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Analysis of pair matched dataAnalysis of pair matched data
HRP 261 2/24/03 10HRP 261 2/24/03 10--11 am11 am
Pair MatchingPair MatchingRecall question 7 on homework 2—dependent proportions. We’re returning to this concept today.
Pairing can control for extraneous sources of variability and increase the power of a statistical test.Match 1 control to 1 case based on potential confounders, such as age and gender.
ExampleExampleJohnson and Johnson (NEJM 287: 1122-1125, 1972) selected 85 Hodgkin’s patients who had a sibling of the same sex who was free of the disease and whose age was within 5 years of the patient’s…they presented the data as….
Hodgkin’s
Sib control
Tonsillectomy None
41 44
33 52
OR=1.47; chi-square=1.53 (NS)
From John A. Rice, “Mathematical Statistics and Data Analysis.
ExampleExampleBut several letters to the editor pointed out that those investigators had made an error by ignoring the pairings. These are not independent samples because the sibs are paired…better to analyze data like this:
OR=2.14; chi-square=2.91 (p=.09)
Tonsillectomy
None
Tonsillectomy None
37 7
15 26
CaseControl
From John A. Rice, “Mathematical Statistics and Data Analysis.
Pair Matching: Pair Matching: Agresti Agresti exampleexample
Match each MI case to an MI control based on age and gender.
Ask about history of diabetes to find out if diabetes increases your risk for MI.
Pair Matching: Pair Matching: AgrestiAgresti exampleexampleJust the discordant cells are
informative!
Diabetes
No diabetes
25 119
Diabetes No Diabetes
9 37
16 82
46
98
144
MI cases
MI controls
Which cells are informative?
Pair MatchingPair Matching
Diabetes
No diabetes
25 119
Diabetes No Diabetes
9 37
16 82
46
98
144
MI cases
MI controls
OR estimate comes only from discordant pairs!
The question is: among the discordant pairs, what proportion are discordant in the direction of the case vs. the control. If more discordant pairs “favor” the case, this indicates OR>1.
Diabetes
No diabetes
25 119
Diabetes No Diabetes
9 37
16 82
46
98
144
MI cases
MI controls
P(“favors” case/discordant pair) =
)~/(*)/(~)~/(~*)/()~/(~*)/(
DEPDEPDEPDEPDEPDEP
+
=the probability of observing a case-control pair with only the control exposed
=the probability of observing a case-control pair with only the case exposed
Diabetes
No diabetes
25 119
Diabetes No Diabetes
9 37
16 82
46
98
144
MI cases
MI controls
P(“favors” case/discordant pair) =
5337
163737ˆ =
+=
+=
cbbp
Diabetes
No diabetes
25 119
Diabetes No Diabetes
9 37
16 82
46
98
144
MI cases
MI controls
odds(“favors” case/discordant pair) =
1637
==cbOR
Diabetes
No diabetes
25 119
Diabetes No Diabetes
9 37
16 82
46
98
144
MI cases
MI controls
OR estimate comes only from discordant pairs!!
OR= 37/16 = 2.31
Makes Sense!
McNemar’s McNemar’s TestTest
Diabetes
No diabetes
Diabetes No Diabetes
9 37
16 82
MI casesMI controls
Null hypothesis: P(“favors” case / discordant pair) = .5(note: equivalent to OR=1.0 or cell b=cell c)
...)5(.)5(.3953
)5(.)5(.3853
)5(.)5(.3753 143915381637 +
+
+
=− valuep
By normal approximation to binomial:
01.;88.264.3
5.10)5)(.5(.53
)2
53(37<==
−= pZ
McNemar’s McNemar’s Test: generallyTest: generally
By normal approximation to binomial:
exp
No exp
exp No exp
a b
c d
casescontrols
cbcb
cb
cb
cb
cbbZ
+−
=+
−=
+
+−
=
4
22)5)(.5)(.(
)2
(
Equivalently:
cbcb
cbcb
+−
=+−
=2
221
)()(χ
95% CI for difference in 95% CI for difference in dependent proportionsdependent proportions
Diabetes
No diabetes
25 119
Diabetes No Diabetes
9 37
16 82
46
98
144
MI cases
MI controls
24.05.)0024.(96.115.17.- 32. : CI %95
0024.144
)11.*26.57.*06(.2)83)(.17(.)68)(.32(.
