selection of important variables and determination of functional form for continuous predictors in multivariable model building in developing regression models data analysts are often faced with many predictor variables that may influence an outcome variable after more than half a century of research the best way of selecting a multivariable model is still unresolved it is generally agreed that subject matter knowledge when available should guide model building however such knowledge is often limited and data dependent model building is required we limit the scope of the modelling exercise to selecting important predictors and choosing interpretable and transportable functions for continuous predictors assuming linear functions stepwise selection and all subset strategies are discussed the key tuning parameters are the nominal p value for testing a variable for inclusion and the penalty for model complexity respectively we argue that stepwise procedures perform better than a literature based assessment would suggest concerning selection of functional form for continuous predictors the principal competitors are fractional polynomial functions and various types of spline techniques we note that a rigorous selection strategy known as multivariable fractional polynomials mfp has been developed no spline based procedure for simultaneously selecting variables and functional forms has found wide acceptance results of fp and spline modelling are compared in two data sets it is shown that spline modelling while extremely flexible can generate fitted curves with uninterpretable wiggles particularly when automatic methods for choosing the smoothness are employed we give general recommendations to practitioners for carrying out variable and function selection while acknowledging that further research is needed we argue why mfp is our preferred approach for multivariable model building with continuous covariates

some aspects of the spline smoothing approach to non parametric regression curve fitting non parametric regression using cubic splines is an attractive flexible and widely applicable approach to curve estimation although the basic idea was formulated many years ago the method is not as widely known or adopted as perhaps it should be the topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology particular subjects covered include the basic principles of the method the relation with moving average and other smoothing methods the automatic choice of the amount of smoothing and the use of residuals for diagnostic checking and model adaptation the question of providing inference regions for curves and for relevant properties of curves is approached via a finite dimensional bayesian formulation

nonparametric modeling of neural point processes via stochastic gradient boosting regression statistical nonparametric modeling tools that enable the discovery and approximation of functional forms e g tuning functions relating neural spiking activity to relevant covariates are desirable tools in neuroscience in this article we show how stochastic gradient boosting regression can be successfully extended to the modeling of spiking activity data while preserving their point process nature thus providing a robust nonparametric modeling tool we formulate stochastic gradient boosting in terms of approximating the conditional intensity function of a point process in discrete time and use the standard likelihood of the process to derive the loss function for the approximation problem to illustrate the approach we apply the algorithm to the modeling of primary motor and parietal spiking activity as a function of spiking history and kinematics during a two dimensional reaching task model selection goodness of fit via the time rescaling theorem model interpretation via partial dependence plots ranking of covariates according to their relative importance and prediction of peri event time histograms are illustrated and discussed additionally we use the tenfold cross validated log likelihood of the modeled neural processes 67 cells to compare the performance of gradient boosting regression to two alternative approaches standard generalized linear models glms and bayesian p splines with markov chain monte carlo mcmc sampling in our data set gradient boosting outperformed both bayesian p splines in approximately 90 of the cells and glms 100 because of its good performance and computational efficiency we propose stochastic gradient boosting regression as an off the shelf nonparametric tool for initial analyses of large neural data sets e g more than 50 cells more than 105 samples per cell with corresponding multidimensional covariate spaces e g more than four covariates in the cases where a functional form might be amenable to a more compact representation gradient boosting might also lead to the discovery of simpler parametric models

asymptotic properties of solutions of difference equations with several delays and volterra summation equations we study a scalar linear difference equation with several delays by transforming it to a system of volterra equations without delays the results obtained for this system are then used to establish oscillation criteria and asymptotic properties of solutions of the considered equation

frechet spaces with nuclear kothe quotients each separable frechet non banach space x with a continuous norm is shown to have a quotient y with a continuous norm and a basis if in addition y can be chosen to be nuclear we say that x has a nuclear kothe quotient we obtain a slightly technical characterization of those separable frechet spaces with nuclear kothe quotients in particular separable reflexive frh secondary 46a12 46a14 46a20 46a25 46a35 research partially supported by the national science foundation 01982 american mathematical society 0002 9947 82 0000 0090 05 00

local minima in cross validation functions summary the method of least squares cross validation for choosing the bandwidth of a kernel density estimator has been the object of considerable research through both theoretical analysis and simulation studies the method involves the minimization of a certain function of the bandwidth one of the less attractive features of this method which has been observed in simulation studies but has not previously been understood theoretically is that rather often the cross validation function has multiple local minima the theoretical results of this paper provide an explanation and quantification of this empirical observation through modelling the cross validation function as a gaussian stochastic process asymptotic analysis reveals that the degree of wiggliness of the cross validation function depends on the underlying density through a fairly simple functional but dependence on the kernel function is much more complicated a simulation study explores the extent to which the asymptotic analysis describes the actual situation our techniques may also be used to obtain other related results e g to show that spurious local minima of the cross validation function are more likely to occur at too small values of the bandwidth rather than at too large values

discriminant analysis by gaussian mixtures fisher rao linear discriminant analysis lda is a valuable tool for multigroup classification lda is equivalent to maximum likelihood classification assuming gaussian distributions for each class in this paper we fit gaussian mixtures to each class to facilitate effective classification in non normal settings especially when the classes are clustered low dimensional views are an important by product of lda our new techniques inherit this feature we can control the within class spread of the subclass centres relative to the between class spread our technique for fitting these models permits a natural blend with nonparametric versions of lda

estimation and variable selection for generalized additive partial linear models we study generalized additive partial linear models proposing the use of polynomial spline smoothing for estimation of nonparametric functions and deriving quasi likelihood based estimators for the linear parameters we establish asymptotic normality for the estimators of the parametric components the procedure avoids solving large systems of equations as in kernel based procedures and thus results in gains in computational simplicity we further develop a class of variable selection procedures for the linear parameters by employing a nonconcave penalized quasi likelihood which is shown to have an asymptotic oracle property monte carlo simulations and an empirical example are presented for illustration

