Non-nested linear models: A conditional confidence approach

The comparison of nested linear models with normal error is well standardized in the common procedures of the analysis of variance. This article considers the comparison of two non-nested linear models that have the same parameter dimension; the comparison is made on the assumption that the true mean lies somewhere in the linear span of the two models. The analysis leads to a precision-based conditional confidence interval for the unsigned angular direction of the true mean, and this in turn provides a confidence assessment of the two directions that correspond to the two models being compared. The confidence interval is an approximate conditional interval (given the distance of the estimate from the intersection of the hypotheses), and its length as a fraction of π indicates the precision of the confidence procedure. The method provides a conditional-inference alternative to a confidence interval available by Creasy-Fieller analysis.     

A mixture-based approach to robust analysis of generalised linear models

A method for robustness in linear models is to assume that there is a mixture of standard and outlier observations with a different error variance for each class. For generalised linear models (GLMs) the mixture model approach is more difficult as the error variance for many distributions has a fixed relationship to the mean. This model is extended to GLMs by changing the classes to one where the standard class is a standard GLM and the outlier class which is an overdispersed GLM achieved by including a random effect term in the linear predictor. The advantages of this method are it can be extended to any model with a linear predictor, and outlier observations can be easily identified. Using simulation the model is compared to an M-estimator, and found to have improved bias and coverage. The method is demonstrated on three examples.     

Best linear unbiased estimators of parameters of a simple linear regression model based on ordered ranked set samples

As an alternative to an estimation based on a simple random sample (BLUE-SRS) for the simple linear regression model, Moussa-Hamouda and Leone [E. Moussa-Hamouda and F.C. Leone, The o-blue estimators for complete and censored samples in linear regression, Technometrics, 16 (3) (1974), pp. 441–446.] discussed the best linear unbiased estimators based on order statistics (BLUE-OS), and showed that BLUE-OS is more efficient than BLUE-SRS for normal data. Using the ranked set sampling, Barreto and Barnett [M.C.M. Barreto and V. Barnett, Best linear unbiased estimators for the simple linear regression model using ranked set sampling. Environ. Ecoll. Stat. 6 (1999), pp. 119–133.] derived the best linear unbiased estimators (BLUE-RSS) for simple linear regression model and showed that BLUE-RSS is more efficient for the estimation of the regression parameters (intercept and slope) than BLUE-SRS for normal data, but not so for the estimation of the residual standard deviation in the case of small sample size. As an alternative to RSS, this paper considers the best linear unbiased estimators based on order statistics from a ranked set sample (BLUE-ORSS) and shows that BLUE-ORSS is uniformly more efficient than BLUE-RSS and BLUE-OS for normal data.     

A generalized linear model approach to seasonal aspects of wind speed modeling

The aim of the article is to identify the intraday seasonality in a wind speed time series. Following the traditional approach, the marginal probability law is Weibull and, consequently, we consider seasonal Weibull law. A new estimation and decision procedure to estimate the seasonal Weibull law intraday scale parameter is presented. We will also give statistical decision-making tools to discard or not the trend parameter and to validate the seasonal model.     

Comparing search and estimation performances of designs based on a compound criterion

Since the introduction of the search design by Srivastava [Designs for searching non-negligible effects. In: Srivastava, editor. A survey of statistical design and linear models. Amsterdam: North-Holland, Elsevier; 1975. p. 507–519], construction of such designs has been considered by many researchers. The efficient performances of constructed search designs in terms of parameter estimation and search ability of parameters have also been investigated by several authors. They have proposed suitable optimality measures such as DD- and AD-optimality for estimation in the early stage of search design construction. Moreover, since 1990s, some criteria have been developed to evaluate search performance of a design. Although these criteria are useful none of them is able to evaluate both estimation and search efficiency of a design simultaneously. In this paper, we propose dual-task criteria to deal with searching and estimating performances of search designs. These compound criteria are weighted multiplication of estimation and search suitable criteria. They will be used for design comparison and the results will be presented.     

Densities and distribution functions of the ratio of two linear functions and the product of Gamma variates: A saddlepoint approach

Saddlepoint approximations for the densities and the distribution functions of the ratio of two linear functions of gamma random variables and the product of gamma random variables are derived. Ratios of linear functions with positive and negative weights and non identical gamma variables are considered. The saddlepoint approximations are very accurate in the tails as in the center of the distribution. Extensive simulation studies are used to evaluate the accuracy of the proposed methods.     

A theorem on invariance of estimability of linear parametric functions in linear models

Consider the general linear model Y = Xβ + ? , where E[??'] = σ²I and rank of X is less than or equal to the number of columns of X. It is well known that the linear parametric function λ'β is estimable if and only if λ' is in the row space of X. This paper characterizes all orthogonal matrices P such that the row space of XP is equal to the row space of X, i.e. the estimability of λ'β is invariant under P. An additional property of these matrices is the invariance of the spectrum of the information matrix X'X. An application of the results is also given.     

