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Categorical

McNemar's Test — Paired Binary Data

Paired-proportions test for before/after certification pass rates (n=20). Contrasts the uncorrected chi-square (p=.035) against SPSS/R's default continuity-corrected result (p=.070).

📊 χ²(1) = 3.27 (corrected), p = .070
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🗂️ 32 · McNemar's Test — Paired Binary Data

Did a Training Course Change the Certification Pass Rate?
Categorical
Research Question: 20 candidates took a certification exam (pass/fail), completed a 6-week training course, then took the exam again. Did the pass rate change? Because the same 20 people were measured twice, a chi-square test of independence is wrong here — McNemar's test is the correct paired alternative.
After: Pass After: Fail
Before: Pass6 (a)2 (b)
Before: Fail9 (c)3 (d)
Only the discordant cells (b, c) carry information
χ 2 = ( b − c ) 2 b + c (uncorrected) , χ c c 2 = ( | b − c | − 1 ) 2 b + c (continuity-corrected)
  1. 1
    Identify the discordant pairs: b = 2 candidates went pass→fail; c = 9 went fail→pass. The concordant cells (a=6 pass/pass, d=3 fail/fail) contribute no information about change and are dropped.
  2. 2
    Uncorrected χ²:
    χ 2 = ( 2 − 9 ) 2 2 + 9 = 49 11 = 4.455 , p = .035
  3. 3
    Continuity-corrected χ² (b+c = 11 < 25 → apply it):
    χ c c 2 = ( | 2 − 9 | − 1 ) 2 2 + 9 = 36 11 = 3.273 , p = .070
    SPSS and R's stats::mcnemar.test() apply this correction by default whenever b+c is small, precisely because it matters here: the correction flips the conclusion from p = .035 to p = .070.
  4. 4
    Decision rule: Report the corrected result: p = .070 > .05 → fail to reject H₀ → the pass-rate change is not statistically significant, despite 9 candidates improving against only 2 regressing. With this few discordant pairs, the uncorrected test (p = .035) overstates the evidence.
3.27
χ² (corrected)
.070
p (cc)
4.5
OR (b/c)
20
N
🟢 Not significant (corrected) — χ²(1) = 3.27, p = .070. The uncorrected χ²(1) = 4.46, p = .035 overstates the evidence for change.
APA-7
A McNemar's test with continuity correction indicated no statistically significant change in certification pass rate following training, χ²(1) = 3.27, p = .070 (OR = 4.50). The uncorrected test, χ²(1) = 4.46, p = .035, would have overstated this evidence.

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