Non-Normal Bivariate Distributions with Normal Marginals

The purpose of this note is to indicate that Fieller's Theorem can be expressed in the matrix formulation of the general linear model. The practical consequence is that one general computer program which can estimate the parameters and test the validity of a pertinent model, can also compute confidence limits for the ratios of any linear combinations of the parameters.     

Understanding increments in model performance metrics

The area under the receiver operating characteristic curve (AUC) is the most commonly reported measure of discrimination for prediction models with binary outcomes. However, recently it has been criticized for its inability to increase when important risk factors are added to a baseline model with good discrimination. This has led to the claim that the reliance on the AUC as a measure of discrimination may miss important improvements in clinical performance of risk prediction rules derived from a baseline model. In this paper we investigate this claim by relating the AUC to measures of clinical performance based on sensitivity and specificity under the assumption of multivariate normality. The behavior of the AUC is contrasted with that of discrimination slope. We show that unless rules with very good specificity are desired, the change in the AUC does an adequate job as a predictor of the change in measures of clinical performance. However, stronger or more numerous predictors are needed to achieve the same increment in the AUC for baseline models with good versus poor discrimination. When excellent specificity is desired, our results suggest that the discrimination slope might be a better measure of model improvement than AUC. The theoretical results are illustrated using a Framingham Heart Study example of a model for predicting the 10-year incidence of atrial fibrillation.     

INFLUENCE FUNCTIONS IN FUNCTIONAL MEASUREMENT ERROR MODELS WITH REPLICATED DATA

We consider the construction and properties of influence functions in the context of functional measurement error models with replicated data. In these models estimates of the parameters can be affected both by the individual observations and the means of replicated observations. We show that influence function of the means of replicates on the estimate of regression coefficients can be only derived under the assumption that the variances of the errors are known, while one for the individual observations can be only derived simultaneously with their influence function on the estimators of the variances of the errors.     

Geometric characterization of planar regression

The geometric characterization of linear regression in terms of the ‘concentration ellipse’ by Galton [Galton, F., 1886, Family likeness in stature (with Appendix by Dickson, J.D.H.). Proceedings of the Royal Society of London, 40, 42–73.] and Pearson [Pearson, K., 1901, On lines and planes of closest fit to systems of points in space. Philosophical Magazine, 2, 559–572.] was extended to the case of unequal variances of the presumably uncorrelated errors in the experimental data [McCartin, B.J., 2003, A geometric characterization of linear regression. Statistics, 37(2), 101–117.]. In this paper, this geometric characterization is further extended to planar (and also linear) regression in three dimensions where a beautiful interpretation in terms of the concentration ellipsoid is developed.     

A note on ∞manque in variance covariance components models

Stimualted by C.R. Rao's MINQUE J. Focke and G. Dewess introduced the so called r-and ∞ MINQUE. Although they developed a unique charecterization of ∞-MINQUE, they did not give explicite formulas for its computation. The goal this paper is to close this lack and to etend the concept to more general models.     

Book reviews

S.M.Kendall:Multivariate Analysis.Charles Griffin & Co. Ltd., London and High Wycombe 1975, 210 pp

C.T.Leondes (ed.):Control and Dynamic Systems, Advances in Theory and Applications. Vol. 11, Academic Press, New York and London 1974, 516 pp., $ 24.50.

CH. R.Nelson:Applied Time Series Analysis for Managerial Forecasting. Holden Day, Inc., San Francisco 1973, 231 pp., $ 14.95.

B.DE Finetti:Theory of Probability. Vol. 1, 2, John Wiley & Sons, New York, London, Sydney, Toronto. Vol. 1, 1974, 300 pp., £ 7.50 - Vol. 2 1975, 375 pp., £ 10.50.

P.Erdördos, J. Spencer:Probabilistic Methods in Combinatorics. Akadémic Press, New York and London; Akadémiai Kiadó, Budapest 1974, 106 pp., $ 11.75.

J.S.R.Ustagi:Variational Methods in Statistics.Academic Press, New York and London 1975.

J.S.Rustagi:Optimizing Methods in Statistics.Academic Press, New York and London 1971,488 pp., $ 17.00.

Karl V. Bury:Statistical Models in Applied Science. John Wiley & Sons, New York-London-Sydney-Toronto 1975,.625 pp., £ 15.60; $ 28.00.

Michael R. Anderberg:Cluster Analysis for Applications. Academic Press, New York-San Francisco-London 1973, 359 pp., $ 27,–.

