numerical solutions of variable order time fractional 1 1 and 1 2 dimensional advection dispersion and diffusion models abstract a numerical scheme based on haar wavelets coupled with finite differences is suggested to study variable order time fractional partial differential equations tfpdes the technique is tested on 1 1 dimensional advection dispersion and 1 2 dimensional advection diffusion equations in the proposed scheme time fractional derivative is firstly approximated by quadrature formula and then finite differences are combined with one and two dimensional haar wavelets with the help of suggested method the tfpdes convert to a system of algebraic equations which is easily solvable also convergence of the proposed scheme has been discussed which is an important part of the present work for validation the obtained results are matched with earlier work and exact solutions computations illustrate that the proposed scheme has better outcomes

a new method based on haar wavelet for the numerical solution of two dimensional nonlinear integral equations a new numerical method based on haar wavelet is proposed for two dimensional nonlinear fredholm volterra and volterra fredholm integral equations of first and second kind the proposed method is an extension of the haar wavelet method aziz and siraj ul islam 2013 siraj ul islam et al 2013 and siraj ul islam et al 2014 from one dimensional nonlinear integral equations fredholm and volterra to two dimensional nonlinear integral equations fredholm volterra and volterra fredholm the main characteristic of the method is that unlike several other methods it does not involve numerical integration which results in an improved accuracy of the method in order to show the effectiveness of the method it is applied to several benchmark problems the numerical results are compared with other methods existing in the recent literature

a non uniform haar wavelet method for numerically solving two dimensional convection dominated equations and two dimensional near singular elliptic equations abstract a non uniform haar wavelet based collocation method has been developed in this paper for two dimensional convection dominated equations and two dimensional near singular elliptic partial differential equations in which traditional haar wavelet method produces oscillatory solutions or low accurate solutions the main idea behind the proposed method is to transform the computation of numerical solution of considered partial differential equations to computation of solution of a linear system of equations this process is done by discretizing space variables with non uniform haar wavelets to confirm efficiency of the proposed method seven benchmark problems are solved and the obtained results are compared with exact solutions and with local meshless methods finite element method finite difference method and polynomial collocation method numerical experiments show that the proposed method gives convincing results even in less number of collocation nodes

statistical analysis of field experiments using neighbouring plots the paper describes and provides examples of four different applications of the use of neighbouring plot values in the analysis of agricultural field experiments the topics include the use of check plots to control environmental variation in large unreplicated variety trials spatial models based on first differences to accommodate fertility effects in trials that have some degree of replication adjustment for interplot competition using a simultaneous equations formulation and the analysis of trials in which plot values are affected by the particular treatments on neighbouring plots

causal diagrams for empirical research summary the primary aim of this paper is to show how graphical models can be used as a mathematical language for integrating statistical and subject matter information in particular the paper develops a principled nonparametric framework for causal inference in which diagrams are queried to determine if the assumptions available are sufficient for identifying causal effects from nonexperimental data if so the diagrams can be queried to produce mathematical expressions for causal effects in terms of observed distributions otherwise the diagrams can be queried to suggest additional observations or auxiliary experiments from which the desired inferences can be obtained

laplace s approximation for nonlinear mixed models summary an approximation to laplace s method for integrals is applied to marginal distributions of data arising from models in which both fixed and random effects enter nonlinearly the approach provides alternative derivations of some recent algorithms for fitting such models and it has direct ties with gaussian restricted maximum likelihood and the accompanying mixed model equations

an application of single term haar wavelet series in the solution of nonlinear oscillator equations in this paper a novel single term haar wavelet series sthws method is implemented for the solution of the duffing equation and painleve s transcendents pi and pii the results in the form of a block pulse and a discrete solution are presented unlike classical numerical schemes the sthws method has no restrictions on the coefficients of the duffing equation as regards its solution pi and pii are analysed as regards their solutions up to nearest singularities poles using the sthws also an efficient computational implementation shows the remarkable features of wavelet based techniques

