Influential Observations In Linear Regression Research Articles

Estimation of grass biomass consumed by rabbits housed in movable paddocks

Biomass allowance is a key feature in pasture-based rabbit production systems. It conditions not only the stock density (rabbits/m²) and/or the number of grazing days, it also influences the grazing behaviour of animals. When herbage restriction occurs, pelleted feed and/or cereal intake goes up. Inadequate pasture management may also impair the biomass quantity and quality if overgrazing occurs. To avoid the undesirable effects of overgrazing and better manage pellet and cereal intake, information on both biomass availability and rabbits’ grazing capacity are needed. Here, we present an adaptation of the rising plate meter method (developed for biomass intake measures for ruminants) for use in rabbit. To this end, we designed an experiment where two groups of 12 rabbits each were kept in two different fields: under an apple orchard (AO) or on fallow land (FL). We followed the animals for 5 consecutive weeks (from 45 to 80 d old). Rabbits lived in 25 m² movable paddocks, and every week a new paddock location (called paddock-spot) was made available for them. At each new paddock-spot, we measured the herbage height inside the paddocks and performed samplings of the available biomass (i.e. herbage cut after herbage height measurement) outside the paddocks. From this data we estimated the available biomass inside each paddock-spot by fitting linear regression equations of biomass to herbage height. Overall, rabbits in the AO and FL had access to 1328±65.7 and 1386±58.6 kg of dry matter (DM) per ha, respectively. In every field and paddock-spot, the biomass available was lower than the rabbits’ grazing capacity; overgrazing was the rule. Roughly, and under a restricted herbage allowance, rabbits in the AO ingested 45.2 g DM/d and rabbits in the FL 43.4 g DM/d. In the last week (64 to 80 d old), the biomass intake of rabbits in the AO and AL represented 26.4 and 23.5% of the total DM intake, respectively. These values, however, does not represent the real grazing capacity of growing rabbits. In this study, we provide some advice on the sampling method to obtain reliable biomass estimations and we mention two methods for handling influential observations in linear regression.

Identification and classification of multiple outliers, high leverage points and influential observations in linear regression

Detection of multiple unusual observations such as outliers, high leverage points and influential observations (IOs) in regression is still a challenging task for statisticians due to the well-known masking and swamping effects. In this paper we introduce a robust influence distance that can identify multiple IOs, and propose a sixfold plotting technique based on the well-known group deletion approach to classify regular observations, outliers, high leverage points and IOs simultaneously in linear regression. Experiments through several well-referred data sets and simulation studies demonstrate that the proposed algorithm performs successfully in the presence of multiple unusual observations and can avoid masking and/or swamping effects.

Procedures for the identification of multiple influential observations in linear regression

Since the seminal paper by Cook (1977) in which he introduced Cook's distance, the identification of influential observations has received a great deal of interest and extensive investigation in linear regression. It is well documented that most of the popular diagnostic measures that are based on single-case deletion can mislead the analysis in the presence of multiple influential observations because of the well-known masking and/or swamping phenomena. Atkinson (1981) proposed a modification of Cook's distance. In this paper we propose a further modification of the Cook's distance for the identification of a single influential observation. We then propose new measures for the identification of multiple influential observations, which are not affected by the masking and swamping problems. The efficiency of the new statistics is presented through several well-known data sets and a simulation study.

Sufficient jackknife-after-bootstrap method for detection of influential observations in linear regression models

In this study, we adapt sufficient bootstrap into the jackknife-after-bootstrap (JaB) algorithm. The performances of the sufficient and conventional JaB methods have been compared for detecting influential observations in linear regression. Comparison is based on two real-world examples and an extensive designed simulation study. Design includes different sample sizes and various modeling scenarios. The results reveal that proposed method is a good competitor for conventional JaB method with less standard error and amount of computation.

A Diagnostic Measure for Influential Observations in Linear Regression

In linear regression it is a common practice of measuring influence of an observation is to delete the case from the analysis and to investigate the change in the parameters or in the vector of forecasts resulting from this deletion. Pena (2005) introduced a new idea to measure the influence of an observation based on how this observation is being influenced by the rest of the data. In this article we propose a new influence measure extending the idea of Pena to group deletion for identifying multiple influential observations in linear regression. We investigate the usefulness of the proposed technique by two well-referred data sets, an artificial large data with high-dimension and heterogeneous sample points and by reporting a Monte Carlo simulation experiment.

