Likelihood ratios: Clinical application in day-to-day practice

In this article we provide an introduction to the use of likelihood ratios in clinical ophthalmology. Likelihood ratios permit the best use of clinical test results to establish diagnoses for the individual patient. Examples and step-by-step calculations demonstrate the estimation of pretest probability, pretest odds, and calculation of pos tt est odds and pos tt est probability using likelihood ratios. The bene (cid:222) ts and limitations of this approach are discussed.

In an earlier article, we explained how we use sensitivity, speciÞ city, positive and negative predictive values in day-today practice. [1] While the clinical application of sensitivity and speciÞ city is useful, they have limitations. With sensitivity and speciÞ city, we use a cutoff point to divide the test into (only) two results: positive or negative. In real life, diseases present in gradations of severity; by limiting a test result to "positive" or "negative", we stand to lose important diagnostic information. Sensitivity and speciÞ city are independent of disease prevalence, but are dependent on disease severity. [2,3] In early disease, it is diffi cult to diff erentiate between health and illness and the sensitivity decreases; it increases in severe disease. Accordingly, the reported sensitivity and speciÞ city for a disease may not always reß ect the sensitivity and speciÞ city for the individual patient. Additionally, disease prevalence has a signiÞ cant impact on the positive predictive value (PPV) and negative predictive values (NPV). [4] Therefore, despite a high sensitivity and speciÞ city of a given test, the PPV will be very low in a disease with a very low prevalence. [1] As a "bad" example of clinical practice, suppose a colleague measures an intraocular pressure (IOP) of 22 mm Hg (recorded once) on a 45-year-old male and orders a nerve fiber layer imaging with glaucoma diagnostics variable corneal compensator (GDx VCC, scanning laser polarimetry), or any other imaging modality. (We chose the GDx VCC as its parameter "NFI (nerve Þ ber index)" is easy to use in the examples that follow). We consider this a "bad" example of clinical practice, because such tests obtained aft er a comprehensive examination (and conÞ rmation of the IOP values) have far more signiÞ cance. Nevertheless, the machine provides a nice printout and a value for the "NFI" of say 48.
The literature indicates that an NFI of 50 is highly suggestive for glaucoma [5] but does that establish our diagnosis? The NFI value varies from 0 -100. Dividing this continuous scale into two zones at an arbitrary cutoff value of 30 (or 50 as is suggested for "deÞ nite" pathology) has limitations.
(Multilevel) likelihood ratios (LRs) overcome the disadvantage of a single cutoff and allow us to best apply the results of diagnostic tests to the individual patient. [4] In this article we will explain LRs as well and use clinical examples to illustrate their use. In order to understand LR, we need to revise the concepts of sensitivity and speciÞ city discussed in an earlier article. [1] . We will brieß y summarize these terms here and move on.
In the conventional 2x2 Table shown in Table 1, Sensitivity = a / a+c = a (true positive) / a+c (true positive + false negative) SpeciÞ city = d / b+d = d (true negative) / b+d (true negative + false positive) Positive Predictive Value (PPV): = a / a+b = a (true positive) / a+b (true positive + false positive) = Probability that the patient has disease when test is positive. NPV Negative Predictive Value (NPV): = d / c +d = d (true negatives) / c+d (false negative + true negative) = Probability that the patient does not have disease when test is negative. Prevalence (in a clinical situation this is referred to as pretest probability) of a disease is the proportion of patients with the target disorder in the population tested.
= a+c / (a+b+c+d) Notice in Table 1 that sensitivity and speciÞ city are calculated "vertically" while the PPVs are calculated horizontally. Accordingly, PPVs are inß uenced by the number of patients in the columns; we can alter this by changing the number of diseased and controls. In other words, the PPV is aff ected by prevalence (pretest probability) of the disease; sensitivity and speciÞ city are not.