maximum likelihood variance components estimation for binary data abstract we consider a class of probit normal models for binary data and describe ml and reml estimation of variance components for that class as well as best prediction for the realized values of the random effects ml estimates are calculated using an em algorithm for complicated models em includes a gibbs step the computations are illustrated through two examples

on the generalization of the likelihood function and the likelihood principle abstract the parametric likelihood and likelihood principle lp play a central role in parametric methodology and in the foundations of statistics the main purpose of this article is to extend the concepts of likelihood and lp to general inferential aims and models covering for example prediction and empirical bayes models the likelihood function is the joint distribution of the data and the unobserved variables of inferential interest considered as a function of the parameters and these inferential variables lp is based on this likelihood and the principles of sufficiency sp and conditionality cp are modified such that the equivalence sp and cp lp is valid generalizing birnbaum s theorem

a two dimensional haar wavelets method for solving systems of pdes in this paper we modify the idea of haar wavelets method to obtain semi analytical solutions for the systems of three dimensional nonlinear partial differential equations theoretical considerations are discussed to demonstrate the efficiency of the method a test problem is presented the approximate solutions of the system of three dimensional nonlinear partial differential equations are compared with the exact solutions as well as existing numerical solutions found in the literature the numerical solutions which are obtained using the suggested method show that numerical solutions are in a very good coincidence with the exact solutions

hierarchical generalized linear models we consider hierarchical generalized linear models which allow extra error components in the linear predictors of generalized linear models the distribution of these components is not restricted to be normal this allows a broader class of models which includes generalized linear mixed models we use a generalization of henderson s joint likelihood called a hierarchical or h likelihood for inferences from hierarchical generalized linear models this avoids the integration that is necessary when marginal likelihood is used under appropriate conditions maximizing the h likelihood gives fixed effect estimators that are asymptotically equivalent to those obtained from the use of marginal likelihood at the same time we obtain the random effect estimates that are asymptotically best unbiased predictors an adjusted profile h likelihood is shown to give the required generalization of restricted maximum likelihood for the estimation of dispersion components a scaled deviance test for the goodness of fit a model selection criterion for choosing between various dispersion models and a graphical method for checking the distributional assumption of random effects are proposed the ideas of quasi likelihood and extended quasi likelihood are generalized to the new class we give examples of the poisson gamma binomial beta and gamma inverse gamma hierarchical generalized linear models a resolution is proposed for the apparent difference between population averaged and subject specific models a unified framework is provided for viewing and extending many existing methods

estimation in generalized linear models with random effects summary a conceptually very simple but general algorithm for the estimation of the fixed effects random effects and components of dispersion in generalized linear models with random effects is proposed conditions are described under which the algorithm yields approximate maximum likelihood or quasi maximum likelihood estimates of the fixed effects and dispersion components and approximate empirical bayes estimates of the random effects the algorithm is applied to two data sets to illustrate the estimation of components of dispersion and the modelling of overdispersion

a case cohort design for epidemiologic cohort studies and disease prevention trials summary suppose that a cohort of individuals is to be followed in order to relate failure rates to preceding covariate histories a design is proposed which involves covariate data only for cases experiencing failure and for members of a randomly selected subcohort odds ratio and relative risk estimation procedures are presented for such a case cohort design a small simulation study compares case cohort relative risk estimation procedures to full cohort and synthetic case control analyses relevance to epidemiologic cohort studies and disease prevention trials is discussed

regression analysis of longitudinal data in the presence of informative observation and censoring times longitudinal data frequently occur in many studies such as longitudinal follow up studies to develop statistical methods and theory for the analysis of these data independent or noninformative observation and censoring times are typically assumed which naturally leads to inference procedures conditional on observation and censoring times but in many situations this may not be true or realistic that is longitudinal responses may be correlated with observation times as well as censoring times this article considers the analysis of longitudinal data where these correlations may exist and proposes a joint modeling approach that uses some latent variables to characterize the correlations for inference about regression parameters estimating equation approaches are developed and both large sample and final sample properties of the proposed estimators are established in addition some graphical and numerical procedures are presented for model checking the methodology is applied to a bladder cancer stu

