on model diagnostics using varying coefficient models summary the paper motivates the use of varying coefficient models for diagnostics in regression models with continuous and factorial covariates the varying coefficient model which is restricted only by assumptions concerning the smoothness of the effects is considered as the alternative to a parametric generalised linear model estimation is based on local likelihood smoothing and asymptotic properties of the estimators are derived for the investigation of the discrepancy between the parametric model and the smooth alternative a simple graphical procedure is proposed moreover tests are derived which allow for overall and componentwise investigation the componentwise tests may be used to examine interaction effects of continuous and factorial regressors

efficient estimation and inferences for varying coefficient models abstract this article deals with statistical inferences based on the varying coefficient models proposed by hastie and tibshirani local polynomial regression techniques are used to estimate coefficient functions and the asymptotic normality of the resulting estimators is established the standard error formulas for estimated coefficients are derived and are empirically tested a goodness of fit test technique based on a nonparametric maximum likelihood ratio type of test is also proposed to detect whether certain coefficient functions in a varying coefficient model are constant or whether any covariates are statistically significant in the model the null distribution of the test is estimated by a conditional bootstrap method our estimation techniques involve solving hundreds of local likelihood equations to reduce the computational burden a one step newton raphson estimator is proposed and implemented the resulting one step procedure is shown to save computational cost on an order of tens with no

variance component testing in generalised linear models with random effects summary there is considerable interest in testing for overdispersion correlation and heterogeneity across groups in biomedical studies in this paper we cast the problem in the framework of generalised linear models with random effects we propose a global score test for the null hypothesis that all the variance components are zero this test is a locally asymptotically most stringent test and is robust in the special sense that the test does not require specifying the joint distribution of the random effects we also propose individual score tests and their approximations for testing the variance components separately both tests can be easily implemented using existing statistical software we illustrate these tests with an application to the study of heterogeneity of mating success across males and females in an experiment on salamander matings and evaluate their performance through simulation

the rigidity of embedded constant mean curvature surfaces we study the rigidity of complete embedded constant mean curvature surfaces in r 3 among other things we prove that when such a surface has finite genus then intrinsic isometries of the surface extend to isometries of r 3 or its isometry group contains an index two subgroup of isometries that extend mathematics subject classification primary 53a10 secondary 49q05 53c42

a goodness of fit test for binary regression models based on smoothing methods a new global test statistic for models with continuous covariates and binary response is introduced the test statistic is based on nonparametric kernel methods explicit expressions are given for the mean and variance of the test statistic asymptotic properties are considered and approximate corrections due to parameter estimation are presented properties of the test statistic are studied by simulation the goodness of fit method is illustrated on data from a dutch follow up study on preterm infants recommendations for practitioners are given

varying coefficient models we explore a class of regression and generalized regression models in which the coefficients are allowed to vary as smooth functions of other variables general algorithms are presented for estimating the models flexibly and some examples are given this class of models ties together generalized additive models and dynamic generalized linear models into one common framework when applied to the proportional hazards model for survival data this approach provides a new way of modelling departures from the proportional hazards assumption

tests of homogeneity for generalized linear models abstract we propose two tests for testing homogeneity among clustered data adjusting for the effects of covariates the first is a score test for a generalized linear model with random effect in which the distribution of the response variable given the random effect is entirely defined in contrast to the likelihood ratio test however the score test does not require estimation of the parameters of a mixed effects model nor specification of the mixing distribution the second test is proposed in the framework of the generalized estimating equation gee approach in deriving this test we need only the specification of the marginal expectation and variance of the response variable and the fourth moment for the overdispersion term whereas for deriving the score test for mixed effects models the entire conditional distribution must be specified we demonstrate that the two tests are identical when the covariance matrix assumed in the gee approach is that of the random effects model in both approaches

testing the fit of a regression model via score tests in random effects models this paper considers testing the goodness of fit of regression models emphasis is on a goodness of fit test for generalized linear models with canonical link function and known dispersion parameter the test is based on the score test for extra variation in a random effects model by choosing a suitable form for the dispersion matrix a goodness of fit test statistic is obtained which is quite similar to test statistics based on non parametric kernel methods we consider the distribution of the test statistic and discuss the choice of the dispersion matrix the testing method can handle models with continuous and discrete covariates corrections for bias when parameters are estimated are available and extensions to models with unknown dispersion parameters and more general nonlinear models are discussed the proposed goodness of fit method is demonstrated in a simulation study and on real data of bone marrow transplant patients the individual contributions of observations to the test statistic are used to perform residual analyses

testing the parametric null model hypothesis in semiparametric partial and generalized spline models on etend les resultats de cox et koh 1986 aux modeles generalises de wahba 1985 et a divers autres modeles splines partiels on montre que la statistique de test a une relation etroite avec le comportement de la fonction de validation croisee generalisee

