### What is a Likelihood Ratio?

Likelihood ratios (LR) are used to assess two things: 1) the potential utility of a particular diagnostic test, and 2) how likely it is that a patient has a disease or condition. LRs are basically a ratio of the probability that a test result is correct to the probability that the test result is incorrect. The sensitivity and specificity of the test are the numbers used to generate a LR, which is calculated for both positive and negative test results and is expressed as ‘LR+’ and ‘LR-‘, respectively. The calculations are based on the following formulas:

- LR+ = sensitivity / 1- specificity
- LR- = 1- sensitivity / specificity

In its simplest expression, LR+ is equivalent to the probability that a person with the disease tested negative for the disease (false negative) divided by the probability that a person without the disease tested negative for the disease (true negative). LR- is equivalent to the probability that a person with the disease tested negative for the disease (true negative) divided by the probability that a person without the disease tested negative for the disease (true positive).

### Bayes’ Theorem and Pre-Test Probability

LRs are commonly used in decision-making based on Bayes’ Theorem. Bayes’ Theorem is basically a mathematical recognition of **context** as an important factor in decision making. In other words no diagnostic test is perfect, and because every test will be wrong sometimes the likelihood that a test is right will depend heavily upon its context. This approach requires an estimate of the probability of a disease *before any test is ordered* (i.e. the ‘pre-test probability’).

As an example, imagine a positive pregnancy test on a man. This will certainly be a wrong result, and the context of the test (the zero probability of a man being pregnant) was more helpful than the test itself. This is the essence of Bayes’ Theorem: the pre-test probability of zero was critical in understanding how to interpret the result. For a second example imagine a 19-year old girl who has a ‘positive’ stress test. This is much like the man tested for pregnancy. A teenager has a virtually zero chance of significant coronary artery disease, but might well have a ‘positive’ stress test. Now imagine a 65-year old diabetic man who smokes and has been having chest pain with exertion. His positive stress test is quite likely to be correct. This is because his pre-test probability of disease was in the range of 50%. After a positive stress test his probability moves to roughly 80% or more—a number we can determine by mathematically applying LRs to the pre-test probabiliy.

Thus, the calculation of how likely it is that someone has a disease is based on a pre-test probability (typically estimated by the clinician), with LRs applied to this number. A critical, and often discussed, part of this process is the estimation of pre-test probability of a disease, something that can vary by clinician, by setting, by location, and many other factors. Technically, the pre-test probability for a population of patients is the same thing as the prevalence of the disease in that population. But this will shift based on patient presentation and risk factors. For instance, the prevalence of appendicitis in the general population at any given moment is low, far less than 1%. But the prevalence of appendicitis in those who arrive to an emergency department with right lower quadrant abdominal tenderness is considerably higher, in many studies about 30%. Patients can therefore become members of a ‘population’ simply by experiencing pain in a certain area of the body.

However, the estimation of pre-test probability is typically subjective, based on the clinician’s experience and gestalt. Thus the pre-test probability estimate will often vary based on the clinician, which means that clinical judgment remains a critical part of the process of diagnosis, even when LRs for a given test are known. In extreme cases there will be no variation (everyone will agree that a man has a zero pre-test probability) however in some cases there will be wide variation. Thus the ‘context’ that Bayes’ Theorem asks us to respect and use as a part of clinical decision making inevitably injects an element of subjectivity and judgment into the process.

### Affecting Probability

After estimating a pre-test probability, the clinician may determine that the pre-test probability is low enough or high enough to obviate the need for further testing. However, when pre-test probability is not low or high enough to rule-in or rule-out the disease, the clinician may perform or order a test or try to obtain more information. Importantly, a ‘test’ in this case can mean a physical finding (e.g. reproducible chest tenderness), a pertinent feature of the medical history (e.g. radiation to both arms), or an electrocardiogram, laboratory assay, or imaging test. When these test results become available, the clinician can apply LR+ or LR- (for a positive test or negative test results, respectively) and arrive at a new probability for the disease (i.e. post-test probability).

Thus applying the likelihood ratios could move the probability upward or downward, resulting in higher or lower post-test probabilities. If after performing a test (or obtaining new information) the probability is low enough to rule out the disease, no further testing is necessary. Alternatively, if the probability is high enough to secure the diagnosis, treatment could be started. However, when the post-test probability of the disease falls in an area where the disease cannot be ruled-in or out with enough certainty, the clinician may start gathering more information and may consider more testing to shift the probability farther in (hopefully) one direction.