a comparative study of numerical integration based on haar wavelets and hybrid functions a quadrature rule based on uniform haar wavelets and hybrid functions is proposed to find approximate values of definite integrals the wavelet based algorithm can be easily extended to find numerical approximations for double triple and improper integrals the main advantage of this method is its efficiency and simple applicability error estimates of the proposed method alongside numerical examples are given to test the convergence and accuracy of the method

sampling bias and logistic models summary in a regression model the joint distribution for each finite sample of units is determined by a function px y depending only on the list of covariate values x x u1 x un on the sampled units no random sampling of units is involved in biological work random sampling is frequently unavoidable in which case the joint distribution p y x depends on the sampling scheme regression models can be used for the study of dependence provided that the conditional distribution p y x for random samples agrees with px y as determined by the regression model for a fixed sample having a non random configuration x the paper develops a model that avoids the concept of a fixed population of units thereby forcing the sampling plan to be incorporated into the sampling distribution for a quota sample having a predetermined covariate configuration x the sampling distribution agrees with the standard logistic regression model with correlated components for most natural sampling plans such as sequential or simple random sampling the conditional distribution p y x is not the same as the regression distribution unless px y has independent components in this sense most natural sampling schemes involving binary random effects models are biased the implications of this formulation for subject specific and population averaged procedures are explored

random effects models for serial observations with binary response this paper presents a general mixed model for the analysis of serial dichotomous responses provided by a panel of study participants each subject s serial responses are assumed to arise from a logistic model but with regression coefficients that vary between subjects the logistic regression parameters are assumed to be normally distributed in the population inference is based upon maximum likelihood estimation of fixed effects and variance components and empirical bayes estimation of random effects exact solutions are analytically and computationally infeasible but an approximation based on the mode of the posterior distribution of the random parameters is proposed and is implemented by means of the em algorithm this approximate method is compared with a simpler two step method proposed by korn and whittemore using data from a panel study of asthmatics originally described in that paper one advantage of the estimation strategy described here is the ability to use all of the data including that from subjects with insufficient data to permit fitting of a separate logistic regression model as required by the korn and whittemore method however the new method is computationally intensive

ignorability for categorical data we study the problem of ignorability in likelihood based inference from incomplete categorical data two versions of the coarsened at random assumption car are distinguished their compatibility with the parameter distinctness assumption is investigated and several conditions for ignorability that do not require an extra parameter distinctness assumption are established it is shown that car assumptions have quite different implications depending on whether the underlying complete data model is saturated or parametric in the latter case car assumptions can become inconsistent with observed data

computation of eigenvalues and solutions of regular sturm liouville problems using haar wavelets the paper presents a novel method for the computation of eigenvalues and solutions of sturm liouville eigenvalue problems sleps using truncated haar wavelet series this is an extension of the technique proposed by hsiao to solve discretized version of variational problems via haar wavelets the proposed method aims to cover a wider class of problems by applying it to historically important and a very useful class of boundary value problems thereby enhancing its applicability to demonstrate the effectiveness and efficiency of the method various celebrated sturm liouville problems are analyzed for their eigenvalues and solutions also eigensystems are investigated for their asymptotic and oscillatory behavior the proposed scheme unlike the conventional numerical schemes such as rayleigh quotient and rayleigh ritz approximation gives eigenpairs simultaneously and provides upper and lower estimates of the smallest eigenvalue and it is found to have quadratic convergence with increase in resolution

ignorability and coarse data we present a general statistical model for data coarsening which includes as special cases rounded heaped censored partially categorized and missing data formally with coarse data observations are made not in the sample space of the random variable of interest but rather in its power set grouping is a special case in which the degree of coarsening is known and nonstochastic we establish simple conditions under which the possible stochastic nature of the coarsening mechanism can be ignored when drawing bayesian and likelihood inferences and thus the data can be validly treated as grouped data the conditions are that the data be coarsened at random a generalization of the condition missing at random and that the parameters of the data and the coarsening process be distinct applications of the general model and the ignorability condition are illustrated in a numerical example and described briefly in a variety of special cases

