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Standard use of Cox regression requires collection of covariate information for all individuals in a cohort even when only a small fraction of them experiences the event of interest (fail). This may be very expensive for large cohorts. Further in biomarker studies, it will imply a waste of valuable biological material that one may want to save for future studies. A nested case-control study offers a useful alternative. For this design, covariate information is only needed for the failing individuals (cases) and a sample of controls selected from the cases' at-risk sets. Methods based on martingale residuals are useful for checking the fit of Cox's regression model for cohort data. But similar methods have so far not been developed for nested case-control data. In this article, it is described how one may define martingale residuals for nested case-control data, and it is shown how plots and tests based on cumulative sums of martingale residuals may be used to check model fit. The plots and tests may be obtained using available software., (© 2015, The International Biometric Society.)

The assessment of regression models with discrete outcomes is challenging and has many fundamental issues. With discrete outcomes, standard regression model assessment tools such as Pearson and deviance residuals do not follow the conventional reference distribution (normal) under the true model, calling into question the legitimacy of model assessment based on these tools. To fill this gap, we construct a new type of residuals for regression models with general discrete outcomes, including ordinal and count outcomes. The proposed residuals are based on two layers of probability integral transformation. When at least one continuous covariate is available, the proposed residuals closely follow a uniform distribution (or a normal distribution after transformation) under the correctly specified model. One can construct visualizations such as QQ plots to check the overall fit of a model straightforwardly, and the shape of QQ plots can further help identify possible causes of misspecification such as overdispersion. We provide theoretical justification for the proposed residuals by establishing their asymptotic properties. Moreover, in order to assess the mean structure and identify potential covariates, we develop an ordered curve as a supplementary tool, which is based on the comparison between the partial sum of outcomes and of fitted means. Through simulation, we demonstrate empirically that the proposed tools outperform commonly used residuals for various model assessment tasks. We also illustrate the workflow of model assessment using the proposed tools in data analysis. for this article are available online. [ABSTRACT FROM AUTHOR]

Semicontinuous outcomes commonly arise in a wide variety of fields, such as insurance claims, healthcare expenditures, rainfall amounts, and alcohol consumption. Regression models, including Tobit, Tweedie, and two-part models, are widely employed to understand the relationship between semicontinuous outcomes and covariates. Given the potential detrimental consequences of model misspecification, after fitting a regression model, it is of prime importance to check the adequacy of the model. However, due to the point mass at zero, standard diagnostic tools for regression models (eg, deviance and Pearson residuals) are not informative for semicontinuous data. To bridge this gap, we propose a new type of residuals for semicontinuous outcomes that is applicable to general regression models. Under the correctly specified model, the proposed residuals converge to being uniformly distributed, and when the model is misspecified, they significantly depart from this pattern. In addition to in-sample validation, the proposed methodology can also be employed to evaluate predictive distributions. We demonstrate the effectiveness of the proposed tool using health expenditure data from the US Medical Expenditure Panel Survey. [ABSTRACT FROM AUTHOR]

The simple linear regression model and the associated goodness-of-fit measure, the coefficient of determination, R2, are only appropriate when all measurement errors are associated with the measurement of the data in the dependent variable. When measurement errors are assumed in both variables, a Deming regression can be used; however, there is no associated R2-type measure for this specific type of regression. In this paper, we propose a measure, R g 2 which utilizes the minimum percentage improvement of the Deming regression over either the horizontal or the vertical line through the centroid of the data. We investigate some properties of this measure and its relation to R2. We also consider other candidate methodologies for a generalized R2 measure for a Deming regression model and investigate strengths and weaknesses of each as a way of beginning the conversation about which measure is the best and for which applications it is most suited. [ABSTRACT FROM AUTHOR]

In a multiple regression model setup, we examine the effects of trimming and winsorization of regressors on the regression estimates and their efficiency. The exact results developed here to enable us to evaluate the effects of winsorizing one regressor on the estimates of the coefficients associated with another regressor. These results have important implications in accounting and finance literature wherein trimming or winsorization is frequently adopted before performing regression analysis. [ABSTRACT FROM AUTHOR]

The financial market is a complex system with chaotic behavior that can lead to wild swings within the financial system. This can drive the system into a variety of interesting phenomenon such as phase transitions, bubbles, and crashes, and so on. Of interest in financial modelling is identifying the distribution and the stylized facts of a particular time series, as the distribution and stylized facts can determine if volatility is present, resulting in financial risk and contagion. Regression modelling has been used within this study as a methodology to identify the goodness-of-fit between the original and generated time series model, which serves as a criterion for model selection. Different time series modelling methods that include the common Box–Jenkins ARIMA, ARMA-GARCH type methods, the Geometric Brownian Motion type models and Tsallis entropy based models when data size permits, can use this methodology in model selection. Determining the time series distribution and stylized facts has utility, as the distribution allows for further modelling opportunities such as bivariate regression and copula modelling, apart from the usual forecasting. Determining the distribution and stylized facts also allows for the identification of the parameters that are used within a Geometric Brownian Motion forecasting model. This study has used the Carbon Emissions Futures price between the dates of 1 May 2012 and 1 May 2022, to highlight this application of regression modelling. [ABSTRACT FROM AUTHOR]