low rank scale invariant tensor product smooths for generalized additive mixed models summary a general method for constructing low rank tensor product smooths for use as components of generalized additive models or generalized additive mixed models is presented a penalized regression approach is adopted in which tensor product smooths of several variables are constructed from smooths of each variable separately these marginal smooths being represented using a low rank basis with an associated quadratic wiggliness penalty the smooths offer several advantages i they have one wiggliness penalty per covariate and are hence invariant to linear rescaling of covariates making them useful when there is no natural way to scale covariates relative to each other ii they have a useful tuneable range of smoothness unlike single penalty tensor product smooths that are scale invariant iii the relatively low rank of the smooths means that they are computationally efficient iv the penalties on the smooths are easily interpretable in terms of function shape v the smooths can be generated completely automatically from any marginal smoothing bases and associated quadratic penalties giving the modeler considerable flexibility to choose the basis penalty combination most appropriate to each modeling task and vi the smooths can easily be written as components of a standard linear or generalized linear mixed model allowing them to be used as components of the rich family of such models implemented in standard software and to take advantage of the efficient and stable computational methods that have been developed for such models a small simulation study shows that the methods can compare favorably with recently developed smoothing spline anova methods

generalized additive and index models with shape constraints summary we study generalized additive models with shape restrictions e g monotonicity convexity and concavity imposed on each component of the additive prediction function we show that this framework facilitates a non parametric estimator of each additive component obtained by maximizing the likelihood the procedure is free of tuning parameters and under mild conditions is proved to be uniformly consistent on compact intervals more generally our methodology can be applied to generalized additive index models here again the procedure can be justified on theoretical grounds and like the original algorithm has highly competitive finite sample performance practical utility is illustrated through the use of these methods in the analysis of two real data sets our algorithms are publicly available in the r package scar short for shape constrained additive regression

stable and efficient multiple smoothing parameter estimation for generalized additive models representation of generalized additive models gam s using penalized regression splines allows gam s to be employed in a straightforward manner using penalized regression methods not only is inference facilitated by this approach but it is also possible to integrate model selection in the form of smoothing parameter selection into model fitting in a computationally efficient manner using well founded criteria such as generalized cross validation the current fitting and smoothing parameter selection methods for such models are usually effective but do not provide the level of numerical stability to which users of linear regression packages for example are accustomed in particular the existing methods cannot deal adequately with numerical rank deficiency of the gam fitting problem and it is not straightforward to produce methods that can do so given that the degree of rank deficiency can be smoothing parameter dependent in addition models with the potential flexibility of gam s can also present

dose response analyses using restricted cubic spline functions in public health research taking into account a continuous exposure in regression models by using categorization when non linear dose response associations are expected have been widely criticized as one alternative restricted cubic spline rcs functions are powerful tools i to characterize a dose response association between a continuous exposure and an outcome ii to visually and or statistically check the assumption of linearity of the association and iii to minimize residual confounding when adjusting for a continuous exposure because their implementation with sas software is limited we developed and present here an sas macro that i creates an rcs function of continuous exposures ii displays graphs showing the dose response association with 95 per cent confidence interval between one main continuous exposure and an outcome when performing linear logistic or cox models as well as linear and logistic generalized estimating equations and iii provides statistical tests for overall and non linear associations we illustrate the sas macro using the third national health and nutrition examination survey data to investigate adjusted dose response associations with different models between calcium intake and bone mineral density linear regression folate intake and hyperhomocysteinemia logistic regression and serum high density lipoprotein cholesterol and cardiovascular mortality cox model

periodic and almost periodic solutions of nonlinear discrete volterra equations with unbounded delay the existence of periodic and almost periodic solutions of nonlinear discrete volterra equations with unbounded delay is obtained by using stability properties of a bounded solution

asymptotic behavior of volterra difference equations abstract in this paper we investigate boundedness and stability properties of some classes of discrete volterra equations the main tool in this work is the use of a representation formula which allows us to express the solution of discrete volterra equations in terms of the resolvent matrix of the corresponding system of volterra difference equations

automatic smoothing of regression functions in generalized linear models abstract we consider the penalized likelihood method for estimating nonparametric regression functions in generalized linear models nelder and wedderburn 1972 and present a generalized cross validation procedure for empirically assessing an appropriate amount of smoothing in these estimates asymptotic arguments and numerical simulations are used to show that the generalized cross validatory procedure preforms well from the point of view of a weighted mean squared error criterion the methodology adds to the battery of graphical tools for model building and checking within the generalized linear model framework included are two examples motivated by medical and horticultural applications

a boosting approach to flexible semiparametric mixed models in linear mixed models the influence of covariates is restricted to a strictly parametric form with the rise of semi and non parametric regression also the mixed model has been expanded to allow for additive predictors the common approach uses the representation of additive models as mixed models an alternative approach that is proposed in the present paper is likelihood based boosting boosting originates in the machine learning community where it has been proposed as a technique to improve classification procedures by combining estimates with reweighted observations likelihood based boosting is a general method which may be seen as an extension of l2 boost in additive mixed models the advantage of boosting techniques in the form of componentwise boosting is that it is suitable for high dimensional settings where many explanatory variables are present it allows to fit additive models for many covariates with implicit selection of relevant variables and automatic selection of smoothing parameters moreover boosting techniques may be used to incorporate the subject specific variation of smooth influence functions by specifying random slopes on smooth effects this results in flexible semiparametric mixed models which are appropriate in cases where a simple random intercept is unable to capture the variation of effects across subjects

convergence of arithmetic means of operators in frechet spaces abstract every kothe echelon frechet space x that is montel and not isomorphic to a countable product of copies of the scalar field admits a power bounded continuous linear operator t such that i t does not have closed range but the sequence of arithmetic means of the iterates of t converges to 0 uniformly on the bounded sets in x on the other hand if x is a frechet space which does not have a quotient isomorphic to a nuclear kothe echelon space with a continuous norm then the sequence of arithmetic means of the iterates of any continuous linear operator t for which 1 n t n converges to 0 on the bounded sets converges uniformly on the bounded subsets of x i e t is uniformly mean ergodic if and only if the range of i t is closed this result extends a theorem due to lin for such operators on banach spaces the connection of browder s equality for power bounded operators on frechet spaces to their uniform mean ergodicity is exposed an analysis of the mean ergodic properties of the classical cesaro operator on banach sequence spaces is also given

stability analysis of discrete recurrence equations of volterra type with degenerate kernels abstract stability criteria are derived for difference equations of volterra type with degenerate kernels the main tool in this analysis is the use of the new representation formula which allows us to express the solution of discrete volterra equation with degenerate kernel in terms of the fundamental matrix of the corresponding first order system of the difference equations

asymptotically periodic solutions of volterra system of difference equations a volterra system of difference equations of the form x s n 1 a s n b s n x s n email protected email protected i 0nk s p n i x p i where email protected n a s b s x s n r and k s p nxn r s 1 2 r is studied sufficient conditions for the existence of asymptotically periodic solutions of this system are derived

