## 5 Discussion
Overall, the SAD(1)-structured approach performs best for this particular simulated longitudinal data, followed by the unstructured and AR(1)-structured approach in an order. If the covariance does not have a structure, an unstructured approach, such as one based on the Wishart prior, should be used. In D, we describe a different approach for estimating the full matrix based on a reference prior. The model was used to reanalyze a published data set on the growth of body mass in mice, confirming the discovery of a few QTL detected by conventional ML-based functional mapping. Yet, the new model provides estimates of confidence intervals of curve parameters, thus allowing better statistical inferences about the genetic control of dynamic QTLs. Simulation studies show that Bayesian-based functional mapping is more robust than ML-based functional mapping, in that the former provides more reasonable \begin{table}
\begin{tabular}{l r r r r r r} \hline Parameter & \(QQ\) & \multicolumn{3}{c}{\(Qq\)} & \multicolumn{1}{c}{\(qq\)} \\ \hline \multicolumn{6}{c}{**Unstructured approach**} \\ Location & & & 32.74 & (28.02, 38.16) & & \\ \(\alpha\) & 36.67 & (35.81, 37.46) & 36.03 & (35.47, 36.63) & 33.67 & (32.67, 34.31) \\ \(\beta\) & 11.83 & (11.95, 12.64) & 11.22 & (10.97, 11.50) & 11.30 & (10.94, 11.60) \\ \(\gamma\) & 0.66 & (0.64, 0.67) & 0.64 & (0.63, 0.65) & 0.64 & (0.63, 0.66) \\ \multicolumn{6}{c}{**SAD(1)-structured approach**} \\ Location & & & 34.63 & (33.01, 36.30) & & \\ \(\alpha\) & 36.58 & (35.85, 37.36) & 35.88 & (35.37, 36.39) & 33.81 & (33.09, 34.55) \\ \(\beta\) & 12.04 & (11.65, 12.45) & 11.27 & (11.01, 11.54) & 11.46 & (11.09, 11.83) \\ \(\gamma\) & 0.66 & (0.65, 0.67) & 0.64 & (0.63, 0.65) & 0.65 & (0.64, 0.65) \\ \multicolumn{6}{c}{**AR(1)-structured approach**} \\ Location & & & 33.54 & (25.54, 41.39) & & \\ \(\alpha\) & 36.60 & (35.59, 37.66) & 35.57 & (34.88, 36.35) & 33.61 & (32.65, 34.54) \\ \(\beta\) & 12.04 & (11.61, 12.48) & 11.23 & (10.99, 11.56) & 11.42 & (11.05, 11.83) \\ \(\gamma\) & 0.65 & (0.64, 0.66) & 0.63 & (0.62, 0.65) & 0.66 & (0.64, 0.67) \\ \hline \end{tabular}
\end{table} 
Table 2: Bayesian estimates of QTL locations and genotype-specific growth curves for an assumed QTL from the simulated data set for an F\({}_{2}\) population of 450 individuals based on different covariance-structuring approaches. Numbers in parentheses are the \(95\%\) equal-tail confidence intervals. estimates of QTL positions and dynamic QTL effects when the heritability of growth curves is modest (0.1) compared to the latter. However, a unique issue related to the Bayesian approach is about its stability or sensitivity in parameter estimation to different priors. One approach for testing the model's stability is to initiate the chain with several different starting points and/or different priors under the same MCMC sampling scheme [41], and further examine how the estimates depend on the choices of initial values and priors. The model proposed in this article can be modified by considering a network of genetic control. As a basic Bayesian framework, our single-QTL interval mapping model is not adequate to explore the effects of interactions on developmental variation for a growth trait between different QTLs from the same genome [43] or different genomes [44], and between QTLs and environments [24]. One of the significant advantages of Bayesian approaches lies in the estimation of the optimal number of QTLs involved. Variable selection via stepwise regression is used in ML mapping, but it is highly computationally expensive. Corresponding to this variable selection procedure, reversible jump MCMC is proposed in Bayesian analysis [45; 46], although it is subject to poor mixing and a slow convergence to the stationary distribution [47; 48]. More efficient methods based on Bayesian shrinkage analysis [16] and stochastic search variable selection [17] have now been proposed.