We have developed hierarchical Poisson and binomial models for mapping multiple interacting QTL for count phenotypes based upon the unified generalized linear model framework of

Yi and Banerjee (2009). Our method can fit a large number of effects, including covariates, main effects of numerous loci, and epistatic and gene-environment interactions, and can accommodate the correlation among the variables. Many complex traits, including reproductive phenotypes in mice, were measured as Poisson or binomial data. However, statistical methods for mapping multiple interacting QTL for such phenotypes have not been fully developed previously.

To better characterize the genetic architecture of reproductive traits in mice, we applied our methods to two counted reproductive traits, live fetuses (

*LF*) and dead fetuses (

*DF*), in a F

_{2} female mouse population. Since the observed number of OR provides important information about the variation of

*LF* and

*DF*, the numbers of

*LF* and

*DF* were modeled using the Poisson model with

*OR* as an offset, and as the events of “success” out of total “trials” (

*OR*) using the binomial model. As described earlier, the Poisson and binomial models capture different properties of the trait, and allow us to detect some different QTL. In contrast, since the Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the expected number of successes remains fixed, the two models can lead to similar results (

Casella and Berger 2001). In the present study, however, since the sample size (

*n* = 439) is not large enough and the proportions of

*LF* and

*DF* (0.75 and 0.10, respectively) are not small, it is not surprising that the QTL detected in the Poisson models are not in full agreement with those detected in the binomial models. Furthermore, since the proportion of

*DF* is lower than that of

*LF*, there is more coincidence between two models for

*DF* than that for

*LF*.

More than 10 QTL involved in the main and epistatic effects were identified for

*LF* and

*DF* respectively, which exhibit a complex pattern of genetic influence on

*LF* and

*DF*. Most QTL show a very weak main effect but a strong epistatic effect on

*LF* and

*DF*. Compared to the initial study (

Rocha *et al.* 2004b), in which only 3 and 1 QTL were identified for

*LF* and

*DF* respectively, and no epistasis was evaluated, the current study not only identified additional QTL but also provided new information about the genetic architecture of

*LF* and

*DF* through the epistasis. The results also demonstrated that the models incorporating analysis of epistasis can identify QTL which might have a weak main effect but a strong epistatic effect with other QTL. Moreover, based on the number of QTL involved in the epistatic effects, the absolute values of estimated epistatic effects, and the total proportion of phenotypic variances accounted for by the epistatic effects, one conclusion can be drawn that the epistasis plays a more crucial role than does the main effect in regulating the genetic variation of

*LF* and

*DF*. Given the importance of epistasis in the genetic architecture of complex traits, appropriate statistical analyses should accommodate epistatic effects (

Manolio et al. 2009).

Several main and epistatic effects are shared by

*LF* and

*DF*, which suggests that the pleiotropy plays an important role in the reproductive traits in this particular context of F

_{2} mice. Among all of the chromosomes on which some QTL were detected for

*LF* and

*DF* in the present study, the most active one is chromosome 2, on which about 7 QTL involved in the main and epistatic effects were detected. Furthermore, the QTL on chromosome 2 has the strongest main or epistatic effects and contribute the highest proportion to the overall phenotypic variance in some fitted models, which suggests that chromosome 2 has potentially biological relevance to

*LF* and

*DF*. The result is consistent with that in the initial study (

Rocha *et al.* 2004b). Other frequently involved chromosomes include chromosomes 1, 6, 9, and 10.