Root-n-consistent semiparametric estimation of partially linear models based on k-nn method

In the context of the partially linear semiparametric model examined by Robinson (1988), we show that root-n-consisten estimation results established using kernel and series methods can also be obtained by using k-nearest-neighbor (k-nn) method.     

Tests of hypotheses based on ranks in the general linear model

A unified approach is developed for testing hypotheses in the general linear model based on the ranks of the residuals. It complements the nonparametric estimation procedures recently reported in the literature. The testing and estimation procedures together provide a robust alternative to least squares. The methods are similar in spirit to least squares so that results are simple to interpret. Hypotheses concerning a subset of specified parameters can be tested, while the remaining parameters are treated as nuisance parameters. Asymptotically, the test statistic is shown to have a chi-square distribution under the null hypothesis. This result is then extended to cover a sequence of contiguous alternatives from which the Pitman efficacy is derived. The general application of the test requires the consistent estimation of a functional of the underlying distribution and one such estimate is furnished.     

Root-n-consistent semiparametric estimation of partially linear models based on k-nn method

In the context of the partially linear semiparametric model examined by Robinson (1988), we show that root-n-consisten estimation results established using kernel and series methods can also be obtained by using k-nearest-neighbor (k-nn) method.     

A Bayesian approach for generalized linear models with explanatory biomarker measurement variables subject to detection limit: an application to acute lung injury

Biomarkers have the potential to improve our understanding of disease diagnosis and prognosis. Biomarker levels that fall below the assay detection limits (DLs), however, compromise the application of biomarkers in research and practice. Most existing methods to handle non-detects focus on a scenario in which the response variable is subject to the DL; only a few methods consider explanatory variables when dealing with DLs. We propose a Bayesian approach for generalized linear models with explanatory variables subject to lower, upper, or interval DLs. In simulation studies, we compared the proposed Bayesian approach to four commonly used methods in a logistic regression model with explanatory variable measurements subject to the DL. We also applied the Bayesian approach and other four methods in a real study, in which a panel of cytokine biomarkers was studied for their association with acute lung injury (ALI). We found that IL8 was associated with a moderate increase in risk for ALI in the model based on the proposed Bayesian approach.     

Estimation of patterned covariance in the multivariate linear models: an outer product least-squares approach

In this article, we present a framework of estimating patterned covariance of interest in the multivariate linear models. The main idea in it is to estimate a patterned covariance by minimizing a trace distance function between outer product of residuals and its expected value. The proposed framework can provide us explicit estimators, called outer product least-squares estimators, for parameters in the patterned covariance of the multivariate linear model without or with restrictions on regression coefficients. The outer product least-squares estimators enjoy the desired properties in finite and large samples, including unbiasedness, invariance, consistency and asymptotic normality. We still apply the framework to three special situations where their patterned covariances are the uniform correlation, a generalized uniform correlation and a general q-dependence structure, respectively. Simulation studies for three special cases illustrate that the proposed method is a competent alternative of the maximum likelihood method in finite size samples.     

Testing for lack of fit in linear multiresponse models based on exact or near replicates

Three procedures for testing the adequacy of a proposed linear multiresponse regression model against unspecified general alternatives are considered. The model has an error structure with a matrix normal distribution which allows the vector of responses for a particular run to have an unknown covariance matrix while the responses for different runs are uncorrelated. Furthermore, each response variable may be modeled by a separate design matrix. Multivariate statistics corresponding to the classical univariate lack of fit and pure error sums of squares are defined and used to determine the multivariate lack of fit tests. A simulation study was performed to compare the power functions of the test procedures in the case of replication. Generalizations of the tests for the case in which there are no independent replicates on all responses are also presented.     

A predictive approach to measuring the strength of statistical evidence for single and multiple comparisons

The normalized maximum likelihood (NML) is a recent penalized likelihood that has properties that justify defining the amount of discrimination information (DI) in the data supporting an alternative hypothesis over a null hypothesis as the logarithm of an NML ratio, namely, the alternative hypothesis NML divided by the null hypothesis NML. The resulting DI, like the Bayes factor but unlike the P‐value, measures the strength of evidence for an alternative hypothesis over a null hypothesis such that the probability of misleading evidence vanishes asymptotically under weak regularity conditions and such that evidence can support a simple null hypothesis. Instead of requiring a prior distribution, the DI satisfies a worst‐case minimax prediction criterion. Replacing a (possibly pseudo‐) likelihood function with its weighted counterpart extends the scope of the DI to models for which the unweighted NML is undefined. The likelihood weights leverage side information, either in data associated with comparisons other than the comparison at hand or in the parameter value of a simple null hypothesis. Two case studies, one involving multiple populations and the other involving multiple biological features, indicate that the DI is robust to the type of side information used when that information is assigned the weight of a single observation. Such robustness suggests that very little adjustment for multiple comparisons is warranted if the sample size is at least moderate. The Canadian Journal of Statistics 39: 610–631; 2011. © 2011 Statistical Society of Canada     