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

asymptotically periodic solutions of second order difference equations abstract in this paper we obtain sufficient conditions for the existence of asymptotically periodic solutions of the second order difference equations of the form r n x n a n f x n b n moreover we give sufficient conditions under which every bounded solution of the above equation is asymptotically periodic our approach allows us to control the degree of approximation

comparison between splines and fractional polynomials for multivariable model building with continuous covariates a simulation study with continuous response in observational studies many continuous or categorical covariates may be related to an outcome various spline based procedures or the multivariable fractional polynomial mfp procedure can be used to identify important variables and functional forms for continuous covariates this is the main aim of an explanatory model as opposed to a model only for prediction the type of analysis often guides the complexity of the final model spline based procedures and mfp have tuning parameters for choosing the required complexity to compare model selection approaches we perform a simulation study in the linear regression context based on a data structure intended to reflect realistic biomedical data we vary the sample size variance explained and complexity parameters for model selection we consider 15 variables a sample size of 200 1000 and r 2 0 2 0 8 is the scenario with the smallest largest amount of information for assessing performance we consider prediction error correct and incorrect inclusion of covariates qualitative measures for judging selected functional forms and further novel criteria from limited information a suitable explanatory model cannot be obtained prediction performance from all types of models is similar with a medium amount of information mfp performs better than splines on several criteria mfp better recovers simpler functions whereas splines better recover more complex functions for a large amount of information and no local structure mfp and the spline procedures often select similar explanatory models

on weighted inductive limits of spaces of continuous functions the general problem of a projective description of weighted inductive limits of spaces of continuous functions was raised by bierstedt eise and summers in 3 similar questions were studied in 4 in the setting of sequence spaces on a general index set i the study was recently continued by bierstedt and meise in 5 the aim of the present article is to treat some questions proposed to the author by k d bierstedt on this subject our notation for locally convex spaces 1 c s is standard and we refer the reader to 13 and 14 if e is a 1 c s cs e denotes the family of all the continuous seminorms on e following floret 10 a 1 c s is said to satisfy the countable neighbourhood property c n p if given p ces e there are a 0 and p cs e such that p x 0 such that u 0 there is v v such that avi 0 i e if given any x x there is v v with v x 0 then v will be refered to as a nachbin family on x in what follows if e is a 1 c s c x e indicates the collection of all continuous e valued functions on x and cc x e the subspace of c x e of all the functions with compact support corresponding to a nachbin family v and to a 1 c s e we associate the following nachbin spaces

a new approach to modelling interactions between treatment and continuous covariates in clinical trials by using fractional polynomials we consider modelling interaction between a categoric covariate t and a continuous covariate z in a regression model here t represents the two treatment arms in a parallel group clinical trial and z is a prognostic factor which may influence response to treatment known as a predictive factor generalization to more than two treatments is straightforward the usual approach to analysis is to categorize z into groups according to cutpoint s and to analyse the interaction in a model with main effects and multiplicative terms the cutpoint approach raises several well known and difficult issues for the analyst we propose an alternative approach based on fractional polynomial fp modelling of z in all patients and at each level of t other prognostic variables can also be incorporated by first constructing a multivariable adjustment model which may contain binary covariates and fp transformations of continuous covariates other than z the main step involves fp modelling of z and testing equality of regression coefficients between treatment groups in an interaction model adjusted for other covariates extensive experience suggests that a two term fractional polynomial fp2 function may describe the effect of a prognostic factor on a survival outcome quite well in a controlled trial this fp2 function describes the prognostic effect averaged over the treatment groups we refit this function in each treatment group to see if there are substantial differences between groups allowing different parameter values for the chosen fp2 function is flexible enough to detect such differences within the same algorithm we can also deal with the conceptually different cases of a predefined hypothesis of interaction or searching for interactions we demonstrate the ability of the approach to detect and display treatment covariate interactions in two examples from controlled trials in cancer

periodic solutions of non linear discrete volterra equations with finite memory in this paper we discuss the existence of periodic solutions of discrete and discretized non linear volterra equations with finite memory the literature contains a number of results on periodic solutions of non linear volterra integral equations with finite memory of a type that arises in biomathematics the summation equations studied here can arise as discrete models in their own right but are as we demonstrate of a type that arise from the discretization of such integral equations our main results are in two parts i results for discrete equations and ii consequences for quadrature methods applied to integral equations the first set of results are obtained using a variety of fixed point theorems the second set of results address the preservation of properties of integral equations on discretizing them the effect of weak singularities is addressed in a final section the detail that is presented which is supplemented using appendices reflects the differing prerequisites of functional analysis and numerical analysis that contribute to the outcomes

modelling continuous exposures with a spike at zero a new procedure based on fractional polynomials a common task in epidemiology is to estimate the dose response function for a continuous exposure often a proportion of subjects is unexposed typical examples are cigarette consumption alcohol intake or occupational exposures the question arises as to how to model such variables statistically the fractional polynomial method of modelling continuous exposure variables is extended to allow for a proportion unexposed a binary variable for the unexposed fraction is added to the model in a two stage procedure we assess whether the binary variable and or the continuous function for the exposed individuals is required for a good fit to the data extension to the multivariable situation is described three data sets with different characteristics are used as illustrations the analyses of the three studies using the proposed procedure give differing results in one example only the binary variable seems to be required in the other two examples the binary variable and fractional polynomial functions of the exposure variable are needed one function is monotonic and the other has a minimum in the third example adjusting for confounders has almost no effect on the function selected in conclusion the new procedure offers a worthwhile extension of dose response modelling with an unexposed fraction it is simple to carry out with standard software

