Triangular systems for symmetric binary variables

We introduce and study distributions of sets of binary vari- ables that are symmetric, that is each has equally probable levels. The joint distribution of these special types of binary variables, if generated by a recursiveprocess of linear main e!ects is essentially par ametrized in terms of marginal correlations. This contrasts with the log-linear formulation of joint probabilities in which parameters measure conditional associations given all remaining variables. The new formulation permits useful compar- isons of di!erent types of graphical Markov models and leads to a close approximation of Gaussian orthant probabilities.

[1]  Stephen E. Fienberg,et al.  The Analysis of Multidimensional Contingency Tables , 1970 .

[2]  M. Bartlett.  Contingency Table Interactions , 1935 .

[3]  N. Wermuth,et al.  Partial inversion for linear systems and partial closure of independence graphs , 2006 .

[4]  K. R. Johnson,et al.  A GENERALIZATION OF A RESULT OF , 2008 .

[5]  B. Streitberg.  Lancaster Interactions Revisited , 1990 .

[6]  D. Cox.  On a generalization of a result of W. G. Cochran , 2007 .

[7]  Nanny Wermuth,et al.  A general condition for avoiding effect reversal after marginalization , 2003 .

[8]  B. Streitberg.  Exploring interactions in high-dimensional tables: a bootstrap alternative to log-linear models , 1999 .

[9]  J. McFadden.  URN Models of Correlation and a Comparison with the Multivariate Normal Integral , 1955 .

[10]  J. Darroch,et al.  Generalized Iterative Scaling for Log-Linear Models , 1972 .

[11]  M. C. Cheng.  The Orthant Probabilities of Four Gaussian Variates , 1969 .

[12]  Arthur E. Miller,et al.  A Multiple Integral , 1961 .

[13]  N. Wermuth,et al.  Matrix representations and independencies in directed acyclic graphs , 2009, 0904.0333.

[14]  G. M. Marchetti,et al.  Chain graph models of multivariate regression type for categorical data , 2009, 0906.2098.

[15]  F. Yates.  Design and Analysis of Factorial Experiments , 1958 .

[16]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[17]  N. Wermuth,et al.  Joint response graphs and separation induced by triangular systems , 2004 .

[18]  N. Wermuth.  Linear Recursive Equations, Covariance Selection, and Path Analysis , 1980 .

[19]  W. G. Cochran.  The Omission or Addition of an Independent Variate in Multiple Linear Regression , 1938 .

[20]  Nanny Wermuth,et al.  Multivariate Dependencies: Models, Analysis and Interpretation , 1996 .

[21]  Leo A. Goodman,et al.  The analysis of multidimensional contingency tables when some variables are posterior to others: a modified path analysis approach , 1973 .

[22]  N. Wermuth.  Analogies between Multiplicative Models in Contingency Tables and Covariance Selection , 1976 .

[23]  Nanny Wermuth,et al.  A note on the quadratic exponential binary distribution , 1994 .

[24]  B. Qaqish,et al.  Multivariate logistic models , 2006 .

[25]  R. A. FISHER,et al.  The Design and Analysis of Factorial Experiments , 1938, Nature.

[26]  W. R. Buckland,et al.  Distributions in Statistics: Continuous Multivariate Distributions , 1974 .

[27]  Stephen E. Fienberg,et al.  The analysis of cross-classified categorical data , 1980 .

[28]  P. Rosenbaum,et al.  Conditional Association and Unidimensionality in Monotone Latent Variable Models , 1985 .

[29]  Z. Geng,et al.  Collapsibility of distribution dependence , 2006 .

[30]  The numerical evaluation of a class of integrals , 1956 .

[31]  M. W. Birch.  Maximum Likelihood in Three-Way Contingency Tables , 1963 .

[32]  N. Wermuth,et al.  Covariance chains , 2006 .

[33]  W. Sheppard.  On the geometrical treatment of the 'normal curve' of statistics, with especial reference to correlation and to the theory of error , 1898, Proceedings of the Royal Society of London.

[34]  M. Frydenberg.  The chain graph Markov property , 1990 .

[35]  S. Orbom,et al.  When Can Association Graphs Admit A Causal Interpretation? , 1993 .

[36]  B. Everitt.  Log-linear models for contingency tables , 1992 .

[37]  M. Drton.  Discrete chain graph models , 2009, 0909.0843.

[38]  David R. Cox,et al.  PRINCIPLES OF STATISTICAL INFERENCE , 2017 .

[39]  Nanny Wermuth,et al.  On association models defined over independence graphs , 1998 .

[40]  M. Frydenberg.  Multivariate Dependencies. Models, Analysis and Interpretation. D. R. Cox and N. Wermuth, Chapman & Hall, London, 1996, No. of pages: 272. Price: £39. ISBN 0‐412‐75410‐X , 2000 .

[41]  M. Manser,et al.  Chi-Squared Distribution , 2010 .

[42]  S. Wright.  The Method of Path Coefficients , 1934 .

[43]  An approximation for the symmetric, quadrivariate normal integral , 1956 .

[44]  Irving John Good,et al.  The Interaction Algorithm and Practical Fourier Analysis , 1958 .