Exploring gillnet catch efficiency of sardines in the coastal waters of Sri Lanka by means of three statistical techniques: a comparison of linear and nonlinear modelling techniques

The present investigation was undertaken to study the gillnet catch efficiency of sardines in the coastal waters of Sri Lanka using commercial catch and effort data. Commercial catch and effort data of small mesh gillnet fishery were collected in five fisheries districts during the period May 1999–August 2002. Gillnet catch efficiency of sardines was investigated by developing catch rates predictive models using data on commercial fisheries and environmental variables. Three statistical techniques [multiple linear regression, generalized additive model and regression tree model (RTM)] were employed to predict the catch rates of trenched sardine Amblygaster sirm (key target species of small mesh gillnet fishery) and other sardines (Sardinella longiceps, S. gibbosa, S. albella and S. sindensis). The data collection programme was conducted for another six months and the models were tested on new data. RTMs were found to be the strongest in terms of reliability and accuracy of the predictions. The two operational characteristics used here for model formulation (i.e. depth of fishing and number of gillnet pieces used per fishing operation) were more useful as predictor variables than the environmental variables. The study revealed a rapid tendency of increasing the catch rates of A. sirm with increased sea depth up to around 32 m.     

Addressing the effect of parameter estimation on phase II monitoring of multivariate multiple linear profiles via a new cluster-based approach

Abstract

Profile monitoring is applied when the quality of a product or a process can be determined by the relationship between a response variable and one or more independent variables. In most Phase II monitoring approaches, it is assumed that the process parameters are known. However, it is obvious that this assumption is not valid in many real-world applications. In fact, the process parameters should be estimated based on the in-control Phase I samples. In this study, the effect of parameter estimation on the performance of four Phase II control charts for monitoring multivariate multiple linear profiles is evaluated. In addition, since the accuracy of the parameter estimation has a significant impact on the performance of Phase II control charts, a new cluster-based approach is developed to address this effect. Moreover, we evaluate and compare the performance of the proposed approach with a previous approach in terms of two metrics, average of average run length and its standard deviation, which are used for considering practitioner-to-practitioner variability. In this approach, it is not necessary to know the distribution of the chart statistic. Therefore, in addition to ease of use, the proposed approach can be applied to other type of profiles. The superior performance of the proposed method compared to the competing one is shown in terms of all metrics. Based on the results obtained, our method yields less bias with small-variance Phase I estimates compared to the competing approach.     

A Monte Carlo simulation study on partially adaptive estimators of linear regression models

This paper presents a comprehensive comparison of well-known partially adaptive estimators (PAEs) in terms of efficiency in estimating regression parameters. The aim is to identify the best estimators of regression parameters when error terms follow from normal, Laplace, Student's t, normal mixture, lognormal and gamma distribution via the Monte Carlo simulation. In the results of the simulation, efficient PAEs are determined in the case of symmetric leptokurtic and skewed leptokurtic regression error data. Additionally, these estimators are also compared in terms of regression applications. Regarding these applications, using certain standard error estimators, it is shown that PAEs can reduce the standard error of the slope parameter estimate relative to ordinary least squares.     

Generalized linear spatial models in epidemiology: A case study of zoonotic cutaneous leishmaniasis in Tunisia

Generalized linear spatial models (GLSM) are used here to study spatial characters of zoonotic cutaneous leishmaniasis (ZCL) in Tunisia. The response variable stands for the number of affected by district during the period 2001–2002. The model covariates are: climates (temperature and rainfall), humidity and surrounding vegetation status. As the environmental and weather data are not available for all the studied districts, Kriging based on linear interpolation was used to estimate the missing data. To account for unexplained spatial variation in the model, we include a stationary Gaussian process S with a powered exponential spatial correlation function. Moran coefficient, DIC criterion and residuals variograms are used to show the high goodness-of-fit of the GLSM. When compared with the statistical tools used in the previous ZCL studies, the optimal GLSM found here yields a better assessment of the impact of the risk factors, a better prediction of ZCL evolution and a better comprehension of the disease transmission. The statistical results show the progressive increase in the number of affected in zones with high temperature, low rainfall and high surrounding vegetation index. Relative humidity does not seem to affect the distribution of the disease in Tunisia. The results of the statistical analyses stress the important risk of misleading epidemiological conclusions when non-spatial models are used to analyse spatially structured data.