This study introduces a coefficient of discrimination for use with nominal and ordinal regression models. Computation of the coefficient is demonstrated with data from the Pew Research Center's 25th Anniversary of the Web Omnibus Survey pertaining to cell/home phone ownership, where the coefficient of discrimination indicates that respondent age and gender increased the probability of a correct versus incorrect classification by 13.9%. Additionally, the coefficient is compared to existing coefficients. [ABSTRACT FROM AUTHOR]

Statistical mediation analysis is used to investigate mechanisms through which a randomized intervention causally affects an outcome variable. Mediation analysis is often carried out in a pretest-posttest control group design because it is a common choice for evaluating experimental manipulations in the behavioral and social sciences. There are four different two-wave (i.e., pretest-posttest) mediation models that can be estimated using either linear regression or a Latent Change Score (LCS) specification in Structural Equation Modeling: Analysis of Covariance, difference and residualized change scores, and a cross-sectional model. Linear regression modeling and the LCS specification of the two-wave mediation models provide identical mediated effect estimates but the two modeling approaches differ in their assumptions of model fit. Linear regression modeling assumes each of the four two-wave mediation models fit the data perfectly whereas the LCS specification allows researchers to evaluate the model constraints implied by the difference score, residualized change score, and cross-sectional models via model fit indices. Therefore, the purpose of this paper is to provide a conceptual and statistical comparison of two-wave mediation models. Models were compared on the assumptions they make about time-lags and cross-lagged effects as well as statistically using both standard measures of model fit (χ2, RMSEA, and CFI) and newly proposed T -size measures of model fit for the two-wave mediation models. Overall, the LCS specification makes clear the assumptions that are often implicitly made when fitting two-wave mediation models with regression. In a Monte Carlo simulation, the standard model fit indices and newly proposed T -size measures of model fit generally correctly identified the best fitting two-wave mediation model. [ABSTRACT FROM AUTHOR]

The dependence of statistical validation parameters was investigated on the size of the sample taken in fit of multivariate linear curves. We observed that R2 and related internal parameters were misleading as they overestimated the goodness-of-fit of models at small sample size. Cross-validation metrics showed correct trends. It was possible to scale the leave-one-out and the leave-many-out results close to identical by correcting the degrees of freedom of the models. y and x-randomized validation parameters were calculated and the methods provided close to identical results. We suggest to use the simplest methods in both cases. The external parameters followed correct trends with respect to the sample size, but their sensitivity differed. We plotted the Roy-Ojha metrics in 2D and we coloured them with respect to other external parameters to provide an easy classification of models. The rank correlations were calculated between the performance parameters. Up to a sample size, goodness-of-fit and robustness were distinguishable, but above a certain sample size, the parameters were redundant. The external-internal pairs were weakly correlated. Our data show that all the three aspects of validation are necessary at small sample sizes, but the internal check of robustness is not informative above a given sample size. [ABSTRACT FROM AUTHOR]

It has been observed that certain economic and financial variables commonly exhibit switching behavior depending on their magnitude. This phenomenon in general cannot be naturally captured by the linear regression (LR), which assumes a linear relationship between the dependent and explanatory variables. To decipher investor behavior more appropriately by accounting for this observation, a switching-regime regression (SRR) is proposed and applied to the S&P 500 market return with respect to seven explanatory variables. It is shown that, compared with LR, the new regression results in a significantly improved adjusted R 2 , increasing from less than 4 % to over 50 % . In addition, SRR yields better out-of-sample forecasting performance, besides that the fitted values from the new regression even resemble the dip during the 2008 financial crisis, while those from LR do not. The study thus indicates that the switching-regime regression improves significantly the statistical properties including the goodness of fit as well as conforms more to investor behavior theory. [ABSTRACT FROM AUTHOR]

An exponentiated Weibull-geometric distribution is defined and studied. A new count data regression model, based on the exponentiated Weibull-geometric distribution, is also defined. The regression model can be applied to fit an underdispersed or an over-dispersed count data. The exponentiated Weibull-geometric regression model is fitted to two numerical data sets. The new model provided a better fit than the fit from its competitors. [ABSTRACT FROM AUTHOR]

