The math of decision in Radiology - MIR

THE MATH OF DECISION IN RADIOLOGY
Prof. Utku Şenol, MD, PhD of medical informatics
Akdeniz University, School of Medicine,,
Department of Radiology, Division of Neuroradiology
ANTALYA, TURKEY
utkusenol@gmail.com

Contents
• Decision making in radiology
• Test characteristics
• Likelihood ratios
• Predictive values
• Calculating post-test probability
• Bayesian approach
• Probability revision
• Nomograms, Probability graphs

Question 1
A lymphadenopathy in mediastinum was reported
Sensitivity 90%
Specificity 80%
What is the actual probability of the lymphadenopathy?
A) 2-3%
B) 20-25%
C) 50-60%
D) 80-90%
E) Can not be calculated

Question 2
• A Breast cancer screening program
• Prevalence of the Breast Ca is 0.1%
• A positive mammogram was reported.
sensitivity 95%
specificity 95%
What is the actual probability of the patient having a cancer?
A) 2%
B) 10-20%
C) 50-60%
D) 90-95%
E) Can not be calculated

When Do Radiologists Make Decision ?
• While
– Diagnosing
– Excluding disease
– Giving probability for a certain disease
– Choosing the proper exam
– Planning a survey program
– …

How Do Radiologists Make Decision?
Do They Use Probability or Math?

• Virtually always
• Often
• Not often
• Virtually never
AJR 174:2000 letter from Kliewer

AJR 174:2000 letter from Kliewer

Sir William Osler
1849 -1919
“Medicine is a science of
uncertainty and an art
of probability”

Do not treat
Further tests
or follow-up
Treat
0%
Very unlikely
to have disease
25%
Probably doesn’t
have disease
50%
Don’t know
75%
Probably does
have
100 %
Very likely
to have disease

Probability in Orders
• Thunderclap headache
• Lower quadrant pain, leukocytosis,
womiting, ribaund
• Lung Ca, recent vertigo, Rull out brain
metastasis
• 35 year age woman, pain and visual loss in
right eye and hemiparesis

Probability in Reports
• Hemorrhagic density with sulcal pattern
• Well circumscribed lytic lesion with a
sclerotic rim in distal femur
• An ill defined solitary nodule in right apex

Subjective probability of estimates are not statistical
and are highly prone to error!!
Those errors may be as;
• Pesudodiagnosticity
• Premature diagnosis
• Errors Caused by Heuristics
– Availability
– Representativeness
– Anchoring and adjustment
– Value-induced bias
– Affect heuristics
– …..

How do we use tests to define
probability?

Diagnostic tests are not perfect.
False positive and false negative results cannot be eliminated…
Sorry, you can not have baby
But, We had
Tests say
you can’t.
Give
them to
me…

Sensitivity and Specificity
Sensitivity (TPR) = a/a+c
Specificity (TNR) = d/b+d
Test +
Disease
+
a
Disease
-
b
Total
a+b
TP
FP
Test -
c
d
c+d
FN
TN
Total
a+c
b+d
a+b+c+d
TP+FN
TN+FP

Which Test is the Strongest?
Test Sensitivity Specificity
A 80% 75%
B 60% 85%
C 70% 80%
D 78% 78%

Likelihood Ratios
A unique value generated by using both
specificity and sensitivity
A perfect indicator for defining strength of tests

Positive Likelihood Ratio
LR+ = sensitivity/(1-specificity)
Sensitivity = 90%
Specificity = 95%
LR+ = .90/.05 = 18 =
The probability of a positive result in a patient with disease
The probability of a positive result in a patient without disease

Positive Likelihood Ratio
• 10 or
strong
• 5-10
medium
• 2-5
weak
• 2 or
too weak

Test Sensitivity Specificity
A 80% 75%
B %65 %85
C %70 %80
D %78 %78

Test Sensitivity Specificity LR+
A 80% 75%
3.2
B %65 %85
4.3
C %70 %80
3.5
D %78 %78
3.54

