STA 3024 – Overview of Topics for Exam 2
Part 1 – Comparing 2 Proportions (IIb,c,d)
Measures of Association for Comparing 2 Population Proportions


Relative Risk: RR=p1/p2
Odds Ratio: OR=odds1/odds2 where oddsk = pk / (1-pk)
Test for equal proportions based on small, independent samples (or when
there are very few successes or failures in one or both groups)

Fisher’s Exact Test
Test for equal proportions for the same characteristic in independent samples

McNemar’s Test
Measures of Association for 2 Proportions:
 Study Designs: Prospective vs Retrospective
 Prospective studies: Subjects sampled based on group
(independent variable), outcome (dependent variable) observed
subsequently
 Retrospective studies: Subjects sampled based on outcome
(dependent variable), group/risk factor (independent variable) is
ascertained “after the fact”
 “Success” & “Failure” are levels of the dependent variable
 Group id’s are levels of independent variable
 P1 = Proportion of “Successes” in group 1
 P2 = Proportion of “Successes” in group 2
Relative Risk:
Population Relative Risk:  
p1
p2
(Note: This is new notation)
Sample Data collected:
n1 individuals from from group 1, X1 “Successes”
n2 individuals from from group 2, X2 “Successes”
^
p1 
X1
n1
^
p2 
X2
n2
^
Sample Relative Risk : RR 
p1
^
(Natural) log(RR)  ln( RR )
p2
^
^
1  p1 1  p 2
Std. Error of ln( RR ) : SE (ln( RR ))  v 

X1
X2
Approximat e 95%CI for ln(  ) : ln( RR )  1.96 v

Approximat e 95%CI for  : e
LBln(RR )
,e
UBln(RR )


 ( LBln(RR ) , UBln(RR ) )
( RR )e
1.96 v
, ( RR )e 1.96
v

Conceptual Questions:
 When we conclude the Success probability is higher for group 1? For group




2? When can we not conclude they differ?
Is Relative Risk a meaningful measure based on prospective studies? Why or
why not?
Is Relative Risk a meaningful measure based on retrospective studies? Why
or why not?
Why construct a confidence interval for the (Natural) log of the relative
risk, then “transform it back” into original units?
How would you simulate the sampling distribution of the sample relative
risk?
Odds Ratio:
Population Relative Risk:  
p1 /(1  p1 )
p 2 /(1  p2 )
(Note: This is new notation)
Sample Data collected:
n1 individuals from from group 1, X1 “Successes”
n2 individuals from from group 2, X2 “Successes”
^
^
p1
p2
X1
odds1 

^
1  p 1 n1  X 1
Sample Odds Ratio : OR 
odds 2 
^

1 p2
X2
1 X 2
odds1 X 1 (n2  X 2 )

odds 2
X 2 (n1  X 1 )
1
1
1
1



X 1 n1  X 1 X 2 n2  X 2
Std. Error of ln( OR ) : SE (ln( OR ))  v 
Approximat e 95%CI for ln(  ) : ln( OR )  1.96 v

Approximat e 95%CI for  : e
LBln(OR )
,e
UBln(OR )
(Natural) log(OR)  ln( OR )


 ( LBln(OR) , UBln(OR) )
(OR )e
1.96 v
, (OR )e 1.96
v

Conceptual Questions:
 When we conclude the Success probability is higher for group 1? For group




2? When can we not conclude they differ?
Is Odds Ratio a meaningful measure based on prospective studies? Why or
why not?
Is Odds Ratio a meaningful measure based on retrospective studies? Why or
why not?
Why construct a confidence interval for the (Natural) log of the odds ratio,
then “transform it back” into original units?
How would you simulate the sampling distribution of the sample odds ratio?
Fisher’s Exact Test (Independent Samples, Small Cell counts or
sample sizes):


Set up contingency table, fix the row and margin totals
Obtain all tables that show as strong as stronger evidence of an
association (depends on 1-sided or 2-sided alternative)
Obtain the probability of each table above, probabilities can be
obtained based on the hypergeometric distribution
 Sum the probabilities to obtain the P-value of the Test

a
b
c
d
ac bd
ab
cd
abcd

 a  b  c  d 



a  c 

Probabilit y 
a  b  c  d 


ac


 y
y!
where   
 x  x!( y  x)!
McNemar’s Test (Dependent Samples/Matched Pairs):




Identify the pairs of “individuals”/“cases”/ “occasions”
Identify the 2 possible outcomes
Identify the
Set up the contingency table:
Individual 1 \ 2
Outcome A
Outcome B
Outcome A
nAA
nBA
Outcome B
nAB
nBB

Compute the test statistic:
z obs 
n AB  n BA
n AB  n BA
 Decision Rule (2-sided test, =0.05):
 Conclude P(Outcome A) is higher for individual 1 if zobs  1.96
 Conclude P(Outcome A) is lower for individual 1 if zobs  -1.96
 Don’t conclude P(Outcome A) differs for the individuals otherwise
NOTE: This a actually a test of a binomial proportion being equal to
0.5, and is based on a normal approximation. An exact test can be
obtained based on the binomial distribution:
P-value =2P(Xmin(nAB,nBA)) when X~Binomial(n=nAB+nBA,p=0.5).
See Table C, pp. T-6—T-10 for n20