Likelihood
ratios tell us how much
we should shift our
suspicion for a particular
test result. Because tests can be positive or negative, there are at
least two
likelihood ratios for each test. The
“positive likelihood ratio” (LR+) tells us how much
to increase the
probability of disease if the test is positive. On the other
hand, the
“negative likelihood
ratio” (LR-) tells us how much to decrease it if the test is
negative. The general formula for calculating
the likelihood ratio is:
probability
that an individual with disease
has the test result
LR
=
probability that an
individual without disease
has the test result
Thus,
the positive and likelihood ratio are:
probability that an
individual with disease
has a positive test
Positive LR = LR+
=
probability that an
individual without disease
has a positive test
probability
that an
individual with
disease has
a negative test
Negative LR = LR-
=
probability that an
individual without
disease has a negative test
You
can also define the LR+ and LR- in terms of sensitivity and
specificity. Notice that the numerator for LR+ is the definition
for sensitivity (probably that an individual with disease has a
positive test), and the denominator is the converse of specificity. For
LR-, the numerator is the converse of
sensitivity and the
denominator is specificity. So:
LR+
= sensitivity / (1-specificity)
LR-
= (1-sensitivity) / specificity
Let’s
consider an example. In
a study of the ability of rapid antigen tests ("strep screens") to
diagnose strep
pharyngitis, 80%
of patients with strep pharyngitis have a positive rapid antigen test,
while 95% of those without strep pharyngitis have a negative test. Thus, the sensitivity is 80% and the
specificity is 95%.
The LR+ for the ability of
rapid antigen tests to diagnose
strep
pharyngitis is:
LR+ = 80% / (100%-95%) = 80% / 5% = 18
The negative likelihood ratio is:Likelihood ratios have unique properties that make them particularly useful for clinicians and healthcare decision-makers, which we'll discuss shortly. Perhaps most important is that:
The LR- corresponds to the clinical concept of "ruling-out disease"
The LR+ and LR- don't change as the underlying probability of disease changes (predictive values do change, as you just learned)
The
first thing to realize about LR’s is that a LR greater than
1 increases the
probability that the target disorder is present, and a LR less than
1 decreases the probability that the target disorder is
present.
The following
are general guidelines, which must be correlated with the clinical
scenario:
LR |
Interpretation |
> 10 |
Large and often conclusive increase in the
likelihood of disease |
5 - 10 |
Moderate increase in the likelihood of
disease |
2 - 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 |
0.5 - 1.0 |
Minimal decrease in the likelihood of disease |
0.2 - 0.5 |
Small decrease in the likelihood of disease |
0.1 - 0.2 |
Moderate decrease in the likelihood of
disease |
< 0.1 |
Large and often conclusive decrease in the
likelihood of disease |
Here is a collection of likelihood ratios for the diagnosis of appendicitis, from the Essential Evidence database:
Sensitivity | Specificity | LR+ | LR- | |
Adults | ||||
CT scan | 94 | 95 | 18.8 | 0.06 |
Ultrasound | 86 | 81 | 4.5 | 0.17 |
C-reactive protein > 1.0 mg/dl | 64 | 72 | 2.3 | 0.5 |
Children | ||||
Ultrasound | 86 | 95 | 17.2 | 0.15 |
CT scan (non-contrast) | 97 | 93 | 13.9 | 0.03 |
Ultrasound followed by CT if indeterminate (non-contrast) | 99 | 89 | 9.0 | 0.01 |
C-reactive protein > 1.0 mg/dl | 64 | 82 | 3.6 | 0.44 |
The decision to order a test is also based on our initial assessment of the likelihood of the target disorder, and how important it is to rule-in or rule-out disease. For example, a CT scan might have a good likelihood ratio for ruling in (or out) appendicitis in a child with abdominal pain. But if you believe a patient has a simple gastroenteritis, and that appendicitis is very unlikely, CT shouldn’t be ordered given the cost, radiation exposure, and fact that a positive scan is likely a false positive. So, beginning with ultrasound and only getting CT if the results remain unclear or the patient doesn't improve might be the best option. Clearly, there is more to the evaluation of diagnostic tests than a simple assessment of accuracy. More on that later.
Not all tests are dichotmous (yes/no, positive/negative). In the next section we will learn about tests that are "polytomous".
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