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Dataplot Vol 1 Auxillary Chapter

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 > CHIPPF(1-alpha,1)

      T2 ≤ BINPPF(alpha/2,0.5,b+c)
      T2 ≥ BINPPF(alpha/2,0.5,b+c)

      where CHIPPF(alpha,nu) 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: 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
    .
    read matrix m
    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
        

Date created: 1/15/2008
Last updated: 1/15/2008
Please email comments on this WWW page to alan.heckert@nist.gov.