Appendix C: R Functions Reference

Introduction

This appendix provides a concise reference guide to all R functions used throughout the 15-week linear models course. Functions are organized by functional category (not alphabetically) to help you quickly locate related operations.

Purpose and Scope

This reference is designed to:

Provide quick lookup of function syntax and key parameters
Show which weeks each function is introduced or used heavily
Distinguish between base R and package functions
Offer troubleshooting guidance for common issues

How to Use This Appendix

Quick lookup: Use the table of contents to jump to a specific category
Function search: Use your browser’s search function (Ctrl+F or Cmd+F) to find a specific function name
Week review: Check the “Week(s)” column to see where concepts are taught
Troubleshooting: Jump to Common Pitfalls or Quick Troubleshooting Guide for common issues

Relationship to Other Appendices

Appendix A: Mathematical notation reference
Appendix B: In-depth matrix algebra tutorial with derivations
Appendix C (this): Quick reference for R functions

Package Installation and Setup

Most functions in this appendix are from base R and require no additional installation. For package functions, install as needed:

# Essential packages
install.packages("MASS")      # ginv(), boxcox()
install.packages("car")       # Anova(), vif()
install.packages("emmeans")   # emmeans(), contrast(), pairs()
install.packages("Matrix")    # rankMatrix(), sparse matrices

# Optional packages (used in specific weeks)
install.packages("lme4")      # lmer() for mixed models preview (Week 14)
install.packages("multcomp")  # Multiple comparisons (Week 8)
install.packages("ggplot2")   # Enhanced graphics

Loading Packages

Load required packages at the start of your R session:

library(MASS)
library(car)
library(emmeans)
library(Matrix)

Base R vs Packages

Base R functions (like lm(), solve(), matrix()) are always available
Package functions (like ginv(), Anova()) require loading the package first with library()

Basic Matrix Operations

Functions for creating and manipulating matrices.

Function	Package	Syntax	Description	Key Parameters	Week(s)
`matrix()`	base R	`matrix(data, nrow, ncol)`	Create a matrix from vector	`data=`, `nrow=`, `ncol=`, `byrow=`	1-15
`t()`	base R	`t(X)`	Transpose matrix	`x=`	1-15
`%*%`	base R	`A %*% B`	Matrix multiplication	-	1-15
`solve()`	base R	`solve(A)` or `solve(A, b)`	Compute matrix inverse or solve linear system	`a=`, `b=`	2-15
`ginv()`	MASS	`ginv(X)`	Generalized (Moore-Penrose) inverse	`X=`, `tol=`	2,12-13
`cbind()`	base R	`cbind(v1, v2, ...)`	Combine vectors/matrices by columns	`...`	3-15
`rbind()`	base R	`rbind(v1, v2, ...)`	Combine vectors/matrices by rows	`...`	Throughout
`c()`	base R	`c(x1, x2, ...)`	Combine values into vector	`...`	1-15

Matrix Multiplication vs Element-wise

%*% performs matrix multiplication (inner dimensions must match: (m×n) × (n×p) → (m×p))
* performs element-wise multiplication (dimensions must match exactly)

Efficiency Tip

For computing $\mathbf{X}'\mathbf{X}$, use crossprod(X) instead of t(X) %*% X - it’s faster and more numerically stable.

Matrix Properties and Decompositions

Functions for examining matrix characteristics and performing decompositions.

Function	Package	Syntax	Description	Key Parameters	Week(s)
`qr()`	base R	`qr(X)`	QR decomposition; use `$rank` to get rank	`x=`, `tol=`	2,12
`det()`	base R	`det(A)`	Compute determinant	`x=`	2
`diag()`	base R	`diag(X)` or `diag(n)`	Extract diagonal or create diagonal matrix	`x=`, `nrow=`, `ncol=`	2-15
`eigen()`	base R	`eigen(A)`	Eigenvalue decomposition	`x=`, `symmetric=`	2,5
`svd()`	base R	`svd(X)`	Singular value decomposition	`x=`	2,13
`rankMatrix()`	Matrix	`rankMatrix(X)`	Compute rank (more reliable than `qr()$rank`)	`x=`, `method=`	2,12
`crossprod()`	base R	`crossprod(X)` or `crossprod(X, Y)`	Compute $\mathbf{X}'\mathbf{X}$ or $\mathbf{X}'\mathbf{Y}$ efficiently	`x=`, `y=`	5-15
`tcrossprod()`	base R	`tcrossprod(X)`	Compute $\mathbf{X}\mathbf{X}'$ efficiently	`x=`, `y=`	5-15
`norm()`	base R	`norm(X, type)`	Compute matrix norm	`x=`, `type=`	2

Rank Determination

For rank deficiency detection, use rankMatrix() from the Matrix package - it’s more robust than qr()$rank for near-singular matrices. See Week 12 for details on non-full rank models.

Linear Model Fitting

Functions for fitting linear models and building design matrices.

