N. Reid

The Theory of the Design of Experiments

SOME GENERAL CONCEPTS Types of Investigation Observational Studies Some Key Terms Requirements in Design Interplay between Design and Analysis Key Steps in Design A Simplified Model A Broader View AVOIDANCE OF BIAS General Remarks Randomization Retrospective Adjustment for Bias Some More on Randomization More on Causality CONTROL OF HAPHAZARD VARIATION General Remarks Precision Improvement by Blocking Matched Pairs Randomized Block Design Partitioning Sums of Squares Retrospective Adjustment for Improving Precision Special Models of Error Variation SPECIALIZED BLOCKING TECHNIQUES Latin Squares Incomplete Block Designs Cross-Over Designs FACTORIAL EXPERIMENTS: BASIC IDEAS General Remarks Example Main Effects and Interactions Example: Continued Two-Level Factorial Systems Fractional Factorials Example FACTORIAL EXPERIMENTS: FURTHER DEVELOPMENTS General Remarks Confounding in 2k Designs Other Factorial Systems Split Plot Designs Nonspecific Factors Designs for Quantitative Factors Taguchi Methods Conclusion OPTIMAL DESIGN General Remarks Some Simple Examples Some General Theory Other Optimality Criteria Algorithms for Design Construction Nonlinear Design Space-Filling Designs Bayesian Design Optimality of Traditional Designs SOME ADDITIONAL TOPICS Scale of Effort Adaptive Designs Sequential Regression Design Designs for One-Dimensional Error Structure Spatial Designs APPENDIX A: Statistical Analysis APPENDIX B: Some Algebra APPENDIX C: Computational Issues Each chapter also contains Bibliographic Notes plus Further Results and Exercises

Approximations to noncentral distributions

Two methods for approximating the distribution of a noncentral random variable by a central distribution in the same family are presented. The first consists of relating a stochastic expansion of a random variable to a corresponding asymptotic expansion for its distribution function. The second approximates the cumulant generating function and is used to provide central x2 and gamma approximations to the noncentral X2 and gamma distributions.

Estimating the Number of Faults in a System

Abstract We consider a reliability system, with an unknown number of faults, which is observed for a fixed length of time. Failures are assumed to follow an exponential distribution with an unknown mean, and we concentrate on estimation of the number of faults v. The likelihood function, considered as a function of v, has its maximum at one of the boundary points (the observed number of faults or infinity) with substantial probability. Likelihood interval estimates and improved point estimates are discussed. The harmonic mean of the endpoints of the likelihood interval turns out to be a reasonable point estimate. We comment on some extensions of this problem suggested in the software-reliability literature.

Molecular classification of oral cancer by cDNA microarrays identifies overexpressed genes correlated with nodal metastasis

Our purpose was to classify OSCCs based on their gene expression profiles, to identify differentially expressed genes in these cancers and to correlate genetic deregulation with clinical and histopathologic data and patient outcome. After conducting proof‐of‐principle experiments utilizing 6 HNSCC cell lines, the gene expression profiles of 20 OSCCs were determined using cDNA microarrays containing 19,200 sequences and the BTSVQ method of data analysis. We identified 2 sample clusters that correlated with the T3‐T4 category of disease (p = 0.035) and nodal metastasis (p = 0.035). BTSVQ analysis identified a subset of 23 differentially expressed genes with the lowest QE scores in the cluster containing more advanced‐stage tumors. Expression of 6 of these differentially expressed genes was validated by quantitative real‐time RT‐PCR. Statistical analysis of quantitative real‐time RT‐PCR data was performed and, after Bonferroni correction, CLDN1 overexpression was significantly correlated with the cluster containing more advanced‐stage tumors (p = 0.007). Despite the clinical heterogeneity of OSCC, molecular subtyping by cDNA microarray analysis identified distinct patterns of gene expression associated with relevant clinical parameters. Application of this methodology represents an advance in the classification of oral cavity tumors and may ultimately aid in the development of more tailored therapies for oral carcinoma.

Likelihood and higher‐order approximations to tail areas: A review and annotated bibliography

Recent developments in higher-order asymptotic theory for statistical inference have emphasized the fundamental role of the likelihood function in providing accurate approximations to cumulative distribution functions. This paper summarizes the main results, with an emphasis on classes of problems for which relatively easily implemented solutions exist. A survey of the literature indicates the large number of problems solved and solvable by this method. Generalizations and extensions, with suggestions for further development, are considered.