a new homotopy perturbation method for solving systems of partial differential equations abstract in this paper a new homotopy perturbation method nhpm is introduced for obtaining solutions of systems of non linear partial differential equations theoretical considerations are discussed to illustrate the capability and reliability of the method three examples are provided comparison of the results of applying nhpm with those of applying hpm reveal the effectiveness and convenience of the new technique

methods for the analysis of sampled cohort data in the cox proportional hazards model methods are provided for regression parameter and cumulative baseline hazard estimation in the cox proportional hazards model when the cohort is sampled according to a predictable sampling probability law the key to the methodology is to define counting processes which count joint failure and sampled risk sets occurrences and to derive the appropriate intensities for these counting processes leading to estimation methods parallel to those for full cohort data these techniques are illustrated for a number of sampling designs including three novel designs counter matching with additional randomly sampled controls quota sampling of controls and nested case control sampling with number of controls dependent on the failure s exposure status general asymptotic theory is developed for the maximum partial likelihood estimator and cumulative baseline hazard estimator and is used to derive the asymptotic distributions for estimators from a large class of designs

an efficient numerical algorithm for multi dimensional time dependent partial differential equations abstract an efficient and robust numerical scheme based on haar wavelets and finite differences is suggested for the solution of two dimensional time dependent linear and nonlinear partial differential equations pdes excellent feature of the scheme is the conversion of linear and non linear pdes to algebraic equations which are comparatively easy to handle convergence of the scheme which guarantees small error norm as the resolution level increases is also an important part of this work different error norms are computed to check efficiency of the technique computations verify accuracy flexibility and low computational cost of the method

marginally specified logistic normal models for longitudinal binary data summary likelihood based inference for longitudinal binary data can be obtained using a generalized linear mixed model breslow n and clayton d g 1993 journal of the american statistical association88 9 25 wolfinger r and o connell m 1993 journal of statistical computation and simulation48 233 243 given the recent improvements in computational approaches alternatively fitzmaurice and laird 1993 biometrika80 141 151 molenberghs and lesaffre 1994 journal of the american statistical association89 633 644 and heagerty and zeger 1996 journal of the american statistical association91 1024 1036 have developed a likelihood based inference that adopts a marginal mean regression parameter and completes full specification of the joint multivariate distribution through either canonical and or marginal higher moment assumptions each of these marginal approaches is computationally intense and currently limited to small cluster sizes in this manuscript an alternative parameterization of the logistic normal random effects model is adopted and both likelihood and estimating equation approaches to parameter estimation are studied a key feature of the proposed approach is that marginal regression parameters are adopted that still permit individual level predictions or contrasts an example is presented where scientific interest is in both the mean response and the covariance among repeated measurements

sub and supercritical stochastic quasi geostrophic equation in this paper we study the 2d stochastic quasi geostrophic equation on t2 for general parameter 0 1 and multiplicative noise we prove the existence of weak solutions and markov selections for multiplicative noise for all 0 1 in the subcritical case 1 2 we prove existence and uniqueness of probabilistically strong solutions moreover we prove ergodicity for the solution of the stochastic quasi geostrophic equations in the subcritical case driven by possibly degenerate noise the law of large numbers for the solution of the stochastic quasi geostrophic equations in the subcritical case is also established in the case of nondegenerate noise and 2 3 in addition exponential ergodicity is proved

asymptotic theory for nested case control sampling in the cox regression model by providing a probabilistic model for nested case control sampling in epidemiologic cohort studies consistency and asymptotic normality of the maximum partial likelihood estimator of regression parameters in a cox proportional hazards model can be derived using process and martingale theory as in andersen and gill a general expression for the asymptotic variance is given and used to calculate asymptotic relative efficiencies relative to the full cohort variance in some important special cases

