## 4 Monte Carlo Simulation  
Simulation studies were performed to study the statistical behavior of Bayesian-based functional mapping and demonstrated its applicability and utilization. The simulation design is based on an F\({}_{2}\) population containing 450 individuals. A linkage group of length 100 cM was simulated with 11 equally-spaced, ordered codominant markers. A QTL that affects growth curves was assumed at 34 cM from the first marker. The true values of curve parameters for three QTL genotypes are given as:  
\[\mathbf{\Theta}_{0}=(33.4,11.2,0.65),\quad\mathbf{\Theta}_{2}=(36.7,11.9,0.65), \quad\mathbf{\Theta}_{1}=(35.6,11.2,0.64),\]  
which are close to the estimates for body mass growth from the real mouse example used above. A total of 10 evenly spaced time points were assumed to measure growth curves. The residual covariance matrix among different time points was set to be the same as the sample covariance matrix estimated from the mouse example. Given this hypothesized covariance matrix and curves parameters for the three QTL genotypes, the heritability of the simulated growth trait at a middle time point is about \(0.1\). The time-dependent phenotypic values of the growth trait are assumed to follow a multivariate normal distribution. The priors for curve parameters \(\mathbf{\Theta}_{j}\) (\(j=0,1,2\)) are assumed to be a multivariate normal distribution centered at \((30,10,0.7)\) with covariance matrix \(\mathbf{\Lambda}=diag\{10,5,4\}\). We used a uniform distribution as a prior for the QTL location over the simulated linkage group. Three different approaches were used to estimate the residual covariance matrix \(\mathbf{\Sigma}\): (1) estimating the unstructured covariance matrix based on an inverse Whishart prior with \(T=10\) degrees of freedom, (2) estimating the structured covariance matrix based on the SAD(1) model by imposing \(IG(1,1)\) as a prior for innovation variance \(\
u^{2}\) and  
Figure 1: A profile of Estimated marginal posterior distribution of the QTL location by assuming that exactly one QTL is located on one of the chromosome respectively. \(Normal(0,10)\) as a prior for antedependence parameter \(\phi\), and (3) estimating the structured covariance matrix based on the AR(1) model by imposing \(IG(10,1)\) as a prior for variance \(\sigma^{2}\) and \(Uniform(-1,1)\) as a prior for correlation \(\rho\). For each MCMC-implemented run, 10,000 initial \"burn-in\" iterations were discarded and, after then, samples are collected once for every 60 cycles so that a working set of 1,000 states were used to estimate the posteriors of parameters. The MCMC experiment was repeated several times with the same simulated data set to ensure the chain to converge to a stationary distribution. Table 2 gives the Bayesian estimates of the QTL location and growth curves for different QTL genotypes, along with the \(95\%\) empirical highest posterior density (HPD) confidence regions, under three different matrix-estimating approaches described above, respectively. It can be seen that our Bayesian-based functional mapping provides reasonably good estimates of the parameters in terms of accuracy and precision. The estimates of parameters, especially the QTL location (Figure 4), are affected by matrix-estimating approaches. Overall, the SAD(1)-structured approach is more precise in parameter estimation than the unstructured approach, whereas the unstructured approach is more precise than the AR(1)-structured approach. The SAD(1)-structured approach gives a narrowest confidence interval [33.01, 36.30] for the localization of  
Figure 2: Fitted growth curves for the three QTL genotypes (\(qq\), red; \(Qq\), blue; \(QQ\), black) assuming a single QTL is located on mouse chromosome 6, 7 and 10. QTL, followed by the unstructured [28.02, 38.16] and AR(1)-structured approach [25.54, 41.39]. Also, the AR(1)-structured approach generates a small peak for the marginal posterior distribution at a wrong location (Figure 4), thus increasing a risk to detect a spurious QTL.