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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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CellObserved FreqExpected Freq 1+,2- 1-,2+ McNemar chi-square is equivalent to goodness-of-fit chi-square computed from the table below.

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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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2+ 2- 1+ 1- Expected cell probabilities for an unconstrained two-class latent class model + 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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2+ 2- 1+ 1- Expected cell probabilities for a constrained two-class latent class model + 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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“She is married and does not want any more children” “She is not married and does not want to marry the man” Exemplary analyses for two abortion items from GSS for six years: 1993 – 1998 Sample sizes varied from 856 to 1750

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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 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]

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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 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]

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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 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]

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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.

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