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

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Stratified McNemar Tests C. Mitchell Dayton University of Maryland

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Stratified McNemar Tests

C. Mitchell Dayton

University of Maryland

Table 1

Theoretic Proportions for 2X2 Table

McNemar Statistic computed from 2x2 table

DF = 1

Correction for continuity is available

McNemar chi-square is equivalent to goodness-of-fit

chi-square computed from the table below.

C-Class Latent-Class Model

is a latent class proportion

is a conditional probability for an item

Expected cell probabilities for an unconstrained two-class

latent class model

+ coded as 1

- coded as 2

Model for 2x2 Table: Unrestricted

Model for 2x2 Table: Restricted = Proctor Error Model

Expected cell probabilities for a constrained two-class

latent class model

+ coded as 1

- coded as 2

Class 1 = {+,+}

Class 2 = {-,-}

Maximum Likelihood Estimates

This model yields the same expected frequencies, DF,

and chi-square goodness-of-fit statistic as the McNemar test

Same restricted latent class model written conditional

on grouping on basis of manifest variable, y

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”

Homogeneous subsets of years for fitted models

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

X|Y

D|XY eq2

H|XY 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]

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

X|Y

D|XY eq2

H|XY 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]

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

X|Y

D|XY eq2

H|XY 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]

References

Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In B.N. Petrov and F. Csake (eds.),

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.