Function	Package	Syntax	Description	Key Parameters	Week(s)
`lm()`	base R	`lm(formula, data)`	Fit linear model using ordinary least squares	`formula=`, `data=`, `weights=`	4-15
`model.matrix()`	base R	`model.matrix(formula, data)`	Build design matrix $\mathbf{X}$ from formula	`object=`, `data=`	3-15
`formula()`	base R	`formula(model)`	Extract formula from fitted model	`x=`	Throughout
`update()`	base R	`update(model, new_formula)`	Update and refit model with new formula	`object=`, `formula=`, `data=`	6-10
`lmer()`	lme4	`lmer(formula, data)`	Fit mixed model (preview only)	`formula=`, `data=`, `REML=`	14

Design Matrix Construction

Always verify your design matrix before fitting:

X <- model.matrix(~ breed + sex, data=mydata)
head(X)  # Check structure
qr(X)$rank  # Check rank

Model Summaries and Extraction

Functions for extracting information from fitted models.

Function	Package	Syntax	Description	Key Parameters	Week(s)
`summary()`	base R	`summary(model)`	Comprehensive model summary (coefficients, R², F-test)	`object=`	4-15
`coef()`	base R	`coef(model)`	Extract coefficient estimates $\mathbf{b}$	`object=`	4-15
`residuals()`	base R	`residuals(model)` or `resid(model)`	Extract residuals $\mathbf{e} = \mathbf{y} - \hat{\mathbf{y}}$	`object=`, `type=`	4-15
`fitted()`	base R	`fitted(model)`	Extract fitted values $\hat{\mathbf{y}} = \mathbf{Xb}$	`object=`	4-15
`predict()`	base R	`predict(model, newdata)`	Predict for new observations	`object=`, `newdata=`, `interval=`	4,6
`vcov()`	base R	`vcov(model)`	Variance-covariance matrix of estimates $(\mathbf{X}'\mathbf{X})^{-1}\hat{\sigma}^2$	`object=`	5-15
`sigma()`	base R	`sigma(model)`	Extract residual standard error $\hat{\sigma}$	`object=`	5-15
`deviance()`	base R	`deviance(model)`	Residual sum of squares (SSE)	`object=`	5-15
`logLik()`	base R	`logLik(model)`	Log-likelihood of fitted model	`object=`	6,14
`AIC()`	base R	`AIC(model)`	Akaike Information Criterion	`object=`, `k=`	6,14
`confint()`	base R	`confint(model, level=0.95)`	Confidence intervals for parameters	`object=`, `parm=`, `level=`	5-6

Residuals vs Fitted Values

residuals(model): Returns $\mathbf{e} = \mathbf{y} - \hat{\mathbf{y}}$
fitted(model): Returns $\hat{\mathbf{y}} = \mathbf{Xb}$
Verify: all.equal(y, fitted(model) + residuals(model)) should be TRUE

Hypothesis Testing and ANOVA

Functions for testing hypotheses about model parameters.

Function	Package	Syntax	Description	Key Parameters	Week(s)
`anova()`	base R	`anova(model)` or `anova(model1, model2)`	ANOVA table with Type I (sequential) SS	`object=`	6-10
`Anova()`	car	`Anova(model, type=3)`	Type II or Type III SS for unbalanced data	`mod=`, `type=`	9-10,12
`emmeans()`	emmeans	`emmeans(model, specs)`	Estimated marginal means (LSMEANS)	`object=`, `specs=`, `by=`	8-10
`contrast()`	emmeans	`contrast(emm, method)`	Test contrasts among means	`object=`, `method=`, `adjust=`	8-10
`pairs()`	emmeans	`pairs(emm)`	Pairwise comparisons	`x=`, `adjust=`	8-9
`drop1()`	base R	`drop1(model, test="F")`	Test dropping each term	`object=`, `scope=`, `test=`	6,9
`add1()`	base R	`add1(model, scope, test="F")`	Test adding each term	`object=`, `scope=`, `test=`	6
`vif()`	car	`vif(model)`	Variance inflation factors (detect collinearity)	`mod=`	6

Type I vs Type III SS

Type I SS (sequential): Order matters! Each term adjusted for previous terms only
- Use anova(model) - default in R
- Appropriate for balanced designs
Type III SS (partial): Each term adjusted for all other terms
- Use car::Anova(model, type=3)
- Appropriate for unbalanced designs and recommended for general use

See Weeks 9-10 for detailed discussion.

Model Diagnostics

Functions for checking model assumptions and identifying influential observations.

Function	Package	Syntax	Description	Key Parameters	Week(s)
`plot()`	base R	`plot(lm_object)`	Create 4 diagnostic plots automatically	`x=`, `which=`	11
`hatvalues()`	base R	`hatvalues(model)`	Extract leverage values (diagonal of hat matrix $\mathbf{H}$)	`model=`	11
`cooks.distance()`	base R	`cooks.distance(model)`	Compute Cook’s distance for each observation	`model=`	11
`rstandard()`	base R	`rstandard(model)`	Standardized residuals	`model=`, `type=`	11
`rstudent()`	base R	`rstudent(model)`	Studentized deleted residuals	`model=`	11
`influence.measures()`	base R	`influence.measures(model)`	Comprehensive influence diagnostics	`model=`	11
`dffits()`	base R	`dffits(model)`	Change in fitted value when observation deleted	`model=`	11
`dfbetas()`	base R	`dfbetas(model)`	Change in coefficients when observation deleted	`model=`	11
`qqnorm()`	base R	`qqnorm(residuals)`	Normal Q-Q plot to check normality	`y=`	11
`boxcox()`	MASS	`boxcox(model)`	Box-Cox transformation for normality/homoscedasticity	`object=`, `lambda=`	11,14

Rule-of-Thumb Thresholds

High leverage: $h_{ii} > 2p/n$ or $3p/n$ where $p$ = number of parameters, $n$ = sample size
Influential observation: Cook’s D > 1 or Cook’s D > 4/n
Outlier: |studentized residual| > 3

Week 11 provides detailed guidance on interpretation and remedial measures.

