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In this paper, we extend the LASSO method (Tibshirani 1996) for simultaneously fitting a regression model and identifying important interaction terms. Unlike most of the existing variable selection methods, our method automatically enforces the heredity constraint, that is, an interaction term can be included in the model only if the corresponding main terms are also included in the model. Furthermore, we extend our method m generalized linear models, and show that it performs as well as if the true model were given in advance, that is, the oracle properly as in Fun and Li (2001) and Fan and Peng (2004). The proof of the oracle property is given in online supplemental materials, Numerical results on both simulation data and real data indicate that our method lends lo remove irrelevant variables more effectively and provide better prediction performance than previous work (Yuan. Joseph, and Lin 2007 and Zhao. Rocha, and Yu 2009 as well as the classical LASSO method). KEY WORDS: Heredity structure; LASSO; Regularization.

Theoretical advantages of nonparametric logarithm of odds to map polygenic diseases are supported by tests of the [Beta] model that depends on a single logistic parameter and is the only model under which paternal and maternal transmissions to sibs of specified phenotypes are independent. Although it does not precisely describe recurrence risks in monozygous twins, the [Beta] model has greater power to detect family resemblance or linkage than the more general [Delta] model which describes the probability of 0, 1, or 2 alleles identical by descent (ibd) with two parameters. Available data on ibd in sibs are consistent with the [Beta] model, but not with the equally parsimonious but less powerful [Gamma] model that assumes a fixed probability of 1/2 for 1 allele ibd. Additivity of loci on the liability scale is not disproven. A simple equivalence extends the [Beta] model to multipoint analysis.

Maximum likelihood methodology was applied to determine the mode of inheritance of rare binary traits with data structures typical for swine populations. The genetic models considered included a monogenic, a digenic, a polygenic, and three mixed polygenic and major gene models. The main emphasis was on the detection of major genes acting on a polygenic background. Deterministic algorithms were employed to integrate and maximize likelihoods. A simulation study was conducted to evaluate model selection and parameter estimation. Three designs were simulated that differed in the number of sires/number of dams within sires (10/10, 30/30, 100/30). Major gene effects of at least one SD of the liability were detected with satisfactory power under the mixed model of inheritance, except for the smallest design. Parameter estimates were empirically unbiased with acceptable standard errors, except for the smallest design, and allowed to distinguish clearly between the genetic models. Distributions of the likelihood ratio statistic were evaluated empirically, because asymptotic theory did not hold. For each simulation model, the Average Information Criterion was computed for all models of analysis. The model with the smallest value was chosen as the best model and was equal to the true model in almost every case studied.

The theory of genetic dominance has has resulted in numerous hypothetical explanations on its viability. Fisher (1928) proposed an evolutionary pattern for the expression of hereditary genes. Fischer's theory was challenged by Wright (1934) who suggested a sequential series of reactions between alleles and gene expression with metabolism. H. Kacser's research on gene expression based on enzymatic reactions and metabolism has remained as one of the most influencing theories of genetics.

Populations of progeny testing such as F3, B1s and B2s obtained from F2, which bridges two inbred lines, have been obtained by an extension of the Elston model. Hypotheses testing can give rise to slow simplification of models, and this can raise the viability of the resultant analysis. Interaction between dwarf genes and polygenic surroundings is evident.

Haldane's rule (1922) states that 'when in the [F.sub.1] offspring of two different animal types one sex is absent, rare, or sterile, that sex is the heterozygous (heterogametic or XY) [...]

The dual genetic system in animals consisting of unclear genome derived from both parents and cytoplasmic genome derived from the mother shows that animal traits may be affected by both genes and that the nuclear inheritance model may be inappropriate to explain the traits. The interaction between nuclear and cytoplasmic genomes will influence both behavioral and psychological interactions among individuals. The inadequate model, based only on the nuclear genome, developed from the idea that nuclear DNA regulated cytoplasmic function.

We develop quantitative-genetic models for the evolution of multiple traits under maternal inheritance, in which traits are transmitted through non-Mendelian as well as Mendelian mechanisms, and maternal selection, in which the fitness of offspring depends on their mother's phenotype as well as their own. Maternal inheritance results in time lags in the evolutionary response to selection. These cause a population to evolve for an indefinite number of generations after selection ceases and make the rate and direction of evolution change even when the strength of selection and parameters of inheritance remain constant. The rate and direction of evolution depend on the inheritance of traits that are not under selection, unlike under classical Mendelian inheritance. The models confirm earlier findings that the response to selection can be larger or smaller than what is possible with simple Mendelian inheritance, and even in a direction opposite to what selection favors. Maternal selection, in which a mother's phenotype influences her offspring's fitness, is frequency-dependent and can cause a population to evolve maladaptively away from a fitness peak, regardless of whether traits are transmitted by Mendelian or maternal inheritance. Maternal selection differs from other forms of selection in that its force depends not only on the fitness function but also on the phenotypic resemblance of parents and offspring.

Brookfield (1993) suggests five tests that he claims establish the statistical significance of Haldane's Rule, contrary to our conclusion (Read and Nee 1991). We reject the basis of all five. [...]

De Koning et al. (PNAS, 2000) detected gametic imprinting for QTL in a swine F2 cross based on a comparison of levels of significance of paternal and maternal imprinting effects against a no-QTL model. They did not test imprinting against a Mendelian model, which is needed to identify deviations from Mendelian inheritance. The objective of this study was to develop such tests. Phenotypes on 2 traits (last rib back fat and marbling) and genotypes from 7 markers on chromosome 1 from 525 F2's from a swine breed cross were used. Analyses were based on line cross regression interval mapping using the following QTL models: Mendelian (additive and dominance effects), full imprinting (separate maternal and paternal allele effects plus dominance), and paternal (no maternal and dominance effects) and maternal imprinting. Each model was tested against the no-QTL model and the full imprinting model was tested against the Mendelian and paternal and maternal imprinting models. Chromosome-wise 5% significance thresholds were derived from 10,000 data permutations. For tests against the Mendelian model paternal and maternal coefficients were permuted within individual. For tests against the paternal (maternal) imprinting model the sign of the maternal (paternal) and dominance coefficients was randomly changed. Thresholds against the no-QTL model for the two traits were 5.1 and 5.0 for the Mendelian model, 4.1 and 4.0 for full imprinting, and 7.2 and 7.1 for maternal or paternal imprinting. Thresholds for full imprinting were 4.6 and 4.0 against the Mendelian model, and 3.2 and 3.8 against maternal or paternal imprinting. In our data, backfat showed significant QTL at similar positions under the Mendelian and full and paternal imprinting models. Full imprinting was significant over Mendelian inheritance but not over paternal imprinting, indicating a paternally expressed QTL. Marbling showed significant QTL under all models but imprinting was not significant over Mendelian inheritance. The models and tests developed in this study allow detection and evaluation of imprinted QTL. Key Words: QTL mapping, Gametic imprinting, Swine