E non pelleted feed e pelleted feed controls e e

E+ = non-pelleted 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 flock-size 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 non-pelleted feed than control farms
Analysis of matched case-
control studies
Hypothesis testing
H
O
: OR = 1 vs. H
a
: OR ≠ 1
McNemar’s chi-square 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 chi-square 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
Mantel-Haenszel (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 case-control contingency
table layout
l
Sets of observations with discordant (different)
exposures contribute to the overall MH estimate
Analysis of pair-matched 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 risk-factor + and
control is risk-factor – (“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 risk-factor - and
control is risk-factor + (“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