Dataplot Vol 1 Vol 2

# MCNEMAR TEST

Name:
MCNEMAR TEST
Type:
Analysis Command
Purpose:
Perform a McNemar test for independence in a 2x2 table.
Description:
Given two paired variables where each variable has exactly two possible outcomes (coded as 0 and 1), the McNemar test can be used to test if there is a statistically significant difference between the probability of a (0,1) pair and the probability of a (1,0) pair. For example, this test is often used for the situation where we are testing for the prescence (= 1) or absence (= 0) of something and variable 1 is the state before an experiment and variable 2 is the state after the experiment (i.e., did the experiment have an effect?).

We can summarize the data in the following table. We call variable 1 X and variable 2 Y.

Yi = 0 Yi = 1

Xi = 0 a b
Xi = 1 c d

The McNemar test has the the following assumptions:

1. The pairs (Xi,Yi) are mutually independent.

2. Each Xi and Yi can be assigned to one of two possible categories.

3. The difference

P(Xi = 0, Yi = 1) - P(Xi = 1, Yi = 0)

is negative for all i or zero for all i or positive for all i.

If we let P1 = P(Xi = 0, Yi = 1) and P2 = P(Xi = 1, Yi = 0), then the McNemar test can be formulated as follows.

 H0: P1 = P2       for all i (this is equivalent to "new equal to old" or "before equal to after") Ha: P1 ≠ P2       for all i (this is equivalent to "new not equal to old" or "before not equal to after") Test Statistic: If b + c > 20, T1 = (b - c)2/(b + c) If b + c ≤ 20, T2 = b There is also a continuity corrected version of the T1: T1' = (|b - c| - 1)2/(b + c) Significance Level: $$\alpha$$ Critical Region: T1 > $$\chi_{1-\alpha,1}^{2}$$ T2 ≤ BINPPF($$\alpha/2$$,0.5,b+c) T2 ≥ BINPPF($$\alpha/2$$,0.5,b+c) where $$\chi_{\alpha,\nu}^{2}$$ and BINPPF denote the percent point functions of the chi-square and binomial distributions, respectively Conclusion: Reject the null hypothesis if the test statistic is in the critical region
Syntax 1:
MCNEMAR TEST <y1> <y2>             <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
<y2> is the second response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax is used for the raw data case (i.e., <y1> and <y2> are variable containing 0's and 1's).

Syntax 2:
MCNEMAR TEST <m>             <SUBSET/EXCEPT/FOR qualification>
where <m> is a matrix containing the two-way table;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax is used for the case where we the data have already been cross-tabulated into a two-way table.

Syntax 3:
MCNEMAR TEST <n11> <n12> <n21> <n22>
where <n11> is a parameter containing the value for row 1, column 1 of a 2x2 table (i.e, a);
<n12> is a parameter containing the value for row 1, column 2 of a 2x2 table (i.e., b);
<n21> is a parameter containing the value for row 2, column 1 of a 2x2 table (i.e., c);
<n22> is a parameter containing the value for row 2, column 2 of a 2x2 table (i.e., d).

This syntax is used for the special case where you have a 2x2 table. In this case, you can enter the 4 values directly, although you do need to be careful that the parameters are entered in the order expected above.

Examples:
MCNEMAR TEST Y1 Y2
MCNEMAR TEST M
MCNEMAR TEST N11 N12 N21 N22
Note:
The McNemar test is essentially a sign test. Conover discusses how to transform a McNemar test to an explicit sign test.

The McNemar test is also a special case of the Cochran test with c = 2 (enter HELP COCHRAN TEST for details).

Note:
Dataplot saves the following internal parameters:

STATVAL = the value of the McNemar test statistic
STATCDF = the cdf for the McNemar test statistic
Default:
None
Synonyms:
None
Related Commands:
 SIGN TEST = Perform a sign test. COCHRAN TEST = Perform a Cochran test. CHI-SQUARE INDEPENDENCE TEST = Perform a chi-square test for independence. FISHER EXACT TEST = Perform Fisher's exact test. ASSOCIATION PLOT = Generate an association plot. SIEVE PLOT = Generate a sieve plot. ROSE PLOT = Generate a Rose plot. BINARY TABULATION PLOT = Generate a binary tabulation plot. ROC CURVE = Generate a ROC curve. ODDS RATIO = Compute the bias corrected odds ratio. LOG ODDS RATIO = Compute the bias corrected log(odds ratio).
Reference:
Conover (1999), Practical Nonparametric Statistics, Third Edition, Wiley, pp. 166-169.

Fleiss, Levin, and Paik (2003), Statistical Methods for Rates and Proportions, Third Edition, p. 375.

