stochastic control for linear systems with additive cauchy noises an optimal predictive controller for linear vector state dynamic systems driven by cauchy measurement and process noises is developed for the vector state system only the characteristic function of the conditional probability density function pdf and not the pdf itself can be expressed analytically in a closed form consequently the conditional performance index formulated for the controller design can also be evaluated only in the spectral domain in particular by taking the conditional expectation of an objective function that is a product of functions resembling cauchy pdfs the conditional performance index is obtained in closed form by using parseval s identity and integrating over the spectral vector this forms a deterministic non convex function of the control signal and the measurement history that must be optimized numerically at each time step a two state example is used to expose the interesting robustness characteristics of the proposed controller

semiparametric estimation of censored selection models with a nonparametric selection mechanism abstract in this paper estimation of the coefficients in a single index selectivity bias model is considered under the assumption that the selection correction function depends on the conditional mean of some observable selection variable the estimation method follows a familiar two step strategy the first step uses a nonparametric regression estimator for the selection variable while the second step uses a weighted instrumental variables estimator for the coefficients in the equation of interest the paper gives conditions under which the proposed estimator is root n consistent and asymptotically normal the proposed method is applied to data on labor supply

distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution summary a fairly general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry and to generate a whole set of non symmetric densities the approach is sufficiently general to encompass some recent proposals in the literature variously related to the skew normal distribution the special case of skew elliptical densities is examined in detail establishing connections with existing similar work the final part of the paper specializes further to a form of multivariate skew t density likelihood inference for this distribution is examined and it is illustrated with numerical examples

on power and sample size calculations for likelihood ratio tests in generalized linear models summary a direct extension of the approach described in self mauritsen and ohara 1992 biometrics 48 31 39 for power and sample size calculations in generalized linear models is presented the major feature of the proposed approach is that the modification accommodates both a finite and an infinite number of covariate configurations furthermore for the approximation of the noncentrality of the noncentral chi square distribution for the likelihood ratio statistic a simplification is provided that not only reduces substantial computation but also maintains the accuracy simulation studies are conducted to assess the accuracy for various model configurations and covariate distributions

a heckman selection t model sample selection arises often in practice as a result of the partial observability of the outcome of interest in a study in the presence of sample selection the observed data do not represent a random sample from the population even after controlling for explanatory variables that is data are missing not at random thus standard analysis using only complete cases will lead to biased results heckman introduced a sample selection model to analyze such data and proposed a full maximum likelihood estimation method under the assumption of normality the method was criticized in the literature because of its sensitivity to the normality assumption in practice data such as income or expenditure data often violate the normality assumption because of heavier tails we first establish a new link between sample selection models and recently studied families of extended skew elliptical distributions then this allows us to introduce a selection t slt model which models the error distribution using a st

sample size determination for logistic regression revisited there is no consensus on the approach to compute the power and sample size with logistic regression some authors use the likelihood ratio test some use the test on proportions some suggest various approximations to handle the multivariate case we advocate the use of the wald test since the z score is routinely used for statistical significance testing of regression coefficients the null variance formula became popular from early studies which contradicts modern software which utilizes the method of maximum likelihood estimation mle when the variance of the mle is estimated at the mle not at the null we derive general wald based power and sample size formulas for logistic regression and then apply them to binary exposure and confounder to obtain a closed form expression these formulas are applied to minimize the total sample size in a case control study to achieve a given power by optimizing the ratio of controls to cases approximately the optimal number of controls to cases is equal to the square root of the alternative odds ratio our sample size and power calculations can be carried out online at www dartmouth edu approximately eugened

estimating the proportion of treatment effect explained by a surrogate marker in this paper we measure the extent to which a biological marker is a surrogate endpoint for a clinical event by the proportional reduction in the regression coefficient for the treatment indicator due to the inclusion of the marker in the cox regression model we estimate this proportion by applying the partial likelihood function to two cox models postulated on the same failure time variable we show that the resultant estimator is asymptotically normal with a simple variance estimator one can construct confidence intervals for the proportion by using the direct normal approximation to the point estimator or by using fieller s theorem extensive simulation studies demonstrate that the proposed methods are appropriate for practical use we provide applications to hiv aids clinical trials

