Stratified McNemar Tests

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Stratified McNemar Tests C. Mitchell
Dayton University of Maryland
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Table 1 Theoretic
Proportions for 2X2 Table
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McNemar Statistic computed from 2x2 table DF
1 Correction for continuity is available
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McNemar chi-square is equivalent to
goodness-of-fit chi-square computed from the
table below.
Cell Observed Freq Expected Freq
1,2-
1-,2

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C-Class Latent-Class Model
is a latent class proportion
is a conditional probability for an item
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Expected cell probabilities for an unconstrained
two-class latent class model
2 2-
1
1-
coded as 1 - coded as 2
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Model for 2x2 Table Unrestricted
Model for 2x2 Table Restricted Proctor Error
Model
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Expected cell probabilities for a constrained
two-class latent class model
2 2-
1
1-
coded as 1 - coded as 2
Class 1 , Class 2 -,-
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Maximum Likelihood Estimates
This model yields the same expected frequencies,
DF, and chi-square goodness-of-fit statistic as
the McNemar test
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Same restricted latent class model written
conditional on grouping on basis of manifest
variable, y
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Exemplary analyses for two abortion items
from GSS for six years 1993 1998 Sample
sizes varied from 856 to 1750
She is married and does not want any more
children She is not married and does not want
to marry the man
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(No Transcript)
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Homogeneous subsets of years for fitted models
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LEM input file for Homogeneous model Six years
of abortion data Item No More, Not Married
Stratified McNemar test Homogeneous Model lat
1 man 3 dim 2 6 2 2 lab X Y D H X latent
variable Y year D No More H Not Married
mod Y XY DXY eq2
HXY eq2 des 0 2 0 2 0 2 0 2 0 2 0 2 2 0 2 0 2
0 2 0 2 0 2 0 0 2 0 2 0 2 0 2 0 2 0 2 2
0 2 0 2 0 2 0 2 0 2 0 dat 342 45 47 422 376 42
41 475 429 43 44 476 829 90 78 903 725
109 75 867 672 68 69 941
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LEM input file for Heterogeneous model Six
years of abortion data Item No More, Not
Married Stratified McNemar test
Heterogeneous Model lat 1 man 3 dim 2 6 2 2 l
lab X Y D H X latent variable Y year D
No More H Not Married mod Y XY
DXY eq2 HXY eq2 des 0 2 0 4
0 6 0 8 0 10 0 12 2 0 4 0 6 0 8 0 10 0 12 0
0 2 0 4 0 6 0 8 0 10 0 12 2 0 4 0 6 0 8 0 10 0
12 0 dat 342 45 47 422 376 42 41 475 429 43 44
476 829 90 78 903 725 109 75 867 672 68
69 941
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LEM input file for Part Heterogeneous C model
Six years of abortion data Item No More, Not
Married Stratified McNemar test Part
Heterogeneous Model C lat 1 man 3 dim 2 6 2
2 lab X Y D H X latent variable Y year D
No More H Not Married mod Y XY
DXY eq2 HXY eq2 des 0 2 0
4 0 4 0 4 0 2 0 6 2 0 4 0 4 0 4 0 2 0 6 0
0 2 0 4 0 4 0 4 0 2 0 6 2 0 4 0 4 0 4 0 2 0 6
0 dat 342 45 47 422 376 42 41 475 429 43 44
476 829 90 78 903 725 109 75 867 672 68
69 941
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References Akaike, H. (1973). Information theory
and an extension of the maximum likelihood
principle. In B.N. Petrov and F. Csake (eds.),
Second International Symposium on Information
Theory. Budapest Akademiai Kiado,
267-281. Bishop, Y. M. M., Fienberg, S. E.
Holland, P. W. (1975) Discrete Multivariate
Analysis Theory and Practice, Cambridge MIT
Press Dayton, C. M. (1999) Latent Class Scaling
Analysis. Sage Publications. Dayton, C. M.
Macready, G. B. (1983) Latent structure analysis
of repeated classifications with dichotomous
data. British Journal of Mathematical
Statistical Psychology, 36, 189-201. Fleiss, J.
L. (1981) Statistical Methods for Rates and
Proportions. New York Wiley Haberman, S. J.
(1979), Analysis of Qualitative Data, Volume 2
New Developments, New York Academic Press.
McNemar Q. (1947) Note on the sampling error of
the difference between correlated proportions or
percentages. Psychometrika, 12, 153-157. Maxwell
A. E. (1970) Comparing the classification of
subjects by two independent judges. British
Journal of Psychiatry, 116, 651-655. Schwarz, G.
(1978). Estimating the dimension of a model.
Annals of Statistics, 6, 461-464. Stuart A. A.
(1955) A test for homogeneity of the marginal
distributions in a two-way classification.
Biometrika, 42, 412-416. Vermunt, J. K. (1993).
Log-linear event history analysis with missing
data using the EM algorithm. WORC Paper,
Tilburg University, The Netherlands.