Quick Diagnostic Check

par(mfrow=c(2,2))
plot(model)  # Creates all 4 diagnostic plots at once

Residuals vs Fitted: Check linearity and homoscedasticity
Q-Q Plot: Check normality assumption
Scale-Location: Check homoscedasticity
Residuals vs Leverage: Identify influential points

Advanced Model Tools

Functions for working with contrasts, constraints, and model parameterizations.

Function	Package	Syntax	Description	Key Parameters	Week(s)
`contr.treatment()`	base R	`contr.treatment(n)`	Treatment (reference cell) contrasts	`n=`, `base=`	3,8,13
`contr.sum()`	base R	`contr.sum(n)`	Sum-to-zero contrasts	`n=`	8,13
`contrasts()`	base R	`contrasts(factor)`	Get or set contrasts for a factor	`x=`, `value=`	8,13
`options()`	base R	`options(contrasts=c(...))`	Set default contrast system	`contrasts=`	13
`relevel()`	base R	`relevel(factor, ref)`	Change reference level of factor	`x=`, `ref=`	8-9
`interaction.plot()`	base R	`interaction.plot(f1, f2, y)`	Plot interaction between two factors	`x.factor=`, `trace.factor=`, `response=`	9
`all.equal()`	base R	`all.equal(x, y)`	Compare objects with tolerance (for floating-point)	`target=`, `current=`, `tolerance=`	Throughout
`is.estimable()`	estimability	`is.estimable(contrast, X)`	Check if linear function is estimable	`x=`, `X=`	8,12-13

Contrast Systems

Set-to-zero (treatment) contrasts:

options(contrasts = c("contr.treatment", "contr.poly"))

Default in R
First level is reference (coefficient = 0)
Other coefficients are differences from reference

Sum-to-zero (effects) contrasts:

options(contrasts = c("contr.sum", "contr.poly"))

Coefficients sum to zero: $\sum \alpha_i = 0$
Each coefficient is deviation from overall mean
More symmetric, preferred for balanced designs

See Week 13 for detailed comparison and implications.

Data Manipulation

Functions for organizing and summarizing data.

Function	Package	Syntax	Description	Key Parameters	Week(s)
`tapply()`	base R	`tapply(X, INDEX, FUN)`	Apply function to subgroups	`X=`, `INDEX=`, `FUN=`	7-10
`aggregate()`	base R	`aggregate(x, by, FUN)`	Compute summary statistics by groups	`x=`, `by=`, `FUN=`	7-10
`factor()`	base R	`factor(x, levels)`	Create or modify factor variable	`x=`, `levels=`, `labels=`	3-15
`mean()`	base R	`mean(x)`	Compute arithmetic mean	`x=`, `na.rm=`	1-15
`sum()`	base R	`sum(x)`	Compute sum	`x=`, `na.rm=`	1-15
`length()`	base R	`length(x)`	Number of elements in vector	`x=`	Throughout
`nrow()`	base R	`nrow(X)`	Number of rows in matrix/data frame	`x=`	Throughout
`ncol()`	base R	`ncol(X)`	Number of columns in matrix/data frame	`x=`	Throughout
`subset()`	base R	`subset(data, condition)`	Extract subset of data	`x=`, `subset=`, `select=`	Throughout
`transform()`	base R	`transform(data, ...)`	Add or modify variables	`_data=`, `...`	Throughout
`with()`	base R	`with(data, expr)`	Evaluate expression in data environment	`data=`, `expr=`	Throughout
`round()`	base R	`round(x, digits)`	Round to specified decimal places	`x=`, `digits=`	Throughout
`sprintf()`	base R	`sprintf(fmt, ...)`	Format output with C-style formatting	`fmt=`, `...`	Throughout
`cat()`	base R	`cat(...)`	Print output (useful in R Markdown)	`...`, `sep=`, `fill=`	Throughout

Computing Group Statistics

Three common approaches:

1. Using tapply() - Returns vector/array:

tapply(y, group, mean)  # Mean of y for each group

2. Using aggregate() - Returns data frame:

aggregate(y ~ group, data=mydata, FUN=mean)

3. Using emmeans() - Returns model-based estimates:

emmeans(lm(y ~ group), specs="group")

Choose based on your needs: - tapply(): Quick calculations - aggregate(): Multiple variables at once - emmeans(): Adjusted means from models

Common Pitfalls

Common issues encountered when working with linear models in R, their symptoms, and solutions.