statistical applications of the multivariate skew normal distribution azzalini and dalla valle have recently discussed the multivariate skew normal distribution which extends the class of normal distributions by the addition of a shape parameter the first part of the present paper examines further probabilistic properties of the distribution with special emphasis on aspects of statistical relevance inferential and other statistical issues are discussed in the following part with applications to some multivariate statistics problems illustrated by numerical examples finally a further extension is described which introduces a skewing factor of an elliptical density

adjusting power for a baseline covariate in linear models the analysis of covariance provides a common approach to adjusting for a baseline covariate in medical research with gaussian errors adding random covariates does not change either the theory or the computations of general linear model data analysis however adding random covariates does change the theory and computation of power analysis many data analysts fail to fully account for this complication in planning a study we present our results in five parts i a review of published results helps document the importance of the problem and the limitations of available methods ii a taxonomy for general linear multivariate models and hypotheses allows identifying a particular problem iii we describe how random covariates introduce the need to consider quantiles and conditional values of power iv we provide new exact and approximate methods for power analysis of a range of multivariate models with a gaussian baseline covariate for both small and large samples the new results apply to the hotelling lawley test and the four tests in the univariate approach to repeated measures unadjusted huynh feldt geisser greenhouse box the techniques allow rapid calculation and an interactive graphical approach to sample size choice v calculating power for a clinical trial of a treatment for increasing bone density illustrates the new methods we particularly recommend using quantile power with a new satterthwaite style approximation

robust statistical modeling using the t distribution abstract the t distribution provides a useful extension of the normal for statistical modeling of data sets involving errors with longer than normal tails an analytical strategy based on maximum likelihood for a general model with multivariate t errors is suggested and applied to a variety of problems including linear and nonlinear regression robust estimation of the mean and covariance matrix with missing data unbalanced multivariate repeated measures data multivariate modeling of pedigree data and multivariate nonlinear regression the degrees of freedom parameter of the t distribution provides a convenient dimension for achieving robust statistical inference with moderate increases in computational complexity for many models estimation of precision from asymptotic theory and the bootstrap is discussed and graphical methods for checking the appropriateness of the t distribution are presented

sample size considerations for the evaluation of prognostic factors in survival analysis when the role of a new prognostic factor is investigated careful planning of an appropriate study is required this includes an assessment of the power of the study in terms of sample sizes an adequate analysis of the independent prognostic effect of a new factor has to be adjusted for the existing standard factors with survival time as endpoint this will usually be done with the cox proportional hazards model sample size and power formulae in survival analysis have been developed by schoenfeld for randomized treatment comparisons in the analysis of prognostic factors the covariates included are expected to be correlated with the factor of primary interest in this situation the existing sample size and power formulae are not valid and may not be applied in this paper schoenfeld s formula is first extended to the situation where a correlated factor is included in the analysis the validity of the resulting approximate asymptotic formula is investigated for its asymptotic behaviour by numerical integration and for its finite behaviour by simulation second an approximate formula for sample size and power is provided to detect an interaction between the interesting and a second correlated factor this extends the formula for independent effects finally the approach is illustrated by an example on the prognostic impact of dna ploidy and other factors in advanced ovarian cancer

a simple method of sample size calculation for linear and logistic regression a sample size calculation for logistic regression involves complicated formulae this paper suggests use of sample size formulae for comparing means or for comparing proportions in order to calculate the required sample size for a simple logistic regression model one can then adjust the required sample size for a multiple logistic regression model by a variance inflation factor this method requires no assumption of low response probability in the logistic model as in a previous publication one can similarly calculate the sample size for linear regression models this paper also compares the accuracy of some existing sample size software for logistic regression with computer power simulations an example illustrates the methods