• We introduce a new package for checking key assumptions in the Cox regression and Fine and Gray regression models. • It implements tests based on cumulative sums of residuals including Anderson-Darling type statistics and is available from CRAN R repositories. • Approximations to the null distribution were obtained using Monte-Carlo methods, and validated using extensive numerical studies. • It may help the spread of recent goodness-of-fit methods in medical applications and other fields, such as finance, criminology and industrial engineering studies. In this paper, we introduce a new R package goftte for goodness-of-fit assessment based on cumulative sums of model residuals useful for checking key assumptions in the Cox regression and Fine and Gray regression models. Monte-Carlo methods are used to approximate the null distribution of cumulative sums of model residuals. To limit the computational burden, the main routines used to approximate the null distributions are implemented in a parallel C++ programming environment. Numerical studies are carried out to evaluate the empirical type I error rates of the different testing procedures. The package and the documentation are available to users from CRAN R repositories. Results from simulation studies suggested that all statistical tests implemented in goftte yielded excellent control of the type I error rate even with modest sample sizes with high censoring rates. As compared to other R packages goftte provides new useful method for testing functionals, such as Anderson-Darling type test statistics for checking assumptions about proportional (sub-) distribution hazards. Approximations for the null distributions of test statistics have been validated through simulation experiments. Future releases will provide similar tools for checking model assumptions in multiplicative intensity models for recurrent data. The package may help to spread the use of recent advocated goodness-of-fit techniques in semiparametric regression for time-to-event data. [ABSTRACT FROM AUTHOR]

Motivated by the question of identifying the start time τ of financial bubbles, we propose an improved calibration approach for time series in which the inception of the latest regime of interest is unknown. By taking into account the tendency of a given model to overfit data, we introduce the Lagrange regularisation of the normalised sum of the squared residuals, χ n p 2 (Φ) , to endogenously detect the optimal fitting window size ≔ w ∗ ∈ [ τ : t ̄ 2 ] that should be used for calibration, assuming a fixed pseudo present time t ̄ 2. The Lagrange regularisation of χ n p 2 (Φ) defines the Lagrange regularised sum of the squared residuals, χ λ 2 (Φ). Its performance is exemplified on a simple Linear Regression problem with a change point and compared against the performances of the Residual Sum of Squares (RSS) ≔ χ 2 (Φ) and RSS/(N-p) ≔ χ n p 2 (Φ) , where N is the sample size, p is the number of degrees of freedom and Φ is the parameter vector. Applied to synthetic models of financial bubbles with a well-defined transition regime and to a number of financial time series (US S&P500, Brazil IBovespa and China SSEC Indices), χ λ 2 (Φ) is found to provide well-defined reasonable determinations of the starting times for major bubbles such as the bubbles ending with the 1987 Black-Monday, the 2008 Sub-prime crisis and minor speculative bubbles on other Indexes, without any further exogenous information. The application of the method thus allows one to endogenise the determination of the starting time of bubbles, a problem that has yet not received a systematic objective solution. Moreover, the technique appears as a practical solution for comparing goodness-of-fit across unbalanced sample sizes. [ABSTRACT FROM AUTHOR]

Weighted regression approach is one of the popular problems in robust regression analysis. Recently, robust fuzzy regression models have proven to be alternative approaches to fuzzy regression models attempting to identify, down-weight and/or ignore unusual points (outliers). This paper proposes a new robust fuzzy regression modeling technique known as weighted least squares (LS) fuzzy regression to construct a model for crisp input-fuzzy output data. We introduce a new weighted objective function to overcome the disadvantages of the ordinary LS approach in the presence of outliers. We derive and describe an iterative reweighted algorithm for minimization of the objective function. The algorithm is presented to approximate the weighted estimators of the fuzzy regression by solving the weighted optimization problem. The proposed algorithm decreases the affect of outliers on the model fit attempting to identify/down-weight them. To this end, experiments on datasets with different numbers of outliers are performed. The accuracy of our approach in a real setting is also tested on establishing a predictive model for evaluation of suspended load based on a real world dataset in hydrology engineering. The numerical results show that in the presence of unusual points the proposed weighted fit tracks the main body of the data considerably better than the ordinary LS fuzzy regression fit both in terms of the selected performance criteria and in terms of identifying and down weighting unusual data (outliers). The results of the numerical examples show that this approach has the capability to examine the behavior of value changes in the goodness-of-fit criteria of the fuzzy regression models when the downweighted observations are omitted. [ABSTRACT FROM AUTHOR]

A methodology is proposed to compare the power of normality tests with a wide variety of alternative unimodal distributions. It is based on the representation of a distribution mosaic in which kurtosis varies vertically and skewness horizontally. The mosaic includes distributions such as exponential, Laplace or uniform, with normal occupying the centre. Simulation is used to determine the probability of a sample from each distribution in the mosaic being accepted as normal. We demonstrate our proposal by applying it to the analysis and comparison of some of the most well-known tests. [ABSTRACT FROM AUTHOR]

Summary: In recent years, M‐quantile regression has been applied to small area estimation to obtain reliable and outlier robust estimators without recourse to strong parametric assumptions. In this paper, after a review of M‐quantile regression and its application to small area estimation, we cover several topics related to model specification and selection for M‐quantile regression that received little attention so far. Specifically, a pseudo‐R2 goodness‐of‐fit measure is proposed, along with likelihood ratio and Wald type tests for model specification. A test to assess the presence of actual area heterogeneity in the data is also proposed. Finally, we introduce a new estimator of the scale of the regression residuals, motivated by a representation of the M‐quantile regression estimation as a regression model with Generalised Asymmetric Least Informative distributed error terms. The Generalised Asymmetric Least Informative distribution, introduced in this paper, generalises the asymmetric Laplace distribution often associated to quantile regression. As the testing procedures discussed in the paper are motivated asymptotically, their finite sample properties are empirically assessed in Monte Carlo simulations. Although the proposed methods apply generally to M‐quantile regression, in this paper, their use ar illustrated by means of an application to Small Area Estimation using a well known real dataset. [ABSTRACT FROM AUTHOR]