DeÞ nition of LR
The LR is the probability of a given test result in a patient with the target disorder divided by the probability of that same result in a person without the target disorder. The components of the LR are calculated vertically, and like the sensitivity and speciÞ city are immune to the prevalence. In other words, LR+ = True positivity rate / False positivity rate Which is the same as sensitivity / 1-speciÞ city It is appropriate at this stage to introduce and explain certain terms we need to use LR: Pretest Probability is deÞ ned as the probability of the target disorder before a diagnostic test is ordered. In the "general" population, it would be called the prevalence of the disease. For example, the prevalence of primary open angle glaucoma (POAG) in the south Indian population above 40 years of age is approximately 2.5%. [6] Accordingly, any patient in this age group who walks into a general ophthalmology clinic has a 2.5% probability of glaucoma even before the history or examination. In the clinic, the "prevalence" of a disease before ordering a test is called the pretest probability. If this is done without taking a history or examining the patient, the pretest probability is about the same as the prevalence of the disease in the population.
Pretest odds: The odds that the patient has the target disorder, before the test is carried out. It is slightly diff erent from and is calculated from pretest probability.
Postt est odds: (pretest odds x LR): The odds that the patient has the target disorder, aft er the test results are known. It is calculated by multiplying the pretest odds by the likelihood of a positive or negative test (as we will show).
Postt est probability is deÞ ned, as the probability of the target disorder aft er a diagnostic test result is known. It is slightly diff erent from and calculated from postt est odds.
The advantage of the LR is that we can multiply the pretest odds that the patient has disease by the LR of a positive test to obtain the postt est odds that the patient has disease. This of course means that we must Þ rst estimate the pretest probability (and pretest odds) of disease. As mentioned, prior to the history or examination, the pretest probability of disease is the same as the prevalence of the disease in the general population. Once the history and examination are completed, the pretest probability may remain the same, decrease, or is revised upwards. This seems complicated, but clinicians do this intuitively all the time; with a litt le experience they can learn to quantify their gut feeling. The pretest probability is converted to the pretest odds. The pretest odds are multiplied by the LR to provide the postt est odds.
Postt est odds = pretest odds * LR F inally the posttest odds are converted to the posttest probability of disease.