Issue	Symptom	Solution
Singular matrix	`Error in solve.default(X): system is computationally singular`	Use `ginv()` from MASS package instead of `solve()`. Occurs when $\mathbf{X}'\mathbf{X}$ is not full rank (Week 12).
Object not found	`Error: object 'variable_name' not found`	Add `data=` argument to `lm()`: `lm(y ~ x, data=mydata)`. Ensures variables are found in the data frame.
Wrong SS type	Type I SS gives different results than expected in unbalanced data	Use `car::Anova(model, type=3)` instead of base `anova()` for unbalanced designs (Weeks 9-10).
Non-estimable contrasts	Different g-inverses give different parameter estimates	Verify contrast is estimable: check $\sum c_i = 0$ for treatment contrasts. Only interpret estimable functions (Week 8, 12-13).
Uncentered covariate	Intercept not interpretable as group mean at $\bar{x}$	Center covariate: `lm(y ~ group + I(x - mean(x)))` (Week 10).
Rank deficiency warnings	`Coefficients: (1 not defined because of singularities)`	Either use cell means model (`~ factor - 1`) or apply explicit constraints. Check design matrix rank: `qr(X)$rank` (Week 12).
Factor vs numeric	`contrasts can be applied only to factors with 2 or more levels`	Convert numeric variable to factor: `factor(variable)` or include in formula as `factor(variable)`.
Dimension mismatch	`non-conformable arrays` or `arguments imply differing number of rows`	Check dimensions with `dim(X)`, `length(y)`. Matrix multiplication requires matching inner dimensions (Week 1-2).

Debugging Strategy

When encountering errors, follow this systematic approach:

Check your data: str(data), summary(data), head(data)
Check your formula: formula(model)
Check design matrix: X <- model.matrix(model), then dim(X), qr(X)$rank
Verify dimensions: Ensure $n \times p$ design matrix matches $n \times 1$ response
Check for rank deficiency: Compare qr(X)$rank to ncol(X)
Compare with lm(): Verify manual calculations match lm() output
Test estimability: For contrasts, verify $\mathbf{c}'\boldsymbol{\beta}$ is estimable

See Week 11 (Diagnostics) and Week 12 (Non-Full Rank) for detailed troubleshooting.

Quick Troubleshooting Guide

Common Error Messages and Solutions

“Error in solve.default(): system is computationally singular”

Cause: Matrix $\mathbf{X}'\mathbf{X}$ is not full rank (determinant ≈ 0)

Solution:

library(MASS)
b <- ginv(t(X) %*% X) %*% t(X) %*% y  # Use generalized inverse

Related: Week 2 (Linear Algebra), Week 12 (Non-Full Rank Models)

“Error: object ‘variable_name’ not found”

Cause: Variable not in current environment or data frame not specified

Solution:

model <- lm(y ~ x, data=mydata)  # Always use data= argument

Related: Week 4 (Simple Regression)

“Coefficients: (1 not defined because of singularities)”

Cause: Design matrix is rank deficient (overparameterized model)

Solution:

# Option 1: Cell means model
model <- lm(y ~ breed - 1, data=mydata)

# Option 2: Check which parameters are aliased
alias(model)

Related: Week 12 (Non-Full Rank Models)

“contrasts can be applied only to factors with 2 or more levels”

Cause: Categorical variable coded as numeric instead of factor

Solution:

mydata$breed <- factor(mydata$breed)
# Or in formula:
model <- lm(y ~ factor(breed), data=mydata)

Related: Week 3 (Design Matrices), Week 7 (ANOVA)

“non-conformable arrays”

Cause: Matrix dimensions don’t match for the operation

Solution:

dim(X)  # Check dimensions
dim(y)
# For matrix multiplication A %*% B, need ncol(A) == nrow(B)

Related: Week 1-2 (Matrix Operations)

Getting Help in R

Function documentation:

?lm           # Quick help
help(lm)      # Same as ?
??regression  # Search all help files

Examples:

example(lm)           # Run examples from help file
demo("graphics")      # Interactive demonstrations

Package information:

help(package="MASS")  # Package overview and function list

Vignettes (detailed tutorials):

vignette()                    # List all available vignettes
vignette("emmeans")           # View specific vignette
browseVignettes("emmeans")    # Open in browser

Online resources:

R Documentation: https://www.rdocumentation.org/
Stack Overflow: https://stackoverflow.com/questions/tagged/r
R-bloggers: https://www.r-bloggers.com/
Quick-R: https://www.statmethods.net/

Verification Strategy

When building your own least squares solvers, always verify against lm():

# 1. Manual calculation
X <- model.matrix(~ breed + sex, data=mydata)
y <- mydata$weight
b_manual <- solve(t(X) %*% X) %*% t(X) %*% y

# 2. Using lm()
model <- lm(weight ~ breed + sex, data=mydata)
b_lm <- coef(model)

# 3. Compare (should be near-zero)
max(abs(b_manual - b_lm))

# 4. Use all.equal() for floating-point comparisons
all.equal(as.vector(b_manual), b_lm)
# Should return TRUE (or very small tolerance message)

# 5. Verify fitted values
y_hat_manual <- X %*% b_manual
y_hat_lm <- fitted(model)
all.equal(as.vector(y_hat_manual), y_hat_lm)

# 6. Verify residuals
e_manual <- y - y_hat_manual
e_lm <- residuals(model)
all.equal(as.vector(e_manual), e_lm)

# 7. Verify SSE
SSE_manual <- sum(e_manual^2)
SSE_lm <- deviance(model)
all.equal(SSE_manual, SSE_lm)

Floating-Point Comparison

Never use == to compare floating-point numbers! Use all.equal() instead:

# BAD
b_manual == b_lm  # May fail due to rounding errors

# GOOD
all.equal(b_manual, b_lm)  # Allows small tolerance (default: 1.5e-8)

Week-by-Week Function Index

Quick reference showing which functions are introduced or used heavily in each week.