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

on haar wavelet operational matrix of general order and its application for the numerical solution of fractional bagley torvik equation abstract in the present analysis the motion of an immersed plate in a newtonian fluid described by torvik and bagley s fractional differential equation 1 has been considered this bagley torvik equation has been solved by operational matrix of haar wavelet method the obtained result is compared with analytical solution suggested by podlubny 2 haar wavelet method is used because its computation is simple as it converts the problem into algebraic matrix equation

numerical solutions of two dimensional sobolev and generalized benjamin bona mahony burgers equations via haar wavelets abstract this work addresses numerical method based on haar wavelets and finite differences to solve two dimensional linear nonlinear sobolev and non linear generalized benjamin bona mahony burgers ngbbmb equations the temporal part is discretized using finite differences while spatial part is approximated by two dimensional haar wavelets with this strategy computing solution of two dimensional pdes reduces to computing solution of linear system of algebraic equations collocation approach is then applied to determine the wavelet coefficients from linear system this paper shows that two dimensional haar wavelets are suitable and effective for two dimensional linear and non linear pdes for validation of the proposed scheme different problems have been solved and error norms l l 2 are computed computation verifies that current scheme has good outcome

haar wavelet operational methods for the numerical solutions of fractional order nonlinear oscillatory van der pol system in this paper an efficient numerical method for the solution of nonlinear damped van der pol equation based on the haar wavelets approach is proposed the proposed scheme can be used for solving one of the widely studied and challenging equation in nonlinear dynamics like the damped van der pol equation we have applied a numerical procedure involving haar series for the highest order of derivative moreover the numerical result shows that the proposed mechanism is quite reasonable when compare to exact solution

a comparison of gcv and gml for choosing the smoothing parameter in the generalized spline smoothing problem the partially improper prior behind the smoothing spline model is used to obtain a generalization of the maximum likelihood gml estimate for the smoothing parameter then this estimate is compared with the generalized cross validation gcv estimate both analytically and by monte carlo methods the comparison is based on a predictive mean square error criteria it is shown that if the true unknown function being estimated is smooth in a sense to be defined then the gml estimate undersmooths relative to the gcv estimate and the predictive mean square error using the gml estimate goes to zero at a slower rate than the mean square error using the gcv estimate if the true function is rough then the gcv and gml estimates have asymptotically similar behavior a monte carlo experiment was designed to see if the asymptotic results in the smooth case were evident in small sample sizes mixed results were obtained for n 32 gcv was somewhat better than gml for n 64 and gcv was decidedly superior for n 128 in the n 32 case gcv was better for smaller sigma 2 and the comparison close for larger sigma 2 the theoretical results are shown to extend to the generalized spline smoothing model which includes the estimate of functions given noisy values of various integrals of them

numerical solution of nonlinear fredholm integral equations of the second kind using haar wavelets in this work we present a computational method for solving nonlinear fredholm integral equations of the second kind which is based on the use of haar wavelets error analysis is worked out that shows efficiency of the method finally we also give some numerical examples

causal inference using potential outcomes design modeling decisions causal effects are defined as comparisons of potential outcomes under different treatments on a common set of units observed values of the potential outcomes are revealed by the assignment mechanism a probabilistic model for the treatment each unit receives as a function of covariates and potential outcomes fisher made tremendous contributions to causal inference through his work on the design of randomized experiments but the potential outcomes perspective applies to other complex experiments and nonrandomized studies as well as noted by kempthorne in his 1976 discussion of savage s fisher lecture fisher never bridged his work on experimental design and his work on parametric modeling a bridge that appears nearly automatic with an appropriate view of the potential outcomes framework where the potential outcomes and covariates are given a bayesian distribution to complete the model specification also this framework crisply separates scientific inference for causal effects and decisions based on s

haar wavelet method for solving fisher s equation in this paper we develop an accurate and efficient haar wavelet solution of fisher s equation a prototypical reaction diffusion equation the solutions of fisher s equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient there is an ongoing effort to better adapt haar wavelet methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations moreover the use of haar wavelets is found to be accurate simple fast flexible convenient small computation costs and computationally attractive

haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations haar wavelet operational matrix has been widely applied in system analysis system identification optimal control and numerical solution of integral and differential equations in the present paper we derive the haar wavelet operational matrix of the fractional order integration and use it to solve the fractional order differential equations including the bagley torvik ricatti and composite fractional oscillation equations the results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply

an improved method based on haar wavelets for numerical solution of nonlinear integral and integro differential equations of first and higher orders in this paper a novel technique is being formulated for the numerical solution of integral equations ies as well as integro differential equations ides of first and higher orders the present approach is an improved form of the haar wavelet methods aziz and siraj ul islam 2013 siraj ul islam et al 2013 the proposed modifications resulted in computational efficiency and simple applicability of the earlier methods aziz and siraj ul islam 2013 siraj ul islam et al 2013 in addition to this the new approach is being extended from ides of first order to ides of higher orders with initial and boundary conditions unlike the methods aziz and siraj ul islam 2013 siraj ul islam et al 2013 where the kernel function is being approximated by two dimensional haar wavelet the kernel function in the present case is being approximated by one dimensional haar wavelet the modified approach is easily extendable to higher order ides numerical examples are being included to show the accuracy and efficiency of the new method

new algorithms for the numerical solution of nonlinear fredholm and volterra integral equations using haar wavelets two new algorithms based on haar wavelets are proposed the first algorithm is proposed for the numerical solution of nonlinear fredholm integral equations of the second kind and the second for the numerical solution of nonlinear volterra integral equations of the second kind these methods are designed to exploit the special characteristics of haar wavelets in both one and two dimensions formulae for calculating haar coefficients without solving the system of equations have been derived these formulae are then used in the proposed numerical methods in contrast to other numerical methods the advantage of our method is that it does not involve any intermediate numerical technique for evaluation of the integral present in integral equations the methods are validated on test problems and numerical results are compared with those from existing methods in the literature

analysis of longitudinal data with irregular outcome dependent follow up summary a frequent problem in longitudinal studies is that subjects may miss scheduled visits or be assessed at self selected points in time as a result observed outcome data may be highly unbalanced and the availability of the data may be directly related to the outcome measure and or some auxiliary factors that are associated with the outcome if the follow up visit and outcome processes are correlated then marginal regression analyses will produce biased estimates building on the work of robins rotnitzky and zhao we propose a class of inverse intensity of visit process weighted estimators in marginal regression models for longitudinal responses that may be observed in continuous time this allows us to handle arbitrary patterns of missing data as embedded in a subject s visit process we derive the large sample distribution for our inverse visit intensity weighted estimators and investigate their finite sample behaviour by simulation our approach is illustrated with a data set from a health services research study in which homeless people with mental illness were randomized to three different treatments and measures of homelessness as percentage days homeless in the past 3 months and other auxiliary factors were recorded at follow up times that are not fixed by design

models for longitudinal data a generalized estimating equation approach this article discusses extensions of generalized linear models for the analysis of longitudinal data two approaches are considered subject specific ss models in which heterogeneity in regression parameters is explicitly modelled and population averaged pa models in which the aggregate response for the population is the focus we use a generalized estimating equation approach to fit both classes of models for discrete and continuous outcomes when the subject specific parameters are assumed to follow a gaussian distribution simple relationships between the pa and ss parameters are available the methods are illustrated with an analysis of data on mother s smoking and children s respiratory disease

statistics and causal inference abstract problems involving causal inference have dogged at the heels of statistics since its earliest days correlation does not imply causation and yet causal conclusions drawn from a carefully designed experiment are often valid what can a statistical model say about causation this question is addressed by using a particular model for causal inference holland and rubin 1983 rubin 1974 to critique the discussions of other writers on causation and causal inference these include selected philosophers medical researchers statisticians econometricians and proponents of causal modeling

connection coefficients on an interval and wavelet solutions of burgers equation a definition of connection coefficients is introduced and techniques of computation are presented we use semi implicit time difference scheme to solve burgers equation by applying the evaluations of connection coefficients in calculating the integrals of the variational form comparisons of accuracy and robustness of numerical solutions are mentioned in the examples