Package: ALDEx3 1.4.0

Motivation | What BLMM Approximates | Conceptual Formulation | Implemented Profiled Formulation | The parameter vector (\theta_s) determines the random-effects covariancerelative to the residual variance. As in lme4, BLMM represents this relativecovariance by a lower-triangular factor(\Lambda_ | \Lambda_{\theta_s}\Lambda_{\theta_s}^T,\qquad\operatorname{Var}(u_s) | The response-dependent part of the draw-specific criterion is the profiledweighted residual sum of squares[\mathrm{PWRSS}_s(\theta_s) | Let (\ell_s) denote the profiled REML objective that BLMM minimises. Up to anadditive constant that does not depend on the covariance parameters, it is[\ell_s(\theta_s) | More precisely, the anchor fit targets the average of the per-draw profiledREML criteria:[\bar{\ell}(\phi) | BLMM then takes a local second-order approximation around the anchor(\bar{\phi}). Let[H=\nabla^2\bar{\ell}(\bar{\phi})]denote the observed Hessian of the averaged objective at that anchor. For draw(s), BLMM computes the score correction[g_s | Conditional on (\phi_s), BLMM then solves the fixed effects exactly bygeneralized least squares:[\hat{\beta}_s | \left(X^T V_s^{-1}X\right)^{-1}X^T V_s^{-1} y_s,]where the marginal covariance implied by the updated parameters is[V_s | \sigma_s^2\left(Z\Sigma_{\theta_s}Z^T+I_N\right) | Assumptions and Scope | Validation Strategy | Worked Example | Example Comparison: lme4 Versus blmm | Runtime Scaling

ALDEx3 | Overview | Installation | Quickstart (Inference with Relative Abundances) | Mixed Effects Models | Inference with Absolute Abundances: Defining Scale Models | Prior Knowledge About a Group Difference | External Measurements for Individual Samples | CLR with Scale Uncertainty | Model Outputs | Plotting pairwise analyses