Abstract: In statistical modelling, it is often of interest to evaluate non‐negative quantities that capture heterogeneity in the population such as variances, mixing proportions and dispersion parameters. In instances of covariate‐dependent heterogeneity, the implied homogeneity hypotheses are nonstandard and existing inferential techniques are not applicable. In this paper, we develop a quasi‐score test statistic to evaluate homogeneity against heterogeneity that varies with a covariate profile through a regression model. We establish the limiting null distribution of the proposed test as a functional of mixtures of chi‐square processes. The methodology does not require the full distribution of the data to be entirely specified. Instead, a general estimating function for a finite dimensional component of the model, that is, of interest is assumed but other characteristics of the population are left completely unspecified. We apply the methodology to evaluate the excess zero proportion in zero‐inflated models for count data. Our numerical simulations show that the proposed test can greatly improve efficiency over tests of homogeneity that neglect covariate information under the alternative hypothesis. An empirical application to dental caries indices demonstrates the importance and practical utility of the methodology in detecting excess zeros in the data. [ABSTRACT FROM AUTHOR]

A new five-parameter model called the exponentiated Marshall-Olkin Frèchet distribution is studied. Various of its mathematical properties including ordinary and incomplete moments, quantile and generating functions and order statistics are investigated. The proposed density function can be expressed as a linear mixture of Frèchet densities. The maximum likelihood method is used to estimate the model parameters. The flexibility of the new distribution is proved empirically using two real data sets. [ABSTRACT FROM AUTHOR]

We provide a simple and practical, yet flexible, penalized estimation method for a Cox proportional hazards model with current status data. We approximate the baseline cumulative hazard function by monotone B-splines and use a hybrid approach based on the Fisher-scoring algorithm and the isotonic regression to compute the penalized estimates. We show that the penalized estimator of the nonparametric component achieves the optimal rate of convergence under some smooth conditions and that the estimators of the regression parameters are asymptotically normal and efficient. Moreover, a simple variance estimation method is considered for inference on the regression parameters. We perform 2 extensive Monte Carlo studies to evaluate the finite-sample performance of the penalized approach and compare it with the 3 competing R packages: C1.coxph, intcox, and ICsurv. A goodness-of-fit test and model diagnostics are also discussed. The methodology is illustrated with 2 real applications. [ABSTRACT FROM AUTHOR]

We propose two new tests for symmetry based on well-known characterisations of symmetric distributions. The performance of the new tests is evaluated and compared to that of other existing tests by means of a Monte Carlo study. All tests are carried out in a regression setup where we test whether the error distribution in a linear regression model is symmetric. It is found that the newly proposed tests perform favourably compared to the other tests. [ABSTRACT FROM AUTHOR]

The article presents the discussion on accounting for non stationarity in both spatial and spatio-temporal processes. Topics include justifying more complicated nonstationary approaches showing stationarity being assumed for the sake of mathematical convenience; and propose partitioning the spatial or spatio-temporal data into subregions using the K-means algorithm based on a set of appropriate geographic features.

Consider the following semiparametric transformation model Λ θ ( Y ) = m ( X ) + ε , where X is a d -dimensional covariate, Y is a univariate dependent variable and ε is an error term with zero mean and which is independent of X . We assume that m is an unknown regression function and that { Λ θ : θ ∈ Θ } is a parametric family of strictly increasing functions. We use a profile likelihood estimator for the parameter θ and a local polynomial estimator for m . Our goal is to develop a new test for the parametric form of the regression function m , which has power against local directional alternatives that converge to the null model at parametric rate, and to compare its performance to that of the test proposed by Colling and Van Keilegom (2016). The idea of the new test is to compare the integrated regression function estimated in a semiparametric way to the integrated regressionfunction estimated under the null hypothesis. We consider two different test statistics, a Kolmogorov–Smirnov and a Cramér–von Mises type statistic, and establish the limiting distributions of these two test statistics under the null hypothesis and under a local alternative. We use a bootstrap procedure to approximate the critical values of the test statistics under the null hypothesis. Finally, a simulation study is carried out to illustrate the performance of our testing procedure, to compare this new test to the previous one and to see under which model conditions which test behaves the best. We also apply both methods on a real data set. [ABSTRACT FROM AUTHOR]

