved. Some Advantages of Likelihood Ratios Simplicity Numerically expressed as a single figure, LRs provide information regarding the possibility that a disease is or is not present. 3 Additionally, a practitioner also may compare several LRs from different laboratory tests or clinical signs and choose the strongest combination to rule in or rule out a diagnosis. Level of Risk Another advantage of LRs is that they work with diagnostic test results presented on categoric scales (eg, ordinals 0, 1, 2, 3, and so forth) or continuous scales (ie, absolute values on a continuum, such as plasma bilirubin levels). This strategy can be used for categorizing results into different levels to determine the LR in each of the levels. For each level, the ratio of patients having the studied disease at a given level is divided by the ratio of patients who do not have the studied disease at the same level. The result is the LR for each individual level.

4 88 SIERRA ET AL CLINICAL GASTROENTEROLOGY AND HEPATOLOGY Vol. 3, No. 1 Figure 2. The art of clinical diagnosis. Combination of Risks LRs allow physicians to work in a cumulative and combined way because each of the tools obtained from analyzing the clinical history, the physical examination, or laboratory tests and their respective LRs can be multiplied, so that the posttest probability obtained from each one of them becomes the pretest possibility of the following diagnostic test and its respective LR. 3,6 Range of Values It should be stressed that similar to any other statistical figure, a level of range of values can be calculated for each specific LR by using confidence intervals. Examining a range of values for a continuous variable is more powerful than just looking at the LR for a cut-off point. In this way, it is easier to be more precise in determining the probability of having a particular disease after performing the test. Clinical Scenario Showing Use of Likelihood Ratios A 53-year-old man with alcohol-induced cirrhosis (Child B) with ascites and portal hypertension presents with a 10-day history of malaise, weakness, and crampy abdominal pain. Vital signs reveal a blood pressure of 100/70 mm Hg, a heart rate of 94 bpm, and a temperature 37.7 C. On examination, the patient has jaundice, moderate ascites, and lower-extremity edema. There are no features of hepatic encephalopathy. Initial laboratory data reveal a white blood cell count of 7000 L, a hemoglobin level of 10 g/dl, a platelet count of 125,000/L, a sodium level of 132 mmol/l, a creatinine level of 1.2 mg/dl, an aspartate transaminase level of 85 U/L, an alanine transaminase level of 45 U/L, a serum bilirubin level of 108 m/l, an alkaline phosphatase level of 120 U/L, and international normalized ratio of 1.5. Diagnostic Approach Given his history of alcohol abuse, cirrhosis, ascites, and presenting symptoms of malaise and abdominal pain, spontaneous bacterial peritonitis (SBP) is a plausible diagnosis. The possibility that this is a correct diagnosis (pretest probability) can be presumed as being around 20% 50% based on the given clinical features. 13 In SBP, 68% of patients present with fever, 54% of patients present with changes in mental status, 59% of patients present with abdominal pain, and 39% of patients present with abdominal tenderness. 13 In this scenario, the pretest probability must be placed on a line of decisions (Figure 2). As shown in Figure 2, this line of certainty connects 2 points marking action (diagnostic threshold and treatment threshold). If this percentage crosses the diagnostic intervention threshold (for SBP, 40%), then further diagnostic tests must be ordered; but if it were to reach 80%, then it would cross the therapeutic threshold and treatment would be administered. In this setting, the diagnostic threshold was crossed and, therefore, the analysis of ascitic fluid could be used to make the diagnosis of SBP. A diagnostic paracentesis revealed a polymorphonuclear (PMN) cell count of 1250 cells/mm 3. This information leads to revising the role of the LR based on the PMN count in the ascitic fluid for making the diagnosis of SBP (Table 2). 14 With this information, using the Fagan 5 nomogram (Figure 1), an 85% posttest probability of having SBP is calculated (ie, an important change from the initial 20% 40%). To further improve the usefulness of the LR as a handy diagnostic tool, some posttest probabilities can be calculated from the same pretest probability of.2 but with different LRs, depending of the number of PMNs found in the ascitic fluid (Figure 3). Summary The LR represents a useful tool that the clinician has available for changing an initial clinical impression (pretest probability of a specific diagnosis) into a definitive and accurate final diagnosis (posttest probability of a specific diagnosis). This tool uses odds rather than probabilities. The LR expresses the odds of a test occurring in patients with a disease versus those without disease. This means that a positive test will have one LR Table 2. Operative Characteristics of PMN Cell Count in the Ascitic Fluid of a Patient With Cirrhosis and SBP PMN/mm 3 ascitic fluid Present % Absent % LR Total Data from Akriviadis and Runyon. 13

