An Epidemiological Approach to Diagnostic Process. Steve Doucette, BSc, MSc Email: sdoucette@ohri.ca Ottawa Health Research Institute Clinical Epidemiology Program The Ottawa Hospital (General Campus).
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An Epidemiological Approach to Diagnostic Process
Steve Doucette, BSc, MSc
Email: sdoucette@ohri.ca
Ottawa Health Research Institute
Clinical Epidemiology Program
The Ottawa Hospital (General Campus)
Screening  the proportion of affected persons is likely to be small (Breast Cancer)
Early detection of disease is helpful only if early intervention is helpful
Diagnostic tests  many patients have medical problems that require investigation
Usually to diagnosis disease for immediate treatment
NO
NO
NO
sensitive
specific
predictive
accurate
P(A) = a / n
P(AB)
These are all conditional probabilities!!
+
Test Result

Truth

+
A
B
A+B
C
D
C+D
B+D
A+B+C+D
A+C
Formulas:
Truth
+

Sensitivity = a / a+c
A
B
+
Test
Result
Specificity = d / b+d

C
D
Accuracy= a+d / a+b+c+d
Prevalence = a+c / a+b+c+d
Predictive Value:
Positive Test = a+b / a+b+c+d
positive = a / a+b
Negative Test = c+d / a+b+c+d
negative = d / c+d
Diseased = a+c / a+b+c+d
Not Diseased = b+d / a+b+c+d
A method for testing whether an individual is a carrier of hemophilia (a bleeding disorder) takes the ratio of factor VIII activity to factor VIII antigen. This ratio tends to be lower in carriers thus providing a basis for diagnostic testing. In this example, a ratio < 0.8 gives a positive test result.
Results:38 tested positive, 6 incorrectly.
28 tested negative, 2 incorrectly.
The 2 X 2 Table
Carrier State
Carrier
NonCarrier
+
32
38
6
F8 < 0.8
Test
Result

30
2
28
F8 > 0.8
68
34
34
Carrier State
Exercise:
NonCarrier
Carrier
6
32
38
+
Sensitivity
=
Test
Result
Specificity
=
28
2

30
Accuracy
=
68
34
34
Prevalence
=
Predictive Value:
Positive Test
=
positive
=
Negative Test
=
negative
=
Diseased
=
Not Diseased
=
A method for testing whether an individual is a digoxin toxic measures serum digoxin levels. A cut off value for serum concentration provides a basis for diagnostic testing.
Results:39 tested positive, 14 incorrectly.
96 tested negative, 18 incorrectly.
The 2 X 2 Table
Toxicity
D+
D
T+
25
39
14
Test
Result
T
96
18
78
135
43
92
Toxicity
Exercise:
D
D+
25
14
39
T+
Sensitivity
=
Test
Result
Specificity
=
78
18
T
96
Accuracy
=
135
43
92
Prevalence
=
Predictive Value:
Positive Test
=
positive
=
Negative Test
=
negative
=
Diseased
=
Not Diseased
=
Non
Carrier
Non
Carrier
Carrier
Carrier
32
6
13
33
38
46
+
+
Test
Result
Test
Result
28
2
1
21


30
22
68
68
34
34
34
34
Exercise:
Exercise:
Sensitivity
= 33/(33+1) = 0.97
Sensitivity
= 32/(32+2) = 0.94
Specificity
= 21/(21+13) = 0.62
Specificity
= 28/(28+6) = 0.82
Predictive Value:
Predictive Value:
positive
= 33/(33+13) = 0.72
positive
= 32/(32+6) = 0.84
negative
= 21/(21+1) = 0.95
negative
= 28/(28+2) = 0.93
Non
Carrier
Non
Carrier
Carrier
Carrier
32
6
600
32
38
632
+
+
Test
Result
Test
Result
28
2
2
2800