model building in nonproportional hazard regression recent developments of statistical methods allow for a very flexible modeling of covariates affecting survival times via the hazard rate including also the inspection of possible time dependent associations despite their immediate appeal in terms of flexibility these models typically introduce additional difficulties when a subset of covariates and the corresponding modeling alternatives have to be chosen that is for building the most suitable model for given data this is particularly true when potentially time varying associations are given we propose to conduct a piecewise exponential representation of the original survival data to link hazard regression with estimation schemes based on of the poisson likelihood to make recent advances for model building in exponential family regression accessible also in the nonproportional hazard regression context a two stage stepwise selection approach an approach based on doubly penalized likelihood and a componentwise functional gradient descent approach are adapted to the piecewise exponential regression problem these three techniques were compared via an intensive simulation study an application to prognosis after discharge for patients who suffered a myocardial infarction supplements the simulation to demonstrate the pros and cons of the approaches in real data analyses

functional generalized linear models with images as predictors functional principal component regression fpcr is a promising new method for regressing scalar outcomes on functional predictors in this article we present a theoretical justification for the use of principal components in functional regression fpcr is then extended in two directions from linear to the generalized linear modeling and from univariate signal predictors to high resolution image predictors we show how to implement the method efficiently by adapting generalized additive model technology to the functional regression context a technique is proposed for estimating simultaneous confidence bands for the coefficient function in the neuroimaging setting this yields a novel means to identify brain regions that are associated with a clinical outcome a new application of likelihood ratio testing is described for assessing the null hypothesis of a constant coefficient function the performance of the methodology is illustrated via simulations and real data analyses with positron emission tomography images as predictors

stochastic approximation boosting for incomplete data problems boosting is a powerful approach to fitting regression models this article describes a boosting algorithm for likelihood based estimation with incomplete data the algorithm combines boosting with a variant of stochastic approximation that uses markov chain monte carlo to deal with the missing data applications to fitting generalized linear and additive models with missing covariates are given the method is applied to the pima indians diabetes data where over half of the cases contain missing values

mixed binary continuous copula regression models with application to adverse birth outcomes bivariate copula regression allows for the flexible combination of two arbitrary continuous marginal distributions with regression effects being placed on potentially all parameters of the resulting bivariate joint response distribution motivated by a study examining the risk factors of adverse birth outcomes we consider mixed binary continuous responses that extend this framework to the situation where one response variable is discrete more precisely binary while the other response remains continuous utilizing the latent continuous representation of binary regression models we implement a penalized likelihood based approach for the resulting class of copula regression models and employ it in the context of modelling jointly gestational age and the presence absence of low birth weight the analysis strongly benefits from the flexible specification of regression effects including nonlinear effects of continuous covariates and spatial effects our results imply that racial and spatial inequalities in the risk factors for infant mortality are even greater than previously suggested

examples of non archimedean frechet spaces without nuclear kothe quotients let k be a spherically complete non archimedean valued field we prove that the dual space l of the banach space c0 has a total strongly non norming subspace m using this subspace m we construct a non normable frechet space f of countable type with a continuous norm such that its strong dual fb is a strict lb space next we show that f has no nuclear kothe quotient

strengthening analytical thinking for observational studies the stratos initiative the validity and practical utility of observational medical research depends critically on good study design excellent data quality appropriate statistical methods and accurate interpretation of results statistical methodology has seen substantial development in recent times unfortunately many of these methodological developments are ignored in practice consequently design and analysis of observational studies often exhibit serious weaknesses the lack of guidance on vital practical issues discourages many applied researchers from using more sophisticated and possibly more appropriate methods when analyzing observational studies furthermore many analyses are conducted by researchers with a relatively weak statistical background and limited experience in using statistical methodology and software consequently even standard analyses reported in the medical literature are often flawed casting doubt on their results and conclusions an efficient way to help researchers to keep up with recent methodological developments is to develop guidance documents that are spread to the research community at large these observations led to the initiation of the strengthening analytical thinking for observational studies stratos initiative a large collaboration of experts in many different areas of biostatistical research the objective of stratos is to provide accessible and accurate guidance in the design and analysis of observational studies the guidance is intended for applied statisticians and other data analysts with varying levels of statistical education experience and interests in this article we introduce the stratos initiative and its main aims present the need for guidance documents and outline the planned approach and progress so far we encourage other biostatisticians to become involved

linearized stability analysis of discrete volterra equations abstract various different types of stability are defined in a unified framework for discrete volterra equations of the type x n f n n j 0 k n j x n n 0 under appropriate assumptions stability results are obtainable from those valid in the linear case k n j x n b n j x j and a linearized stability theory is studied here by using the fundamental and resolvent matrices several necessary and sufficient conditions for stability are obtained for solutions of the linear equation by considering the equations in various choices of banach space b the elements of which are sequences of vectors x n f n e d b n j e d e d n j 0 etc we show that the theory including a number of new results as well as results already known can be presented in a systematic framework in which results parallel corresponding results for classical volterra integral equations of the second kind

mixed effects smoothing spline analysis of variance we propose a general family of nonparametric mixed effects models smoothing splines are used to model the fixed effects and are estimated by maximizing the penalized likelihood function the random effects are generic and are modelled parametrically by assuming that the covariance function depends on a parsimonious set of parameters these parameters and the smoothing parameter are estimated simultaneously by the generalized maximum likelihood method we derive a connection between a nonparametric mixed effects model and a linear mixed effects model this connection suggests a way of fitting a nonparametric mixed effects model by using existing programs the classical two way mixed models and growth curve models are used as examples to demonstrate how to use smoothing spline analysis of variance decompositions to build nonparametric mixed effects models similarly to the classical analysis of variance components of these nonparametric mixed effects models can be interpreted as main effects and interactions the penalized likelihood estimates of the fixed effects in a two way mixed model are extensions of james stein shrinkage estimates to correlated observations in an example three nested nonparametric mixed effects models are fitted to a longitudinal data set