testing for overdispersion in poisson and binomial regression models abstract in this article a method for obtaining tests for overdispersion with respect to a natural exponential family is derived the tests are designed to be powerful against arbitrary alternative mixture models where only the first two moments of the mixed distribution are specified various tests for extra poisson and extra binomial variation are obtained as special cases the use of a particular test may be motivated by a consideration of the mechanism through which the overdispersion may arise the common occurrence of extra poisson and extra binomial variation has been noted by several authors however the poisson and binomial models remain valid in many instances and because of their simplicity and appeal it is of real interest to ascertain when they apply this paper develops a unifying theory for testing for overdispersion and generalizes tests previously derived including those by fisher 1950 collings and margolin 1985 and prentice 1986 it also shows the pearson statistic to be a sc

comparing nonparametric versus parametric regression fits in general there will be visible differences between a parametric and a nonparametric curve estimate it is therefore quite natural to compare these in order to decide whether the parametric model could be justified an asymptotic quantification is the distribution of the integrated squared difference between these curves we show that the standard way of bootstrapping this statistic fails we use and analyse a different form of bootstrapping for this task we call this method the wild bootstrap and apply it to fitting engel curves in expenditure data analysis

a locally most powerful test for homogeneity with many strata summary a locally most powerful test is proposed for homogeneity of nuisance parameters from many strata in the mixed model setting the test is based on the difference between the square of the score and the observed information about the nuisance parameters all evaluated at the null hypothesis several examples including constant normal mean with possibly different variances and matched case control studies are presented in detail the relative efficiencies of the unstratified estimator of parameter of interest versus the stratified one are calculated two data sets on matched case control studies are used for illustration

restricted likelihood ratio lack of fit tests using mixed spline models penalized regression spline models afford a simple mixed model representation in which variance components control the degree of non linearity in the smooth function estimates this motivates the study of lack of fit tests based on the restricted maximum likelihood ratio statistic which tests whether variance components are 0 against the alternative of taking on positive values for this one sided testing problem a further complication is that the variance component belongs to the boundary of the parameter space under the null hypothesis conditions are obtained on the design of the regression spline models under which asymptotic distribution theory applies and finite sample approximations to the asymptotic distribution are provided test statistics are studied for simple as well as multiple regression models

testing the goodness of fit of a linear model via nonparametric regression techniques abstract this article investigates the use of nonparametric regression methodology to test the adequacy of a parametric linear model the large sample properties of parametric goodness of fit tests for linearity are considered the inadequacies of such tests lead to the proposal of new tests that are constructed from nonparametric regression fits to the residuals from linear regression large sample theory is derived for two variants of this type of statistic the results demonstrate that such tests are consistent against all fixed smooth alternatives to linearity but are incapable of detecting local alternatives converging to a linear model at the parametric rate n 1 2 simulation experiments involving a test based on fitting cubic smoothing splines to residuals reveals that this test has good power properties against several reasonable alternatives

generalized score test of homogeneity based on correlated random effects models summary we present a family of tests based on correlated random effects models which provides a synthesis and a generalization of recent work on homogeneity testing in these models each subject has a particular random effect but the random effects between subjects are correlated we derive the general form of the score statistic for testing that the random effects have a variance equal to 0 we apply this result to both parametric and semiparametric models in both cases we show that under certain conditions the score statistic has an asymptotic normal distribution we consider several applications of this theory including overdispersion heterogeneity between groups spatial correlations and genetic linkage

smoothing spline based score tests for proportional hazards models we propose score type tests for the proportional hazards assumption and for covariate effects in the cox model using the natural smoothing spline representation of the corresponding nonparametric functions of time or covariate the tests are based on the penalized partial likelihood and are derived by viewing the inverse of the smoothing parameter as a variance component and testing an equivalent null hypothesis that the variance component is zero we show that the tests have a size close to the nominal level and good power against general alternatives and we apply them to data from a cancer clinical trial

a goodness of fit test for cox s proportional hazards model based on martingale residuals a goodness of fit test for cox s proportional hazards model is proposed on the basis of martingale residuals the test is derived as the score test for the presence of an extra random effect in the exponential part of the hazard function assumptions about the distribution of the random effects are not made only the correlation matrix must be specified this can be done on the basis of an existing division of individuals into groups such as families or on the basis of the distance between observed covariates the variance of the test statistic is derived using a counting process approach corrections for the estimation of parameters are available the test is demonstrated in a simulated data set as well as in a real data set on ovarian cancer

testing goodness of fit in regression a general approach for specified alternatives when fitting generalized linear models or the cox proportional hazards model it is important to have tools to test for lack of fit because lack of fit comes in all shapes and sizes distinguishing among different types of lack of fit is of practical importance we argue that an adequate diagnosis of lack of fit requires a specified alternative model such specification identifies the type of lack of fit the test is directed against so that if we reject the null hypothesis we know the direction of the departure from the model the goodness of fit approach of this paper allows to treat different types of lack of fit within a unified general framework and to consider many existing tests as special cases connections with penalized likelihood and random effects are discussed and the application of the proposed approach is illustrated with medical examples tailored functions for goodness of fit testing have been implemented in the r package globaltest copyright 2012 john wiley sons ltd