In this paper we propose influence measures for two basic goodness-of-fit statistics, the coefficient of determination R2 and test statistic F in the linear regression model using the deletion method. Some useful lemmas are provided.We also express the influence measures in terms of basic building blocks such as residual, leverage, and deviation that showed them as increasing function of residuals and a decreasing function of deviation. Further, the proposed measure reduces computational burden from O(n) to O(1). As illustrative examples, we applied the proposed measures to the stackloss data sets.We verified that deletion of one or few influential observations may result in big change in R2 and F-statistic. [ABSTRACT FROM AUTHOR]

Birnbaum–Saunders (BS) models are receiving considerable attention in the literature. Multivariate regression models are a useful tool of the multivariate analysis, which takes into account the correlation between variables. Diagnostic analysis is an important aspect to be considered in the statistical modeling. In this paper, we formulate multivariate generalized BS regression models and carry out a diagnostic analysis for these models. We consider the Mahalanobis distance as a global influence measure to detect multivariate outliers and use it for evaluating the adequacy of the distributional assumption. We also consider the local influence approach and study how a perturbation may impact on the estimation of model parameters. We implement the obtained results in theRsoftware, which are illustrated with real-world multivariate data to show their potential applications. [ABSTRACT FROM AUTHOR]

Objective: Multiple sclerosis (MS) is a chronic illness involving both controllable and uncontrollable stressors. The goodness-of-fit hypothesis posits that managing stressors effectively requires the use of different coping approaches in the face of controllable and uncontrollable stressors. To test the applicability of the goodness-of-fit model in a sample of adults with MS, we evaluated the ratio of 2 types of coping (an active problem-solving approach and an emotion-based meaning-focused approach) as a moderator of the relations between stress uncontrollability and mental health outcomes. Methods: Participants were veterans with MS (N = 90) receiving medical services through the Veterans Health Administration who completed telephone-based interviews. Regression analyses tested the interaction of stress uncontrollability and the problem- and meaning-focused coping ratio on anxious and depressive symptoms. Significant interactions were probed at 1 SD above the mean of coping (use of predominantly problem-focused coping) and 1 SD below the mean of coping (use of predominantly meaning-focused coping). Results: Findings largely supported the goodness-of-fit hypothesis. Anxiety and depressive symptoms were elevated when participants used more problem-focused strategies relative to meaning-focused strategies in the face of perceived uncontrollable stress. Conversely, symptoms of anxiety and depression were lower when uncontrollable stress was met with predominantly meaning-focused coping; however, the relations did not reach statistical significance. Conclusions: The impact of uncontrollable stressors on mental health outcomes for individuals with MS may vary depending on the degree to which problem-focused versus meaning-focused coping strategies are employed, lending support to the goodness-of-fit model. [ABSTRACT FROM AUTHOR]

Recently, Fine and Gray (J Am Stat Assoc 94:496-509, ) proposed a semi-parametric proportional regression model for the subdistribution hazard function which has been used extensively for analyzing competing risks data. However, failure of model adequacy could lead to severe bias in parameter estimation, and only a limited contribution has been made to check the model assumptions. In this paper, we present a class of analytical methods and graphical approaches for checking the assumptions of Fine and Gray's model. The proposed goodness-of-fit test procedures are based on the cumulative sums of residuals, which validate the model in three aspects: (1) proportionality of hazard ratio, (2) the linear functional form and (3) the link function. For each assumption testing, we provide a $$p$$ -values and a visualized plot against the null hypothesis using a simulation-based approach. We also consider an omnibus test for overall evaluation against any model misspecification. The proposed tests perform well in simulation studies and are illustrated with two real data examples. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of proper lifetime model selection in the context of statistical reliability analysis. Namely, we consider regression models describing the dependence of failure intensities on a covariate, for instance, a stressor. Testing the model fit is standardly based on the so-called martingale residuals. Their analysis has already been studied by many authors. Nevertheless, the Bayes approach to the problem, in spite of its advantages, is just developing. We shall present the Bayes procedure of estimation in several semi-parametric regression models of failure intensity. Then, our main concern is the Bayes construction of residual processes and goodness-of-fit tests based on them. The method is illustrated with both artificial and real-data examples. [ABSTRACT FROM AUTHOR]

Abstract: Purpose: Progressive disease (PD) per Response Evaluation Criteria in Solid Tumours (RECIST) 1.1 is defined as growth of measurable target lesions, presence of new lesions or unequivocal progression of non-target disease. In this manuscript we explored whether a more refined categorisation of tumour response and/or these components of progression, varying over time, can improve prediction of overall survival (OS) in the RECIST database. Methods: Data were randomly selected from 13 randomised clinical trials (3758 patients with breast, lung or colorectal cancer). A maximum of five target lesions contributed to the sum of longest diameters. At each measurement time we determined: best target response as best % improvement from baseline; tumour growth of target lesions as worst % change and worst rate of increase (mm/week) from nadir; presence of new lesions and occurrence of non-target PD. OS was analysed by tumour type using Cox regression, adjusting for baseline sum and including these parameters as time-dependent covariates. Results: 36% of patients had new lesions, 28% non-target PD and 49% experienced target lesion growth (median strongest growth 1.5mm/week). Regardless of tumour type, presence of new lesions (hazard ratio (HR) ranging 1.5–2.3) and non-target PD (HR 1.5–2.0) were strongly associated with worse OS. The explanatory value of tumour growth for OS was low compared to the other components. Conclusion: Modelling target lesion tumour growth did not show a marked improvement in OS prediction over and above the other components. These analyses enable a better understanding of the role of each component in PD evaluation. Work is ongoing to incorporate this information into an updated version of RECIST with enhanced prediction of subsequent survival. [Copyright &y& Elsevier]

