Note 17
Paired and Matched Binary Data
The data here are from a clinical trial of an allergy medication. Participants recorded use of decongestants before starting on study medication and then after using study medication for four weeks.
deco0 records usage (0/1) before medication, deco1 after medication.
[17.1] . use hepp-patient, clear
. list
+--------------------------+
| id
deco0
deco1
n |
|--------------------------|
1. | 1
0
0
237 |
2. | 2
0
1
57 |
3. | 3
1
0
71 |
4. | 4
1
1
14 |
+--------------------------+
The first command below shows an incorrect analysis, since the data are actually paired. (That
is, the test for independence here takes no note of the fact that there are actually 379 pairs of
measurements represented here.)
The correct analysis uses McNemar’s test, which looks only at the diagonals of the 2 × 2 table,
the only cells which are informative about how the before and after probabilities differ.
[17.2] . ta deco0 deco1 [fw=n], chi2 exact
** WRONG **
Decongesta |
nt before | Decongestant after rx
rx |
0
1 |
Total
-----------+----------------------+---------0 |
237
57 |
294
1 |
71
14 |
85
-----------+----------------------+---------Total |
308
71 |
379
Pearson chi2(1) =
Fisher’s exact =
1-sided Fisher’s exact =
0.3686
Pr = 0.544 ** WRONG **
0.637 ** WRONG **
0.332 ** WRONG **
The following commands do a correct analysis. The first does McNemar’s test using the exact binomial probabilities, while the next two use the normal approximation. When the z value computed
below is squared, it is very close to McNemar’s χ2 test, which is calculated in the last display.
1
NOTE 17. PAIRED AND MATCHED BINARY DATA
2
[17.3] . bitesti 128 71 .5
N
Observed k
Expected k
Assumed p
Observed p
-----------------------------------------------------------128
71
64
0.50000
0.55469
Pr(k >= 71)
= 0.125220
Pr(k <= 71)
= 0.907659
Pr(k <= 57 or k >= 71) = 0.250440
(one-sided test)
(one-sided test)
(two-sided test)
[17.4] . display "z = " (71/128 - 0.5)/sqrt(71*57/(128*128*128))
z = 1.2449056
[17.5] . display (1.2449056)^2 "
1.54979
.21316645
" 2*(1-normprob(1.2449056))
[17.6] . display "McNemar’s chi-squared statistic = " ((71-57)^2)/(71+57)
McNemar’s chi-squared statistic = 1.53125
The mcc command analyzes a matched case-control study, and is the way to calculate McNemar’s
test for a paired binomial data set in Stata. Unfortunately, one must reinterpret the labels that
Stata uses, as this is obviously not a case-control study. You can think of treatment status as
defining case vs. control, so that cases correspond to the “after” measurement and controls to the
“before.”
[17.7] . mcc deco1 deco0 [fw=n]
| Controls
|
Cases
|
Exposed
Unexposed |
Total
-----------------+------------------------+-----------Exposed |
14
57 |
71
Unexposed |
71
237 |
308
-----------------+------------------------+-----------Total |
85
294 |
379
McNemar’s chi2(1) =
1.53
Prob > chi2 = 0.2159
Exact McNemar significance probability
= 0.2504
Proportion with factor
Cases
.1873351
Controls
.2242744
--------difference -.0369393
ratio
.8352941
rel. diff. -.047619
odds ratio
.8028169
[95% Conf. Interval]
--------------------.0979673
.0240887
.6278769
1.111231
-.1248173
.0295792
.5564015
1.153877
(exact)
NOTE 17. PAIRED AND MATCHED BINARY DATA
3
More generally, we could have several “controls” for each “case”, in which case the approach
above cannot readily be used. The clogit command does conditional logistic regression, which is a
generalization of McNemar’s test.
The data need to be in long format for this command, so we start by having Stata reshape the
data set.
[17.8] . reshape long deco, i(id) j(after)
(note: j = 0 1)
Data
wide
->
long
----------------------------------------------------------------------------Number of obs.
4
->
8
Number of variables
4
->
4
j variable (2 values)
->
after
xij variables:
deco0 deco1
->
deco
----------------------------------------------------------------------------[17.9] . list, sep(0)
1.
2.
3.
4.
5.
6.
7.
8.
[17.10]
+-------------------------+
| id
after
deco
n |
|-------------------------|
| 1
0
0
237 |
| 1
1
0
237 |
| 2
0
0
57 |
| 2
1
1
57 |
| 3
0
1
71 |
| 3
1
0
71 |
| 4
0
1
14 |
| 4
1
1
14 |
+-------------------------+
. clogit after deco [fw=n], group(id) nolog or
Conditional (fixed-effects) logistic regression
Log likelihood = -261.93562
Number of obs
LR chi2(1)
Prob > chi2
Pseudo R2
=
=
=
=
758
1.53
0.2155
0.0029
-----------------------------------------------------------------------------after | Odds Ratio
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------deco |
.8028169
.1427759
-1.23
0.217
.5665467
1.13762
------------------------------------------------------------------------------
NOTE 17. PAIRED AND MATCHED BINARY DATA
4
There is yet one more way to carry out these calculations. McNemar’s test can be thought of as a
test of symmetry in the 2 × 2 table about the main diagonal. Sometimes we have paired observations
of ordered categories. In the example below, antihistamine use before and after treatment is recorded
in patients receiving an allergy treatment.
[17.11]
. use hepp-anti-full, clear
[17.12]
. symmetry before after [fw=n]
----------------------------------------------------------------------|
after
before |
None
Occasional
Daily low Daily full
Total
-----------+----------------------------------------------------------None |
199
56
8
5
268
Occasional |
34
16
2
0
52
Daily low |
15
6
3
1
25
Daily full |
25
7
2
4
38
|
Total |
273
85
15
10
383
----------------------------------------------------------------------chi2
df
Prob>chi2
-----------------------------------------------------------------------Symmetry (asymptotic)
|
30.17
6
0.0000
Marginal homogeneity (Stuart-Maxwell) |
30.11
3
0.0000
-----------------------------------------------------------------------In this case, considerable information is lost if the data are simply reduced to antihistamine use
(any or none). In the 2 × 2 case, both symmetry tests are equal to the McNemar’s χ2 test.
[17.13]
. symmetry antibefore antiafter [fw=n], exact
------------------------------antibefor |
antiafter
e
|
0
1
Total
----------+-------------------0 | 199
69
268
1 |
74
41
115
|
Total | 273
110
383
------------------------------chi2
df
Prob>chi2
-----------------------------------------------------------------------Symmetry (asymptotic)
|
0.17
1
0.6759
Marginal homogeneity (Stuart-Maxwell) |
0.17
1
0.6759
-----------------------------------------------------------------------Symmetry (exact significance probability)
0.7381