Dichotomous Tests (Tom)
•Their results change the probability of disease
Negative test Positive test
Reassurance TreatmentOrder a Test
•A good test moves us across action thresholds.
0% 100%
T+T-
•The best tests are definitive
What tests do
Post-Test Probability of Disease Depends on 2 Things
1. Where you started from (low, medium, high)
2. Length and direction of the “arrow” Basic paradigm:
What we thought before + test result = what we think now
Prior probability + LR from test = post-test probability
LR = P(Result|Disease)/P(Result|No Disease)
Assessing information from dichotomous tests (review):
Disease + Disease - Total
Test + a b a+b
True Positives False Positives Total Positives
Test - c d c+d
False Negatives True Negatives Total Negatives
Total a+c b+d Total N
Total With Disease
Total without Disease
Sensitivity=a/(a+c) Specificity =d/(b+d)Positive predictive value (PPV) = a/(a+b); Negative predictive value (NPV) d/(c+d)Prior probability = P(D); Posterior probability = P(D|test result)
False-negative confusion
Sensitivity of rapid strep test is 85%
Therefore, false negative rate is 15%
15% is too high, so always culture to confirm negative rapid strep tests
What’s wrong?Strep No StrepTotal
Rapid Test + TP FP TP+FPRapid Test - FN TN TN+FN
TP+FN FP+TN 2 definitions of “false negative rate”
1-sensitivity = FN/(TP+FN). This one is easier because it’s (assumed to be) constant.
1 - negative predictive value = FN/(FN+TN). This one is harder because it depends on prior probability, but it is the one that should determine clinical decisions.
If prior probability of strep = 20%
False negative rate (def #2) = 15/407 = 3.7%
NNC (number needed to culture) = 1/.037 = 27 to identify 1 false negative rapid test. (Pre-test probability of 20%)
At some prior probability of strep, culture after negative quick test is not indicated.
Strep No Strep TotalRapid test + 85 8 95Rapid test - 15 392 407Total 100 400 500
(Assumes 98% specificity)
Similar examples:
Sensitivity of UA for UTI is only 80%, therefore always culture after a negative UA
Sensitivity of CT scan for subarachnoid hemorrhage is only 90%, therefore always do LP after a negative CT
Importance of Sampling Scheme
If sampling separately from Disease+ and Disease– groups (case-control sampling), cannot calculate prevalence, positive predictive value, or negative predictive value.
Dx Test:Case-Control SamplingDisease +Sampled
Separately
Disease –Sampled
Separately
Test +a
True Positivesb
False Positives
Test -c
False Negatives
dTrue
Negatives
Total
a + cTotal With
Disease
b + dTotal
WithoutDiseaseSensitivity = a/(a + c)
Specificity = d/(b + d)
Dx Test: Cross-sectional Sampling
Prevalence = (a + c)/NPositive Predictive Value = a/(a + b)Negative Predictive Value = d/(c + d)
Disease + Disease - Total
Test + aTrue Positives
bFalse Positives
a + bTotal
Positives
Test - cFalse
Negatives
dTrue
Negatives
c + dTotal
Negatives
Total a + cTotal With
Disease
b + dTotal
WithoutDisease
a + b + c + dTotal N
R. henselae titers and Cat Scratch Disease*
Case Control
R. henselae titer
Positive 38 4 42
Negative 4 108 122
45 112
*Zangwill, N Engl J Med. 1993;329:8-13. EBD Problem 3.2
Authors stated negative predictive value = 38/42 = 90.5%. Is there a problem?
Example from Chapter 3
65-year-old woman with mammogram suspicious for malignancy
Pre-test probability ≈ 0.015LR(“suspicious for malignancy”) ≈
100Post-test probability = ?
Update Pre-Test Probability Using LR(test result)
1) Convert pre-test probability (P) to pre-test odds. Pre-Test Odds = P/(1-P)
2) Calculate LR. P(result|D+)/P(result|D-). 3) Post-Test Odds = Pre-Test Odds × LR4) Convert post-test odds to post-test
probability. Prob = Odds/(1+Odds)
Update Pre-Test Probability Using LR(test result)
1) Pre-test probability P = 0.015Pre-test odds = P/(1-P) ≈ 0.015
2) LR(“Suspicious for Malignancy”) = 100
3) Post-Test Odds = 0.015 × 100 = 1.54) Post-test probability =
Odds/(1+Odds) = 1.5/2.5 = 0.60
Can Use Slide Rule
Threshold Model
Single disease (D+,D-) with single treatment (no further testing available)
Cost of failing to treat D+ = B Cost of treating D- unnecessarily = C Treat if P(D) > C/(C+B) C/(C+B) = Treatment Threshold
Probability = PTTPauker SG, Kassirer JP.. N Engl J Med. 1975 Jul
31;293(5):229-34.
Pauker SG, Kassirer JP.. N Engl J Med. 1975 Jul 31;293(5):229-34.
Define Costs B and C
“X-Graph”
Introduce a Dichotomous (+/-) Test
P(+|D+) = Probability of positive test “given” D+ = Sensitivity
P(-|D-) = Probability of negative test “given” D- = Specificity
P(+|D-) = 1 – Specificity or “False Positive Rate”
P(-|D+) = 1 – Sensitivity of “False Negative Rate”
T = Cost of Test
Pauker SG, Kassirer JP. N Engl J Med. 1980 May 15;302(20):1109-17.
“X-Graph”
New “X-Graph”
Threshold Formulas
Assumptions in the Threshold ModelThreshold Model: One disease One dichotomous test Only two post-test options: treat and no treat
Real world: Multiple possible diseases Multiple possible test results (not just
+/-) Multiple possible tests Multiple post-test options including
observation and additional testing
2) Multilevel Tests (Michael)
Likelihood ratios for results other than “+” or “-”