the robust inference for the cox proportional hazards model abstract we derive the asymptotic distribution of the maximum partial likelihood estimator for the vector of regression coefficients under a possibly misspecified cox proportional hazards model as in the parametric setting this estimator converges to a well defined constant vector in addition the random vector n 1 2 is asymptotically normal with mean 0 and with a covariance matrix that can be consistently estimated the newly proposed robust covariance matrix estimator is similar to the so called sandwich variance estimators that have been extensively studied for parametric cases for many misspecified cox models the asymptotic limit or part of it can be interpreted meaningfully in those circumstances valid statistical inferences about the corresponding covariate effects can be drawn based on the aforementioned asymptotic theory of and the related results for the score statistics extensive studies demonstrate that the proposed robust tests and interval estimation procedures

multivariate cauchy estimator with scalar measurement and process noises the conditional mean estimator for a n state linear system with additive cauchy measurement and process noises is developed it is shown that although the cauchy densities which model the initial state the process noise and the measurement noise have undefined first moments and an infinite second moment the probability density function conditioned on the measurement history does have a finite conditional mean and conditional variance for the multivariable system state the characteristic function of the unnormalized conditional probability density function is sequentially propagated through measurement updates and dynamic state propagation while expressing the resulting characteristic function in a closed analytical form once the characteristic function of the unnormalized conditional probability density function is obtained the probability density function of the measurement history the conditional mean and conditional variance are easily computed from the characteristic function and its conti

power sample size calculations for generalized linear models an approach for estimating power sample size is described within the framework of generalized linear models this approach is based on an asymptotic approximation to the power of the score test under contiguous alternatives and is applicable to tests of composite null hypotheses an implementation is described for the special case of logistic regression models simulation studies are presented which indicate that the asymptotic approximation to the finite sample situation is good over a range of parameter configurations

sample size formula for the proportional hazards regression model a formula is derived for determining the number of observations necessary to test the equality of two survival distributions when concomitant information is incorporated this formula should be useful in designing clinical trials with a heterogeneous patient population schoenfeld 1981 biometrika 68 316 319 derived the asymptotic power of a class of statistics used to test the equality of two survival distributions that result is extended to the case where concomitant information is available for each individual and where the proportional hazards model holds the loss of efficiency caused by ignoring concomitant variables is also computed

sample size for poisson regression summary for the poisson regression model an exact expression for fisher s information matrix based upon the moment generating function of the distribution of covariates is calculated this parallels a similar approximate calculation by whittemore 1981 for logistic regression the resulting asymptotic variance of the maximum likelihood estimate of the parameters is used to calculate the sample size required to test hypotheses about the parameters at a specified significance and power methods for calculating sample size are derived for various distributions of a single covariate and for a family of multivariate exponential type distributions of multiple covariates the procedures are illustrated with two examples

stochastic estimation for two state linear dynamic systems with additive cauchy noises an efficient recursive state estimator is developed for two state linear systems driven by cauchy distributed process and measurement noises for a general vector state system the estimator is based on recursively propagating the characteristic function of the conditional probability density function cpdf where the number of terms in the sum that expresses this characteristic function grows with each measurement update both the conditional mean and the conditional error variance are functions of the measurement history for systems with two states the proposed estimator reduces substantially the number of terms needed to express the characteristic function of the cpdf by taking advantage of relationships not yet developed in the general vector state case further by using a fixed sliding window of the most recent measurements the improved efficiency of the proposed two state estimator allows an accurate approximation for real time computation in this way the computational complexity of each measurement update eventually becomes constant and an arbitrary number of measurements can be processed the numerical performance of the cauchy estimator in both cauchy and gaussian simulations was demonstrated and compared to the kalman filter

an estimation approach for linear stochastic systems based on characteristic functions abstract this paper presents an alternative characteristic function based approach for the bayesian design of estimators for dynamic linear systems and linear detection problems for a measurement update the a posteriori characteristic function of the unnormalized conditional probability density function ucpdf of the state given the measurement history is obtained as a convolution of the a priori characteristic function of the ucpdf with the characteristic function of the measurement noise it is shown that this convolution holds for a general measurement structure time propagation involves the product of the updated characteristic function of the ucpdf and the characteristic function of the process noise some estimation problems are found to be naturally tractable using only characteristic functions such as the multivariable linear system with additive cauchy measurement and process noise it is shown that even the derivation of the kalman filter algorithm has advantages when formulated using the characteristic function approach finally in some instances the estimation problem can only be formulated in terms of characteristic functions this is illustrated by a one update scalar example for symmetric stable distributions