Economic significance is frequently assessed through statistical hypothesis testing, which however, does not always correspond to the implicit economical questions being addressed. In this article we propose using mean value decomposition to assess economic significance. Unlike most previously suggested methods the proposed one is intuitive and simple to conduct. The technique is demonstrated and contrasted with hypothesis tests by an empirical example involving the income of Mexican children, which shows that the two inference approaches provide different and supplementary pieces of information. [ABSTRACT FROM PUBLISHER]

Abstract: The modeling of relationships between motor vehicle crashes and underlying factors has been investigated for more than three decades. Recently, many highway safety studies have documented the use of negative binomial (NB) regression models. On rare occasions, the Poisson model may be the only alternative especially when crash sample mean is low. Pearson's X 2 and the scaled deviance (G 2) are two common test statistics that have been proposed as measures of goodness-of-fit (GOF) for Poisson or NB models. Unfortunately, transportation safety analysts often deal with crash data that are characterized by low sample mean values. Under such conditions, the traditional test statistics may not perform very well. This study has three objectives. The first objective is to examine all the traditional test statistics and compare their performance for the GOF of accident models subjected to low sample means. Secondly, this study proposes a new test statistic that is not dependent on the sample size for Poisson regression model, as opposed to the grouped G 2 method. The proposed method is easy to use and does not require grouping data, which is time consuming and may not be feasible to use if the sample size is small. Moreover, the proposed method can be used for lower sample means than documented in previous studies. Thirdly, this study provides guidance on how and when to use appropriate test statistics for both Poisson and negative binomial (NB) regression models. [Copyright &y& Elsevier]

We give chi-squared goodness-of fit tests for parametric regression models such as accelerated failure time, proportional hazards, generalized proportional hazards, frailty models, transformation models, and models with cross-effects of survival functions. Random right censored data are used. Choice of random grouping intervals as data functions is considered. [ABSTRACT FROM AUTHOR]

A class of semiparametric regression models, called probabilistic index models, has been recently proposed. Because these models are semiparametric, inference is only valid when the proposed model is consistent with the underlying data-generating model. However, no fortnal goodness-of-fit methods for these probabilistic index models exist yet. We propose a test and a graphical tool for assessing the model adequacy. Simulation results indicate that both methods succeed in detecting lack-of-fit. The methods are also illustrated on a case study. [ABSTRACT FROM AUTHOR]

Willmott et al. [Willmott CJ, Robeson SM, Matsuura K. 2012. A refined index of model performance. International Journal of Climatology, forthcoming. DOI:10.1002/joc.2419.] recently suggest a refined index of model performance ( d r) that they purport to be superior to other methods. Their refined index ranges from − 1.0 to 1.0 to resemble a correlation coefficient, but it is merely a linear rescaling of our modified coefficient of efficiency ( E1) over the positive portion of the domain of d r. We disagree with Willmott et al. (2012) that d r provides a better interpretation; rather, E1 is more easily interpreted such that a value of E1 = 1.0 indicates a perfect model (no errors) while E1 = 0.0 indicates a model that is no better than the baseline comparison (usually the observed mean). Negative values of E1 (and, for that matter, d r < 0.5) indicate a substantially flawed model as they simply describe a 'level of inefficacy' for a model that is worse than the comparison baseline. Moreover, while d r is piecewise continuous, it is not continuous through the second and higher derivatives. We explain why the coefficient of efficiency ( E or E2) and its modified form ( E1) are superior and preferable to many other statistics, including d r, because of intuitive interpretability and because these indices have a fundamental meaning at zero. We also expand on the discussion begun by Garrick et al. [Garrick M, Cunnane C, Nash JE. 1978. A criterion of efficiency for rainfall-runoff models. Journal of Hydrology 36: 375-381.] and continued by Legates and McCabe [Legates DR, McCabe GJ. 1999. Evaluating the use of 'goodness-of-fit' measures in hydrologic and hydroclimatic model validation. Water Resources Research 35(1): 233-241.] and Schaefli and Gupta [Schaefli B, Gupta HV. 2007. Do Nash values have value? Hydrological Processes 21: 2075-2080. DOI: 10.1002/hyp.6825.]. This important discussion focuses on the appropriate baseline comparison to use, and why the observed mean often may be an inadequate choice for model evaluation and development. Copyright © 2012 Royal Meteorological Society [ABSTRACT FROM AUTHOR]