Estimating the Relative Risk with Censored Data

Abstract We investigate a class of nonparametric estimators of relative risk in the two-sample case of the proportional hazards model for censored data. The asymptotic distribution of these estimators is derived using influence functions. The optimal estimator in this class has the same influence functions and the same asymptotic distribution as the maximum partial likelihood estimator of Cox (1972). The behavior of the influence functions is discussed briefly, and the last section presents two examples from the literature.

Ancillary Information for Statistical Inference

Inference for a scalar interest parameter in the presence of nuisance parameters is obtained in two steps: an initial ancillary reduction to a variable having the dimension of the full parameter and a subsequent maxginalization to a scalar pivotal quantity for the component parameter of interest. Recent asymptotic likelihood theory has provided highly accurate third order approximations for the second maxginalization to a pivotal quantity, but no general procedure for the initial ancillary conditioning. We develop a second order location type ancillary generalizing (1995) and a first order affine type ancillary generalizing (1980). The second order ancillary leads to third order p-values and the first order ancillary leads to second order p-values. For an n dimensional variable with p dimensional parameter, we also show that the only information needed concerning the ancillary is an array V = (v 1 ... v p ) of p linearly independent vectors tangent to the ancillary surface at the observed data. For the two types of ancillarity simple expressions for the array V are given; these are obtained from a full dimensional pivotal quantity. A brief summary describes the use of V to obtain second and third order p-values and some examples are discussed.

Strong matching of frequentist and Bayesian parametric inference

We define a notion of strong matching of frequentist and Bayesian inference for a scalar parameter, and show that for the special case of a location model strong matching is obtained for any interest parameter linear in the location parameters. Strong matching is defined using one-sided interval estimates constructed by inverting test quantities. A brief survey of methods for choosing a prior, of principles relating to the Bayesian paradigm, and of confidence and related procedures leads to the development of a general location reparameterization. This is followed by a brief survey of recent likelihood asymptotics which provides a basis for examining strong matching to third order in general continuous statistical methods. It is then shown that a flat prior with respect to the general location parameterization gives third-order strong matching for linear parameters; and for nonlinear parameters the strong matching requires an adjustment to the flat prior which is based on the observed Fisher information. A computationally more accessible approach then uses full dimensional pivotal quantities to generate default priors for linear parameters; this leads to second-order matching. A concluding section describes a confidence, fiducial, or default Bayesian inversion relative to the location parameterization. This provides a method to adjust interval estimates by means of a personal prior taken relative to the flat prior in the location parameterization.

Statistical Theory and Modeling: In Honour of Sir David Cox, FRS.

Part 1 Statistical theory - theoretical concepts significance tests theory of optimal tests estimation asymptotic theory. Part 2 Applied statistics - preliminary considerations statistical models statistical inference model adequancy exploratory and robust methods multi-variate methods use of computers. Part 3 Generalized linear models - a class of non-linear regression models likelihood functions estimation analysis of deviance checking the model. Part 4 Residuals and diagnostics - general remarks normal theory linear model a general definition general regression models. Part 5 Life table analysis - survival distributions inference for a single sample dependence on explanatory variables inference for models involving explanatory variables graphical methods - goodness of fit several types of failure: competing risks multi-variate failure distributions. Part 6 Sequential methods - Walds theory of the SPRT sequential tests with nuisance parameters sequential estimation sequential clinical trials decision theory and optimality in clinical trials inference and decisions sequential design and related topics. Part 7 Time series methods - stationary models frequency domain parametric models non-parametric estimation regression with correlated errors. Part 8 Modelling stochastic phenomena - independence and the Markov property semi-Markov and Markov renewal processes point processes spatial processes applications. Part 9 Optimal design of experiments - convex design theory numerical methods some specific designs for standard problems non-standard problems. Part 10 Likelihood theory - the primary theory some refinements pseudo-likelihoods. Part 11 Quasi-likelihood and estimating functions - least squares quasi-likelihood estimation quasi-likelihood function estimating functions confidence sets an open problem. Part 12 Approximations and asymptotics - basic theory and notation edgeworth expansion saddlepoint expansion Laplace approximations stochastic asymptotic expansion.

Information, ancillarity, and sufficiency in the presence of nuisance parameters†

The Fisher information about parameters of interest (P-information) is invariant with respect to nuisance parameters, and induces an information inequality associated with likelihood factorization. This information inequality provides a natural basis for measuring information loss due to using only a sublikelihood function for inference. In contrast with the global reparametrization of some previous concepts in the literature, the concepts of P-ancillarity and P-sufficiency proposed in this article are characterized by the notion of no pointwise information loss with respect to the parameters of interest. A conditional version of P-sufficiency is also proposed. The asymptotic efficiency of likelihood inference under P-ancillarity or P-sufficiency is outlined.