For the mathematically inclined one can arrive at post-test probability by multiplying pre-test odds (O = P ⁄ [1 – P]) by LR+ or LR- and converting the resultant post-test odds to post-test probability (P = O/[1 + O]). Alternatively, and much more commonly, the Fagan nomogram can show the post-test probability of a disease if the pre-test probability and the LR are known (Figure).

Keeping this model in mind, it is obvious that a test could only be useful if its LR can *significantly alter the probability of a diagnosis*. Thus if the LR value is 1 then the value of the diagnostic test is of no practical significance. The further away the LR value is from 1, the more useful it will be.

### Summary

In summary, likelihood ratios are statements about how a given diagnostic test may be used in clinical practice to offer a probability of disease on which to base decisions. As the quantitative value of a calculated likelihood ratio is further away from 1 in either direction, there is increasing utility of a diagnostic test to point toward, or away from, a diagnosis.

### What An LR Cannot Do

It is important to understand the limitations of the LR. First, and perhaps most obviously, the accuracy of a LR depends entirely upon the relevance and quality of the studies that generated the numbers (sensitivity and specificity) that inform that LR.

Second, humans typically perform clinical decision making by absorbing multiple factors and generating impressions simultaneously. This is the essence of pattern recognition, the most common instinctive form of clinical diagnosis. However, LRs demand that we consider one element of diagnosis at a time, judiciously and individually. Even when a LR is used in this fashion there are definite limits to the accuracy that we can presume underlies the number, for example the sensitivity and specificity evidence as originally generated may be flawed and the pre-test probability judgment can vary widely which mean that there are margins of error that should considered even under ideal circumstances. In addition, LRs have never been validated for use in series or in parallel. In other words there is no precedent to suggest that LRs can be used one after the other (i.e. using one LR to generate a post-test probability, and then using this as a pre-test probability for application of a different LR) or simultaneously, to arrive at a more accurate probability or diagnosis.

It is important to keep these limitations in mind when using LRs because in many ways it is quite counter-intuitive to imagine that only one question at a time can be addressed when seeing a patient in the clinical environment with all of its inherent complexity. Despite this seemingly narrow use, LRs remain an invaluable and unique tool, as there is no other established method for adjusting a probability of disease based on known diagnostic test properties.

Lastly, some clinicians use one LR to generate a post-test probability, and then use the new post-test probability as a pre-test probability for application of the next LR related to a different test. Although this approach seems intuitive and practical, the reader should keep in mind that LRs have never been validated for use in series or in parallel. In other words, there is no evidence to support or refute the use of LRs one after the other.

### How to Use Our Web-Based LR Tool

Each disease/condition has a corresponding page with LRs for both positive and negative findings, as extracted from the highest quality available literature on the topic. The LRs are posted on this page in differing color fonts. The white values represent LRs that are sufficiently far from zero that they should be considered potentially useful for assisting in diagnosis. The grey values represent LRs that either include 1 in the margin of error (i.e. the confidence interval) or else are so close to 1 in their value that they should considered diagnostically not useful (i.e. they cannot significantly impact the probability of disease). Thus we do not provide probability calculators for these grey colored values.

When you click on a white LR that you have chosen to apply you will see a box appear with three slide rules. These represent the three numbers involved in the LR calculation: the pre-test probability, the LR, and the post-test probability. The first should be adjusted by you to whatever number you have judge to be a reasonable estimate of the likelihood that the patient has the disease before considering the test in question. The second is the LR, which is set based on the LR that we have determined to be most accurate for that test based on the best available evidence. The third number, the post-test probability, will slide as you adjust your pre-test probability judgment. This is the patient’s likelihood of having the disease after application of that particular test.

As an example, imagine that you are seeing a patient you believe to potentially be suffering from an aortic dissection. You note that the EKG shows LVH and would like to know how much this finding should alter your sense of the probability of disease. When you click on the number ‘3.2x’ on the Diagnosis webpage for aortic dissection, the three-slide box will appear. The default pre-test probability is 50% and you should adjust this to you best estimate of the likelihood that your patient has an aortic dissection. If the patient seems to you, in virtually every way, to be very likely to have a dissection (perhaps they have severe new hypertension and acute onset tearing pain with radiation to the back) then an estimate as high as 50% will be appropriate. In many cases the estimate will be much lower than this, and therefore the slide should be dragged to the lower number. As the slide is dragged the post-test probability will automatically adjust based on the LR+ of 3.2. If the pre-test probability estimate is 10%, for instance, the third slider should show 26.2% as a post-test probability estimate. If the pre-test probability remained at 50% then the final probability will be 76.2%.