Applications:
Categorical Data Analysis
Implementation Date:
2007/3
Program:

let n11 = 63
let n21 = 4
let n12 = 21
let n22 = 12
.
2 4
4 6
end of data
.
mcnear test n11 n21 n12 n22
mcnemar test m

The following output is generated.
           MCNEMAR TEST FOR NEW BETTER THAN OLD (2X2 TABLE)

NULL HYPOTHESIS: NEW AND OLD ARE EQUAL
ALTERNATIVE HYPOTHESIS: NEW AND OLD ARE NOT EQUAL

|       VARIABLE TWO       |      ROW
VARIABLE ONE  |  SUCCESSES     FAILURES  |      TOTAL
=====================================================
SUCCESSES     |         63           21  |         84
|   0.940298     0.636364  |
FAILURES      |          4           12  |         16
|   0.059701     0.363636  |
=====================================================
COLUMN TOTAL  |         67           33  |        100

LARGE SAMPLE CASE (N >= 20)
(CRITICAL VALUES BASED ON CHI-SQUARE WITH ONE DEGREE OF FREEDOM)

WITHOUT CONTINUITY CORRECTION
VALUE OF TEST STATISTIC                  =    11.56000
CDF OF TEST STATISTIC                    =   0.9993260

WITH CONTINUITY CORRECTION
VALUE OF TEST STATISTIC                  =    10.24000
CDF OF TEST STATISTIC                    =   0.9986256

WITHOUT CONTINUITY CORRECTION
NULL HYPOTHESIS   NULL
NULL          CONFIDENCE    CRITICAL  ACCEPTANCE        HYPOTHESIS
HYPOTHESIS    LEVEL         VALUE     INTERVAL          CONCLUSION
===================================================================
NEW/OLD EQUAL    50.0%        0.45     (0,0.500)        REJECT
NEW/OLD EQUAL    80.0%        1.64     (0,0.800)        REJECT
NEW/OLD EQUAL    90.0%        2.71     (0,0.900)        REJECT
NEW/OLD EQUAL    95.0%        3.84     (0,0.950)        REJECT
NEW/OLD EQUAL    99.0%        6.63     (0,0.990)        REJECT

WITH CONTINUITY CORRECTION
NULL HYPOTHESIS   NULL
NULL          CONFIDENCE    CRITICAL  ACCEPTANCE        HYPOTHESIS
HYPOTHESIS    LEVEL         VALUE     INTERVAL          CONCLUSION
===================================================================
NEW/OLD EQUAL    50.0%        0.45     (0,0.500)        REJECT
NEW/OLD EQUAL    80.0%        1.64     (0,0.800)        REJECT
NEW/OLD EQUAL    90.0%        2.71     (0,0.900)        REJECT
NEW/OLD EQUAL    95.0%        3.84     (0,0.950)        REJECT
NEW/OLD EQUAL    99.0%        6.63     (0,0.990)        REJECT

MCNEMAR TEST FOR NEW BETTER THAN OLD (2X2 TABLE)

NULL HYPOTHESIS: NEW AND OLD ARE EQUAL
ALTERNATIVE HYPOTHESIS: NEW AND OLD ARE NOT EQUAL

|       VARIABLE TWO       |      ROW
VARIABLE ONE  |  SUCCESSES     FAILURES  |      TOTAL
=====================================================
SUCCESSES     |          2            4  |          6
|   0.333333     0.400000  |
FAILURES      |          4            6  |         10
|   0.666667     0.600000  |
=====================================================
COLUMN TOTAL  |          6           10  |         16

SMALL SAMPLE CASE (N < 20)
(CRITICAL VALUES BASED ON BINOMIAL WITH
P = 0.5 AND N =  8)

VALUE OF TEST STATISTIC                  =    4.000000
CDF OF TEST STATISTIC                    =   0.6367188

LOWER     UPPER     NULL HYPOTHESIS         NULL
NULL          CONFIDENCE    CRITICAL  CRITICAL  ACCEPTANCE        HYPOTHESIS
HYPOTHESIS    LEVEL         VALUE     VALUE     INTERVAL          CONCLUSION
=============================================================================
NEW/OLD EQUAL    50.0%        3.00      5.00     (0.250,0.750)        ACCEPT
NEW/OLD EQUAL    80.0%        2.00      6.00     (0.100,0.900)        ACCEPT
NEW/OLD EQUAL    90.0%        2.00      6.00     (0.050,0.950)        ACCEPT
NEW/OLD EQUAL    95.0%        1.00      7.00     (0.025,0.975)        ACCEPT
NEW/OLD EQUAL    98.0%        1.00      7.00     (0.010,0.990)        ACCEPT
NEW/OLD EQUAL    99.0%        1.00      7.00     (0.005,0.995)        ACCEPT


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Date created: 01/15/2008
Last updated: 11/03/2015