bayesian analysis of cross section and clustered data treatment models this paper is concerned with the problem of determining the effect of a categorical treatment variable on a response given that the treatment is non randomly assigned and the response on any given subject is observed for one setting of the treatment we consider classes of models that are designed for such problems these models are subjected to a fully bayesian analysis based on markov chain monte carlo methods the analysis of the treatment effect is then based on amongst other things the posterior distribution of the potential outcomes counter factuals at the subject level which is obtained as a by product of the mcmc simulation procedure the analysis is extended to models with categorical treatments and binary and clustered outcomes the problem of model comparisons is also considered different aspects of the methodology are illustrated through two data examples

power calculations for likelihood ratio tests in generalized linear models summary recently self and mauritsen 1988 biometrics 44 79 86 described an approach for sample size power calculations within the framework of generalized linear models that is based on a noncentral chi square approximation to the distribution of score statistics in this work we present another approach to this same problem that is based on a noncentral chi square approximation to the distribution of the likelihood ratio statistic this approach is easier to implement than the previous one and simulation studies indicate that it is accurate over a much wider range of parameter values and data configurations than the earlier method the new method is also applied to the problem of power calculations for 1 k matched case control studies which is related to but formally outside of the generalized linear model framework

sample size calculations for evaluating mediation this paper presents sample size calculations for testing the mediation of the effect of a primary predictor by an intermediate variable this problem is related to validating surrogate markers and to testing the effect of a primary predictor in the presence of confounders for those problems proposals for sample size calculation exist in the literature and can be adapted to the problem of mediation methods based on the variance inflation factor in linear regression provide exact sample size calculations for the linear model and approximations for the logistic poisson and cox models we propose another procedure based on simulation of the underlying data structure with applications to the logistic and cox models for the poisson model a new analytic method is also proposed the behavior of the different proposals is investigated by means of simulation studies copyright 2008 john wiley sons ltd

a practical approach to computing power for generalized linear models with nominal count or ordinal responses data analysts facing study design questions on a regular basis could derive substantial benefit from a straightforward and unified approach to power calculations for generalized linear models many current proposals for dealing with binary ordinal or count outcomes are conceptually or computationally demanding limited in terms of accommodating covariates and or have not been extensively assessed for accuracy assuming moderate sample sizes here we present a simple method for estimating conditional power that requires only standard software for fitting the desired generalized linear model for a non continuous outcome the model is fit to an appropriate expanded data set using easily calculated weights that represent response probabilities given the assumed values of the parameters the variance covariance matrix resulting from this fit is then used in conjunction with an established non central chi square approximation to the distribution of the wald statistic alternatively the model can be re fit under the null hypothesis to approximate power based on the likelihood ratio statistic we provide guidelines for constructing a representative expanded data set to allow close approximation of unconditional power based on the assumed joint distribution of the covariates relative to prior proposals the approach proves particularly flexible for handling one or more continuous covariates without any need for discretizing we illustrate the method for a variety of outcome types and covariate patterns using simulations to demonstrate its accuracy for realistic sample sizes

sample size for logistic regression with small response probability abstract the fisher information matrix for the estimated parameters in a multiple logistic regression can be approximated by the augmented hessian matrix of the moment generating function for the covariates the approximation is valid when the probability of response is small with its use one can obtain a simple closed form estimate of the asymptotic covariance matrix of the maximum likelihood parameter estimates and thus approximate sample sizes needed to test hypotheses about the parameters the method is developed for selected distributions of a single covariate and for a class of exponential type distributions of several covariates it is illustrated with an example concerning risk factors for coronary heart disease