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

Introduction | Overview of PhILR Analysis | Loading and Preprocessing Dataset | Data preparation: TreeSE | Filter Extremely Low-Abundance OTUs | Process Phylogenetic Tree | Investigate Dataset Components | Transform Data using PhILR | Ordination in PhILR Space | Visualization with TreeSE | Identify Balances that Distinguish Human/Non-Human | Name Balances | Visualize Results | Use Balances for Dimension Reduction | Package versions | References

An introduction to fido | The pibble model | Example analysis of microbiome data | More Technical Details | A few notes on model inference and parameter collapsing | References

fido::basset, the model | Non-linear time-series modeling | Updated Functionality: Additive Gaussian Processes | References

fido::orthus, the model | Joint modeling of Microbial 16S data and Metabolomics | Investigate Model Results | Qualifications and Caution | References

Overview | Background on the Matrix Normal | The Vec Operation | Kronker Products | The Matrix Normal | The prior for $\Sigma$ | Background on the Prior | Choosing $\upsilon$ and $\Xi$. | The prior for $\Lambda$ | Choosing $\Theta$ | Choosing $\Gamma$ | The Prior for $\eta$ | How the Choice of $\upsilon$ and $\Xi$ Interacts With the Choice of $\Gamma$

Overview | An Example | Using Maximum Marginal Likelihood to estimate a scale of Gamma

A Simple Example | A More Complicated Example | Custom Functions