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Interactions in the Analysis of Variance


  • P. L. Davies


The standard model for the analysis of variance is over-parameterized. The resulting identifiability problem is typically solved by placing linear constraints on the parameters. In the case of the interactions, these require that the marginal sums be zero. Although seemingly neutral, these conditions have unintended consequences: the interactions are of necessity connected whether or not this is justified, the minimum number of nonzero interactions is four, and, in particular, it is not possible to have a single interaction in one cell. There is no reason why nature should conform to these constraints. The approach taken in this article is one of sparsity: the linear factor effects are chosen so as to minimize the number of nonzero interactions subject to consistency with the data. The resulting interactions are attached to individual cells making their interpretation easier irrespective of whether they are isolated or form clusters. In general, the calculation of a sparse solution is a difficult combinatorial problem but the special nature of the analysis of variance simplifies matters considerably. In many cases, the sparse L 0 solution coincides with the L 1 solution obtained by minimizing the sum of the absolute residuals and that can be calculated quickly. The identity of the two solutions can be checked either algorithmically or by applying known sufficient conditions for equality.

Suggested Citation

  • P. L. Davies, 2012. "Interactions in the Analysis of Variance," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(500), pages 1502-1509, December.
  • Handle: RePEc:taf:jnlasa:v:107:y:2012:i:500:p:1502-1509 DOI: 10.1080/01621459.2012.726895

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    References listed on IDEAS

    1. Maindonald, John, 2006. "Generalized Additive Models: An Introduction with R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 16(b03).
    2. Nikolay Nenovsky & S. Statev, 2006. "Introduction," Post-Print halshs-00260898, HAL.
    3. Jianqing Fan, 2000. "Simultaneous Confidence Bands and Hypothesis Testing in Varying-coefficient Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(4), pages 715-731.
    4. H rdle, Wolfgang & Huet, Sylvie & Mammen, Enno & Sperlich, Stefan, 2004. "Bootstrap Inference In Semiparametric Generalized Additive Models," Econometric Theory, Cambridge University Press, vol. 20(02), pages 265-300, April.
    5. Jing Wang & Lijian Yang, 2009. "Efficient and fast spline-backfitted kernel smoothing of additive models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(3), pages 663-690, September.
    6. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521785167, March.
    7. Osmani, R.S., 1990. "Food Deprivation and Undernutrition in Rural Bangladesh," Research Paper 82, World Institute for Development Economics Research.
    8. Gerda Claeskens & Tatyana Krivobokova & Jean D. Opsomer, 2009. "Asymptotic properties of penalized spline estimators," Biometrika, Biometrika Trust, vol. 96(3), pages 529-544.
    9. Haerdle,Wolfgang & Bowman,Adrian, 1986. "Bootstrapping in nonparametric regression: Local adaptive smoothing and confidence bands," Discussion Paper Serie A 71, University of Bonn, Germany.
    10. M. Ruth & K. Donaghy & P. Kirshen, 2006. "Introduction," Chapters,in: Regional Climate Change and Variability, chapter 1 Edward Elgar Publishing.
    11. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521780506, March.
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