The following is from Reference (1) Diagnostics and Likelihood Ratios, Explained from the NNT – Quick summaries of evidence-based medicine [At the very end of the post on the NNT site (but not included here) is an important section “How to Use Our Web-Based LR Tool”]:
Diagnostics and Likelihood Ratios, Explained
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 positive for the disease (true positive) divided by the probability that a person without the disease tested positive for the disease (false positive). 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 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.
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.
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.
[At the very end of the above post on the NNT site (but not included here) is an important section “How to Use Our Web-Based LR Tool”]
(1) Diagnostics and Likelihood Ratios, Explained from the NNT – Quick summaries of evidence-based medicine.
(2) Diagnostic tests 4: likelihood ratios [PubMed Link] [Full Text PDF]. BMJ 2004;329:168–9