Clinical application
L et us come back to our example of "bad" clinical practice: Our colleague has recorded an intraocular pressure (IOP) of 21 mm Hg on a 45-year-old patient and ordered a GDx VCC. The machine provides a nerve Þ ber indicator (NFI) score of 48. Does this patient have POAG?
The information that we have is 1. The pretest probability of POAG. Without any other information, this is the same as the prevalence of POAG in 45-year-olds in the given population. We will use published literature from South India for this purpose: 2.5%. [6] 2. Clinical information: IOP: 22 mm Hg in both eyes 3. Sensitivity and speciÞ city of GDx: GDx VCC (NFI score 48): For our patient at this cutoff using our published data, sensitivity is 59.5% and specificity 97.1%. [7] (For calculation purposes we'll use 60% sensitivity and 97% speciÞ city.) We can now calculate the LR for the test result: GDx VCC (for NFI score 48) LR +ve = Sensitivity / 1-speciÞ city = 0.6 / 0.03 = 20 We now have the positive LR ratio as well as the pretest probability (in this case, the population prevalence) of POAG. As the IOP is at the upper limit of normal as per Indian population-based data, [8] has not been rechecked, and we do not know the corneal thickness, we will not take it into account, and go to the next step. The pretest odds = pretest probability / (1-pretest probability) As the pretest probability is the same as the prevalence of POAG in the population, the pretest odds: 0.025 / 0.975 = 0.03. Postt est odds = pretest odds * LR = 0.03 * 20 = 0. 6 We now calculate postt est probability. Postt est probability = postt est odds / (postt est odds+1) Therefore the Postt est probability = 0.6/ 1.6 = 0.375 or 37.5%.
The result means that after the GDx VCC results, the probability of our patient having glaucoma has increased from 2.5% to 38%. A probability of 38% is worse than obtaining a heads or tails on the random toss of a coin; certainly not good enough to make a diagnosis of glaucoma. This example also demonstrates that one test in isolation, even a good one like the GDx , even if strongly positive, may not conÞ rm the diagnosis. A comprehensive eye examination with judicial use of the GDx (or any other imaging technology) is more helpful.
Let's try the same example with a slight diff erence: The IOP is now "high" (24 mm Hg, conÞ rmed on several readings; "corrected" for corneal thickness). GDx VCC printout shows an NFI score of 48 as above.
Sensitivity and speciÞ city of IOP: (50% sensitivity and 92% speciÞ city [9] ) Positive LR of IOP: = sensitivity / 1-speciÞ city = 0.5/ 100 -92 = 0.5 / .08 = 6.25 LR NFI at cutoff 48: 20 From theabove example, we know that the pretest probability of POAG before examination was 2.5% and the pretest odds 0.03. How much does this change with a raised IOP? For that we need to calculate postt est probability of POAG using the LR ratio of a raised IOP.
Postt est odds = pretest odds * LR for IOP So, postt est odds = 0.03 * 6.25 = 0.188 We now calculate postt est probability: Postt est probability = postt est odds / (postt est odds+1) So, postt est probability = 0.188 / 1.188 = 0.16 That means that aft er our "high" IOP measurement, the probability of our patient having POAG has increased from 2.5% to 16%. Not good enough to make a diagnosis of glaucoma.
We now add on the GDx VCC results. Pretest probability before GDX= 16% Pretest odds: 0.16 / 0.84 = 0.19 LR of GDx result of NFI 48= 20 Postt est odds = 0.19 * 20 = 3.8 Postt est probability = 3.8 / 4.8 = 0.792 Aft er IOP measurement and GDx VCC results data, the probability that our patient has glaucoma has increased from 2.5% to 79%. Still not good enough to clinch the diagnosis.
In the third example, we order the test aft er a full clinical examination; the way it should be done. The Þ ndings are an IOP of 24 mm Hg in both eyes, open angles, other pathology has been ruled out and the disc has been examined stereoscopically using biomicroscopy. The cup disc ratio was 0.7:1 disc ratio in a medium-sized disc, with inferior rim thinning with a wedgeshaped inferior retinal nerve Þ ber layer (RNFL) defect. The GDx VCC NFI score is 48. How do we use this information for the patient?
Aft er IOP measurement, the probability that our patient has POAG had increased to 16%. This 0.16 becomes the pretest probability for our second calculation that is optic disc examination. We now incorporate optic disc Þ ndings into the calculation. Aft er IOP measurement and optic disc assessment, the probability that our patient has POAG has increased from 2.5% through 16% (aft er the raised IOP), to 79% aft er incorporating optic disc Þ nding also. To make a diagnosis so as to start treatment, or at least to tell the patient, we may want to be 90 plus % sure. (How sure we need to be of a diagnosis varies with the disease, the examiner and the patients and is beyond the scope of this article. The reader is referred to our clinical bible [4] .) Be that as it may, we now add the GDx VCC Þ ndings. We have calculated the pretest odds and postt est probability for each stage of the case step by step. This was done in order to familiarize ourselves with these calculations. Fortunately, in practice, we do not have to follow this lengthy route. Once, we have LRs for the various signs, symptoms and tests, we can directly calculate the Þ nal postt est odds using the following formula.

Negative likelihood ratio
The LR of a negative test result (LR-) is described in most texts as LR-= probability that an individual with the condition has a negative test /probability than an individual without the condition has a negative test LR -= 1-sensitivity / speciÞ city We prefer to determine the probability of being normal and Vol. 57 No. 3 use the following formula: LR -= SpeciÞ city / 1-sensitivity This to us is more intuitive and symmetrical to the formula for positive LR.

