Originally published as Genetics Published Articles Ahead of Print on September 14, 2008.

Genetics, Vol. 180, 1743-1761, November 2008, Copyright © 2008
doi:10.1534/genetics.108.091058

Mixed Effects Models for Quantitative Trait Loci Mapping With Inbred Strains

Lara E. Bauman^*,,1, Janet S. Sinsheimer^,,, Eric M. Sobel and Kenneth Lange^,,**

^* Department of Genetics, Southwest Foundation for Biomedical Research, San Antonio, Texas 78245-0549, Department of Biomathematics, University of California, Los Angeles, California 90095-1766, Department of Biostatistics, University of California, Los Angeles, California 90095-1772, Department of Human Genetics, University of California, Los Angeles, California 90095-7088 and ^** Department of Statistics, University of California, Los Angeles, California 90095-1554

¹ Corresponding author: Department of Genetics, Southwest Foundation for Biomedical Research, P.O. Box 760549, San Antonio, TX 78245-0549.
E-mail: lbauman{at}sfbrgenetics.org

Fixed effects models have dominated the statistical analysis of genetic crosses between inbred strains. In spite of their popularity, the traditional models ignore polygenic background and must be tailored to each specific cross. We reexamine the role of random effect models in gene mapping with inbred strains. The biggest difficulty in implementing random effect models is the lack of a coherent way of calculating trait covariances between relatives. The standard model for outbred populations is based on premises of genetic equilibrium that simply do not apply to crosses between inbred strains since every animal in a strain is genetically identical and completely homozygous. We fill this theoretical gap by introducing novel combinatorial entities called strain coefficients. With an appropriate theory, it is possible to reformulate QTL mapping and QTL association analysis as an application of mixed models involving both fixed and random effects. After developing this theory, our first example compares the mixed effects model to a standard fixed effects model using simulated advanced intercross line (AIL) data. Our second example deals with hormone data. Here multivariate traits and parameter identifiability questions arise. Our final example involves random mating among eight strains and vividly demonstrates the versatility of our models.

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