On the asymptotic distribution of Cook's distance in logistic regression models

It sometimes occurs that one or more components of the data exert a disproportionate influence on the model estimation. We need a reliable tool for identifying such troublesome cases in order to decide either eliminate from the sample, when the data collect was badly realized, or otherwise take care on the use of the model because the results could be affected by such components. Since a measure for detecting influential cases in linear regression setting was proposed by Cook [Detection of influential observations in linear regression, Technometrics 19 (1977), pp. 15–18.], apart from the same measure for other models, several new measures have been suggested as single-case diagnostics. For most of them some cutoff values have been recommended (see [D.A. Belsley, E. Kuh, and R.E. Welsch, Regression Diagnostics: Identifying Influential Data and Sources of Collinearity, 2nd ed., John Wiley & Sons, New York, Chichester, Brisban, (2004).], for instance), however the lack of a quantile type cutoff for Cook's statistics has induced the analyst to deal only with index plots as worthy diagnostic tools. Focussed on logistic regression, the aim of this paper is to provide the asymptotic distribution of Cook's distance in order to look for a meaningful cutoff point for detecting influential and leverage observations.

Measures of influence for the functional linear model with scalar response

This paper studies how to identify influential observations in the functional linear model in which the predictor is functional and the response is scalar. Measurement of the effects of a single observation on estimation and prediction when the model is estimated by the principal components method is undertaken. For that, three statistics are introduced for measuring the influence of each observation on estimation and prediction of the functional linear model with scalar response that are generalizations of the measures proposed for the standard regression model by [D.R. Cook, Detection of influential observations in linear regression, Technometrics 19 (1977) 15–18; D. Peña, A new statistic for influence in linear regression, Technometrics 47 (2005) 1–12] respectively. A smoothed bootstrap method is proposed to estimate the quantiles of the influence measures, which allows us to point out which observations have the larger influence on estimation and prediction. The behavior of the three statistics and the quantile estimation bootstrap based method is analyzed via a simulation study. Finally, the practical use of the proposed statistics is illustrated by the analysis of a real data example, which show that the proposed measures are useful for detecting heterogeneity in the functional linear model with scalar response.

Identifying multiple influential observations in linear regression

The identification of influential observations has drawn a great deal of attention in regression diagnostics. Most of these identification techniques are based on single case deletion and among them DFFITS has become very popular with the statisticians. But this technique along with all other single case diagnostics may be ineffective in the presence of multiple influential observations. In this paper we develop a generalized version of DFFITS based on group deletion and then propose a new technique to identify multiple influential observations using this. The advantage of using the proposed method in the identification of multiple influential cases is then investigated through several well-referred data sets.

The Multiple Sets of Deletion Measures and Masking in Regression

Single set deletion influence analysis was introduced by Cook and Weisburg (Cook, R. D., Weisberg, S. (1982). Residuals and Influence in Regression. Chapman and Hall: New York); Pena and Yohai (Pena, D., Yohai, V. J. (1995). The detection of influential subsets in linear regression by using an influence matrix. J. R. Statist. Soc. B 57(1): 145–156); Barrett and Ling (Barrett, B. E., Ling, R. F. (1992). General classes of influence measures for multivariate regression. J. Amer. Stiatist. Assoc. 87(417):184–191); and Cook (Cook, R. D. (2000). Detection of influential observations inlinear regression. Technometrics 42(1):65–68). Multiple sets of deletion measures and conditional influence analysis are given in regression models by Wang and Critchley (Wang, D. Q., Critchley, F. (2000). Multiple deletion measures and influence in regression model. Communication in Statistics: Theory and Methods 29(11): 2391–2404). The present paper is concerned with multiple sets of deletion measures and masking problems in linear regression models. Masking problems have been discussed by Atkinson (Atkinson, A. C. (1985). Plot, Transformations and Regression. Clarendon: Oxford) and Lawrance (Lawrance, A. J. (1995). Deletion influence and masking in regression. J. R. Statist. Soc. B 57(1):181–189). In this paper we are particularly interested in conditional influence and masking problems for the detection of two influential subsets.