on the use of nonparametric regression for checking linear relationships summary the problem of checking the linearity of a regression relationship is addressed through the idea of smoothing of a residual plot a pseudolikelihood ratio test statistic which measures the distance between the nonparametric and the parametric models is derived as a ratio of quadratic forms the distribution of this statistic under the null hypothesis of linearity is calculated numerically by using johnson curves a power study shows the new statistic to be more sensitive to non linearity than the durbin watson statistic

exact likelihood ratio tests for penalised splines summary penalised spline based additive models allow a simple mixed model representation where the variance components control departures from linear models the smoothing parameter is the ratio of the random coefficient and error variances and tests for linear regression reduce to tests for zero random coefficient variances we propose exact likelihood and restricted likelihood ratio tests for testing polynomial regression versus a general alternative modelled by penalised splines their spectral decompositions are used as the basis of fast simulation algorithms we derive the asymptotic local power properties of the tests under weak conditions in particular we characterise the local alternatives that are detected with asymptotic probability one confidence intervals for the smoothing parameter are obtained by inverting the tests for a fixed smoothing parameter versus a general alternative we discuss f and r tests and show that ignoring the variability in the smoothing parameter estimator can have a dramatic effect on their null distributions the powers of several known tests are investigated and a small set of tests with good power properties is identified the restricted likelihood ratio test is among the best in terms of power

testing against a high dimensional alternative summary as the dimensionality of the alternative hypothesis increases the power of classical tests tends to diminish quite rapidly this is especially true for high dimensional data in which there are more parameters than observations we discuss a score test on a hyperparameter in an empirical bayesian model as an alternative to classical tests it gives a general test statistic which can be used to test a point null hypothesis against a high dimensional alternative even when the number of parameters exceeds the number of samples this test will be shown to have optimal power on average in a neighbourhood of the null hypothesis which makes it a proper generalization of the locally most powerful test to multiple dimensions to illustrate this new locally most powerful test we investigate the case of testing the global null hypothesis in a linear regression model in more detail the score test is shown to have significantly more power than the f test whenever under the alternative the large variance principal components of the design matrix explain substantially more of the variance of the outcome than do the small variance principal components the score test is also useful for detecting sparse alternatives in truly high dimensional data where its power is comparable with the test based on the maximum absolute t statistic

testing against a high dimensional alternative in the generalized linear model asymptotic type i error control summary testing a low dimensional null hypothesis against a high dimensional alternative in a generalized linear model may lead to a test statistic that is a quadratic form in the residuals under the null model using asymptotic arguments we show that the distribution of such a test statistic can be approximated by a ratio of quadratic forms in normal variables for which algorithms are readily available for generalized linear models the asymptotic distribution shows good control of type i error for moderate to small samples even when the number of covariates in the model far exceeds the sample size

spline based tests in survival analysis this paper examines a method for testing hypotheses on covariate effects in a proportional hazards model and also on how effects change over time in regression analysis of survival data the technique used is very general and can be applied to testing many other aspects of parametric and semiparametric models the basic idea is to formulate a flexible parametric alternative using fixed knot splines together with a penalty function that penalizes noisy alternatives more than smooth ones to focus the power of the tests toward smooth alternatives the test statistics are the analogs of ordinary likelihood based statistics only computed from a penalized likelihood formed by subtracting the penalty function from the ordinary log likelihood large sample approximations to the distributions are found when the number of knots is held fixed as the sample size increases numerical results suggest these approximations may be adequate with moderate sized samples

a goodness of fit test for multinomial logistic regression this article presents a score test to check the fit of a logistic regression model with two or more outcome categories the null hypothesis that the model fits well is tested against the alternative that residuals of samples close to each other in covariate space tend to deviate from the model in the same direction we propose a test statistic that is a sum of squared smoothed residuals and show that it can be interpreted as a score test in a random effects model by specifying the distance metric in covariate space users can choose the alternative against which the test is directed making it either an omnibus goodness of fit test or a test for lack of fit of specific model variables or outcome categories

tests for variation over groups in survival data abstract when survival data can be divided into natural groups such as in a clinical trial with patients from a number of different institutions the question arises whether effects are different for the different groups for situations where one is concerned with a random pattern of variation over groups this article proposes testing the null hypothesis of no variation using statistics based on the sums of martingale residuals over subjects within groups these tests are closely related to the locally most powerful tests of liang for testing for variation across a large number of strata the tests are applied to data from a lung cancer trial conducted by the eastern cooperative oncology group

inference in smoothing spline analysis of variance summary smoothing spline analysis of variance decomposes a multivariate function into additive components this decomposition not only provides an efficient way to model a multivariate function but also leads to meaningful inference by testing whether a certain component equals 0 no formal procedure is yet available to test such a hypothesis we propose an asymptotic method based on the likelihood ratio to test whether a functional component is 0 this test allows us to choose an optimal model and to compare groups of curves we first develop the general theory by exploiting the connection between mixed effects models and smoothing splines we then apply this to compare two groups of curves and to select an optimal model in a two dimensional problem a small simulation is used to assess the finite sample performance of the likelihood ratio test