The rainbow test for lack of fit in regression

A test for lack of fit in regression is presented. Unlike other methods, this one doesn't require replicates or a prior estimate of variance. It can be used for linear or multiple regression, and would be easy to add to existing computer packages. It is based on comparing a fit over low leverage points with a fit over the entire set of data. Distribution theory results are pre¬sented, with examples of power. A discussion of its use for de¬tecting violations of other regression assumptions is also given.     

Testing for lack of fit in regression - a review

Assessment of the adequacy of a proposed linear regression model is necessarily subjective. However, the following three criteria may warrant investigation whether the distributional assumptions for the stochastic portion of the model are satisfied, whether the predictive capability of the model is satisfactory, and whether the deterministic portion of the model is adejuate in a statistical sense. The first two criteria have been reviewed in the literature to some extent. This paper reviews statistical tests and procedures which aid the experimenter in deterrmining lack of fit or functional misspecification associated with the deterministic portion of a proposed linear regression model.     

Nonparametric test for checking lack of fit of the quantité regression model under random censoring

The author proposes a nonparametric test for checking the lack of fit of the quantile function of survival time given the covariates; she assumes that survival time is subjected to random right censoring. Her test statistic is a kemel‐based smoothing estimator of a moment condition. The test statistic is asymptotically Gaussian under the null hypothesis. The author investigates its behavior under local alternative sequences. She assesses its finite‐sample power through simulations and illustrates its use with the Stanford heart transplant data.     

Testing lack of fit in regression without replication

An F-statistic which tests a hypothesized linear regression model against the general alternative is developed. Observations are grouped using “near neighbours” and a generalization of the usual lack of fit test is derived. Two data sets from Daniel and Wood (1971) are used to illustrate the methodology. Power considerations are discussed.     

Goodness‐of‐fit tests for linear regression models with missing response data

The authors show how to test the goodness‐of‐fit of a linear regression model when there are missing data in the response variable. Their statistics are based on the L₂ distance between nonparametric estimators of the regression function and a ‐consistent estimator of the same function under the parametric model. They obtain the limit distribution of the statistics and check the validity of their bootstrap version. Finally, a simulation study allows them to examine the behaviour of their tests, whether the samples are complete or not.     

Using local linear kernel smoothers to test the lack of fit of nonlinear regression models

Herein, we propose a data-driven test that assesses the lack of fit of nonlinear regression models. The comparison of local linear kernel and parametric fits is the basis of this test, and specific boundary-corrected kernels are not needed at the boundary when local linear fitting is used. Under the parametric null model, the asymptotically optimal bandwidth can be used for bandwidth selection. This selection method leads to the data-driven test that has a limiting normal distribution under the null hypothesis and is consistent against any fixed alternative. The finite-sample property of the proposed data-driven test is illustrated, and the power of the test is compared with that of some existing tests via simulation studies. We illustrate the practicality of the proposed test by using two data sets.     

Two-phase regression and goodness of fit

Analysis of two-phase regression has traditionally been carried out using a variety of likelihood approaches. In this paper we present an alternative procedure based on a goodness of fit criterion.

Exact hypothesis tests for a known switch point are developed. Approximate (conservative) tests for an unknown switch point are also obtained     

Testing lack of fit of regression models under heteroscedasticity

A test is proposed for assessing the lack of fit of heteroscedastic nonlinear regression models that is based on comparison of nonparametric kernel and parametric fits. A data-driven method is proposed for bandwidth selection using the asymptotically optimal bandwidth of the parametric null model which leads to a test that has a limiting normal distribution under the null hypothesis and is consistent against any fixed alternative. The resulting test is applied to the problem of testing the lack of fit of a generalized linear model.     

Interval-valued data regression using partial linear model

Semi-parametric modelling of interval-valued data is of great practical importance, as exampled by applications in economic and financial data analysis. We propose a flexible semi-parametric modelling of interval-valued data by integrating the partial linear regression model based on the Center & Range method, and investigate its estimation procedure. Furthermore, we introduce a test statistic that allows one to decide between a parametric linear model and a semi-parametric model, and approximate its null asymptotic distribution based on wild Bootstrap method to obtain the critical values. Extensive simulation studies are carried out to evaluate the performance of the proposed methodology and the new test. Moreover, several empirical data sets are analysed to document its practical applications.     

