How do I know whether a test is really useful—whether it will really shift the probability of disease across a threshold?
A perfect diagnostic test would always be positive in patients with the disease and would always be negative in patients without the disease (Figure 1-6). Since there are no perfect diagnostic tests, some patients with the disease have negative tests (false-negative), and some without the disease have positive tests (false-positive) (Figure 1-7).
A perfect diagnostic test.
A pictorial representation of test characteristics.
The test characteristics help you to know how often false results occur. They are determined by performing the test in patients known to have or not have the disease, and recording the distribution of results (Table 1-3).
Table 1-3.Test characteristics. ||Download (.pdf) Table 1-3. Test characteristics.
| ||Disease Present ||Disease Absent |
|Test positive ||True-positives ||False-positives |
|Test negative ||False-negatives ||True-negatives |
Table 1-4 shows the test characteristics of duplex ultrasonography for the diagnosis of proximal DVT, based on a hypothetical group of 200 patients, 90 of whom have DVT.
Table 1-4.Results for calculating the test characteristics of duplex ultrasonography. ||Download (.pdf) Table 1-4. Results for calculating the test characteristics of duplex ultrasonography.
| ||Proximal DVT Present ||Proximal DVT Absent |
|Abnormal duplex US ||TP = 86 patients ||FP = 2 patients |
|Normal duplex US ||FN = 4 patients ||TN = 108 patients |
| ||Total number of patients with DVT = 90 ||Total number of patients without DVT = 110 |
The sensitivity is the percentage of patients with DVT who have a true-positive (TP) test result:
Sensitivity = TP/total number of patients with DVT = 86/90 = 0.96 = 96%
Since tests with very high sensitivity have a very low percentage of false-negative results (in Table 1-4, 4/90 = 0.04 = 4%), a negative result is likely a true negative.
The specificity is the percentage of patients without DVT who have a true-negative (TN) test result:
Specificity = TN/total number of patients without DVT = 108/110 = 0.98 = 98%
Since tests with very high specificity have a low percentage of false-positive results (in Table 1-4, 2/110 = 0.02 = 2%), a positive result is likely a true positive.
The sensitivity and specificity are important attributes of a test, but they do not tell you whether the test result will change your pretest probability enough to move beyond the test or treatment thresholds; the shift in probability depends on the interactions between sensitivity, specificity, and pretest probability. The likelihood ratio (LR), the likelihood that a given test result would occur in a patient with the disease compared with the likelihood that the same result would occur in a patient without the disease, enables you to calculate how much the probability will shift.
The positive likelihood ratio (LR+) tells you how likely it is that a result is a true-positive (TP), rather than a false-positive (FP):
Positive LRs that are significantly above 1 indicate that a true-positive is much more likely than a false-positive, pushing you across the treatment threshold. An LR+ > 10 causes a large shift in disease probability; in general, tests with LR+ > 10 are very useful for ruling in disease. An LR+ between 5 and 10 causes a moderate shift in probability, and tests with these LRs are somewhat useful. “Fingerprints,” findings that often rule in a disease, have very high positive LRs.
The negative likelihood ratio (LR–) tells you how likely it is that a result is a false-negative (FN), rather than a true-negative (TN):
Negative LRs that are significantly less than 1 indicate that a false-negative is much less likely than a true-negative, pushing you below the test threshold. An LR– less than 0.1 causes a large shift in disease probability; in general, tests with LR– less than 0.1 are very useful for ruling out disease. An LR– between 0.1 and 0.5 causes a moderate shift in probability, and tests with these LRs are somewhat useful.
The closer the LR is to 1, the less useful the test; tests with a LR = 1 do not change probability at all and are useless. The threshold model in Figure 1-8 incorporates LRs and illustrates how tests can change disease probability.
Incorporating likelihood ratios (LRs) into the threshold model.
When you have a specific pretest probability, you can use the LR to calculate an exact posttest probability (see Box, Calculating an Exact Posttest Probability and Figure 1-9, Likelihood Ratio Nomogram). Table 1-5 shows some examples of how much LRs of different magnitudes change the pretest probability.
Likelihood ratio nomogram. Find the patient’s pretest probability on the left, and then draw a line through the likelihood ratio for the test to find the patient’s posttest probability.
Table 1-5.Calculating posttest probabilities using likelihood ratios (LRs) and pretest probabilities. ||Download (.pdf) Table 1-5. Calculating posttest probabilities using likelihood ratios (LRs) and pretest probabilities.
| ||Pretest Probability y = 5% ||Pretest Probability y = 10% ||Pretest Probability y = 20% ||Pretest Probability y = 30% ||Pretest Probability y = 50% ||Pretest Probability y = 70% |
|LR = 10 ||34% ||53% ||71% ||81% ||91% ||96% |
|LR = 3 ||14% ||25% ||43% ||56% ||75% ||88% |
|LR = 1 ||5% ||10% ||20% ||30% ||50% ||70% |
|LR = 0.3 ||1.5% ||3.2% ||7% ||11% ||23% ||41% |
|LR = 0.1 ||0.5% ||1% ||2.5% ||4% ||9% ||19% |
If you are using descriptive pretest probability terms such as low, moderate, and high, you can use LRs as follows:
A test with an LR– of 0.1 or less will rule out a disease of low or moderate pretest probability.
A test with an LR+ of 10 or greater will rule in a disease of moderate or high probability.
Beware if the test result is the opposite of what you expected!
If your pretest probability is high, a negative test rarely rules out the disease, no matter what the LR– is.
If you pretest probability is low, a positive test rarely rules in the disease, no matter what the LR+ is.
In these situations, you need to perform another test.
Mrs. S has a normal duplex ultrasound scan. Since your pretest probability was moderate and the LR– is < 0.1, proximal DVT has been ruled out. Since duplex ultrasound is less sensitive for distal than for proximal DVT, clinical follow-up is particularly important. Some clinicians repeat the duplex ultrasound after 1 week to confirm the absence of DVT, and some clinicians order a D-dimer assay. When she returns for reexamination after 2 days, her leg looks much better, with minimal erythema, no edema, and no tenderness. The clinical response confirms your diagnosis of cellulitis, and no further diagnostic testing is necessary. (See Chapter 15 for a full discussion of the diagnostic approach to lower extremity DVT.)
CALCULATING AN EXACT POSTTEST PROBABILITY
For mathematical reasons, it is not possible to just multiply the pretest probability by the LR to calculate the posttest probability. Instead, it is necessary to convert to odds and then back to probability.
Convert pretest probability to pretest odds.
Pretest odds = pretest probability/(1 − pretest probability).
Multiply pretest odds by the LR to get the posttest odds.
Posttest odds = pretest odds × LR.
Convert posttest odds to posttest probability.
Posttest probability = posttest odds/(1 + posttest odds).
For Mrs. S, the pretest probability of DVT was 17%, and the LR− for duplex ultrasound was 0.04.
Step 1: pretest odds = pretest probability/(1 − pretest probability) = 0.17/(1 − 0.17) = 0.17/0.83 = 0.2
Step 2: posttest odds = pretest odds × LR = 0.2 × 0.04 = 0.008
Step 3: posttest probability = posttest odds/(1 + posttest odds) = 0.008/(1 + 0.008) = 0.008/1.008 = 0.008
So Mrs. S’s posttest probability of proximal DVT is 0.8%.