MCNEMAR TEST

MCNEMAR TEST

Analysis Command

Perform a McNemar test for independence in a 2x2 table.

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 (X_i,Y_i) are mutually independent.

2. Each X_i and Y_i can be assigned to one of two possible categories.

3. The difference
   P(X_i = 0, Y_i = 1) - P(X_i = 1, Y_i = 0)

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

If we let P₁ = P(X_i = 0, Y_i = 1) and P₂ = P(X_i = 1, Y_i = 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

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).

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.

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.

MCNEMAR TEST Y1 Y2
MCNEMAR TEST M
MCNEMAR TEST N11 N12 N21 N22

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).

Dataplot saves the following internal parameters:
STATVAL = the value of the McNemar test statistic
STATCDF = the cdf for the McNemar test statistic

None

None

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).

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.

Categorical Data Analysis

2007/3

 
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