Week	Topic	Key Functions
1	Overview & Foundations	`matrix()`, `t()`, `%*%`, `c()`, `mean()`, `sum()`, `length()`
2	Linear Algebra	`solve()`, `ginv()`, `qr()`, `det()`, `diag()`, `eigen()`, `svd()`, `rankMatrix()`
3	Design Matrices	`model.matrix()`, `factor()`, `cbind()`, `rbind()`, `formula()`, `contr.treatment()`, `contr.sum()`
4	Simple Regression	`lm()`, `coef()`, `fitted()`, `residuals()`, `predict()`, `summary()`
5	Least Squares Theory	`crossprod()`, `tcrossprod()`, `vcov()`, `sigma()`, `deviance()`, `confint()`
6	Multiple Regression	`anova()`, `drop1()`, `add1()`, `vif()`, `cor()`, `AIC()`, `update()`
7	One-Way ANOVA	`anova()`, `tapply()`, `aggregate()`, `aov()`, `TukeyHSD()`
8	Contrasts	`emmeans()`, `contrast()`, `pairs()`, `contr.sum()`, `contr.treatment()`, `contrasts()`, `relevel()`
9	Two-Way ANOVA	`Anova()` (Type III), `interaction.plot()`, `model.tables()`, `emmeans()` with interactions
10	ANCOVA	`emmeans()` (adjusted means), `anova()` (sequential tests), `Anova()` (partial tests), centering covariates
11	Diagnostics	`plot()`, `hatvalues()`, `cooks.distance()`, `rstandard()`, `rstudent()`, `influence.measures()`, `dffits()`, `dfbetas()`, `qqnorm()`
12	Non-Full Rank	`ginv()`, `rankMatrix()`, `qr()`, `all.equal()`, `alias()`, checking estimability
13	Special Topics I	`options(contrasts)`, `relevel()`, `is.estimable()`, different constraint systems
14	Special Topics II	`poly()`, `lm(weights=)`, `lmer()` (preview), `boxcox()`, `logLik()`
15	Capstone	Integration of all functions above - comprehensive livestock data analysis

Progressive Learning

Weeks 1-3: Foundations (matrix operations, design matrices)
Weeks 4-6: Core regression (simple, theory, multiple)
Weeks 7-10: ANOVA family (one-way, contrasts, two-way, ANCOVA)
Week 11: Diagnostics (checking all assumptions)
Weeks 12-14: Advanced topics (rank deficiency, special cases)
Week 15: Integration (real-world application)

Each week builds on previous concepts - review earlier weeks as needed.

Additional Resources

R Cheat Sheets

Base R: https://iqss.github.io/dss-workshops/R/Rintro/base-r-cheat-sheet.pdf
RStudio: https://www.rstudio.com/resources/cheatsheets/

Textbooks with R Code

Linear Models with R by Julian Faraway
The R Book by Michael Crawley
Applied Linear Statistical Models by Kutner et al. (has R companion)

Online Courses

DataCamp: “Introduction to Linear Modeling in R”
Coursera: “Regression Models” (Johns Hopkins)
R-exercises: https://www.r-exercises.com/

Package Documentation

MASS: https://cran.r-project.org/web/packages/MASS/MASS.pdf
car: https://cran.r-project.org/web/packages/car/car.pdf
emmeans: https://cran.r-project.org/web/packages/emmeans/vignettes/

This completes Appendix C. For mathematical notation, see Appendix A. For in-depth matrix algebra, see Appendix B.

