CHAPTER 3
Probability Theory
• 3.1 - Basic Definitions and Properties
• 3.2 - Conditional Probability and Independence
• 3.3 - Bayes’ Formula
• 3.4 - Applications (biomedical)
More on Sensitivity / Specificity
Examples of Screening Tests for Early Detection of Colorectal Cancer
Highly sensitive and highly
Fecal Occult Blood Test (FOBT)
specific, but expensive.
Cheap, fast, easy, and highly
Cost-effective for males 50+. sensitive, but low specificity… not to
mention TOTALLY DISGUSTING.
“FUITA” Procedure
Very highly sensitive,
routinely administered by
health insurance companies
Some Additional Biomedical
Applications
Case-Control
studies
Cohort
studies
Observational study designs that test for a statistically significant association
between a disease D and exposure E to a potential risk (or protective) factor,
measured via “odds ratio,” “relative risk,” etc. Lung cancer / Smoking
Case-Control
studies
Cohort
studies
PRESENT
PAST
FUTURE
cases
E+ vs. E– ?
controls
reference group
D+ vs. D– E+ vs. E–
 relatively easy and inexpensive
subject to faulty records, “recall bias”
D+ vs. D– ?
 measures direct effect of E on D
expensive, extremely lengthy…
Both types of study yield a 22 “contingency table” of data:
D+
D–
E+
a
b
a+b
E–
c
d
c+d
a+c b+d
n
where a, b, c, d are the
numbers of individuals
in each cell.
Case-Control
studies
PRESENT
PAST
cases
E+ vs. E– ?
Cohort
studies
ref gp
controls
FUTURE
D+ vs. D– E+ vs. E–
D+
D–
E+
a
b
a+b
E–
c
d
c+d
a+c b+d
Cohort studies
D+ vs. D– ?
where a, b, c, d are the
numbers of individuals
in each cell.
n
P ( D  | E  ) a / ( a  b) a


P ( D  | E  ) b / ( a  b) b
P ( D  | E ) c / (c  d ) c


“Odds of Disease, given Not Exposed” = odds(D | E–) =
P ( D  | E ) d / (c  d ) d
“Odds of Disease, given Exposed” = odds(D | E+) =

odds ( D | E ) a / b a d


odds ( D | E –) c / d b c
<1
possible protective factor
=1
No assoc; D, E stat indep
>1
possible risk factor
Case-Control
studies
PRESENT
PAST
cases
E+ vs. E– ?
Example:
Cohort studies
odds(D | E+) =
ref gp
controls
D+ vs. D– E+ vs. E–
D+
D–
E+
500
a
200
b
a700
+b
E–
400
c
300
d
c700
+d
a900
+ c b500
+d
P( D  | E ) a 500
 2.5
 
200
b
P( D  | E )
P( D  | E ) c 400
 
 1.333
odds(D | E–) =
P( D  | E ) d 300

Cohort
studies
FUTURE
D+ vs. D– ?
where a, b, c, d are the
numbers of individuals
in each cell.
1400
n
Among those exposed, the probability of
developing disease is 2.5 times greater than
the probability of not developing disease.
Among those not exposed, the probability of
developing disease is 1.333 times greater
than the probability of not developing disease.
odds ( D | E ) a d
(500)(300)

 1.875

odds ( D | E –) b c
(200)(400)
The odds of disease among those
exposed are 1.875 times greater than the
odds of disease among those not exposed.
Case-Control
studies
PRESENT
PAST
cases
E+ vs. E– ?
Cohort
studies
FUTURE
ref gp
controls
D+ vs. D– E+ vs. E–
Example:
D+
D–
E+
a
b
a+b
E–
c
d
c+d
a+c b+d
Cohort studies

Why not just use
D+ vs. D– ?
n
odds ( D | E ) a d  1.875

odds ( D | E –) b c
P( D  | E )
???
P ( D  | E )
where a, b, c, d are the
numbers of individuals
in each cell.
The odds of disease among exposed
are 1.875 times greater than the
odds of disease among not exposed.
Case-Control
studies
PRESENT
PAST
cases
E+ vs. E– ?
Cohort
studies
FUTURE
ref gp
controls
D+ vs. D– E+ vs. E–
Example:
D+
D–
E+
500
a
200
b
a700
+b
E–
400
c
300
d
c700
+d
a900
+ c b500
+d
Cohort studies

Case-Control studies

D+ vs. D– ?
1400
n
odds ( D | E ) a d  1.875

odds ( D | E –) b c
P( D  | E ) a (c  d )

 1.25
P( D  | E ) c (a  b)
(HW problem)
where a, b, c, d are the
numbers of individuals
in each cell.
odds ( E | D) a d  1.875

odds ( E | D –) b c
The odds of disease among exposed
are 1.875 times greater than the
odds of disease among not exposed.
The probability of disease among
exposed is 1.25 times greater than
the probability of disease among not
exposed.
The odds of exposure among diseased
are 1.875 times greater than the
odds of exposure among not diseased.
Case-Control
studies
PRESENT
PAST
cases
E+ vs. E– ?
Cohort
studies
ref gp
controls
D+ vs. D– E+ vs. E–
Example:
D+
D–
E+
a
b
a+b
E–
c
d
c+d
a+c b+d
Cohort studies
odds ( D | E )  a d

odds ( D | E –) b c
P( D  | E ) a (c  d )

P( D  | E ) c (a  b)
Case-Control studies

odds ( E | D) a d

odds ( E | D –) b c
FUTURE
D+ vs. D– ?
where a, b, c, d are the
numbers of individuals
in each cell.
n
Whereas the Odds Ratio is reliably
approximated from either type of
study using the same formula, the
Relative Risk is not, and is only
appropriately defined for cohort
studies, except…
if the disease is rare in the popul’n…
a is small relative to b, and
c is small relative to d…
then RR ≈ OR.