asymptotic properties of solutions of some volterra difference equations and second order difference equations abstract we give sufficient conditions for the existence of a bounded resp convergent solution of a class of difference equations and volterra difference equation

stability of multivariable fractional polynomial models with selection of variables and transformations a bootstrap investigation sauerbrei and royston have recently described an algorithm based on fractional polynomials for the simultaneous selection of variables and of suitable transformations for continuous predictors in a multivariable regression setting they illustrated the approach by analyses of two breast cancer data sets here we extend their work by considering how to assess possible instability in such multivariable fractional polynomial models we first apply the algorithm repeatedly in many bootstrap replicates we then use log linear models to investigate dependencies among the inclusion fractions for each predictor and among the simplified classes of fractional polynomial function chosen in the bootstrap samples to further evaluate the results we define measures of instability based on a decomposition of the variability of the bootstrap selected functions in relation to a reference function from the original model for each data set we are able to identify large reasonably stable subsets of the bootstrap replications in which the functional forms of the predictors appear fairly stable despite the considerable flexibility of the family of fractional polynomials and the consequent risk of overfitting when several variables are considered we conclude that the multivariable selection algorithm can find stable models

adding local components to global functions for continuous covariates in multivariable regression modeling when global techniques based on fractional polynomials fps are employed for modeling potentially nonlinear effects of several continuous covariates on a response accessible model equations are obtained however local features might be missed therefore a procedure is introduced which systematically checks model fits obtained by the multivariable fractional polynomial mfp approach for overlooked local features statistically significant local polynomials are then parsimoniously added this approach called mfp l is seen to result in an effective control of the type i error with respect to the addition of local components in a small simulation study with univariate and multivariable settings prediction performance is compared with that of a penalized regression spline technique in a setting unfavorable for fps the latter outperforms the mfp approach if there is much information in the data however the addition of local features reduces this performance difference there is only a small detrimental effect in settings where the mfp approach performs better in an application example with children s respiratory health data fits from the spline based approach indicate many local features but mfp l adds only few significant features which seem to have good support in the data the proposed approach may be expected to be superior in settings with local features but retains the good properties of the mfp approach in a large number of settings where global functions are sufficient

a survey stability and boundedness of volterra difference equations abstract volterra difference equations arise in the modeling of many real phenomena in this survey stability and boundedness problems of some volterra nonlinear difference equations are investigated stability conditions and boundedness are formulated in terms of the characteristics of equations

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

generalized additive modeling with implicit variable selection by likelihood based boosting the use of generalized additive models in statistical data analysis suffers from the restriction to few explanatory variables and the problems of selection of smoothing parameters generalized additive model boosting circumvents these problems by means of stagewise fitting of weak learners a fitting procedure is derived which works for all simple exponential family distributions including binomial poisson and normal response variables the procedure combines the selection of variables and the determination of the appropriate amount of smoothing penalized regression splines and the newly introduced penalized stumps are considered as weak learners estimates of standard deviations and stopping criteria which are notorious problems in iterative procedures are based on an approximate hat matrix the method is shown to be a strong competitor to common procedures for the fitting of generalized additive models in particular in high dimensional settings with many nuisance predictor variables it performs very well

flexible regression models for estimating postmortem interval pmi in forensic medicine correct determination of time of death is an important goal in forensic medicine numerous methods have been described for estimating postmortem interval pmi but most are imprecise poorly reproducible and or have not been validated with real data in recent years however some progress in pmi estimation has been made notably through the use of new biochemical methods for quantifying relevant indicator compounds in the vitreous humour the best but unverified results have been obtained with k and hypoxanthine hx using simple linear regression lr models the main aim of this paper is to offer more flexible alternatives to lr such as generalized additive models gams and support vector machines svms in order to obtain improved pmi estimates the present study based on detailed analysis of k and hx in more than 200 vitreous humour samples from subjects with known pmi compared classical lr methodology with gam and svm methodologies both proved better than lr for estimation of pmi svm showed somewhat greater precision than gam but gam offers a readily interpretable graphical output facilitating understanding of findings by legal professionals there are thus arguments for using both types of models r code for these methods is available from the authors permitting accurate prediction of pmi from vitreous humour k hx and u with confidence intervals and graphical output provided copyright 2008 john wiley sons ltd

interaction of treatment with a continuous variable simulation study of significance level for several methods of analysis interactions between treatments and covariates in rcts are a key topic standard methods for modelling treatment covariate interactions with continuous covariates are categorisation or linear functions both approaches are easily criticised but for different reasons multivariable fractional polynomial interactions an approach based on fractional polynomials with the linear interaction model as the simplest special case was proposed four variants of multivariable fractional polynomial interaction flex1 flex4 allowing varying flexibility in functional form were suggested however their properties are unknown and comparisons with other procedures are unavailable additionally we consider various methods based on categorisation and on cubic regression splines we present the results of a simulation study to determine the significance level probability of a type 1 error of various tests for interaction between a binary covariate treatment effect and a continuous covariate in univariate analysis we consider a simplified setting in which the response variable is conditionally normally distributed given the continuous covariate we consider two main cases with the covariate distribution well behaved approximately symmetric or badly behaved positively skewed we construct nine scenarios with different functional forms for the main effect in the well behaved case significance levels are in general acceptably close to nominal and are slightly better for the larger sample size n 250 and 500 were investigated in the badly behaved case departures from nominal are more pronounced for several approaches for a final assessment of these results and recommendations for practice a study of power is needed copyright 2013 john wiley sons ltd

flexible estimation of survival curves conditional on non linear and time dependent predictor effects prognostic studies often estimate survival curves for patients with different covariate vectors but the validity of their results depends largely on the accuracy of the estimated covariate effects to avoid conventional proportional hazards and linearity assumptions flexible extensions of cox s proportional hazards model incorporate non linear nl and or time dependent td covariate effects however their impact on survival curves estimation is unclear our primary goal is to develop and validate a flexible method for estimating individual patients survival curves conditional on multiple predictors with possibly nl and or td effects we first obtain maximum partial likelihood estimates of nl and td effects and use backward elimination to select statistically significant effects into a final multivariable model we then plug the selected nl and td estimates in the full likelihood function and estimate the baseline hazard function and the resulting survival curves conditional on individual covariate vectors the td and nl functions and the log hazard are modeled with unpenalized regression b splines in simulations our flexible survival curve estimates were unbiased and had much lower mean square errors than the conventional estimates in real life analyses of mortality after a septic shock our model improved significantly the deviance likelihood ratio test 84 8 df 20 p 0 0001 and changed substantially the predicted survival for several subjects

