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# Mc Nemar

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### Mc Nemar

1. 1. Comparing Correlated Proportions With McNemar’s Test Suppose you are evaluating the effectiveness of an intervention which is designed to make patients more likely to comply with their physicians’ prescriptions. Prior to introducing the intervention, each of 200 patients is classified as compliant or not. Half (100) are compliant, half (100) are not. After the intervention you reclassify each patient as compliant (120) or not (80). It appears that the intervention has raised compliance from 50% to 65%, but is that increase statistically significant? McNemar’s test is the analysis traditionally used to answer a question like this. To conduct this test you first create a 2 x 2 table where each of the subjects is classified in one cell. In the table below, cell A includes the 45 patients who were compliant at both times, cell B the 55 who were compliant before the intervention but not after the intervention, cell C the 85 who were not compliant prior to the intervention but were after the intervention, and cell D the 15 who were noncompliant at both times. After the Intervention Compliant (1) Not (0) Marginals Prior to the Compliant (1) 45 A 55 B 100 Intervention Not (0) 85 C 15 D 100 Marginals 130 70 200 McNemar’s Chi-square, with a correction for continuity, is computed this way: (| b − c | −1)2 (| 55 − 85 | −1)2 χ = 2 . For the data above, χ = 2 = 6.007 . The chi-square is b+c 55 + 85 evaluated on one degree of freedom, yielding, for these data, a p value of .01425. If you wish not to make the correction for continuity, omit the “−1.” For these data that would yield a chi-square of 6.429 and a p value of .01123. I have not investigated whether or not the correction for continuity provides a better approximation of the exact (binomial) probability or not, but I suspect it does. McNemar Done as an Exact Binomial Test Simply use the binomial distribution to test the null hypothesis that p = q = .5 where the number of successes is either count B or count C from the table and N = B + C. For our data, that is, obtain the probability of getting 55 or fewer failures in 85 + 55 = 140 trials of binomial experiment when p = q = .5. The syntax for doing this with SPSS is COMPUTE p=2*CDF.BINOM(55,140,.5). EXECUTE. and the value computed is .01396.
2. 2. McNemar Done With SAS Data compliance; Input Prior After Count; Cards; 1 1 45 1 0 55 0 1 85 0 0 15 Proc Freq; Tables Prior*After / Agree; Weight Count; run; Statistics for Table of Prior by After McNemar's Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Statistic (S) 6.4286 DF 1 Asymptotic Pr > S 0.0112 Exact Pr >= S 0.0140 The “Statistic” reported by SAS is the chi-square without the correction for continuity. The “Asymptotic Pr” is the p value for that uncorrected chi-square. The “Exact Pr” is an exact p value based on the binomial distribution. McNemar Done With SPSS
3. 3. Analyze, Descriptive Statistics, Crosstabs. Prior into “Rows” and After into “Columns.” Click “Statistics” and select “McNemar.” Continue, OK Prior * After Crosstabulation Count After 0 1 Total Prior 0 15 85 100 1 55 45 100 Total 70 130 200 Chi-Square Tests Exact Sig. (2- Value sided) McNemar Test .014a N of Valid Cases 200 a. Binomial distribution used.
4. 4. Notice that SPSS does not give you a chi-square approximate p value, but rather an exact binomial p value. McNemar Done With Vasser Stats Go to http://faculty.vassar.edu/lowry/propcorr.html Enter the counts into the table and click “Calculate.” Very nice, even an odds ratio with a confidence interval. I am impressed.
5. 5. More Than Two Blocks The design above could be described as one-way, randomized blocks, two levels of the categorical variable (prior to intervention, after intervention). What if there were more than two levels of the treatment – for example, prior to intervention, immediately after intervention, six months after intervention. An appropriate analysis here might be the Cochran test. See http://en.wikipedia.org/wiki/Cochran_test. See also McNemar Tests of Marginal Homogeneity at http://ourworld.compuserve.com/homepages/jsuebersax/mcnemar.htm . Return to Wuensch’s Stats Lessons Karl L. Wuensch, July, 2008