# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Assessing normality in random effects models. (English) Zbl 0672.62081

Summary: When one uses the unbalanced, mixed linear model

${y}_{i}={X}_{i}\alpha +{Z}_{i}{\beta }_{i}+{ϵ}_{i},\phantom{\rule{1.em}{0ex}}i=1,···,n$

to analyze data from longitudinal experiments with continuous outcomes, it is customary to assume ${ϵ}_{i}{\sim }_{ind}𝒩\left(0,{\sigma }^{2}{I}_{i}\right)$ independent of ${\beta }_{i}{\sim }_{iid}𝒩\left(0,{\Delta }\right)$, where ${\sigma }^{2}$ and the elements of an arbitrary ${\Delta }$ are unknown variance and covariance components. In this paper, we describe a method for checking model adequacy and, in particular, the distributional assumption on the random effects ${\beta }_{i}·$

We generalize the weighted normal plot to accommodate dependent, nonidentically distributed observations subject to multiple random effects for each individual unit under study. One can detect various departures from the normality assumption by comparing the expected and empirical cumulative distribution functions of standardized linear combinations of estimated residuals for each of the individual units.

Through application of distributional results for a certain class of estimators to our context, we adjust the estimated covariance of the empirical cumulative distribution function to account for estimation of unknown parameters. Several examples of our method demonstrate its usefulness in the analysis of longitudinal data.

##### MSC:
 62J05 Linear regression 62J10 Analysis of variance and covariance 62F12 Asymptotic properties of parametric estimators 62P10 Applications of statistics to biology and medical sciences