Results presented in the previous section were marginal expectations using the distribution of the estimator of the variance components.
Note that the estimates presented are marginal expectations and therefore cannot be added together (e.g., to get an overall risk of cardiovascular mortality from exposure to PM2.5) in a meaningful way.
The expectation and variance of the test statistics and their marginal expectations and variances according to the distribution of genotypes and estimators of variance components are given as a function of the relationship matrix and of the heritability of the polygenic effect.
The GEE approach, extending the idea of the generalized linear model (GLM), assumes a known function of the marginal expectation of the dependent variables [ 38].
The model is: where μ ijk = E y ijk ), the marginal expectation and the β*'s estimate the population average response parameters.
It is notable however that in the non‐linear case with HWE assumption, even though the marginal expectation of U is zero, the proportion of positive and negative values for U i is not equal in general.
Therefore, the score test statistic U i will have non-zero expectation conditionally on some parental intensities (Table 1), the marginal expectation of U (estimated in 3 by taking the weighted average of U i across all possible parental genotypes, Table 1) will be equal to 0, even if the genotype/intensity relationship is non-linear.
However, although the expectation of the score statistic U i conditionally on parental genotypes may differ from zero in the non-linear case, its marginal expectation is equal to zero provided that the HWE assumption is met for the parental genotypes.
The aim of this section is to derive the marginal expectation and variance of the test statistics, by integrating over the distribution of genotypes and the variance components of the random polygenetic effects, given the relationship matrix and variance components of the true model.
Since both groups are similar at baseline by design, the marginal expectation of the response depends only on a group effect (MTX vs. PBO), a quadratic time trend specific to each period (in order to capture the natural evolution over time), and the DAS44 at baselines (T0 and T +5, see Figure 1), and an interaction period - treatment group [ 16].
We exploit the copula-dependence within the portfolio risks to design a Sequential Monte Carlo Samplers based estimate to the marginal conditional expectations involved in the problem, showing its efficiency through a series of computational examples.
