Summary
Testing that an edge can be excluded from a graphical Gaussian model is an important step in model fitting and the form of the generalised likelihood ratio test statistic for this hypothesis is well known. Herein the modified profile likelihood test statistic for this hypothesis is obtained in closed form and is shown to be a function of the sample partial correlation. Related expressions are given for the Wald and the efficient score statistics. Asymptotic expansions of the exact distribution of this correlation coefficient under the hypothesis of conditional independence are used to compare the adequacy of the chi-squared approximation of these and Fisher’s Z statistics. While no statistic is uniformly best approximated, it is found that the coefficient of the O(n-1) term is invariant to the dimension of the multivariate Normal distribution in the case of the modified profile likelihood and Fisher’s Z but not for the other statistics. This underlines the importance of adjusting test statistics when there are large numbers of variables, and so nuisance parameters in the model.
Similar comparisons are effected for the Normal approximation to the signed square-rooted versions of these statistics.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Amari, S-I. (1982). Geometrical theory of asymptotic ancillarity and conditional inference. Biometrika, 69, 1–17.
Amari, S-I. (1985). Differential-Geometric Methods in Statistics. Lecture Notes in Statistics 28, Springer-Verlag: Heidelberg.
Barndorf -Nielsen, O.E. (1978). Information and Exponential Families in Statistical Theory. Wiley: New York.
Barndorff Nielsen, O.E. (1983). On a formula for the distribution of the maximum likelihood estimator. Biometrika, 70, 343–365.
Barndorff-Nielsen, O.E. (1986). Inference on full or partial parameters based on the standardized signed log likelihood ratio. Biometrika, 73, 307–322.
Barndorff-Nielsen, O.E. (1988). Parametric Statistical Models and Likelihood. Lecture Notes in Statistics 50, Springer-Verlag: Heidelberg.
Barndorff-Nielsen, O.E. (1990a). A note on the standardised signed log likelihood ratio. Scand. J. Statist., 17 157–160.
Barndorff-Nielsen, O.E. (1990b). Approximate probabilities. J. R. Statist. Soc.B, 52, 485–496.
Barndorff-Nielsen, O.E. and Cox, D.R. (1989). Asymptotic Techniques for Use in Statistics. Chapman and Hall: London.
Cox, D.R. and Hinkley, D. V. (1974). Theoretical Statistics. Chapman and Hall: London.
Cox, D.R. and Reid, N. (1987). Parameter orthogonality and approximate conditional inference, (with discussion). J. R. Statist. Soc. B, 49, 1–39.
Cox, D.R. and Wermuth, N. (1990). An approximation to maximum likelihood estimates in reduced models. Biometrika, 77, 747–761.
Davison, A. C. (1988). Approximate conditional inference in generalised linear models. J. R. Statist. Soc. B, 50, 445–462.
Davison, A. C., Smith, P.W.F. and Whittaker, J. (1991). An exact conditional test for covariance selection. Austral. J. Statist., 33, 313–318.
Deemer, W.L. and O1kin, O. (1951). The jacobians of certain matrix trans-formations useful in multivariate analysis. Biometrika, 38, 345–367.
Dempster, A.P. (1972). Covariance selection. Biometrics, 28, 157–175.
Efron, B. (1978). The geometry of exponential families Ann. Statist., 6, 362–376.
Fisher, R.A. (1970). Statistical Methods for Research Workers, 14th Edition. Hafner Press: New York.
Fraser, D.A.S. (1991). Statistical inference: likelihood to significance. J. Amer. Statist. Soc., 86, 258–265.
Frydenburg, M. and Jensen, J.L. (1989). Is the ‘improved likelihood ratio statistic’ really improved for the discrete case? Biometrika, 76, 655–661.
Graybill, F.A. (1983). Matrices with Applications in Statistics. 2nd Edition. Wadsworth: California.
Harris, P. and Peers, H.W. (1980). The local power of the efficient score test statistic. Biometrika, 67, 525–529.
Hayakawa, T. (1975). The likelihood ratio criterion for a composite hypothesis under a local alternative. Biometrika, 62, 451–460.
Isserlis, L. (1918). On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika, 12, 134–139.
Kreiner, S. (1987). Analysis of multi-dimensional contingency tables by exact conditional tests: techniques and strategies. Scand. J. Stat., 14.
Lauritzen, S.L. (1989). Mixed graphical association models. Scand. J. Statist., 16, 273–306.
McCullagh, P. (1987). Tensor Methods in Statistics. Chapman and Hall: London.
Moran, P. A. P. (1970). On asymptotically optimal tests of composite hypotheses. Biometrika, 57, 45–55.
Muirhead, R.J. (1982). Aspects of Multivariate Statistical Theory. Wiley: New York.
Pierce, D.A. and Peters, D. (1992). Practical use of higher order asymptotics for multiparameter exponential families (with discussion). J. R. Statist. Soc. B, 54, 701–737.
Smith, P.W.F. (1990). Edge Exclusion Tests for Graphical Models. Unpublished Ph.D. thesis. Lancaster University.
Speed, T.P. and Kiiveri, H. (1986). Gaussian Markov distributions over finite graphs. Ann. Statist., 14, 138–150.
Wermuth, N. (1976). Analogies between multiplicative models in contingency tables and covariance selection. Biometrics, 32, 95–108.
Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Wiley: Chichester.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Smith, P.W.F., Whittaker, J. (1998). Edge Exclusion Tests for Graphical Gaussian Models. In: Jordan, M.I. (eds) Learning in Graphical Models. NATO ASI Series, vol 89. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5014-9_21
Download citation
DOI: https://doi.org/10.1007/978-94-011-5014-9_21
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6104-9
Online ISBN: 978-94-011-5014-9
eBook Packages: Springer Book Archive