Appendix A: Notation Quick Reference

Vector and Matrix Notation

Symbol	Description	Dimensions
$\mathbf{y}$	Response/observation vector	$n \times 1$
$\mathbf{X}$	Design/incidence matrix	$n \times p$
$\boldsymbol{\beta}$	Parameter vector (fixed effects)	$p \times 1$
$\mathbf{b}$	Estimate vector	$p \times 1$
$\mathbf{e}$	Error/residual vector	$n \times 1$
$\mathbf{I}$	Identity matrix	$n \times n$
$\mathbf{H}$	Hat matrix (projection)	$n \times n$
$\mathbf{C}$	Contrast matrix	$q \times p$

Matrix Operators

Operator	Description
$\mathbf{A}'$ or $\mathbf{A}^\top$	Transpose of $\mathbf{A}$
$\mathbf{A}^{-1}$	Regular inverse of $\mathbf{A}$
$\mathbf{A}^{-}$	Generalized inverse of $\mathbf{A}$
$\mathbf{A}^{+}$	Moore-Penrose inverse
$r(\mathbf{A})$	Rank of $\mathbf{A}$
$tr(\mathbf{A})$	Trace of $\mathbf{A}$
$\det(\mathbf{A})$ or $\|\mathbf{A}\|$	Determinant of $\mathbf{A}$

Scalars and Parameters

Symbol	Description
$n$	Sample size
$p$	Number of parameters
$\sigma^2$	Variance
$\mu$	Population mean
$\alpha_i$	Treatment effect for group $i$

Statistical Quantities

Abbreviation	Full Name	Formula
SSE	Sum of Squares Error	$\mathbf{e}'\mathbf{e}$
SST	Total Sum of Squares	$\sum(y_i - \bar{y})^2$
SSM	Model Sum of Squares	$SST - SSE$
MSE	Mean Square Error	$SSE/(n-p)$
MSM	Mean Square Model	$SSM/(p-1)$
$R^2$	Coefficient of Determination	$1 - SSE/SST$
df	Degrees of Freedom	Various

Mixed Model Extensions (Week 14)

Symbol	Description	Dimensions
$\mathbf{u}$	Random effects vector	$q \times 1$
$\mathbf{Z}$	Incidence matrix for random effects	$n \times q$
$\mathbf{G}$	Variance-covariance for random effects	$q \times q$
$\mathbf{R}$	Variance-covariance for residuals	$n \times n$

Notation Conventions

Vectors: lowercase bold ($\mathbf{v}$, $\mathbf{y}$)
Matrices: uppercase bold ($\mathbf{A}$, $\mathbf{X}$)
Greek parameters: bold ($\boldsymbol{\beta}$, $\boldsymbol{\alpha}$)
Scalars: italic ($n$, $\sigma^2$)
Estimates: hat notation ($\hat{\beta}$, $\hat{\sigma}^2$)

This notation is established in Week 1 and maintained consistently throughout all 15 weeks.

