E non pelleted feed E pelleted feed Controls E E Cases E 22 a 14 b E 2 c 12 d 7

E non pelleted feed e pelleted feed controls e e

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E+ = non-pelleted feed, E- = pelleted feed Controls E+ E- Cases E+ 22 (a*) 14 (b*) E- 2 (c*) 12 (d*)
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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.
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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
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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
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10 Example 1: 3 matching Paired data Unpaired equivalent Controls Cases Risk factor + Risk factor - Risk factor Cases
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