On nonlinear beta regression residuals

We proposed a new residual to be used in linear and nonlinear beta regressions. Unlike the residuals that had already been proposed, the derivation of the new residual takes into account not only information relative to the estimation of the mean submodel but also takes into account information obtained from the precision submodel. This is an advantage of the residual we introduced. Additionally, the new residual is computationally less intensive than the weighted residual. Recall that the computation of the latter involves an matrix, where n is the sample size. Obviously, that can be a problem when the sample size is very large. In contrast, our residual does not suffer from that. It can be easily computed even in large samples. Finally, our residual proved to be able to identify atypical observations as well as the weighted residual. We also propose new thresholds for residual plots and a scheme for the choice of starting values to be used in maximum likelihood point estimation in the class of nonlinear beta regression models. We report Monte Carlo simulation results on the behavior of different residuals. We also present and discuss two empirical applications; one uses the proportion of killed grasshoppers in an assay on the grasshopper Melanopus sanguinipes with the insecticide carbofuran and the synergist piperonyl butoxide, which enhances the toxicity of the insecticide, and the other uses simulated data. The results favor the new methodology we introduce.     

New residuals for Cox regression and their application to outlier screening

The identification of individuals who 'died far too early' or 'lived far too long' as compared to their survival probabilities from a Cox regression can lead to the detection of new prognostic factors. Methods to identify outliers are generally based on residuals. For Cox regression, only deviance residuals have been considered for this purpose, but we show that these residuals are not very suitable. Instead, we develop and propose two new types of residuals: the suggested log-odds and normal deviate residuals are simple and intuitively appealing and their theoretical properties and empirical performance make them very suitable for outlier identification. Finally, various practical aspects of screening for individuals with outlying survival times are discussed by means of a cancer study example.     

Unbalanced regression analysis with residuals having a covariance structure of intra-class form.

Let Yi be an ni X 1 vector of observations, Xi an ni X p matrix of known values, and beta an unknown p X 1 with the structure Yi = Xi beta + epsilon i, where the covariance matrix of epsilon i is of intra-class form, that is Cov (epsilon i) = sigma2[(1 - rho) Ii + rho e i e i'] where Ii is the ni X ni identity matrix and e i is the ni X 1 vector each element of which is unity. This article develops the maximum likelihood estimators of beta, sigma2, and rho when one observes N pairs (Xi, Yi). This situation arises typically in biological problems where one samples clusters of related organisms. The estimation procedure is illustrated in a commonly occurring genetics situation.     

Model-checking techniques based on cumulative residuals

Residuals have long been used for graphical and numerical examinations of the adequacy of regression models. Conventional residual analysis based on the plots of raw residuals or their smoothed curves is highly subjective, whereas most numerical goodness-of-fit tests provide little information about the nature of model misspecification. In this paper, we develop objective and informative model-checking techniques by taking the cumulative sums of residuals over certain coordinates (e.g., covariates or fitted values) or by considering some related aggregates of residuals, such as moving sums and moving averages. For a variety of statistical models and data structures, including generalized linear models with independent or dependent observations, the distributions of these stochastic processes tinder the assumed model can be approximated by the distributions of certain zero-mean Gaussian processes whose realizations can be easily generated by computer simulation. Each observed process can then be compared, both graphically and numerically, with a number of realizations from the Gaussian process. Such comparisons enable one to assess objectively whether a trend seen in a residual plot reflects model misspecification or natural variation. The proposed techniques are particularly useful in checking the functional form of a covariate and the link function. Illustrations with several medical studies are provided.