TY - JOUR

T1 - Some algebra and geometry for hierarchical models, applied to diagnostics

AU - Hodges, James S.

PY - 1998

Y1 - 1998

N2 - Recent advances in computing make it practical to use complex hierarchical models. However, the complexity makes it difficult to see how features of the data determine the fitted model. This paper describes an approach to diagnostics for hierarchical models, specifically linear hierarchical models with additive normal or t-errors. The key is to express hierarchical models in the form of ordinary linear models by adding artificial 'cases' to the data set corresponding to the higher levels of the hierarchy. The error term of this linear model is not homoscedastic, but its covariance structure is much simpler than that usually used in variance component or random effects models. The re-expression has several advantages. First, it is extremely general, covering dynamic linear models, random effect and mixed effect models, and pairwise difference models, among others. Second, it makes more explicit the geometry of hierarchical models, by analogy with the geometry of linear models. Third, the analogy with linear models provides a rich source of ideas for diagnostics for all the parts of hierarchical models. This paper gives diagnostics to examine candidate added variables, transformations, collinearity, case influence arid residuals.

AB - Recent advances in computing make it practical to use complex hierarchical models. However, the complexity makes it difficult to see how features of the data determine the fitted model. This paper describes an approach to diagnostics for hierarchical models, specifically linear hierarchical models with additive normal or t-errors. The key is to express hierarchical models in the form of ordinary linear models by adding artificial 'cases' to the data set corresponding to the higher levels of the hierarchy. The error term of this linear model is not homoscedastic, but its covariance structure is much simpler than that usually used in variance component or random effects models. The re-expression has several advantages. First, it is extremely general, covering dynamic linear models, random effect and mixed effect models, and pairwise difference models, among others. Second, it makes more explicit the geometry of hierarchical models, by analogy with the geometry of linear models. Third, the analogy with linear models provides a rich source of ideas for diagnostics for all the parts of hierarchical models. This paper gives diagnostics to examine candidate added variables, transformations, collinearity, case influence arid residuals.

KW - Bayesian methods

KW - Dynamic linear models

KW - Multilevel models

KW - Random effect models

KW - Spatial data

KW - Time varying regression

KW - Variance components

UR - http://www.scopus.com/inward/record.url?scp=0001750294&partnerID=8YFLogxK

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U2 - 10.1111/1467-9868.00137

DO - 10.1111/1467-9868.00137

M3 - Article

AN - SCOPUS:0001750294

SN - 1369-7412

VL - 60

SP - 497

EP - 536

JO - Journal of the Royal Statistical Society. Series B: Statistical Methodology

JF - Journal of the Royal Statistical Society. Series B: Statistical Methodology

IS - 3

ER -