30
2802
68
3034
34
34
34
3400
Exercise:
Exercise:
Sensitivity
= 32/(32+2) = 0.94
Sensitivity
= 32/(32+2) = 0.94
Specificity
= 2800/(2800+600) = 0.82
Specificity
= 28/(28+6) = 0.82
Predictive Value:
Predictive Value:
positive
= 32/(32+600) = 0.05
positive
= 32/(32+6) = 0.84
negative
= 2800/(2800+2)= 0.999
negative
= 28/(28+2) = 0.93
Cutoff value for test
TP=a
FP=b
FN=c
With Disease
TN=d
Without Disease
TP
TN
FN
FP
0.5
0.6
0.7
0.8
0.9
1.0
1.1
POSITIVE
NEGATIVE
Test Result
Cutoff value for test
With Disease
Without Disease
TP
TN
FP
FN
0.5
0.6
0.7
0.8
0.9
1.0
1.1
POSITIVE
NEGATIVE
Test Result
1
0.8
0.6
0.4
Sensitivity
0.2
0
0
0.2
0.4
0.6
0.8
1
1 Specificity
1
0.8
0.6
0.4
Sensitivity
0.2
0
0
0.2
0.4
0.6
0.8
1
1 Specificity
We are less happy when the ROC curve follows a diagonal path from the lower left hand corner to the upper right hand corner. This means that every improvement in false positive rate is matched by a corresponding decline in the false negative rate
1
0.8
0.6
0.4
Sensitivity
0.2
0
0
0.2
0.4
0.6
0.8
1
1 Specificity
1 = Perfect diagnostic test
0.5 = Useless diagnostic test
1
0.8
0.6
0.4
Test 1
Sensitivity
Test 2
0.2
Test 3
0
0
0.2
0.4
0.6
0.8
1
1 Specificity
P(BA) P(A)
P(AB) =
P(BA) P(A) + P(Bnot A) P(not A)
Note: P(B) = 0
P(T+D+) P(D+)
P(TD) P(D)
P(D+T+) =
P(DT) =
P(T+D+) P(D+) + P(T+D) P(D)
P(TD) P(D) + P(TD+) P(D+)
Positive predictive Value = P(D+T+)
Sensitivity
1 Specificity
Specificity
Negative predictive Value = P(DT)
1 Sensitivity
P(T+D+) P(D+)
P(D+T+) =
P(T+D+) P(D+) + P(T+D) P(D)
P(T+D+) P(D+)
P(D+T+) =
P(T+D+) P(D+) + P(T+D) P(D)
We know P(D+) = 0.6
From before,
1 0. 6 = 0.4
Sensitivity
= 25/(25+18) = 0.58
1 0.85 = 0.15
Specificity
= 78/(78+14) = 0.85
0.58*0.6
P(D+T+) =
= 0.85
0.58*0.6 + 0.15*0.4
P(TD) P(D)
P(DT) =
P(TD) P(D) + P(TD+) P(D+)
We know P(D+) = 0.6
1 0.6 = 0.4
From before,
Sensitivity
= 25/(25+18) = 0.58
1 0.58 = 0.42
Specificity
= 78/(78+14) = 0.85
0.85*0.4
P(DT) =
= 0.57
0.85*0.4 + 0.42*0.6
P(T+D+) P(D+)
P(D+T+) =
P(T+D+) P(D+) + P(T+D) P(D)
We know P(D+) = 0.3
From before,
1 0. 3 = 0.7
Sensitivity
= 25/(25+18) = 0.58
1 0.85 = 0.15
Specificity
= 78/(78+14) = 0.85
0.58*0.3
P(D+T+) =
= 0.62
0.58*0.3 + 0.15*0.7
P(TD) P(D)
P(DT) =
P(TD) P(D) + P(TD+) P(D+)
We know P(D+) = 0.3
1 0.3 = 0.7
From before,
Sensitivity
= 25/(25+18) = 0.58
1 0.58 = 0.42
Specificity
= 78/(78+14) = 0.85
0.85*0.7
P(DT) =
= 0.83
0.85*0.7 + 0.42*0.3
Example: Mrs X. had positive lab results, what is the probability she was a carrier??
P(D+T+)
P(T+D+) P(D+)
P(D+T+) =
P(T+D+) P(D+) + P(T+D) P(D)
From before,
Sensitivity
= 32/(32+2) = 0.94
Specificity
= 28/(28+6) = 0.82
Grandmother was a carrier
Mother was a carrier
0.94*0.25
0.94*0.5
= 0.64
= 0.84
P(D+T+) =
P(D+T+) =
0.94*0.25 + 0.18*0.75
0.94*0.5 + 0.18*0.5
LR+ = Sensitivity
LR = 1Sensitivity
1 Specificity
Specificity
HIGH LR+ and LOW LR imply both sensitivity
and specificity are close to 1
P(A)
2/3
P(A)
1 2/3
1 P(A)
P(NOT A)
Odds in favor of A =
=
(or 2 to 1)
= 2
2
odds
1 + 2
1 + odds
P(A) =
= 2/3
The Odds in favor of heads when a coin is tossed is 1. (Ratio of 1:1)
The Odds in favor of rolling a ‘6’ on any throw of a fair die is 0.2. (Ratio of 1:5)
The Odds AGAINST rolling a ‘6’ on any throw of a fair die is 5. (Ratio of 5:1)
The Odds in favor of drawing an ace from an ordinary deck of playing cards is 1/12. (Ratio of 1:12)
Posterior odds in favor of A
Prior odds in favor of A
Likelihood
ratio
X
=
LR+ if they tested positive
LR if they tested negative
Posterior odds in favor of A
Prior odds in favor of A
Likelihood
ratio
X
=
STEP 1.
Answer: her odds were 1:1, or simply 1.
Answer: her odds were 1:3, or simply 1/3.
LR+ = Sensitivity
1 Specificity
STEP 2.
= 5.3
= 0.94
Answer:
1 0.82
Posterior
odds in
favor of A
Prior odds
in favor of A
Likelihood
ratio
1 X 5.3 = 5.3
=
X
=
The odds are 5.3 to 1 in favor of Mrs. X being a carrier.
Posterior
odds in
favor of A
Prior odds
in favor of A
Likelihood
ratio
(1/3) X 5.3 = 1.8
=
X
=
The odds are 1.8 to 1 in favor of Mrs. X being a carrier.