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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. - PowerPoint PPT Presentation
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CHAPTER 3
Probability Theory• 3.1 - Basic Definitions and Properties• 3.2 - Conditional Probability and Independence• 3.3 - Bayes’ Formula• 3.4 - Applications (biomedical)
Some Additional Biomedical ApplicationsSensitivity and Specificity can change w.r.t. a third variable.
Some Additional Biomedical ApplicationsSensitivity and Specificity can change w.r.t. a third variable.
If ROC is polygonal, then AUC can be
easily calculated via the Trapzoidal Rule.
AUC (Area Under Curve) = P(Test is
correct in a random case-control pair).
Some Additional Biomedical ApplicationsSensitivity and Specificity can change w.r.t. a third variable.
Some Additional Biomedical ApplicationsSensitivity and Specificity can change w.r.t. a third variable.
Test X
Test Y
Test Z
Via a Wilcoxon-like comparison of AUCs, Test X is significantly
higher than that of Y and Z, whose ΔAUC is only marginally statistically significant at best.
Overview of Biostatistical Methods
Case-Control studies
Case-Control studies
Cohort studiesCohort studies
E+ vs. E–
Overview of Biostatistical Methods
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
PRESENT
E+ vs. E– ? D+ vs. D– ?
Case-Control studies
Case-Control studies
Cohort studiesCohort studies
Both types of study yield a 22 “contingency table” of data:
D+ D–
E+ a b a + b
E– c d c + d
a + c b + d n
relatively easy and inexpensive subject to faulty records, “recall bias”
D+ vs. D–
FUTUREPAST
measures direct effect of E on D expensive, extremely lengthy…
where a, b, c, d are the numbers of individuals in each cell.
cases controls reference group
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.
E+ vs. E–
PRESENT
E+ vs. E– ? D+ vs. D– ?
Case-Control studies
Case-Control studies
Cohort studiesCohort studies
D+ vs. D–
FUTUREPASTcases controls
Cohort studies “Odds of Disease, given Exposed” = odds(D | E+) =
( | )
( | )
P D E
P D E
/ ( )
/ ( )
a a b
b a b
a
b
“Odds of Disease, given Not Exposed” = odds(D | E–) =( | )
( | )
P D E
P D E
/ ( )
/ ( )
c c d
d c d
c
d
“ODDS RATIO” OR ( | )
( | –)
D E
D E
oddsodds
/
/
a b
c d
a d
bc= 1 No assoc; D, E stat indep
< 1 possible protective factor
> 1 possible risk factor
ref gp
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.
E+ vs. E–
PRESENT
E+ vs. E– ? D+ vs. D– ?
Case-Control studies
Case-Control studies
Cohort studiesCohort studies
D+ vs. D–
FUTUREPASTcases controls
odds(D | E+) =( | )
( | )
P D E
P D E
a
b
odds(D | E–) =( | )
( | )
P D E
P D E
c
d
OR ( | )
( | –)
D E
D E
oddsodds
a d
bc
ref gp
D+ D–
E+ 500 200 700
E– 400 300 700
900 500 1400
Example:
500
200 2.5
400
300 1.333
(500)(300)
(200)(400) 1.875
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.
The odds of disease among those exposed are 1.875 times greater than the odds of disease among those not exposed.
Cohort studies
D+ D–
E+ a b a + b
E– c d c + d
a + c b + d n
“ODDS RATIO” OR
where a, b, c, d are the numbers of individuals in each cell.
E+ vs. E–
PRESENT
E+ vs. E– ? D+ vs. D– ?
Case-Control studies
Case-Control studies
Cohort studiesCohort studies
D+ vs. D–
FUTUREPASTcases controls
Why not just use ???( | )
( | )
P D E
P D E
ref gp
Example:
( | )
( | –)
D E
D E
oddsodds
a d
bc 1.875
The odds of disease among exposed are 1.875 times greater than the odds of disease among not exposed.
Cohort studies
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.
E+ vs. E–
PRESENT
E+ vs. E– ? D+ vs. D– ?
Case-Control studies
Case-Control studies
Cohort studiesCohort studies
D+ vs. D–
FUTUREPASTcases controls
( | )
( | )
P D E
P D E
ref gp
D+ D–
E+ 500 200 700
E– 400 300 700
900 500 1400
Example:
The odds of disease among exposed are 1.875 times greater than the odds of disease among not exposed.
( | )
( | –)
D E
D E
oddsodds
a d
bc 1.875“ODDS RATIO” OR
( )
( )
a c d
c a b
1.25“RELATIVE RISK” RR
The probability of disease among exposed is 1.25 times greater than the probability of disease among not exposed.
Case-Control studies ( | )
( | –)
E D
E D
oddsodds
a d
bc 1.875“ODDS RATIO” OR
The odds of exposure among diseased are 1.875 times greater than the odds of exposure among not diseased.
(HW problem)
Cohort studies
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.
E+ vs. E–
PRESENT
E+ vs. E– ? D+ vs. D– ?
Case-Control studies
Case-Control studies
Cohort studiesCohort studies
D+ vs. D–
FUTUREPASTcases controls
( | )
( | )
P D E
P D E
ref gp
Example:
( | )
( | –)
D E
D E
oddsodds
a d
bc“ODDS RATIO” OR
( )
( )
a c d
c a b
“RELATIVE RISK” RR
Case-Control studies ( | )
( | –)
E D
E D
oddsodds
a d
bc“ODDS RATIO” OR
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…
then RR ≈ OR.
a is small relative to b, and c is small relative to d…
( )
( )
a c d
c a b
Cohort studies
