© A. Jiratampradab, T. Supapakorn, J. Suntornchost. Article available under the CC BY-SA 4.0 licence
ARTICLE
ABSTRACT
The purpose of this paper is to study and compare the methods for constructing confidence intervals for variance components in an unbalanced one-way random effects model. The methods are based on a classical exact, generalised pivotal quantity, a fiducial inference and a fiducial generalised pivotal quantity. The comparison of criteria involves the empirical coverage probability that maintains at the nominal confidence level of 0.95 and the shortest average length of the confidence interval. The simulation results show that the method based on the generalised pivotal quantity and the fiducial inference perform very well in terms of both the empirical coverage probability and the average length of the confidence interval. The classical exact method performs well in some situations, while the fiducial generalised pivotal quantity performs well in a very unbalanced design. Therefore, the method based on the generalised pivotal quantity is recommended for all situations.
KEYWORDS
variance components, unbalanced one-way random effects model, pivotal quantity, fiducial inference, coverage probability
REFERENCES
Ahrens, H., Pincus, R., (1981). On two measures of unbalancedness in a one-way model and their relation to efficiency. Biometrical Journal, Vol. 23, pp. 227–235.
Arendack´a, B., (2005). Generalized confidence intervals on the variance component in mixed linear models with two variance components. Statistics, Vol. 39, pp. 275–286.
Brownlee, K. A., (1965). Statistical theory and methodology in science and engineering. New York: John Wiley & Sons.
Burch, B. D., (2011). Confidence intervals for variance components in unbalanced one-way random effects model using non-normal distributions. Journal of Statistical Planning and Inference, Vol. 141, pp. 3793–3807.
Demetrashvili, N., Wit, E. C., Van Den Heuvel, E. R., (2016). Confidence intervals for intraclass correlation coefficients in variance components models. Statistical Methods in Medical Research, Vol. 25, pp. 2359–2376.
Graybill, F. A., (1976). Theory and application of the linear model. California: Wadsworth.
Hartung, J., Knapp, G., (2000). Confidence intervals for the between group variance in the unbalanced one-way random effects model of anaylsis of variance. Journal of Statistical Computation and Simulation, Vol. 65, pp. 311–323.
Howe, W. G., (1974). Approximate confidence limits on the mean of X +Y where X and Y are two tabled independent random variables. Journal of the American Statistical Association, Vol. 69, pp. 789–794.
Lamotte, L. R., (1976). Invariant quadratic estimators in the random, one-way anova model. Biometrics, Vol. 32, pp. 793–804.
Li, X., Li, G., (2005). Confidence intervals on sum of variance components with unbalanced designs. Communications in Statistics—Theory and Methods, Vol. 34, pp. 833–845.
Li, X., Li, G., (2007). Comparison of confidence intervals on the among group variance in the unbalanced variance component model. Journal of Statistical Computation and Simulation, Vol. 77, pp. 477–486.
Lidong, E., Hannig, J., Iyer, H., (2008). Fiducial intervals for variance components in an unbalanced two-component normal mixed linear model. Journal of the American Statistical Association, Vol. 103, pp. 854–865.
Liu, X., Xu, X., Hannig, J., (2016). Least squares generalized inferences in unbalanced two-component normal mixed linear model. Computational Statistics, Vol. 31, pp. 973–988.
Milliken, G. A., Johnson, D. E., (2009). Analysis of messy data, volume I: designed experiments. Boca Raton: CRC Press.
Olsen, A., Seely, J., Birkes, D., (1976). Invariant quadratic unbiased estimation for two variance components. The Annals of Statistics, Vol. 4, pp. 878–890.
Park, D. J., Burdick, R. K., (2003). Performance of confidence intervals in regression models with unbalanced one-fold nested error structures. Communications in Statistics- Simulation and Computation, Vol. 32, pp. 717–732.
Searle, S. R., Casella, G., Mcculloch, C. E., (2006). Variance components. New Jersey: John Wiley & Sons.
Thomas, J. D., Hultquist R. A., (1978). Interval estimation for the unbalanced case of the one-way random effects model. The Annals of Statistics, Vol. 6, pp. 582–587.
Ting, N., Burdick, R. K., Graybill, F. A., Gui, R., (1989). One-sided confidence intervals on nonnegative sums of variance components. Statistics & Probability Letters, Vol. 8, pp. 129–135.
Ting, N., Burdick, R. K., Graybill, F. A., Jeyaratnam, S., Lu, T. F. C., (1990). Confidence intervals on linear combinations of variance components that are unrestricted in sign. Journal of Statistical Computation and Simulation, Vol. 35, pp. 135–143.
Wald, A., (1940). A note on the analysis of variance with unequal class frequencies. The Annals of Mathematical Statistics, Vol. 11, pp. 96–100.