The technique applied by the authors to construct consistent and focused tests of fit for i.i.d. samples and regression models is extended to AR models for stationary time series. This approach leads to construct a consistent goodness-of-fit test for the null hypothesis that a stationary series is governed by an autoregressive model of a given order p. In addition of the consistency, the test is focused to detect efficiently the alternative of an AR( p + 1) model. The basic functional statistic conveying the information provided by the series is the process of accumulated sums of the residuals computed under the model of the null hypothesis of fit, reordered as concomitants of the conveniently delayed process. This process is transformed in order to obtain a new process with the same limiting Gaussian law encountered in earlier applications of the technique. Therefore, a Watson type quadratic statistic computed from this process has the same asymptotic laws under the null hypothesis of fit, and also under the alternatives of focusing, than the test statistics used in those applications. As a consequence, the resulting test has the same desirable performance as the tests previously developed by applying the same kind of transformations of processes. [ABSTRACT FROM AUTHOR]

Abstract: We propose a test for shape constraints which can be expressed by transformations of the coordinates of multivariate regression functions. The method is motivated by the constraint of symmetry with respect to some unknown hyperplane but can easily be generalized to other shape constraints of this type or other semi-parametric settings. In a first step, the unknown parameters are estimated and in a second step, this estimator is used in the -type test statistic for the shape constraint. We consider the asymptotic behavior of the estimated parameter and show that it converges with parametric rate if the shape constraint is true. Moreover, we derive the asymptotic distribution of the test statistic under the null hypothesis and furthermore propose a bootstrap test based on the residual bootstrap. In a simulation study, we investigate the finite sample performance of the estimator as well as the bootstrap test. [Copyright &y& Elsevier]

The Weibull proportional hazards model is commonly used for analysing survival data. However, formal tests of model adequacy are still lacking. It is well known that residual-based goodness-of-fit measures are inappropriate for censored data. In this paper, a graphical diagnostic plot of Cox-Snell residuals with a simulated envelope added is proposed to assess the adequacy of Weibull survival models. Both single component and two-component mixture models with random effects are considered for recurrent failure time data. The effectiveness of the diagnostic method is illustrated using simulated data sets and data on recurrent urinary tract infections of elderly women. [ABSTRACT FROM AUTHOR]

Genetic association studies often investigate the effect of haplotypes on an outcome of interest. Haplotypes are not observed directly, and this complicates the inclusion of such effects in survival models. We describe a new estimating equations approach for Cox's regression model to assess haplotype effects for survival data. These estimating equations are simple to implement and avoid the use of the EM algorithm, which may be slow in the context of the semiparametric Cox model with incomplete covariate information. These estimating equations also lead to easily computable, direct estimators of standard errors, and thus overcome some of the difficulty in obtaining variance estimators based on the EM algorithm in this setting. We also develop an easily implemented goodness-of-fit procedure for Cox's regression model including haplotype effects. Finally, we apply the procedures presented in this article to investigate possible haplotype effects of the PAF-receptor on cardiovascular events in patients with coronary artery disease, and compare our results to those based on the EM algorithm. [ABSTRACT FROM AUTHOR]

In his 1985 article ("Projection pursuit"), Huber demonstrates the interest of his method to estimate a density from a data set in a simple given case. He considers the factorization of density through a Gaussian component and some residual density. Huber's work is based on maximizing Kullback-Leibler divergence. Our proposal leads to a new algorithm. Furthermore, we will also consider the case when the density to be factorized is estimated from an i.i.d. sample. We will then propose a test for the factorization of the estimated density. Applications include a new test of fit pertaining to the elliptical copulas. [ABSTRACT FROM AUTHOR]

In this paper, a new bivariate negative binomial regression (BNBR) model allowing any type of correlation is defined and studied. The marginal means of the bivariate model are functions of the explanatory variables. The parameters of the bivariate regression model are estimated by using the maximum likelihood method. Some test statistics including goodness-of-fit are discussed. Two numerical data sets are used to illustrate the techniques. The BNBR model tends to perform better than the bivariate Poisson regression model, but compares well with the bivariate Poisson log-normal regression model. [ABSTRACT FROM AUTHOR]

In this article, we develop a method for checking the estimation equations, which is for joint estimation of the regression parameters and the overdispersion parameters, based on one dimension projected covariate. This method is different from the general testing methods in that our proposed method can be applied to high-dimensional response while the classical testing methods can not be extended to high dimension problem simply to construct a powerful test. Furthermore, the properties of the test statistics are investigated and Nonparametric Monte Carlo Test (NMCT) is suggested to determine the critical values of the test statistics under null hypothesis. [ABSTRACT FROM AUTHOR]

We introduce directed goodness-of-fit tests for Cox-type regression models in survival analysis. “Directed” means that one may choose against which alternatives the tests are particularly powerful. The tests are based on sums of weighted martingale residuals and their asymptotic distributions. We derive optimal tests against certain competing models which include Cox-type regression models with different covariates and/or a different link function. We report results from several simulation studies and apply our test to a real dataset. [ABSTRACT FROM AUTHOR]