J.L.Fleiss:Statistical Methods for Rates and Proportions. John Wiley & Sons, New York-London-Sydney-Toronto 1973. 223 pp., £ 6.50.

J.Tanur et al. (Ed.):Statistics:A Guide to the Unknown. Holden Day, Inc., San Francisco 1972, 430 pp.

H.VÁliaho, T. Pekkonen:A Procedure for Stepwise Regression Analysis.Akademie-Verlag, Berlin 1976, 90 pp., 18,– M.

M.Reinfeldt, U. TrÁnkle:Signifikanztabellen statistischer Testvertellungen. R. Oldenbourg Verlag, Mnchen, Wien 1976, 151 S., DM 44,–.

R.E.Barlow, D.J.Batholomew, J.M.Bremner, H.D.Brunk:Statistical Inference Under Order Restrictions.(The Theory and Applications of Isotonic Regression.) John Wiley & Sons, New York 1972, 388 pp., £ 7.50.

H.J.Larson:Introduction to Probability Theory and Statistical Inference. Wiley, New York 1974, 430 pp., £ 6.85.

R.A.Carlson:Statistics. Holden Day, Inc., San Francisco 1973, 393 pp.

E. Page:Queueing Theory in OR. Butterworths, London 1972, 187 pp., £ 3.60.

H.Krampe, J.Kubat, W.Runge:Bedienungsmodelle. Ein Leitfaden für die praktische Anwendung, Verlag Die Wirtachaft, Berlin 1973, 512 S., 79,– M.

G.S.Fishman:Concepts and Methods in Discrete Events Digital Simulation. John Wiley & Sons, New York 1973, 385 pp., £ 8,75.     

Book reviews

E. Kofler, G. Menges: Entscheidungen bei unvollständiger Information. Lecture Notes in Economics and Mathematical Systems 136. Springer-Verlag, Berlin-Heidelberg-NewYork 1976, 357 S., DM 31.

B. S. Everitt: The Analysis ot Contingency Tables. Chapman and Han, London 1977, 128 S., £ 3.75.

O. Barndorff-Nielsen: Information and Exponential Families in Statistical Theory. J. Wiley & Sons, Chichester-New York-Brisbane-Toronto 1978, 247 S., £ 13.50; $ 28.50.

W. Gilchrist: Statisticai Forecasting. J. Wiley & Sons, London-New York-Sydney-Toronto 1976, 321 S., £ 9.50; $ 19.00.

J. M. Chambers: Computational Methods for Data Analysis. J. Wiley & Sons, New York Chichester-Brisbane-Toronto 1977, 279 pp., £ 11.30.

A. Hughes, D. Grawoig: Statistics: A Foundation for Analysis. Addison-Wesley Publishing Company, Reading-Menlo-Park London-Don Mills 1971, 525 S., $ 11.50.

K.Krickeberg, H.Ziezold: Stochastiche Methoden. Springer Verlag, Berlin-Heidelberg-New York 1977, 201 S., 13 Abb., DM 28.

Ch. Schneeweiss: Inventory-Production Theory. A Linear Policy Approach. Lecture Notes in Economics and Mathematical Systems 151. Springer Verlag, Berlin-Heidelberg-NewYork 1977, 122 S., 13 Abb., 9 Tab., DM 18.

S. Dworatschek: Grundlagen der Datenverarbeltung, 6., völlig neu bearb. u. erw. Aufl., Walter de Gruyter, Berlin-New York 1977, 538 S., 200 Abb., 212 Üb.Aufg., 59 Fotos, DM 38.

H. E. Steinhagen, S. Fuchs:Objekterkennung. Einführung in die mathematischen Methoden der Zeichenerkennung. VEB Verlag Technik, Berlin 1976, 436 S., 165 Abb., 32 Tab., 49,–.M.

G. Tinhofer: Mathematik für Studlenanfänger, Carl Hanser Verlag, München 1977, 464 S., 191 Abb., DM 38.     

Stereological formulas for stationary surface processes

In the paper surface processes are considered, i.e. sets of surfaces distributed at random in the space. For translation invariant surface processes the PALM distribution of the direction of the normal in a typical surface point is defined and related to two roses of intersection. Fibre and point processes are studied which result by intersections with planes and lines respectively. For these processes stereological formulas are proved which connect some quantities with such of the surface process. As a special ease motion invariant, i.e. translation invariant and isotropic, surface processes are treated, Furthermore, weighted surface processes are considered.