Multiple Linear Regression Sample Clauses

Multiple Linear Regression. Now suppose that the GLM model (3.1) is a multiple regression model containing an regression intercept and multiple covariates, expressed as Y = 1β0 + X1β1 + · · · + Xp−1βp−1 + s, (3.15) where the n-vectors of X1, . . . , Xp−1 are regressors, each associated with one of the p − 1 covariates, and β1, . . . , βp−1 are the corresponding regression coefficients. Even though the parameter vector β = (β0, . . . , βp−1)T is not of interest and we treat its elements as nuisance parameters in variance component analysis, it must be accounted for during the analysis. If we estimate these parameters by ordinary least squares (OLS), the resulting OLS estimator for β is expressed as βOLS = (XTX)−1XTY, where X = (1, X1, . . . , Xp−1) is the complete design matrix, and thus the OLS residual is  e = Y − XβOLS = ΣI − X(XTX)−1XTΣY. (3.16) Denote a symmetric and idempotent matrix as R = I − X(XTX)−1XT. The OLS residual e = RY = Rs follows a normal distribution with mean E [e] = 0 and vari- ance Cov(e) = RVR, i.e., e ∼ N(0, RVR), where the projection matrix R projects the unobservable error vector s to its estimate e that is orthogonal to the column space spanned by the columns of design matrix X. Covariates contribute to the between-subject variation, however, these “contribu- tions” may confound the heritability estimation and bias the estimator. Hence it is important to model all known covariate-related effects in X, and subsequently we base all heritability analyses on e, calculated with Equation (3.16). For simplicity of notation going forward, we write Y instead of e for the covariate-adjusted data. This embodies an assumption that we can estimate covariate effects β with high precision, and that n p so that V approximates the actual covariance matrix RVR of the adjusted data; we neglect any correlation induced by removing covari- ates and mean centering, and omit the lost information due to the loss of degrees of freedom caused by OLS estimation. Thus Y has mean 0 during the subsequent linear regression model construction with SD’s. Hence the expectations for different subject pairs are similarly derived as E Σ(Y2j−1,j − Y2j,j )2Σ = Var(Y2j−1,j − Y2j,j ) ≈ 2E for MZ twin pair j (j = 1, . . . , 1 nMZ); E Σ(Y2j−1,j − Y2j,j )2Σ = Var(Y2j−1,j − Y2j,j ) ≈ A + 2E for DZ twin pair j (j = 1 nMZ+1, . . . , 1 (nMZ+nDZ)); and for the remaining unrelated

Multiple Linear Regression. Since k-means is clearly the most efficient pooling method out of those tested in an SLR setting, we now assess its performance in MLR. 10,000 simulations were conducted to mimic the BioCycle dataset, with N = 240 and (X1, log(X2), log(X3), X4) ∼ N4(µX, Σ) where µX = (4.730, 3.174, 2.147, 27.296) (see Table 2.2), and (to match the sample covariance matrix), 0.408 0.008 -0.023 0.140 0.008 0.025 -0.006 0.167 -0.023 -0.006 0.269 0.428 0.140 0.167 0.428 66.64 .,  Σ =