Measures and procedures for the identification of locally influential observations in linear regression

The concept of local influence was introduced by Cook (1986) and modified by Billor & Loynes (1993). Cook's local influence was motivated by the Cook measure (1977); so they only study the local influence on the regression coefficient and they are not resistant to masking and swamping effects. In this article we extend some influence measures to locally influential measures to mitigate these difficulties. We consider, too, the problem of identifying and testing multiple locally influential observations. These measures and procedures are very general methods for assessing the influence if local departures from assumptions in likelihood-based models (not necessarily regression models).

A new measure of overall potential influence in linear regression

In this article we introduce a new influence measure and a graphical display for the detection of influential observations in linear regression. We demonstrate the need for such a new influence measure by giving examples where existing influence measures, numerous as they may be, may fail to detect influential observations. The existing influence measures are procedures each of which is designed to detect the influence of observations on a specific regression result. The proposed diagnostic is a measure of overall potential influence. Fitting a regression model in practice can have many purposes and the user may prefer to begin with an overall measure. This diagnostic possesses several desirable properties that many of the frequently used diagnostics do not generally possess: (a) invariance to location and scale in the response variable. (b) invariance to nonsingular transformations of the explanatory variables, (c) it is an additive function of measures of leverage and of residual error, and (d) it is monotonically increasing in the leverage values and in the squared residuals. The new influence measure is also supplemented by a graphical display which shows the source of influence. This graph is suitable for both single- and multiple-case influence. Analytical and empirical comparisons between the proposed measure and the existing ones indicate that our measure is superior to existing influence measures.

A comparative study on detection of influential observations in linear regression

A large number of statistics are used in the literature to detect outliers and influential observations in the linear regression model. In this paper comparison studies have been made for determining a statistic which performs better than the other. This includes: (i) a detailed simulation study, and (ii) analyses of several data sets studied by different authors. Different choices of the design matrix of regression model are considered. Design A studies the performance of the various statistics for detecting the scale shift type outliers, and designs B and C provide information on the performance of the statistics for identifying the influential observations. We have used cutoff points using the exact distributions and Bonferroni's inequality for each statistic. The results show that the studentized residual which is used for detection of mean shift outliers is appropriate for detection of scale shift outliers also, and the Welsch's statistic and the Cook's distance are appropriate for detection of influential observations.

Assessing the Effects of Multiple Rows on the Condition Number of a Matrix

Abstract Given a matrix X of n observations on k variables, it is known that the singular values of X can change substantially when few rows are omitted from X. Hadi (1988) shows that no general closed-form equation can relate the singular values of X to the singular values of X with one row deleted and gives closed-form approximations to the relationship between the singular values of the two matrices. In this article, we extend Hadi's results to the more general case of multiple-row deletion, carry out systematic numerical investigations to determine the goodness and the speed of the approximation, and give an example using real-life data to illustrate the usefulness of the results in diagnosing jointly influential observations in linear regression. The methods presented in this article deal with the computational as well as the data analytic aspects of problems arising in multivariate data analysis. These methods are applicable in situations where the eigenstructure of a matrix is of interest.

Discussion of the Paper by Bruce and Martin

Discussion of the Paper by Bruce and Martin

Influential Observations in Linear Regression

Abstract Characteristics of observations which cause them to be influential in a least squares analysis are investigated and related to residual variances, residual correlations, and the convex hull of the observed values of the independent variables. It is shown how deleting an observation can substantially alter an analysis by changing the partial F-tests, the studentized residuals, the residual variances, the convex hull of the independent variables, and the estimated parameter vector. Outliers are discussed briefly, and an example is presented.

Influential Observations in Linear Regression

Abstract Characteristics of observations which cause them to be influential in a least squares analysis are investigated and related to residual variances, residual correlations, and the convex hull of the observed values of the independent variables. It is shown how deleting an observation can substantially alter an analysis by changing the partial F-tests, the studentized residuals, the residual variances, the convex hull of the independent variables, and the estimated parameter vector. Outliers are discussed briefly, and an example is presented.