Example
A colleague examines a 54-year-old patient with IOP of 20 mm Hg (recorded twice) and orders a GDx VCC, scanning laser polarimetry. The machine provides an NFI score of 18. What are the chances that this patient is normal? We can now easily perform the calculations for that situation, but as discussed above, using the imaging test alone is not good clinical practice. And as space is limited, we will use an example with the clinical information available aft er a comprehensive eye examination.
What is the information we need? 1) Prevalence of ocular hypertension (OHT) and glaucoma (POAG and primary angle closure glaucoma (PACG)) in a given population. The published literature from south India provides the information.
Prevalence of POAG + OHT + POAG suspect + PACG: 5% (APEDS data) [6,11] 2) Clinical information: Family history of glaucoma: Nil Gonioscopy: open angles IOP: 20 mm Hg in both eyes Optic disc: normal disc size, 0.4 cup to disc ratio, healthy neuroretinal rim (following ISNT rule) 3) Sensitivity and speciÞ city of each test performed IOP: At 21 mm Hg cutoff : Sensitivity: 50%, SpeciÞ city: 92% (Baltimore Survey data) [9] Optic disc (for ISNT rule): 72% sensitivity, 79% speciÞ city10 (reference number should be 10 GDx VCC score of 20: Sensitivity: 90.5%, SpeciÞ city 52.9% [7] We can calculate LR for each test: 1. IOP: LR -= SpeciÞ city / 1-sensitivity. = 0.92 / 1 -0.5 = 0.92/ 0.5 = 1.84 2. Optic disc: Using same formula: LR -= 0.79 / 0.28 = 2.82 3. GDx VCC (for NFI score > 20): LR -= 0.53 / 0.095 = 5.6 How do we use this negative LR ratio? When the patient walked into the clinic the probability of having the disease was the same as a prevalence of a given disease in the population. Here the particular disease is "glaucoma suspect" (which includes ocular hypertension, glaucoma suspect and deÞ nite POAG and PACG) and prevalence is 5%. In other words, the chances that our patient is normal are 1 minus the probability of disease or 100 minus 5% = 95% = 0.95. Pretest odds are pretest probability / 1-pretest probability = 0.95/0.05 = 19. Aft er initial examination and with a normal IOP, what are the chances that the patient is normal? We need to calculate the postt est probability using pretest probability and LR ratio. The formulae are mentioned and explained above.
We will use Postt est Odds = Pretest Odds x LR 1 x LR 2 x LR 3 ... x LR n .
Postt est Odds = 19 X 1.84 X 2.82 X 5.6 = 552 Postt est probability = 552 / 553 = .998 = 99.8% Incidentally, if we had used just the clinical assessment (IOP and optic disc assessment), the probability of our patient being normal would have increased from 95% to 99%. The reader is welcome to calculate that, but that is about as clinically certain as we can get. Using GDx that surety increased to 99.8%. The value of an increase in certainty from 99.5 to 99.8% is debatable.
We don't have to remember all these formulae and deÞ nitely don't need to go through complicated calculations. We can also use a nomogram to calculate the postt est probability. [12] What we need to know is prevalence of disease and the sensitivity and speciÞ city of the test for the value obtained for the individual patient. We only have to calculate LR. LRs are probably the best way to utilize diagnostic data, but do have limitations. One limitation is related to estimation of pretest probabilities. Another is the wide conÞ dence intervals around the LRs, especially the ones that are capable of ruling in or ruling out a diagnosis. This is due to paucity of data at the extremes of the disease spectrum where the LRs are likely to be the most helpful. Finally, as the LRs are calculated from the sensitivity and speciÞ city, like these parameters they too may be aff ected by severity of disease.
One way to work around some of these limitations is to perform a "sensitivity analysis" using different, sensible pretest probabilities. These pretest probabilities can be what members of the clinical team consider to be reasonable aft er a clinical exam. If the lowest "sensible" pretest probability Large and often conclusive increase in the likelihood of disease [5][6][7][8][9][10] Moderate increase in the likelihood of disease [2][3][4][5] Small increase in the likelihood of disease 1-2 Minimal increase in the likelihood of disease 1 No change in the likelihood of disease still provides a postt est probability of 90% or more we can be "sure" of our diagnosis. A similar process can be used to rule out the diagnosis.

Summary
Based on the patient's history and clinical examination we estimate the pretest probability of disease and calculate the pretest odds from that. We then multiply the pretest odds by the LR of the test result for that individual patient (multilevel LR) to obtain the postt est odds. Finally, we convert the postt est odds to the more clinically intuitive postt est probability. Using this information we can get as close to a "rule in" or "rule out" criteria for our individual patient.
This approach of using LR to calculate the posttest probability of disease makes best use of diagnostic information. It is however not required for every case and certainly not for straightforward cases. For example, a patient with IOP of 32 mm Hg and 0.9 cup to disc ratio in a medium-sized disc with a bipolar notch does not need any calculation to make the diagnosis. But the intuitive process underlying this obvious diagnosis is applicable via LRs to diffi cult cases where there is diagnostic dilemma. A patient with suspected pre-perimetric glaucoma, for example, is a good case to use LR. It is also important to remember that there is a limit to testing: we can never be absolutely certain. As our clinical Bible states: "physicians must be content to end not in certainties, bur rather in statistical probabilities. The modern physician has right to be certain, within statistical constraints, but never cocksure. Absolute certainties remain for some theologiansand likeminded physicians." [4,13]