Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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Nonnegative variance component estimation for mixed-effects models

  • Received : 2020.04.20
  • Accepted : 2020.07.26
  • Published : 2020.09.30
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Abstract

This paper suggests three available methods for finding nonnegative estimates of variance components of the random effects in mixed models. The three proposed methods based on the concepts of projections are called projection method I, II, and III. Each method derives sums of squares uniquely based on its own method of projections. All the sums of squares in quadratic forms are calculated as the squared lengths of projections of an observation vector; therefore, there is discussion on the decomposition of the observation vector into the sum of orthogonal projections for establishing a projection model. The projection model in matrix form is constructed by ascertaining the orthogonal projections defined on vector subspaces. Nonnegative estimates are then obtained by the projection model where all the coefficient matrices of the effects in the model are orthogonal to each other. Each method provides its own system of linear equations in a different way for the estimation of variance components; however, the estimates are given as the same regardless of the methods, whichever is used. Hartley's synthesis is used as a method for finding the coefficients of variance components.

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References

  1. Graybill FA (1983). Matrices with Applications in Statistics, Wadsworth, California.
  2. Hartley HO (1967). Expectations, variances and covariances of ANOVA mean squares by "synthesis", Biometrics, 23, 105-114. https://doi.org/10.2307/2528284
  3. Harville DA (1969). Variance component estimation for the unbalanced one-way random classification-a critique, Aerospace Research Laboratories, ARL-69(0180).
  4. Henderson CR (1953). Estimation of variance and covariance components, Biometrics, 9, 226-252. https://doi.org/10.2307/3001853
  5. Hill BM (1965). Inference about variance components in the one-way model, Journal of the American Statistical Association, 60, 806-825. https://doi.org/10.1080/01621459.1965.10480829
  6. Hill BM (1967). Correlated errors in the random model, Journal of the American Statistical Association, 62, 1387-1400. https://doi.org/10.1080/01621459.1967.10500941
  7. Johnson RA and Wichern DW (2014). Applied Multivariate Statistical Analysis, Prentice-Hall, New Jersey.
  8. Milliken GA and Johnson DE (1984). Analysis of Messy Data, Van Nostrand Reinhold, New York.
  9. Montgomery DC (2013). Design and Analysis of Experiments, John Wiley & Sons, New York.
  10. Nelder JA (1954). The interpretation of negative components of variance, Biometrika, 41, 544-548. https://doi.org/10.1093/biomet/41.3-4.544
  11. Searle SR (1971). Linear Models, John Wiley & Sons, New York.
  12. Searle SR, Casella G, and McCulloch CE (2009). Variance Components, John Wiley & Sons, New York.
  13. Searle SR and Fawcett RF (1970). expected mean squares in variance components models having finite populations, Biometrics, 26, 243-254. https://doi.org/10.2307/2529072
  14. Searle SR and Gruber MHG (2016). Linear Models, John Wiley & Sons, New York.
  15. Thompson Jr WA (1961). Negative estimates of variance components: an introduction, Bulletin, International Institute of Statistics, 34, 1-4.
  16. Thompson Jr WA (1963). Non-negative estimates of variance components, Technometrics, 5, 441-450. https://doi.org/10.1080/00401706.1963.10490122