Abstract: We propose a new test for independence of error and covariate in a nonparametric regression model. The test statistic is based on a kernel estimator for the -distance between the conditional distribution and the unconditional distribution of the covariates. In contrast to tests so far available in literature, the test can be applied in the important case of multivariate covariates. It can also be adjusted for models with heteroscedastic variance. Asymptotic normality of the test statistic is shown. Simulation results and a real data example are presented. [Copyright &y& Elsevier]

The least-squares technique has been shown to possess valuable properties as a method of the parameter estimation of classic and fuzzy regression analysis. However, the behavior and properties of the least-squares estimators are affected when outliers arise in the sample and/or by slight changes in the dataset. Robust techniques, on the other hand, provide robust estimators of the parameters which avoid such adverse effects. For this purpose, this paper extends the M -estimation approach to fuzzy regression analysis which provides consistent results in the presence of outliers. The parameters estimation problem is reduced to a reweighted algorithm which is a simple approach both theoretically and computationally. The proposed algorithm decreases the effect of outliers on the model fit by down-weighting them. To show the performances of the proposed method against some commonly used fuzzy regression models simulation studies, and two applicative examples based on real-world datasets in hydrology and atmospheric environment are provided. The sensitivity analysis of the estimated parameters are also reported based on a Monte-Carlo simulation study showing the efficiency of the proposed estimators in comparison with some other well-known methods in fuzzy regression analysis. • The M-estimation approach is proposed for fuzzy regression analysis. • The parameters estimation problem is reduced to an iterative reweighted algorithm. • The algorithm decreases the effect of outliers by down-weighting them. • The numerical results show the model is robust to the presence of outliers in data. • The efficiency of the M-estimators is concluded by running sensitivity analysis. [ABSTRACT FROM AUTHOR]

The authors compare the performance of different regression models for censored survival data in modeling the impact of prognostic factors on all-cause mortality in colon cancer. The data were for 1,951 patients, who were diagnosed in 1977-1991, recorded by the Registry of Digestive Tumors of Cote d'Or, France, and followed for up to 15 years. Models include the Cox proportional hazards model and its three generalizations that allow for hazard ratio to change over time: 1) the piecewise model where hazard ratio is a step function; 2) the model with interaction between a predictor and a parametric function of time; and 3) the non-parametric regression spline model. Results illustrate the importance of accounting for non-proportionality of hazards, and some advantages of flexible non-parametric modeling of time-dependent effects. The authors provide empirical evidence for the dependence of the results of piecewise and parametric models on arbitrary a priori choices, regarding the number of time intervals and specific parametric function, which may lead to biased estimates and low statistical power. The authors demonstrate that a single, a priori selected spline model recovers a variety of patterns of changes in hazard ratio and fits better than other models, especially when the changes are nonmonotonic, as in the case of cancer stages. Am J Epidemiol 1999; 150:1188–1200. [ABSTRACT FROM PUBLISHER]

Increasingly in biomedical studies, health status is inferred through a series of questionnaire item responses. Challenges for analyzing associations between such responses and risk factors include multiplicity many indicators must be combined to derive summary statements about health status, and measurement error persons' self report fluctuates due to causes other than substantive health changes. In order to deal with these challenges, the authors propose a strategy which comprises three methods: 1) score the item responses, then regress the score on predictors; 2) regress each item response on predictors, accounting for within person associations; and 3) summarize and analyze the item responses jointly, using a latent variable model. The authors develop modeling and diagnostic procedures for method 3. They then show how the three method analytic strategy can be used to solve the problem of determining which aspects of vision are associated with self reported functioning in activities that require seeing at a distance. They demonstrate that methods 2 and 3 illuminate basic findings from method 1 by adding specificity, describing patterns as well as severities of health impairments, and identifying isolated items that relate to risk factors differentially than others. They conclude that the three method strategy specifies how risk factors determine questionnaire based health outcomes substantially better than any of the methods in isolation. Am J Epidemiol 1999; 150:1165–78 [ABSTRACT FROM PUBLISHER]

In this paper we consider different approaches for estimation and assessment of covariate effects for the cumulative incidence curve in the competing risks model. The classic approach is to model all cause-specific hazards and then estimate the cumulative incidence curve based on these cause-specific hazards. Another recent approach is to directly model the cumulative incidence by a proportional model (Fine and Gray, J Am Stat Assoc 94:496–509, 1999), and then obtain direct estimates of how covariates influences the cumulative incidence curve. We consider a simple and flexible class of regression models that is easy to fit and contains the Fine–Gray model as a special case. One advantage of this approach is that our regression modeling allows for non-proportional hazards. This leads to a new simple goodness-of-fit procedure for the proportional subdistribution hazards assumption that is very easy to use. The test is constructive in the sense that it shows exactly where non-proportionality is present. We illustrate our methods to a bone marrow transplant data from the Center for International Blood and Marrow Transplant Research (CIBMTR). Through this data example we demonstrate the use of the flexible regression models to analyze competing risks data when non-proportionality is present in the data. [ABSTRACT FROM AUTHOR]