Most current statistical methods developed for mapping quantitative trait loci (QTL) based on inbred line designs apply to crosses from two inbred lines. Analysis of QTL in these crosses is restricted by the parental genetic differences between lines. Crosses from multiple inbred lines or multiple families are common in plant and animal breeding programmes, and can be used to increase the efficiency of a QTL mapping study. A general statistical method using mixture model procedures and the EM algorithm is developed for mapping QTL from various cross designs of multiple inbred lines. The general procedure features three cross design matrices, W, that define the contribution of parental lines to a particular cross and a genetic design matrix, D, that specifies the genetic model used in multiple line crosses. By appropriately specifying W matrices, the statistical method can be applied to various cross designs, such as diallel, factorial, cyclic, parallel or arbitrary-pattern cross designs with two or multiple parental lines. Also, with appropriate specification for the D matrix, the method can be used to analyse different kinds of cross populations, such as F2 backcross, four-way cross and mixed crosses (e.g. combining backcross and F2). Simulation studies were conducted to explore the properties of the method, and confirmed its applicability to diverse experimental designs.

Approaches like multiple interval mapping using a multiple-QTL model for simultaneously mapping QTL can aid the identification of multiple QTL, improve the precision of estimating QTL positions and effects, and are able to identify patterns and individual elements of QTL epistasis. Because of the statistical problems in analytically deriving the standard errors and the distributional form of the estimates and because the use of resampling techniques is not feasible for several linked QTL, there is the need to perform large-scale simulation studies in order to evaluate the accuracy of multiple interval mapping for linked QTL and to assess confidence intervals based on the standard statistical theory. From our simulation study it can be concluded that in comparison with a monogenetic background a reliable and accurate estimation of QTL positions and QTL effects of multiple QTL in a linkage group requires much more information from the data. The reduction of the marker interval size from 10 cM to 5 cM led to a higher power in QTL detection and to a remarkable improvement of the QTL position as well as the QTL effect estimates. This is different from the findings for (single) interval mapping. The empirical standard deviations of the genetic effect estimates were generally large and they were the largest for the epistatic effects. These of the dominance effects were larger than those of the additive effects. The asymptotic standard deviation of the position estimates was not a good criterion for the accuracy of the position estimates and confidence intervals based on the standard statistical theory had a clearly smaller empirical coverage probability as compared to the nominal probability. Furthermore the asymptotic standard deviation of the additive, dominance and epistatic effects did not reflect the empirical standard deviations of the estimates very well, when the relative QTL variance was smaller/equal to 0.5. The implications of the above findings are discussed. mapping / QTL / simulation / asymptotic standard error / confidence interval

The development of molecular genotyping techniques makes it possible to analyze quantitative traits on the basis of individual loci. With marker information, the classical theory of estimating the genetic covariance between relatives can be reformulated to improve the accuracy of estimation. In this study, an algorithm was derived for computing the conditional covariance between relatives given genetic markers. Procedures for calculating the conditional relationship coefficients for additive, dominance, additive by additive, additive by dominance, dominance by additive and dominance by dominance effects were developed. The relationship coefficients were computed based on conditional QTL allelic transmission probabilities, which were inferred from the marker allelic transmission probabilities. An example data set with pedigree and linked markers was used to demonstrate the methods developed. Although this study dealt with two QTLs coupled with linked markers, the same principle can be readily extended to the situation of multiple QTL. The treatment of missing marker information and unknown linkage phase between markers for calculating the covariance between relatives was discussed.

-The development of molecular genotyping techniques makes it possible to analyze quantitative traits on the basis of individual loci. With marker information, the classical theory of estimating the genetic covariance between relatives can be reformulated to improve the accuracy of estimation. In this study, an algorithm was derived for computing the conditional covariance between relatives given genetic markers. Procedures for calculating the conditional relationship coefficients for additive, dominance, additive by additive, additive by dominance, dominance by additive and dominance by dominance effects were developed. The relationship coefficients were computed based on conditional QTL allelic transmission probabilities, which were inferred from the marker allelic transmission probabilities. An example data set with pedigree and linked markers was used to demonstrate the methods developed. Although this study dealt with two QTLs coupled with linked markers, the same principle can be readily extended to the situation of multiple QTL. The treatment of missing marker information and unknown linkage phase between markers for calculating the covariance between relatives was discussed.covariance between relatives / molecular marker / QTL / transmission probability / relationship matrix

A computing simplification was applied to marker-assisted genetic evaluation of quantitative traits including additive and non-additive effects of QTL as well as residual polygenic effects. Different situations including QTL and the residual polygenic effect estimated as a sum or separately, and with or without non-additive effects integrated in models were evaluated. The computing simplification was used in combinations with different models and parameterizations. An example data was adopted to illustrate the simplified computing strategy and was compared with the computing method of direct inversion. Identical results were obtained from both computing strategies. The main advantage of the simplification is that it does not require inversion of nonadditive relationship matrices and relationship matrices of QTL, and the number of random effects in mixed model equations is the same as any animal model with only additive effects.Key Words: molecular markers, gametic models, QTL, non-additive effects, computing simplification Zusammenfassung Titel der Arbeit: Vereinfachungen der markerunterstützten Zuchtwertschätzung und Erfassung nichtadditiver Interaktionseffekte Eine computerunterstützte Vereinfachung der Marker-gestützten genetischen Evaluierung quantitativer Merkmale einschließlich additiver und nicht-additiver Effekte von QTL sowie polygener Restvarianz wurde angewendet. Verschiedene Situationen mit QTL und polygenen Effekten wurden in ihrer Summe betrachtet oder separat sowie mit oder ohne nicht-additive Effekte integriert in den Modellen ausgewertet. Die rechnerische Vereinfachung wurde in den Kombinationen mit unterschiedlichen Modellen und Parametrisierungen verwendet. Beispieldaten wurden angenommen, um die vereinfachte rechnerische Strategie zu veranschaulichen, und wurden mit der Methode der direkten Inversion verglichen. Identische Resultate konnten von beiden rechnerischen Strategien erreicht werden. Der Vorteil der Vereinfachung ist, dass sie eine Inversion der Verwandtschaftsmatrizen für nicht-additive und QTL Effekte nicht erfordert. Die Zahl zufälliger Effekte in den Gleichungen der gemischten Modelle ist identisch mit nur additiven Effekten im Tiermodell.Schlüsselwörter: Molekulare Marker, gametische Modelle, QTL, Nicht-additive Effekte, rechnerische Vereinfachung Introduction In linear model genetic evaluations, the independent variables are usually results in a probability distribution of QTL genotype rather than a specific genotype. Estimation of QTL effects in a linear model requires the use of uncertain QTL genotypes as independent variables. Three types of linear models are often used for the estimation:(1) Mixture model: This type of model is based on a mixture of QTL genotypic distributions. It was proposed by LANDER and BOTSTEIN (1989) for QTL interval mapping and was a direct solution for this kind of evaluation. (2) Regression model: The model was proposed by HALEY and KNOTT (1992). It takes the expectation of