5 January 2005 LR IN THE SBP SETTING 89 and a negative will have another. If a test is positive, the LR is the probability of a positive test in patients with the disease over the probability of a positive test in patients without the disease. If the test is negative, the LR is the probability of a negative test in patients with the disease over the probability of a negative test in patients without the disease. The LR is calculated by dividing the true-positive rate by the false-positive rate. The LR is calculated by dividing the false-positive rate by the true-negative rate. LRs are useful in assessing the accuracy of a diagnostic test; they are also helpful in choosing an appropriate diagnostic test(s). LRs have advantages over sensitivity and specificity. First, they are less prone to change with the prevalence of the disease. Second, they can be calculated for several levels of the symptom or test. Third, they can be used for combining the results of several diagnostic tests and can be used for calculating the posttest probability for a given disease. As shown earlier in the setting of SBP, this tool is useful for determining the probability of having the disease even with different cell counts in the ascitic fluid. 14 The LR is used frequently in the medical literature, particularly in the area of evidence-based medicine; therefore, familiarity with this tool is important for interpreting the medical literature and in clinical diagnosis on a daily basis. Figure 3. Calculation of posttest probabilities based on different LRs. The posttest probability of SBP was calculated from the pretest probability of.2 (20%), but with different LR depending on the number of PMNs found in ascitic fluid. It is important to note that with the same pretest probability, different posttest probabilities are obtained depending on the LR given for the number of PMNs; sometimes it may even decrease the pretest probability of the disease (arrows). This highlights the usefulness of the LR in different clinical scenarios. (T. TEST, threshold to test; T. TREAT, threshold to treatment). References 1. Van Gessel E, Nendaz MR, Vermeulen B, et al. Development of clinical reasoning from the basic sciences to the clerkships: a longitudinal assessment of medical students needs and selfperception after a transitional learning unit. Med Educ 2003;37: Sox HC Jr, Blatt MA, Higgins MC, et al. Clinical decision analysis. Stoneham, MA: Butterworth, Feinstein AR. Principles of medical statistics. Boca Raton, FL: Chapman & Hall/CRC, 2002: Katz DL. Clinical epidemiology & evidence-based medicine. Thousand Oaks, CA: Sage Publications, Inc., Fagan TJ. Normogram for Bayes theorem (letter). N Engl J Med 1975;293: Kramer MS. Clinical epidemiology and biostatistics. Berlin: Springer-Verlag, 1988: Sackett DL, Straus SE, Richardson WS, et al. Evidence-based medicine: how to practice and teach EBM. 2nd ed. Edinburgh: Churchill Livingstone, Reck JR. Touching all the bases in diagnostic test assessment. Am J Med 1990;88: Diamond GA, Denton TA, Berman DS, et al. Prior restraint: a Bayesian perspective on the optimization of technology utilization for diagnosis of coronary artery disease. Am J Cardiol 1995;76: Simel DL, Holleman DR. Should the definition for the negative likelihood ratio be changed? J Clin Epidemiol 1997;50: McGee S. Evidence-based physical diagnosis. Philadelphia: Saunders, 2001: Liddington MI, Thomson WHF. Rebound tenderness test. Br J Surg 1991;78: Akriviadis EA, Runyon BA. Utility of an algorithm in differentiating spontaneous from secondary bacterial peritonitis. Gastroenterology 1990;98: Sackett DL, Straus SE, Richardson WS, et al. Evidence-based medicine: how to practice and teach EBM. Edinburgh: Churchill Livingstone, Accessed at: November 29, Address requests for reprints to: Andrés Cárdenas, MD, MMSc, Instructor in Medicine, Division of Gastroenterology and Hepatology, Beth Israel Deaconess Medical Center, Harvard Medical School, 110 Francis Street, Suite 8E, Boston, Massachusetts acardena@bidmc.harvard.edu; fax: (617)