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

estimation and model selection in generalized additive partial linear models for correlated data with diverging number of covariates this is the publisher s final pdf the published article is copyrighted by the institute of mathematical statistics and can be found at http www imstat org aos

multivariate meta analysis of prognostic factor studies with multiple cut points and or methods of measurement a prognostic factor is any measure that is associated with the risk of future health outcomes in those with existing disease often the prognostic ability of a factor is evaluated in multiple studies however meta analysis is difficult because primary studies often use different methods of measurement and or different cut points to dichotomise continuous factors into high and low groups selective reporting is also common we illustrate how multivariate random effects meta analysis models can accommodate multiple prognostic effect estimates from the same study relating to multiple cut points and or methods of measurement the models account for within study and between study correlations which utilises more information and reduces the impact of unreported cut points and or measurement methods in some studies the applicability of the approach is improved with individual participant data and by assuming a functional relationship between prognostic effect and cut point to reduce the number of unknown parameters the models provide important inferential results for each cut point and method of measurement including the summary prognostic effect the between study variance and a 95 prediction interval for the prognostic effect in new populations two applications are presented the first reveals that in a multivariate meta analysis using published results the apgar score is prognostic of neonatal mortality but effect sizes are smaller at most cut points than previously thought in the second a multivariate meta analysis of two methods of measurement provides weak evidence that microvessel density is prognostic of mortality in lung cancer even when individual participant data are available so that a continuous prognostic trend is examined rather than cut points 2015 the authors statistics in medicine published by john wiley sons ltd

c0 semigroups and mean ergodic operators in a class of frechet spaces abstract it is shown that the generator of every exponentially equicontinuous uniformly continuous c 0 semigroup of operators in the class of quojection frechet spaces x which includes properly all countable products of banach spaces is necessarily everywhere defined and continuous if in addition x is a grothendieck space with the dunford pettis property then uniform continuity can be relaxed to strong continuity two results one of m lin and one of h p lotz both concerned with uniformly mean ergodic operators in banach spaces are also extended to the class of frechet spaces mentioned above they fail to hold for arbitrary frechet spaces

modelling and smoothing parameter estimation with multiple quadratic penalties summary penalized likelihood methods provide a range of practical modelling tools including spline smoothing generalized additive models and variants of ridge regression selecting the correct weights for penalties is a critical part of using these methods and in the single penalty case the analyst has several well founded techniques to choose from however many modelling problems suggest a formulation employing multiple penalties and here general methodology is lacking a wide family of models with multiple penalties can be fitted to data by iterative solution of the generalized ridge regression problem minimise kw 1 2 xp a y k 2 o p m i 1 iip 0 s i p p is a parameter vector x a design matrix si a non negative definite coefficient matrix defining the i th penalty with associated smoothing parameter ii w a diagonal weight matrix y a vector of data or pseudodata and o an overall smoothing parameter included for reasons of computational efficiency this paper shows how smoothing parameter selection can be performed efficiently by applying generalized cross validation to this problem and how this allows non linear generalized linear and linear models to be fitted using multiple penalties substantially increasing the scope of penalized modelling methods examples of non linear modelling generalized additive modelling and anisotropic smoothing are given

fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models summary recent work by reiss and ogden provides a theoretical basis for sometimes preferring restricted maximum likelihood reml to generalized cross validation gcv for smoothing parameter selection in semiparametric regression however existing reml or marginal likelihood ml based methods for semiparametric generalized linear models glms use iterative reml or ml estimation of the smoothing parameters of working linear approximations to the glm such indirect schemes need not converge and fail to do so in a non negligible proportion of practical analyses by contrast very reliable prediction error criteria smoothing parameter selection methods are available based on direct optimization of gcv or related criteria for the glm itself since such methods directly optimize properly defined functions of the smoothing parameters they have much more reliable convergence properties the paper develops the first such method for reml or ml estimation of smoothing parameters a laplace approximation is used to obtain an approximate reml or ml for any glm which is suitable for efficient direct optimization this reml or ml criterion requires that newton raphson iteration rather than fisher scoring be used for glm fitting and a computationally stable approach to this is proposed the reml or ml criterion itself is optimized by a newton method with the derivatives required obtained by a mixture of implicit differentiation and direct methods the method will cope with numerical rank deficiency in the fitted model and in fact provides a slight improvement in numerical robustness on the earlier method of wood for prediction error criteria based smoothness selection simulation results suggest that the new reml and ml methods offer some improvement in mean square error performance relative to gcv or akaike s information criterion in most cases without the small number of severe undersmoothing failures to which akaike s information criterion and gcv are prone this is achieved at the same computational cost as gcv or akaike s information criterion the new approach also eliminates the convergence failures of previous reml or ml based approaches for penalized glms and usually has lower computational cost than these alternatives example applications are presented in adaptive smoothing scalar on function regression and generalized additive model selection

on the existence of solutions of linear volterra difference equations asymptotically equivalent to a given sequence abstract schauder s fixed point technique is applied to asymptotical analysis of solutions of a linear volterra difference equation x n 1 a n b n x n i 0 n k n i x i where n n 0 x n 0 r a n 0 r k n 0 n 0 r and b n 0 r 0 is periodic in the paper sufficient conditions are derived for the validity of a property of solutions that for every admissible constant c r there exists a solution x x n such that x n c i 0 n 1 a i i 1 n where n j 0 n 1 b j for n and inequalities for solutions are derived relevant comparisons and illustrative examples are given as well

