Use of likelihood ratio tests to detect outliers under the variance shift outlier model

ABSTRACT

In this paper, we revisit the alternative outlier model of Thompson [A note on restricted maximum likelihood estimation with an alternative outlier model, J. Roy. Stat. Soc. Ser. B 47 (1985), pp. 53–55] for detecting outliers in the linear model. Gumedze et al. [A variance shift model for detection of outliers in the linear mixed model, Comput. Statist. Data Anal. 54 (2010), pp. 2128–2144] called this model the variance shift outlier model (VSOM). The basic idea behind the VSOM is to detect observations with inflated variance and isolate them for further investigation. The VSOM is appealing because it downweights an outlier in the analysis, with the weighting determined automatically as part of the estimation procedure. We set up the VSOM as a linear mixed model and then use the likelihood ratio test (LRT) statistic as an objective measure for determining whether the weighting is required, i.e. whether the observation is an outlier. We also derived one-step updates of the variance parameter estimates based on observed, expected and average information matrices to obtain one-step LRT statistics which usually require less computation. Both the fully iterated and one-step LRTs are functions of the squared standard residuals from the null model and therefore can be computed directly without the need to fit the VSOM. We investigated the properties of the likelihood ratio tests and compare them. An extension of the model to detect a group of outliers is also given. We illustrate the proposed methodology using simulated datasets and a real dataset.

KEYWORDS:

• Downweighting
• likelihood ratio tests
• linear model
• outliers
• variance shift outlier model

Acknowledgments

The author would like to dedicate this paper to the memory of Professor Tim Dunne who was his mentor. The author also thanks Dr Sue Welham and Professor Robin Thompson of Rothamsted Research, UK and Dr Brian Cullis and Dr Beverley Gogel for the genesis of the ideas presented in this paper.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Freedom N. Gumedze  http://orcid.org/0000-0003-4387-8844

Additional information

Funding

This research was funded by the University of Cape Town and the National Research Foundation of South Africa [91016] and the Newton Advanced Fellowship, The Academy of Medical Sciences, UK.