Checking a Semiparametric Additive Risk Model

McKeague and Sasieni [A partly parametric additive risk model. Biometrika 81 (1994) 501] propose a restriction of Aalen’s additive risk model by the additional hypothesis that some of the covariates have time-independent influence on the intensity of the observed counting process. We introduce goodness-of-fit tests for this semiparametric Aalen model. The asymptotic distribution properties of the test statistics are derived by means of martingale techniques. The tests can be adjusted to detect particular alternatives. As one of the most important alternatives we consider Cox’s proportional hazards model. We present simulation studies and an application to a real data set.     

Standardized Martingale Residuals Applied to Grouped Left Truncated Observations of Dementia Cases

The use of martingale residuals have been proposed for modelchecking and also to get a non-parametric estimate of the effectof an explanatory variable. We apply this approach to an epidemiologicalproblem which presents two characteristics: the data are lefttruncated due to delayed entry in the cohort; the data are groupedinto geographical units (parishes). This grouping suggests anatural way of smoothing the graph of residuals which is to computethe sum of the residuals for each parish. It is also naturalto present a graph with standardized residuals. We derive thevariances of the estimated residuals for left truncated datawhich allows computing the standardized residuals. This methodis applied to the study of dementia in a cohort of old people,and to the possible effect of the concentration of aluminum andsilica in drinking water on the risk of developing dementia.     

RECURSIVE ESTIMATION OF THE GENERAL LINEAR MODEL WITH DEPENDENT ERRORS AND MULTIPLE ADDITIONAL OBSERVATIONS1

The techniques for recursive estimation of the general linear model with dependent errors and known second order properties, is generalised to allow for simultaneous addition of an arbitrary number of additional observations. Computational formulae for recursive updating of parameter estimates are derived, together with a sequence of univariate recursive residuals for testing the constancy of the regression relation over time.     

TESTING THE STABILITY OF REGRESSION COEFFICIENTS USING GENERALIZED RECURSIVE RESIDUALS

Simple but flexible methods to detect deviations from the assumption of constant coefficients in linear regression are presented. Based on recursive residuals a runs test is developed as an alternative to CUSUM- and MOSUM-techniques. Finally a simulation study gives insight into the new method.     

APPLICATIONS OF ROBUST ESTIMATION FOR VARIANCE COMPONENTS MODELS:THE DETECTION OF MAJOR GENE EFFECTS IN FINGER RIDGE COUNTS IN NORMAL AND FRAGILE X FAMILIES

A diagnostic technique is proposed to detect major gene effects and other systematic departures from a model for the trait means in the presence of outliers. The technique is based on the examination of residuals from fitting variance components models to quantitative pedigree data using robust statistical procedures. The approach is demonstrated using the total ridge count and ridge count of the middle finger from 54 extended families affected with the Fragile X syndrome, and a sample of 217 normal pedigrees.     

Checking the censored two-sample accelerated life model using integrated cumulative hazard difference

In this paper, new statistical tests for the censored two-sample accelerated life model are discussed. From the estimating functions using integrated cumulative hazard difference, stochastic processes are constructed. They can be described by martingale residuals, and, given the data, conditional distributions can be approximated by zero mean Gaussian processes. The new methods, based on these processes, provide asymptotically consistent tests against a general departure from the model. A graphical method is also discussed. In various numerical studies, the new tests performed better than the existing method, especially when the hazard curves cross. The proposed procedures are illustrated with two real data sets.     

Diagnosing explainable heterogeneity of variance in random‐effects models

Data‐analytic tools for models other than the normal linear regression model are relatively rare. Here we develop plots and diagnostic statistics for nonconstant variance for the random‐effects model (REM). REMs for longitudinal data include both within‐ and between‐subject variances. A basic assumption is that the two variance terms are constant across subjects. However, we often find that these variances are functions of covariates, and the data set has what we call explainable heterogeneity, which needs to be allowed for in the model. We characterize several types of heterogeneity of variance in REMs and develop three diagnostic tests using the score statistic: one for each of the two variance terms, and the third for a form of multivariate nonconstant variance. For each test we present an adjusted residual plot which can identify cases that are unusually influential on the outcome of the test.     

Theory & Methods: Residual Diagnostic Plots for Checking for Model Mis‐specification in Time Series Regression

This paper considers residuals for time series regression. Despite much literature on visual diagnostics for uncorrelated data, there is little on the autocorrelated case. To examine various aspects of the fitted time series regression model, three residuals are considered. The fitted regression model can be checked using orthogonal residuals; the time series error model can be analysed using marginal residuals; and the white noise error component can be tested using conditional residuals. When used together, these residuals allow identification of outliers, model mis‐specification and mean shifts. Due to the sensitivity of conditional residuals to model mis‐specification, it is suggested that the orthogonal and marginal residuals be examined first.     

Correcting double outward box distributed residuals by WCEV

Double outward box distributed residuals are another type of non monotonic heteroscedasticity that severely violates homoscedasticity assumption. In this study Çelik's (2015 Çelik, R. (2015). Stabilizing heteroscedasticity for butterfly-distributed residuals by the weighting absolute centered external variable. J. Appl. Stat. 42(4):705–721.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) WCEV is applied to double outward box distributed residuals to provide homoscedasticity for simple and multiple regression models.     

Performance of asymmetric links and correction methods for imbalanced data in binary regression

In binary regression, imbalanced data result from the presence of values equal to zero (or one) in a proportion that is significantly greater than the corresponding real values of one (or zero). In this work, we evaluate two methods developed to deal with imbalanced data and compare them to the use of asymmetric links. The results based on simulation study show, that correction methods do not adequately correct bias in the estimation of regression coefficients and that the models with power links and reverse power considered produce better results for certain types of imbalanced data. Additionally, we present an application for imbalanced data, identifying the best model among the various ones proposed. The parameters are estimated using a Bayesian approach, considering the Hamiltonian Monte-Carlo method, utilizing the No-U-Turn Sampler algorithm and the comparisons of models were developed using different criteria for model comparison, predictive evaluation and quantile residuals.