local reflexivity of normed spaces operators and frechet spaces a new and stronger proof of the principle of local reflexivity is given applications to normed spaces local reflexivity of operators and frechet spaces are given a partial answer to a question of pelczynski is obtained the proof is based on an unpublished proof of r c james

quantifying epidemiologic risk factors using non parametric regression model selection remains the greatest challenge logistic regression is widely used to estimate relative risks odds ratios from case control studies but when the study exposure is continuous standard parametric models may not accurately characterize the exposure response curve semi parametric generalized linear models provide a useful extension in these models the exposure of interest is modelled flexibly using a regression spline or a smoothing spline while other variables are modelled using conventional methods when coupled with a model selection procedure based on minimizing a cross validation score this approach provides a non parametric objective and reproducible method to characterize the exposure response curve by one or several models with a favourable bias variance trade off we applied this approach to case control data to estimate the dose response relationship between alcohol consumption and risk of oral cancer among african americans we did not find a uniquely best model but results using linear cubic and smoothing splines were consistent there does not appear to be a risk free threshold for alcohol consumption vis a vis the development of oral cancer this finding was not apparent using a standard step function model in our analysis the cross validation curve had a global minimum and also a local minimum in general the phenomenon of multiple local minima makes it more difficult to interpret the results and may present a computational roadblock to non parametric generalized additive models of multiple continuous exposures nonetheless the semi parametric approach appears to be a practical advance

model averaging for robust assessment of qt prolongation by concentration response analysis assessing the qt prolongation potential of a drug is typically done based on pivotal safety studies called thorough qt studies model based estimation of the drug induced qt prolongation at the estimated mean maximum drug concentration could increase efficiency over the currently used intersection union test however robustness against model misspecification needs to be guaranteed in pivotal settings the objective of this work was to develop an efficient fully prespecified model based inference method for thorough qt studies which controls the type i error and provides satisfactory test power this is achieved by model averaging the proposed estimator of the concentration response relationship is a weighted average of a parametric linear and a nonparametric monotonic i splines estimator with weights based on mean integrated square error the desired properties of the method were confirmed in an extensive simulation study which demonstrated that the proposed method controlled the type i error adequately and that its power was higher than the power of the nonparametric method alone the method can be extended from thorough qt studies to the analysis of qt data from pooled phase i studies

thin plate regression splines discuss the production of low rank smoothers for d greater than or equal to 1 dimensional data which can be fitted by regression or penalized regression methods the smoothers are constructed by a simple transformation and truncation of the basis that arises from the solution of the thin plate spline smoothing problem and are optimal in the sense that the truncation is designed to result in the minimum possible perturbation of the thin plate spline smoothing problem given the dimension of the basis used to construct the smoother by making use of lanczos iteration the basis change and truncation are computationally efficient the smoothers allow the use of approximate thin plate spline models with large data sets avoid the problems that are associated with knot placement that usually complicate modelling with regression splines or penalized regression splines provide a sensible way of modelling interaction terms in generalized additive models provide low rank approximations to generalized smoothing spline models appropriate for use with large data sets provide a means for incorporating smooth functions of more than one variable into non linear models and improve the computational efficiency of penalized likelihood models incorporating thin plate splines given that the approach produces spline like models with a sparse basis it also provides a natural way of incorporating unpenalized spline like terms in linear and generalized linear models and these can be treated just like any other model terms from the point of view of model selection inference and diagnostics

smoothing noisy data with spline functions a procedure for calculating the trace of the influence matrix associated with a polynomial smoothing spline of degree2m 1 fitted ton distinct not necessarily equally spaced or uniformly weighted data points is presented the procedure requires orderm 2 n operations and therefore permits efficient orderm 2 n calculation of statistics associated with a polynomial smoothing spline including the generalized cross validation the method is a significant improvement over an existing method which requires ordern 3 operations

smoothing spline anova for exponential families with application to the wisconsin epidemiological study of diabetic retinopathy the 1994 neyman memorial lecture let y i i 1 n be independent observations with the density of y i of the form h y i f i exp y i f i b f i c y i where b and c are given functions and b is twice continuously differentiable and bounded away from 0 let i t f t i where t t 1 t d t 1 t d t the t are measurable spaces of rather general form and f is an unknown function on t with some assumed smoothness properties given y i t i i 1 n it is desired to estimate f t for t in some region of interest contained in t we develop the fitting of smoothing spline anova models to this data of the form f t c f t f t t the components of the decomposition satisfy side conditions which generalize the usual side conditions for parametric anova the estimate of f is obtained as the minimizer in an appropriate function space of l y f j f j f where l y f is the negative log likelihood of y y 1 y n given f the j j are quadratic penalty functionals and the anova decomposition is terminated in some manner there are five major parts required to turn this program into a practical data analysis tool 1 methods for deciding which terms in the anova decomposition to include model selection 2 methods for choosing good values of the smoothing parameters 3 methods for making confidence statements concerning the estimate 4 numerical algorithms for the calculations and finally 5 public software in this paper we carry out this program relying on earlier work and filling in important gaps the overall scheme is applied to bernoulli data from the wisconsin epidemiologic study of diabetic retinopathy to model the risk of progression of diabetic retinopathy as a function of glycosylated hemoglobin duration of diabetes and body mass index it is believed that the results have wide practical application to the analysis of data from large epidemiologic studies

soap film smoothing summary conventional smoothing methods sometimes perform badly when used to smooth data over complex domains by smoothing inappropriately across boundary features such as peninsulas solutions to this smoothing problem tend to be computationally complex and not to provide model smooth functions which are appropriate for incorporating as components of other models such as generalized additive models or mixed additive models we propose a class of smoothers that are appropriate for smoothing over difficult regions of 2 which can be represented in terms of a low rank basis and one or two quadratic penalties the key features of these smoothers are that they do not smooth across boundary features that their representation in terms of a basis and penalties allows straightforward incorporation as components of generalized additive models mixed models and other non standard models that smoothness selection for these model components is straightforward to accomplish in a computationally efficient manner via generalized cross validation akaike s information criterion or restricted maximum likelihood for example and that their low rank means that their use is computationally efficient

