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Concluding RemarksThe joint model specified in Section 17.4 was developed in order to model the association between MRI and histology, taking into account the disease progression effects on both endpoints. The observation unit that we have used in this chapter is the triplet (Genotype^., MRU, Histology^.). Figure 17.23 illustrates the two sources of association presented in this chapter. For a given age, the effect of the disease on MRI а_{г} and the effect of the disease on histology is represented by the shift in the distribution of both MRI and histology parameters as illustrated in panel b. Panel a illustrates the genotype specific association in the residuals after adjusting for the disease effects а_{г }and вг. We have shown that, using a twostage approach, we can estimate a genotypespecific adjusted association p_{W} and p_{A} using the joint model (17.1) in the first stage, while the prediction of the disease progression effects on histology can be done in the second stage using a linear regression model for в_{г} and а_{г}. Although, the experimental setting discussed in this chapter is completely different from the one discussed in Chapter 4, the same association structure (as illustrated in Figure 17.23a) implies that the same modeling FIGURE 17.23 Illustration of the joint modeling framework: The association between MRI and histology after adjusting for the disease effects. approach can be used in order to evaluate the quality of MRI as a biomarker for histology. We have shown that the use of MRI as a biomarker for histology depends on the brain region, MRI parameters, and histology staining. The case studies presented in this chapter posed two challenges with regard to sample size: (1) there were only five age groups, which implies that estimation of the linear regression line in the second stage is based on only five observations and (2) there were only two control mice at each age group. Therefore, the genotypespecific coefficients in (17.2) are based on two observations, hence they may have higher variability. 
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