# Appendix A: Notation Quick Reference {#sec-appendix-a .unnumbered} ## Vector and Matrix Notation | Symbol | Description | Dimensions | |--------|-------------|------------| | $\mathbf{y}$ | Response/observation vector | $n \times 1$ | | $\mathbf{X}$ | Design/incidence matrix | $n \times p$ | | $\boldsymbol{\beta}$ | Parameter vector (fixed effects) | $p \times 1$ | | $\mathbf{b}$ | Estimate vector | $p \times 1$ | | $\mathbf{e}$ | Error/residual vector | $n \times 1$ | | $\mathbf{I}$ | Identity matrix | $n \times n$ | | $\mathbf{H}$ | Hat matrix (projection) | $n \times n$ | | $\mathbf{C}$ | Contrast matrix | $q \times p$ | ## Matrix Operators | Operator | Description | |----------|-------------| | $\mathbf{A}'$ or $\mathbf{A}^\top$ | Transpose of $\mathbf{A}$ | | $\mathbf{A}^{-1}$ | Regular inverse of $\mathbf{A}$ | | $\mathbf{A}^{-}$ | Generalized inverse of $\mathbf{A}$ | | $\mathbf{A}^{+}$ | Moore-Penrose inverse | | $r(\mathbf{A})$ | Rank of $\mathbf{A}$ | | $tr(\mathbf{A})$ | Trace of $\mathbf{A}$ | | $\det(\mathbf{A})$ or $|\mathbf{A}|$ | Determinant of $\mathbf{A}$ | ## Scalars and Parameters | Symbol | Description | |--------|-------------| | $n$ | Sample size | | $p$ | Number of parameters | | $\sigma^2$ | Variance | | $\mu$ | Population mean | | $\alpha_i$ | Treatment effect for group $i$ | ## Statistical Quantities | Abbreviation | Full Name | Formula | |--------------|-----------|---------| | SSE | Sum of Squares Error | $\mathbf{e}'\mathbf{e}$ | | SST | Total Sum of Squares | $\sum(y_i - \bar{y})^2$ | | SSM | Model Sum of Squares | $SST - SSE$ | | MSE | Mean Square Error | $SSE/(n-p)$ | | MSM | Mean Square Model | $SSM/(p-1)$ | | $R^2$ | Coefficient of Determination | $1 - SSE/SST$ | | df | Degrees of Freedom | Various | ## Mixed Model Extensions (Week 14) | Symbol | Description | Dimensions | |--------|-------------|------------| | $\mathbf{u}$ | Random effects vector | $q \times 1$ | | $\mathbf{Z}$ | Incidence matrix for random effects | $n \times q$ | | $\mathbf{G}$ | Variance-covariance for random effects | $q \times q$ | | $\mathbf{R}$ | Variance-covariance for residuals | $n \times n$ | ## Notation Conventions - **Vectors**: lowercase bold ($\mathbf{v}$, $\mathbf{y}$) - **Matrices**: uppercase bold ($\mathbf{A}$, $\mathbf{X}$) - **Greek parameters**: bold ($\boldsymbol{\beta}$, $\boldsymbol{\alpha}$) - **Scalars**: italic ($n$, $\sigma^2$) - **Estimates**: hat notation ($\hat{\beta}$, $\hat{\sigma}^2$) This notation is established in Week 1 and maintained consistently throughout all 15 weeks.

Symbol	Description	Dimensions
\(\mathbf{y}\)	Response/observation vector	\(n \times 1\)
\(\mathbf{X}\)	Design/incidence matrix	\(n \times p\)
\(\boldsymbol{\beta}\)	Parameter vector (fixed effects)	\(p \times 1\)
\(\mathbf{b}\)	Estimate vector	\(p \times 1\)
\(\mathbf{e}\)	Error/residual vector	\(n \times 1\)
\(\mathbf{I}\)	Identity matrix	\(n \times n\)
\(\mathbf{H}\)	Hat matrix (projection)	\(n \times n\)
\(\mathbf{C}\)	Contrast matrix	\(q \times p\)

Operator	Description
\(\mathbf{A}'\) or \(\mathbf{A}^\top\)	Transpose of \(\mathbf{A}\)
\(\mathbf{A}^{-1}\)	Regular inverse of \(\mathbf{A}\)
\(\mathbf{A}^{-}\)	Generalized inverse of \(\mathbf{A}\)
\(\mathbf{A}^{+}\)	Moore-Penrose inverse
\(r(\mathbf{A})\)	Rank of \(\mathbf{A}\)
\(tr(\mathbf{A})\)	Trace of \(\mathbf{A}\)
\(\det(\mathbf{A})\) or \(\|\mathbf{A}\|\)	Determinant of \(\mathbf{A}\)

Symbol	Description	Dimensions
\(\mathbf{u}\)	Random effects vector	\(q \times 1\)
\(\mathbf{Z}\)	Incidence matrix for random effects	\(n \times q\)
\(\mathbf{G}\)	Variance-covariance for random effects	\(q \times q\)
\(\mathbf{R}\)	Variance-covariance for residuals	\(n \times n\)