inference in generalized additive mixed modelsby using smoothing splines generalized additive mixed models are proposed for overdispersed and correlated data which arise frequently in studies involving clustered hierarchical and spatial designs this class of models allows flexible functional dependence of an outcome variable on covariates by using nonparametric regression while accounting for correlation between observations by using random effects we estimate nonparametric functions by using smoothing splines and jointly estimate smoothing parameters and variance components by using marginal quasi likelihood because numerical integration is often required by maximizing the objective functions double penalized quasi likelihood is proposed to make approximate inference frequentist and bayesian inferences are compared a key feature of the method proposed is that it allows us to make systematic inference on all model components within a unified parametric mixed model framework and can be easily implemented by fitting a working generalized linear mixed model by using existing statistical software a bias correction procedure is also proposed to improve the performance of double penalized quasi likelihood for sparse data we illustrate the method with an application to infectious disease data and we evaluate its performance through simulation

fast stable direct fitting and smoothness selection for generalized additive models summary existing computationally efficient methods for penalized likelihood generalized addi tive model fitting employ iterative smoothness selection on working linear models or working mixed models such schemes fail to converge for a non negligible proportion of models with failure being particularly frequent in the presence of concurvity if smoothness selection is per formed by optimizing whole model criteria these problems disappear but until now attempts to do this have employed finite difference based optimization schemes which are computationally inefficient and can suffer from false convergence the paper develops the first computationally efficient method for direct generalized additive model smoothness selection it is highly sta ble but by careful structuring achieves a computational efficiency that leads in simulations to lower mean computation times than the schemes that are based on working model smoothness selection the method also offers a reliable way of fitting generalized additive mixed models

some asymptotic results on generalized penalized spline smoothing the paper discusses asymptotic properties of penalized spline smoothing if the spline basis increases with the sample size the proof is provided in a generalized smoothing model allowing for non normal responses the results are extended in two ways first assuming the spline coefficients to be a priori normally distributed links the smoothing framework to generalized linear mixed models we consider the asymptotic rates such that the laplace approximation is justified and the resulting fits in the mixed model correspond to penalized spline estimates secondly we make use of a fully bayesian viewpoint by imposing an a priori distribution on all parameters and coefficients we argue that with the postulated rates at which the spline basis dimension increases with the sample size the posterior distribution of the spline coefficients is approximately normal the validity of this result is investigated in finite samples by comparing markov chain monte carlo results with their asymptotic approximation in a simulation study

semiparametric analysis of variance with tensor product thin plate splines we consider fitting multivariate models to response data based on analysis of variance anova like decompositions of functions of several variables f t 1 t d c f t f t t a theory for fitting some components of these models with polynomial smoothing splines exists when each t is in a subset of the real line in this case the various estimated components turn out to be certain tensor sums and products of polynomial splines this approach may not be natural when one or more of the variables are geographic in particular where nature does not know north from east

modeling and forecasting daily electricity load curves a hybrid approach we propose a hybrid approach for the modeling and the short term forecasting of electricity loads two building blocks of our approach are 1 modeling the overall trend and seasonality by fitting a generalized additive model to the weekly averages of the load and 2 modeling the dependence structure across consecutive daily loads via curve linear regression for the latter a new methodology is proposed for linear regression with both curve response and curve regressors the key idea behind the proposed methodology is dimension reduction based on a singular value decomposition in a hilbert space which reduces the curve regression problem to several ordinary i e scalar linear regression problems we illustrate the hybrid method using french electricity loads between 1996 and 2009 on which we also compare our method with other available models including the electricite de france operational model supplementary materials for this article are available online

bayesian generalized additive models for location scale and shape for zero inflated and overdispersed count data frequent problems in applied research preventing the application of the classical poisson log linear model for analyzing count data include overdispersion an excess of zeros compared to the poisson distribution correlated responses as well as complex predictor structures comprising nonlinear effects of continuous covariates interactions or spatial effects we propose a general class of bayesian generalized additive models for zero inflated and overdispersed count data within the framework of generalized additive models for location scale and shape where semiparametric predictors can be specified for several parameters of a count data distribution as standard options for applied work we consider the zero inflated poisson the negative binomial and the zero inflated negative binomial distribution the additive predictor specifications rely on basis function approximations for the different types of effects in combination with gaussian smoothness priors we develop bayesian inference based on markov c

on a system of difference equations abstract we show that the system of difference equations x n 1 ax n 1 by n x n 1 c y n 1 y n 1 x n y n 1 n n 0 where the parameters a b c and initial values x 1 x 0 y 1 y 0 are real numbers can be solved considerably improving the results in the literature

effects of categorization method regression type and variable distribution on the inflation of type i error rate when categorizing a confounding variable the loss of signal associated with categorizing a continuous variable is well known and previous studies have demonstrated that this can lead to an inflation of type i error when the categorized variable is a confounder in a regression analysis estimating the effect of an exposure on an outcome however it is not known how the type i error may vary under different circumstances including logistic versus linear regression different distributions of the confounder and different categorization methods here we analytically quantified the effect of categorization and then performed a series of 9600 monte carlo simulations to estimate the type i error inflation associated with categorization of a confounder under different regression scenarios we show that type i error is unacceptably high 10 in most scenarios and often 100 the only exception was when the variable categorized was a continuous mixture proxy for a genuinely dichotomous latent variable where both the continuous proxy and the categorized variable are error ridden proxies for the dichotomous latent variable as expected error inflation was also higher with larger sample size fewer categories and stronger associations between the confounder and the exposure or outcome we provide online tools that can help researchers estimate the potential error inflation and understand how serious a problem this is copyright 2014 john wiley sons ltd

periodic solutions of difference equations abstract in this paper we discuss the existence of periodic solutions of periodic difference equations including x n 1 f n x n n z where x and f are d vectors and z denotes the set of integers as well as scalar autonomous difference equations including x n f x n 1 n z first we state some recent results concerning periodic solutions of periodic difference equations then we show the existence of periodic solutions of scalar autonomous difference equations by using schauder s fixed point theorem we illustrate the results with some examples