),(2)1()1(
)(),(2)()()(
~//~/~///
~//
212121
−=±=∴
=−−+
=
−−
+−
=
−∴−+=−
==DEDE
controlscases
DEDE
controlscases
DEDE
DEDE
ppCovn
ppn
ppppVar
ppCovpVarpVarppVar
Each pair is it’s own “ageEach pair is it’s own “age--gender” stratumgender” stratum
Diabetes
No diabetes
Case (MI) Control
1 1
0 0
Example: Concordant for
exposure (cell “a” from before)
Diabetes
No diabetes
Case (MI) Control
1 1
0 0
Case (MI) Control
x 9
Diabetes
No diabetes
1 0
0 1x 37
Case (MI) Control
Diabetes
No diabetes
0 1
1 0
Case (MI) Control
x 16
Diabetes
No diabetes
0 0
1 1 x 82
MantelMantel--Haenszel Haenszel for pairfor pair--matched datamatched data
We want to know the relationship between diabetes and MI controlling (very tightly) for age and gender.
Mantel-Haenszel methods apply.
RECALL: The MantelRECALL: The Mantel--HaenszelHaenszelSummary Odds RatioSummary Odds Ratio
∑
∑
=
=k
i i
ii
k
i i
ii
Tcb
Tda
1
1
Exposed
Not Exposed
Case Control
a b
c d
Diabetes
No diabetes
Case (MI) Control
1 1
0 0
Diabetes
No diabetes
Case (MI) Control
1 0
0 1
ad/T = 0
bc/T=0
ad/T=1/2
bc/T=0
Diabetes
No diabetes
Case (MI) Control
0 1
1 0
Diabetes
No diabetes
Case (MI) Control
0 0
1 1
ad/T=0
bc/T=1/2
ad/T=0
bc/T=0
MantelMantel--HaenszelHaenszel Summary ORSummary OR
1637
21*162137
2
2144
1
144
1 ===
∑
∑
=
=x
cb
da
OR
i
ii
i
ii
MH
MantelMantel--HaenszelHaenszel Test StatisticTest Statistic(same as (same as McNemar’sMcNemar’s))
cbcb
cbcbCMH
nVar
nnnnnnVar
nnn
cellsdisc
cellsdisccon cellsdisccase
k
kk
kkkk
k
kk
+−
=+−
=+−
=
=−
===
−=
=
∑∑ ∑
++++
++++
++
++
22
.
..
2
.
21111k
22211
11k
1111k
)()25)(.()](5.)(5[.
25.
]5.5.[41
)12(2)1)(1)(1)(1()(;
21
2)1)(1(
:cells discordant0 contribute cells Concordant
)1()(n
)E(n :recall
µ
Logistic Regression for Logistic Regression for Matched Pairs (1) Matched Pairs (1)
the logisticthe logistic--normal modelnormal modelMixed model; logit=αi+βxWhere αi represents the “stratum effect”– (e.g. different odds of disease for different ages
and genders)– Example of a “random effect”
Allow αi’s to follow a normal distribution with unknown mean and standard deviationGives “marginal ML estimate of β”
Logistic Regression for Matched Logistic Regression for Matched Pairs (2) Pairs (2)
intro. to the conditional likelihoodintro. to the conditional likelihoodThe conditional likelihood is based on….
The conditional probability GIVEN discordant pair =
)/(~*)~/()~/(~*)/()~/(~*)/(
)~/(*)/(~)~/(~*)/()~/(~*)/(
EDPEDPEDPEDPEDPEDP
DEPDEPDEPDEPDEPDEP
+=
+(marginals cancel)
Logistic Regression for Matched Logistic Regression for Matched Pairs (2) Pairs (2)
the conditional likelihoodthe conditional likelihood
∏
∏
+++
++
++
+++
++
++
++
+
+
+
+
+
+
+
casefavor ;discordant stratums
controlfavor ;discordant stratums
1*
11
11*
1
11*
1
11*
11*
11
1*
11
i
i
iii
i
ii
i
ii
i
i
i
i
i
i
i
ee
eeee
eee
x
eee
ee
e
ee
e
α
α
βααβα
βα
αβα
βα
αβα
βα
α
α
βα
α
α
βα
The conditional likelihood=
Conditional Logistic RegressionConditional Logistic Regression
∏∏ ++= +
+
+
casefavor ;discordant stratums
controlfavor ;discordant stratums
1
1
1
1* ii
i
ii
i
eeex
eee
αβα
βα
αβα
α
∏∏ ++=
casefavor ;discordant stratums
controlfavor ;discordant stratums 1
1
1parameter) nuisance of rid (gets !cancel! s' The***
1
1
1 β
β
β
α
eex
e
e i
Conditional Logistic RegressionConditional Logistic Regression
1637
163753)137(
01
53-37dlog(L)
)1log(*5337)log(
1
1
11
1
1
11
=
=
=+
=+
=
+−=
β
β
ββ
β
β
β
β
β
e
eeee
ed
eL