Test Sensitivity Specificity LR+ LR-
A 80% 75%
3.2 0.26
B %65 %85
4.3 0.41
C %70 %80
3.5 0.37
D %78 %78
3.54 0.28

Negative Likelihood Ratio
LR- = (1- sensitivity) / specificity
Sensitivity = %90
Specificity = %95
LR- = 0.10 / 0.95 = 0.105

Negative Likelihood Ratio
• 0.1 ve
strong
• 0.1 - 0.2
medium
• 0.2 - 0.5
weak
• 0.5 ve
too weak

Likelihood Ratios
Strong
medium
medium
Strong

LR helps to predict the post-test
probability

What about Predictive Values?
Positive predictive value
• Probability of
having disease
when the test is
positive
Negative predictive value
• Probability of not
having disease
when the test is
negative

Disease
Disease
Total
+
-
Test +
a
b
a+b
TP
FP
TP+FP
Test -
c
d
c+d
FN
TN
FN+TN
Total
a+c
b+d
a+b+c+d
TP+FN
TN+FP
Positive Predictive Value: a/a+b
Negative Predictive Value: d/c+d
Sensitivity:
Specificity:
a/a+c
d/b+d

Test Characteristics
(sensitivity, specificity)
Predictive values
• May be
generalized to
different
populations
• Can not be
generalized to
other groups
Unique for a
specific study
group

D+ D - Total
T + 63 18 81
T - 37 82 119
Sensitivity: 63%
Specificity: 82%
PPV: 77%
Toplam 100 100 200
D + D - Total
T + 63 162 225
T - 37 738 775
Sensitivity: 63%
Specificity: 82%
PPV: 28%
Total 100 900 1000

How do we calculate the
probability after the test?

Probability revision
The probability after a positive result
Pre test probability
The probability after a negative result

Post-test probability can be
calculated only if there is a pre-test
probability

Post-test probability calculation
methods
• Bayes formulation
• Decision tree
• Contingency table revision
• Nomogram
• Conditional probability graphs
Post-test Odds = Pre-test Odds x LR

Post-test Odds = Pre-test Odds x LR
Post-test probability
Pre-test probability
1- post-test probability = 1- pre-test probability
x LR

Nomogram
Pre-t Prob
LR
Post-t Prob.

Post-test probability
Probability graph:
1
.9
Test +
.8
.7
.6
.5
.4
.3
.2
.1
Test -
0 .1 .2 .3 .4
Pre-test probability
.5 .6 .7 .8 .9 1

Question 1
LR+=4.5
Pre-test probability= ???
A CT report indicates a lymphadenopathy in mediastinum.
Sensitivity 90%
Specificity 80%
What is really the probability of the lymphadenopathy?
A) 2-3%
B) 20-25%
C) 50-60%
D) 80-90%
E) Can not be calculated

Question 2
LR+=19
Pre-test probability=1/1000
• Mammography screening program
• Prevalence of the Breast Ca is 0.1%
• A positive mammogram was reported.
sensitivity 95%
specificity 95%
What is really the probability of the patient having a cancer?
A) 2%
B) 10-20%
C) 50-60%
D) 90-95%
E) Can not be calculated
%1.93

Summary - I
• Radiologists frequently deal with probabilities
while making decision
• To be familiar with the math of the decision
making process is helpful

Summary - II
• Likelihood ratios are used to define;
– strength of test
– post test probability

Summary - III
• Likelihood ratios are calculated by sensitivity
and specificity
LR + = sensitivity/(1-specificity)
LR - = (1-sensitivity)/specificity

Summary - IV
• Post test probability can not be calculated without
pre-test probability.
• So, clinical information is crucial !!

Conclusion-IV
Post test probability can be calculated based on
Bayes principles:
Post-test Odds = Pre-test Odds X LR