Joint impact of multiple observations on a subset of variables in multiple linear regression

In multiple linear regression analysis, each observation affects the fitted regression equation differently and has varying influences on the regression coefficients of the different variables. Chatterjee & Hadi (1988) have proposed some measures such as DSSE_ij (Impact on Residual Sum of Squares of simultaneously omitting the ith observation and the jth variable), F_j (Partial F-test for the jth variable) and F_j(i) (Partial F-test for the jth variable omitting the ith observation) to show the joint impact and the interrelationship that exists among a variable and an observation. In this paper we have proposed more extended form of those measures DSSE_IJ, F_J and F_J(I) to deal with the interrelationships that exist among the multiple observations and a subset of variables by monitoring the effects of the simultaneous omission of multiple variables and multiple observations.     

Heteroscedasticity diagnostics in two-phase linear regression models

In two-phase linear regression models, it is a standard assumption that the random errors of two phases have constant variances. However, this assumption is not necessarily appropriate. This paper is devoted to the tests for variance heterogeneity in these models. We initially discuss the simultaneous test for variance heterogeneity of two phases. When the simultaneous test shows that significant heteroscedasticity occurs in the whole model, we construct two individual tests to investigate whether or not both phases or one of them have/has significant heteroscedasticity. Several score statistics and their adjustments based on Cox and Reid [D. R. Cox and N. Reid, Parameter orthogonality and approximate conditional inference. J. Roy. Statist. Soc. Ser. B 49 (1987), pp. 1–39] are obtained and illustrated with Australian onion data. The simulated powers of test statistics are investigated through Monte Carlo methods.     

An omnibus lack of fit test in logistic regression with sparse data

The usefulness of logistic regression depends to a great extent on the correct specification of the relation between a binary response and characteristics of the unit on which the response is recoded. Currently used methods for testing for misspecification (lack of fit) of a proposed logistic regression model do not perform well when a data set contains almost as many distinct covariate vectors as experimental units, a condition referred to as sparsity. A new algorithm for grouping sparse data to create pseudo replicates and using them to test for lack of fit is developed. A simulation study illustrates settings in which the new test is superior to existing ones. Analysis of a dataset consisting of the ages of menarche of Warsaw girls is also used to compare the new and existing lack of fit tests.     

On the power of the durbin-watson test under high autocorrelation

In the linear regression model without an intercept, it is known that the limiting power of the Durbin-Watson test (as correlation among errors increases) equals either one or zero, depending on the underlying regressor matrix. This paper considers the limiting power in the model with an intercept, and proves that it will never equal one or zero.     

A cubic smoothing spline based lack of fit test for nonlinear regression models

The use of a statistic based on cubic spline smoothing is considered for testing nonlinear regression models for lack of fit. The statistic is defined to be the Euclidean squared norm of the smoothed residual vector obtained from fitting the nonlinear model, The asymptotic distribution of the statistic is derived under suitable smooth local alternatives and a numerical example is presented.     

An F-type small sample simultaneous test for nested linear regression models

A small sample simultaneous testing method is proposed for nested linear regression model. The methodology is based on the generalized likelihood ratio test which is the large sample simultaneous testing method for general nested models. The proposed test is also used for model identification.     

A note on the power of the f-test for testing linear hypothesis with elliptic t errors

The power of the classical .F-test for testing the regression coefficient of a general linear model with elliptic t error variable depends on the degrees of freedom of the t- distribution. In this note it is shown that the power of the F-test based on t-distribution is greater than the normal based test at smaller level of significance.     

Local linear regression with adaptive orthogonal fitting for the wind power application

Short-term forecasting of wind generation requires a model of the function for the conversion of meteorological variables (mainly wind speed) to power production. Such a power curve is nonlinear and bounded, in addition to being nonstationary. Local linear regression is an appealing nonparametric approach for power curve estimation, for which the model coefficients can be tracked with recursive Least Squares (LS) methods. This may lead to an inaccurate estimate of the true power curve, owing to the assumption that a noise component is present on the response variable axis only. Therefore, this assumption is relaxed here, by describing a local linear regression with orthogonal fit. Local linear coefficients are defined as those which minimize a weighted Total Least Squares (TLS) criterion. An adaptive estimation method is introduced in order to accommodate nonstationarity. This has the additional benefit of lowering the computational costs of updating local coefficients every time new observations become available. The estimation method is based on tracking the left-most eigenvector of the augmented covariance matrix. A robustification of the estimation method is also proposed. Simulations on semi-artificial datasets (for which the true power curve is available) underline the properties of the proposed regression and related estimation methods. An important result is the significantly higher ability of local polynomial regression with orthogonal fit to accurately approximate the target regression, even though it may hardly be visible when calculating error criteria against corrupted data.     

Detection of outliers in the multivariate linear regression model

In this article we suggest multivariate kurtosis as a statistic for detection of outliers in a multivariate linear regression model. The statistic has some local optimality properties.