Dichotomous Tests (Tom)
What tests do
•Their results change the probability of
disease
T+
T0%
Negative test
Reassurance
Positive test
Order a Test
100%
Treatment
• A good test moves us across action thresholds.
• The best tests are definitive
Post-Test Probability of Disease Depends
on 2 Things
Where you started from (low, medium,
high)
Length and direction of the “arrow”
Basic paradigm:
1.
2.




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):
Test +
Test -
Disease +
Disease -
Total
a
b
a+b
True Positives
False Positives
Total Positives
c
d
c+d
False Negatives
Total
True Negatives Total Negatives
a+c
b+d
Total With
Disease
Total without
Disease
Total N
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
Rapid Test +
TP
Rapid Test FN
TP+FN

No Strep Total
FP
TP+FP
TN
TN+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%



Strep No Strep Total
Rapid test +
85
8
95
Rapid test 15
392
407
Total
100
400
500
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.
(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 Sampling
Test +
Test Total
Disease +
Sampled
Separately
a
True Positives
Disease –
Sampled
Separately
b
False Positives
c
False Negatives
a+c
Total With
Disease
d
True Negatives
b+d
Total Without
Disease
Sensitivity = a/(a + c)
Specificity = d/(b + d)
Dx Test: Cross-sectional Sampling
Disease +
Disease -
Total
Test +
a
True Positives
b
False Positives
a+b
Total Positives
Test -
c
False Negatives
d
True Negatives
c+d
Total
Negatives
Total
a+c
Total With
Disease
b+d
Total Without
Disease
a+b+c+d
Total N
Prevalence = (a + c)/N
Positive Predictive Value = a/(a + b)
Negative Predictive Value = d/(c + d)
R. henselae titers and Cat Scratch
Disease*
R. henselae titer
Case
Control
Positive
38
4
42
Negative
4
108
122
45
112
Authors stated negative predictive value = 38/42 =
90.5%. Is there a problem?
*Zangwill, N Engl J Med. 1993;329:8-13. EBD
Problem 3.2
Example from Chapter 3
65-year-old woman with mammogram
suspicious for malignancy
Pre-test probability ≈ 0.015
LR(“suspicious for malignancy”) ≈ 100
Post-test probability = ?
Update Pre-Test Probability
Using LR(test result)
1)
2)
3)
4)
Convert pre-test probability (P) to pretest odds. Pre-Test Odds = P/(1-P)
Calculate LR. P(result|D+)/P(result|D-).
Post-Test Odds = Pre-Test Odds × LR
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 = 0.015
Pre-test odds = P/(1-P) ≈ 0.015
2) LR(“Suspicious for Malignancy”) = 100
3) Post-Test Odds = 0.015 × 100 = 1.5
4) Post-test probability = Odds/(1+Odds)
= 1.5/2.5 = 0.60
Can Use Slide Rule
Threshold Model

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
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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
= PTT
Pauker 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




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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
Model
Threshold 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 “-”