E+ = nonpelleted feed,
E = pelleted feed
Controls
E+
E
Cases
E+
22 (a*)
14 (b*)
E
2 (c*)
12 (d*)
7
Analysis of matched case
control studies
Example (1:1 matching)
Odds ratio (OR) = b*/c* = 14/2 = 7
( Note: this estimate may still be confounded because
matching only on flocksize may not remove confounding
from other covariates, so stratification/logistic regression
would still be necessary)
Interpretation:
Case farms were at 7 times higher odds of
being fed nonpelleted feed than control farms
Analysis of matched case
control studies
Hypothesis testing
H
O
:
OR = 1
vs.
H
a
: OR
≠
1
McNemar’s chisquare statistic (1 df)
= (b*  c*  1)
2
/ (b* + c*)
= (14  2  1)
2
/ (14 + 2)
= 7.56 (P < 0.01)
Note: some versions of formula do not include  1 in numerator
Analysis of matched case
control studies
Approximate confidence interval estimation
Approx. 95% CI
=
OR
(1
±
1.96/
χ
)
=
7
(1
±
1.96/2.75)
=
7
(0.287, 1.712)
=
1.75 to 27.98
χ
is the square root of the McNemar’s chisquare statistic
Note: this approach is for illustration of the confidence interval only;
superior approaches are used in statistical software packages.
8
Analysis of matched case
control studies
MantelHaenszel (MH) estimator
l
Applicable to estimation with a varying (by
strata) number of controls per case
l
Results are changed from the table format
above to the traditional casecontrol contingency
table layout
l
Sets of observations with discordant (different)
exposures contribute to the overall MH estimate
Analysis of pairmatched case
control studies
l
Each pair with discordant observations
contributes 1/2 to the overall MH estimate.
l
Concordant (same exposure) pairs contribute
nothing to calculations because ad = bc
= 0.
l
MH weighted odds ratio
=
Σ
a
i
d
i
/n
i
=
Σ
count in b* cell * 1/2
=
b*/c*
Σ
b
i
c
i
/ni
Σ
count in c* cell * 1/2
Table 1. Case and control are
positive for risk factor (“a*” cell)
Paired data
Unpaired
equivalent
Controls
Cases
Risk factor +
Risk factor 
Risk
factor
Cases
Controls
Risk
factor +
1
0
+
1 (a)
1 (b)
Risk
factor 
0
0

0 (c)
0 (d)
n = 2
ad = 0
bc = 0
9
Table 2. Case is riskfactor + and
control is riskfactor – (“b*” cell)
Paired data
Unpaired
equivalent
Controls
Cases
Risk factor +
Risk factor 
Risk
factor
Cases
Controls
Risk
factor +
0
1
+
1 (a)
0 (b)
Risk
factor 
0
0

0 (c)
1 (d)
n = 2
ad = 1
bc = 0
Table 3. Case is riskfactor  and
control is riskfactor + (“c*” cell)
Paired data
Unpaired
equivalent
Controls
Cases
Risk factor +
Risk factor 
Risk
factor
Cases
Controls
Risk
factor +
0
0
+
0 (a)
1 (b)
Risk
factor 
1
0

1 (c)
0 (d)
n = 2
ad = 0
bc = 1
Table 4. Case and control are
negative for risk factor (“d*” cell)
Paired data
Unpaired
equivalent
Controls
Cases
Risk factor +
Risk factor 
Risk
factor
Cases
Controls
Risk
factor +
0
0
+
0 (a)
0 (b)
Risk
factor 
0
1

1 (c)
1 (d)
n = 2
ad = 0
bc = 0
10
Example 1: 3 matching
Paired data
Unpaired
equivalent
Controls
Cases
Risk factor +
Risk factor 
Risk
factor
Cases
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 Winter '16
 Regression Analysis, Epidemiology, Confounding, Holstein, Holsteins, Dr. Ian Gardner