Robust modelling using the t-distribution
by
Julian Taylor
BiometricsSA, Adelaide University and South Australian Research and Development Institute, Australia
Coauthors: Arunas Verbyla

Linear models under the assumption of Gaussian process are vulnerable to outliers in the response. Many diagnostic procedures exist for the detection and deletion of these outliers (see Cook and Weisberg, 1982 and Atkinson, 1985). In some circumstances a robust approach to accommodate for the influence asserted by these outliers while maintaining a parametric general linear model may seem more appropriate. This approach involves the use of the t-distribution for robust regression (see Fraser, 1979; West, 1984 and Lange et al., 1989).

The t-regression model can be extended in a number of ways. Taylor and Verbyla (2002) models the variance associated with the t-distribution and compares it to the Gaussian dispersion model. Pinheiro et al. (2001) includes t-distributed random effects in the mean to extend the t specification to a robust mixed model. This talk is concerned with the extension to Restricted Maximum Likelihood (REML) estimation using the t-distribution. Previous work by James et al (1993) includes modified REML using the Cox and Reid (1987) adjustment to the likelihood. However, Welsh and Richardson (1997) describe its shortcomings and propose new likelihoods for robust estimation of the parameters.

In this presentation, the underlying conditional Normal distribution is exploited to formulate the REML likelihood and the random effects are integrated out using a Laplace approximation. The results suggest stable estimates can be found for the mean effects and the bias associated with the variance diminishes. The inclusion of a log-linear variance possibly dependent on covariates associated with the mean is also discussed.

References. Atkinson, A. C. (1985). Plots, transformations and regressions. Oxford: Clarendon.

Cook, R. D. & Weisberg, S. (1982). Residuals and influence in regression. Chapman and Hall.

Cox, D. R. & Reid, N. (1987). Parameter orthogonality and approximate conditional inference (with discussion). Journal of the Royal Statistical Society B. 49, 1-39.

Fraser, D. A. S. (1979). Inference and linear models. New York: McGraw Hill. James, A. T. & Wiskich, J. T. & Conyers, R. A. (1993). t-REML for robust heteroscedastic regression analysis of mitochondrial power. Biometrics 49, 339-356.

Lange, K. L. & Little, R. J. A. & Taylor, J. M. G. (1989). Robust statistical modelling using the t-distribution. Journal of the American Statistical Association, 84, 881-896.

Pinheiro, J. Liu, C. & Wu, Y. N. (2001). Efficient algorithms for robust estimation in linear mixed-effects models using the multivariate t-distribution. Journal of Computational and Graphical Statistics. 10, 249-276.

Taylor, J. T. & Verbyla, A. P. (2002). Dispersion modelling using the t-distribution. Submitted.

Welsh, A. T. & Richardson, A. M. (1997). Approaches to the robust estimation of mixed models. Handbook of Statistics. 15, 343-384.

West, M. (1984). Outlier models and prior distributions in bayesian linear regression. Journal of the Royal Statistical Society B. 46, 431-439.

Date received: September 6, 2002