## 3 A Worked Example
### Results  
[PERSON] _et al._[23 38] first analyzed [PERSON] _et al._'s [37] data set by using functional mapping with maximum-likelihood based methods. They showed that body masses in the F\({}_{2}\) mice follow a logistic curve which can be described by Equation 2, but display substantial variation in the shape of curves. Bayesian-based functional mapping was used to genome-wide search for the QTLs that control mouse growth curves by estimating the QTL location (\(\lambda\)), genotype-specific curve parameters (\(\Theta\)), and the structure or unstructured covariance matrix (\(\Sigma\)). The prior for the locations of \(s\) QTLs (\(\lambda_{1},...,\lambda_{s}\)) along each chromosome is simply assumed to be uniform over \([0,D_{m}]\). Maximum likelihood estimates of growth curve parameters by [PERSON] _et al._[24,37] provide the information about priors of these parameters. The priors for \(\Theta_{j_{1}j_{2}...j_{s}}\) (\(j_{1},...,j_{s}=0,1,2\)) are a multivariate normal, centered at \((30,10,0.6)^{T}\) with a large dispersion \(\mathrm{diag}(9,4,1)\). The prior for the covariance matrix is set to be inverse-Wishart(\(R^{-1},\rho\)), where \(R\) is given by the sample covariance matrix. We used the distance of 40 cM from the first marker on each chromosome as the initial value of \(\lambda\). The initial values of the other parameters were generated from their priors. For each analysis, the Markov chain was run for 60,000 cycles after discarding the first 2,000 cycles for the burn-in period. In order to reduce the serial correlation among samples, the chain was thinned by saving one iteration in every 60 cycles so that the total number of samples stored for calculating the targeted posterior distributions of the unknowns was 1,000 [26]. Bayesian functional mapping was used to genome-wide scan for the existence of QTLs for body mass growth trajectories in the F\({}_{2}\) mouse population. Estimated marginal posteriors of the QTL locations over all 19 mouse chromosomes are illustrated in Figure 1. from which the posterior peaks were observed on chromosome 6, 7, 10, 11, and 15. By calculating the Bayes factors of the model with equation 28 by assuming one QTL over the model assuming no QTL, chromosomes 6, 7 and 10 were each detected to harbor a QTL given that the logarithmic scaled Bayes factors are 12.91, 13.47 and 7.99 for the three chromosomes, respectively. The estimated locations of these QTLs are between between markers \(D6 nds\) and \(D6 Mit58\) on chromosome 6, marker \(D7 nds1\) and \(D7 Mit17\) on chromosome 7 and marker \(D10 Mit133\) and \(D10 Mit14\) on chromosome 10. Table 1 tabulates Bayesian estimates for the locations of the three QTLs detected and their genotype-specific curve parameters from the marginal posterior means of these parameters. From the estimated \(95\%\) equal-tail confidence intervals as HPD regions, Bayesian estimates are considered to be reasonably precise (Table 1). The estimated curve parameters were used to draw the growth curves of three different genotypes at each of the three detected QTLs (Figure 2). The three QTLs exert an effect on body mass growth curves in a similar pattern. The homozygote for the LG/J allele has the best growth during the time course of measurement, followed by the heterozygote for the LG/J and SM/J alleles and the homozygote for the SM/J allele. Based on quantitative genetic theory, we can partition time-dependent genotypic values into time-dependent additive (\(a(t)\)) and time-dependent dominance effect components (\(d(t)\)) for each QTL, illustrated in Figure 3. The additive effect due to the substitution of the SM/J allele with the LG/J allele is positive and increases with age for all the three detected QTLs. The dominant effect due to the interaction between the LG/J and LG/J alleles is consistently small in time course.