# Appendix C: R Functions Reference {#sec-appendix-c .unnumbered} ## Introduction This appendix provides a concise reference guide to all R functions used throughout the 15-week linear models course. Functions are organized by **functional category** (not alphabetically) to help you quickly locate related operations. ### Purpose and Scope This reference is designed to: - Provide quick lookup of function syntax and key parameters - Show which weeks each function is introduced or used heavily - Distinguish between base R and package functions - Offer troubleshooting guidance for common issues ### How to Use This Appendix - **Quick lookup**: Use the table of contents to jump to a specific category - **Function search**: Use your browser's search function (Ctrl+F or Cmd+F) to find a specific function name - **Week review**: Check the "Week(s)" column to see where concepts are taught - **Troubleshooting**: Jump to @sec-rfunc-pitfalls or @sec-rfunc-troubleshoot for common issues :::{.callout-note} ## Relationship to Other Appendices - **Appendix A**: Mathematical notation reference - **Appendix B**: In-depth matrix algebra tutorial with derivations - **Appendix C** (this): Quick reference for R functions ::: --- ## Package Installation and Setup {#sec-rfunc-setup} Most functions in this appendix are from **base R** and require no additional installation. For package functions, install as needed: ```{r} #| eval: false # Essential packages install.packages("MASS") # ginv(), boxcox() install.packages("car") # Anova(), vif() install.packages("emmeans") # emmeans(), contrast(), pairs() install.packages("Matrix") # rankMatrix(), sparse matrices # Optional packages (used in specific weeks) install.packages("lme4") # lmer() for mixed models preview (Week 14) install.packages("multcomp") # Multiple comparisons (Week 8) install.packages("ggplot2") # Enhanced graphics ``` ### Loading Packages Load required packages at the start of your R session: ```{r} #| eval: false #| message: false #| warning: false library(MASS) library(car) library(emmeans) library(Matrix) ``` :::{.callout-note} ## Base R vs Packages - **Base R functions** (like `lm()`, `solve()`, `matrix()`) are always available - **Package functions** (like `ginv()`, `Anova()`) require loading the package first with `library()` ::: --- ## Basic Matrix Operations {#sec-rfunc-matrix-basic} Functions for creating and manipulating matrices. | Function | Package | Syntax | Description | Key Parameters | Week(s) | |----------|---------|--------|-------------|----------------|---------| | `matrix()` | base R | `matrix(data, nrow, ncol)` | Create a matrix from vector | `data=`, `nrow=`, `ncol=`, `byrow=` | 1-15 | | `t()` | base R | `t(X)` | Transpose matrix | `x=` | 1-15 | | `%*%` | base R | `A %*% B` | Matrix multiplication | - | 1-15 | | `solve()` | base R | `solve(A)` or `solve(A, b)` | Compute matrix inverse or solve linear system | `a=`, `b=` | 2-15 | | `ginv()` | MASS | `ginv(X)` | Generalized (Moore-Penrose) inverse | `X=`, `tol=` | 2,12-13 | | `cbind()` | base R | `cbind(v1, v2, ...)` | Combine vectors/matrices by columns | `...` | 3-15 | | `rbind()` | base R | `rbind(v1, v2, ...)` | Combine vectors/matrices by rows | `...` | Throughout | | `c()` | base R | `c(x1, x2, ...)` | Combine values into vector | `...` | 1-15 | :::{.callout-note} ## Matrix Multiplication vs Element-wise - `%*%` performs **matrix multiplication** (inner dimensions must match: (m×n) × (n×p) → (m×p)) - `*` performs **element-wise multiplication** (dimensions must match exactly) ::: :::{.callout-tip} ## Efficiency Tip For computing $\mathbf{X}'\mathbf{X}$, use `crossprod(X)` instead of `t(X) %*% X` - it's faster and more numerically stable. ::: --- ## Matrix Properties and Decompositions {#sec-rfunc-matrix-props} Functions for examining matrix characteristics and performing decompositions. | Function | Package | Syntax | Description | Key Parameters | Week(s) | |----------|---------|--------|-------------|----------------|---------| | `qr()` | base R | `qr(X)` | QR decomposition; use `$rank` to get rank | `x=`, `tol=` | 2,12 | | `det()` | base R | `det(A)` | Compute determinant | `x=` | 2 | | `diag()` | base R | `diag(X)` or `diag(n)` | Extract diagonal or create diagonal matrix | `x=`, `nrow=`, `ncol=` | 2-15 | | `eigen()` | base R | `eigen(A)` | Eigenvalue decomposition | `x=`, `symmetric=` | 2,5 | | `svd()` | base R | `svd(X)` | Singular value decomposition | `x=` | 2,13 | | `rankMatrix()` | Matrix | `rankMatrix(X)` | Compute rank (more reliable than `qr()$rank`) | `x=`, `method=` | 2,12 | | `crossprod()` | base R | `crossprod(X)` or `crossprod(X, Y)` | Compute $\mathbf{X}'\mathbf{X}$ or $\mathbf{X}'\mathbf{Y}$ efficiently | `x=`, `y=` | 5-15 | | `tcrossprod()` | base R | `tcrossprod(X)` | Compute $\mathbf{X}\mathbf{X}'$ efficiently | `x=`, `y=` | 5-15 | | `norm()` | base R | `norm(X, type)` | Compute matrix norm | `x=`, `type=` | 2 | :::{.callout-important} ## Rank Determination For rank deficiency detection, use `rankMatrix()` from the Matrix package - it's more robust than `qr()$rank` for near-singular matrices. See Week 12 for details on non-full rank models. ::: --- ## Linear Model Fitting {#sec-rfunc-lm-fitting} Functions for fitting linear models and building design matrices. | Function | Package | Syntax | Description | Key Parameters | Week(s) | |----------|---------|--------|-------------|----------------|---------| | `lm()` | base R | `lm(formula, data)` | Fit linear model using ordinary least squares | `formula=`, `data=`, `weights=` | 4-15 | | `model.matrix()` | base R | `model.matrix(formula, data)` | Build design matrix $\mathbf{X}$ from formula | `object=`, `data=` | 3-15 | | `formula()` | base R | `formula(model)` | Extract formula from fitted model | `x=` | Throughout | | `update()` | base R | `update(model, new_formula)` | Update and refit model with new formula | `object=`, `formula=`, `data=` | 6-10 | | `lmer()` | lme4 | `lmer(formula, data)` | Fit mixed model (preview only) | `formula=`, `data=`, `REML=` | 14 | :::{.callout-tip} ## Design Matrix Construction Always verify your design matrix before fitting: ```r X <- model.matrix(~ breed + sex, data=mydata) head(X) # Check structure qr(X)$rank # Check rank ``` ::: --- ## Model Summaries and Extraction {#sec-rfunc-lm-summary} Functions for extracting information from fitted models. | Function | Package | Syntax | Description | Key Parameters | Week(s) | |----------|---------|--------|-------------|----------------|---------| | `summary()` | base R | `summary(model)` | Comprehensive model summary (coefficients, R², F-test) | `object=` | 4-15 | | `coef()` | base R | `coef(model)` | Extract coefficient estimates $\mathbf{b}$ | `object=` | 4-15 | | `residuals()` | base R | `residuals(model)` or `resid(model)` | Extract residuals $\mathbf{e} = \mathbf{y} - \hat{\mathbf{y}}$ | `object=`, `type=` | 4-15 | | `fitted()` | base R | `fitted(model)` | Extract fitted values $\hat{\mathbf{y}} = \mathbf{Xb}$ | `object=` | 4-15 | | `predict()` | base R | `predict(model, newdata)` | Predict for new observations | `object=`, `newdata=`, `interval=` | 4,6 | | `vcov()` | base R | `vcov(model)` | Variance-covariance matrix of estimates $(\mathbf{X}'\mathbf{X})^{-1}\hat{\sigma}^2$ | `object=` | 5-15 | | `sigma()` | base R | `sigma(model)` | Extract residual standard error $\hat{\sigma}$ | `object=` | 5-15 | | `deviance()` | base R | `deviance(model)` | Residual sum of squares (SSE) | `object=` | 5-15 | | `logLik()` | base R | `logLik(model)` | Log-likelihood of fitted model | `object=` | 6,14 | | `AIC()` | base R | `AIC(model)` | Akaike Information Criterion | `object=`, `k=` | 6,14 | | `confint()` | base R | `confint(model, level=0.95)` | Confidence intervals for parameters | `object=`, `parm=`, `level=` | 5-6 | :::{.callout-note} ## Residuals vs Fitted Values - `residuals(model)`: Returns $\mathbf{e} = \mathbf{y} - \hat{\mathbf{y}}$ - `fitted(model)`: Returns $\hat{\mathbf{y}} = \mathbf{Xb}$ - Verify: `all.equal(y, fitted(model) + residuals(model))` should be `TRUE` ::: --- ## Hypothesis Testing and ANOVA {#sec-rfunc-testing} Functions for testing hypotheses about model parameters. | Function | Package | Syntax | Description | Key Parameters | Week(s) | |----------|---------|--------|-------------|----------------|---------| | `anova()` | base R | `anova(model)` or `anova(model1, model2)` | ANOVA table with Type I (sequential) SS | `object=` | 6-10 | | `Anova()` | car | `Anova(model, type=3)` | Type II or Type III SS for unbalanced data | `mod=`, `type=` | 9-10,12 | | `emmeans()` | emmeans | `emmeans(model, specs)` | Estimated marginal means (LSMEANS) | `object=`, `specs=`, `by=` | 8-10 | | `contrast()` | emmeans | `contrast(emm, method)` | Test contrasts among means | `object=`, `method=`, `adjust=` | 8-10 | | `pairs()` | emmeans | `pairs(emm)` | Pairwise comparisons | `x=`, `adjust=` | 8-9 | | `drop1()` | base R | `drop1(model, test="F")` | Test dropping each term | `object=`, `scope=`, `test=` | 6,9 | | `add1()` | base R | `add1(model, scope, test="F")` | Test adding each term | `object=`, `scope=`, `test=` | 6 | | `vif()` | car | `vif(model)` | Variance inflation factors (detect collinearity) | `mod=` | 6 | :::{.callout-important} ## Type I vs Type III SS - **Type I SS** (sequential): Order matters! Each term adjusted for previous terms only - Use `anova(model)` - default in R - Appropriate for **balanced** designs - **Type III SS** (partial): Each term adjusted for all other terms - Use `car::Anova(model, type=3)` - Appropriate for **unbalanced** designs and recommended for general use See Weeks 9-10 for detailed discussion. ::: --- ## Model Diagnostics {#sec-rfunc-diagnostics} Functions for checking model assumptions and identifying influential observations. | Function | Package | Syntax | Description | Key Parameters | Week(s) | |----------|---------|--------|-------------|----------------|---------| | `plot()` | base R | `plot(lm_object)` | Create 4 diagnostic plots automatically | `x=`, `which=` | 11 | | `hatvalues()` | base R | `hatvalues(model)` | Extract leverage values (diagonal of hat matrix $\mathbf{H}$) | `model=` | 11 | | `cooks.distance()` | base R | `cooks.distance(model)` | Compute Cook's distance for each observation | `model=` | 11 | | `rstandard()` | base R | `rstandard(model)` | Standardized residuals | `model=`, `type=` | 11 | | `rstudent()` | base R | `rstudent(model)` | Studentized deleted residuals | `model=` | 11 | | `influence.measures()` | base R | `influence.measures(model)` | Comprehensive influence diagnostics | `model=` | 11 | | `dffits()` | base R | `dffits(model)` | Change in fitted value when observation deleted | `model=` | 11 | | `dfbetas()` | base R | `dfbetas(model)` | Change in coefficients when observation deleted | `model=` | 11 | | `qqnorm()` | base R | `qqnorm(residuals)` | Normal Q-Q plot to check normality | `y=` | 11 | | `boxcox()` | MASS | `boxcox(model)` | Box-Cox transformation for normality/homoscedasticity | `object=`, `lambda=` | 11,14 | :::{.callout-tip} ## Rule-of-Thumb Thresholds - **High leverage**: $h_{ii} > 2p/n$ or $3p/n$ where $p$ = number of parameters, $n$ = sample size - **Influential observation**: Cook's D > 1 or Cook's D > 4/n - **Outlier**: |studentized residual| > 3 Week 11 provides detailed guidance on interpretation and remedial measures. ::: :::{.callout-note} ## Quick Diagnostic Check ```r par(mfrow=c(2,2)) plot(model) # Creates all 4 diagnostic plots at once ``` 1. **Residuals vs Fitted**: Check linearity and homoscedasticity 2. **Q-Q Plot**: Check normality assumption 3. **Scale-Location**: Check homoscedasticity 4. **Residuals vs Leverage**: Identify influential points ::: --- ## Advanced Model Tools {#sec-rfunc-advanced} Functions for working with contrasts, constraints, and model parameterizations. | Function | Package | Syntax | Description | Key Parameters | Week(s) | |----------|---------|--------|-------------|----------------|---------| | `contr.treatment()` | base R | `contr.treatment(n)` | Treatment (reference cell) contrasts | `n=`, `base=` | 3,8,13 | | `contr.sum()` | base R | `contr.sum(n)` | Sum-to-zero contrasts | `n=` | 8,13 | | `contrasts()` | base R | `contrasts(factor)` | Get or set contrasts for a factor | `x=`, `value=` | 8,13 | | `options()` | base R | `options(contrasts=c(...))` | Set default contrast system | `contrasts=` | 13 | | `relevel()` | base R | `relevel(factor, ref)` | Change reference level of factor | `x=`, `ref=` | 8-9 | | `interaction.plot()` | base R | `interaction.plot(f1, f2, y)` | Plot interaction between two factors | `x.factor=`, `trace.factor=`, `response=` | 9 | | `all.equal()` | base R | `all.equal(x, y)` | Compare objects with tolerance (for floating-point) | `target=`, `current=`, `tolerance=` | Throughout | | `is.estimable()` | estimability | `is.estimable(contrast, X)` | Check if linear function is estimable | `x=`, `X=` | 8,12-13 | :::{.callout-important} ## Contrast Systems **Set-to-zero (treatment) contrasts**: ```r options(contrasts = c("contr.treatment", "contr.poly")) ``` - Default in R - First level is reference (coefficient = 0) - Other coefficients are **differences from reference** **Sum-to-zero (effects) contrasts**: ```r options(contrasts = c("contr.sum", "contr.poly")) ``` - Coefficients sum to zero: $\sum \alpha_i = 0$ - Each coefficient is **deviation from overall mean** - More symmetric, preferred for balanced designs See Week 13 for detailed comparison and implications. ::: --- ## Data Manipulation {#sec-rfunc-data} Functions for organizing and summarizing data. | Function | Package | Syntax | Description | Key Parameters | Week(s) | |----------|---------|--------|-------------|----------------|---------| | `tapply()` | base R | `tapply(X, INDEX, FUN)` | Apply function to subgroups | `X=`, `INDEX=`, `FUN=` | 7-10 | | `aggregate()` | base R | `aggregate(x, by, FUN)` | Compute summary statistics by groups | `x=`, `by=`, `FUN=` | 7-10 | | `factor()` | base R | `factor(x, levels)` | Create or modify factor variable | `x=`, `levels=`, `labels=` | 3-15 | | `mean()` | base R | `mean(x)` | Compute arithmetic mean | `x=`, `na.rm=` | 1-15 | | `sum()` | base R | `sum(x)` | Compute sum | `x=`, `na.rm=` | 1-15 | | `length()` | base R | `length(x)` | Number of elements in vector | `x=` | Throughout | | `nrow()` | base R | `nrow(X)` | Number of rows in matrix/data frame | `x=` | Throughout | | `ncol()` | base R | `ncol(X)` | Number of columns in matrix/data frame | `x=` | Throughout | | `subset()` | base R | `subset(data, condition)` | Extract subset of data | `x=`, `subset=`, `select=` | Throughout | | `transform()` | base R | `transform(data, ...)` | Add or modify variables | `_data=`, `...` | Throughout | | `with()` | base R | `with(data, expr)` | Evaluate expression in data environment | `data=`, `expr=` | Throughout | | `round()` | base R | `round(x, digits)` | Round to specified decimal places | `x=`, `digits=` | Throughout | | `sprintf()` | base R | `sprintf(fmt, ...)` | Format output with C-style formatting | `fmt=`, `...` | Throughout | | `cat()` | base R | `cat(...)` | Print output (useful in R Markdown) | `...`, `sep=`, `fill=` | Throughout | :::{.callout-tip} ## Computing Group Statistics Three common approaches: **1. Using `tapply()`** - Returns vector/array: ```r tapply(y, group, mean) # Mean of y for each group ``` **2. Using `aggregate()`** - Returns data frame: ```r aggregate(y ~ group, data=mydata, FUN=mean) ``` **3. Using `emmeans()`** - Returns model-based estimates: ```r emmeans(lm(y ~ group), specs="group") ``` Choose based on your needs: - `tapply()`: Quick calculations - `aggregate()`: Multiple variables at once - `emmeans()`: Adjusted means from models ::: --- ## Common Pitfalls {#sec-rfunc-pitfalls} Common issues encountered when working with linear models in R, their symptoms, and solutions. | Issue | Symptom | Solution | |-------|---------|----------| | **Singular matrix** | `Error in solve.default(X): system is computationally singular` | Use `ginv()` from MASS package instead of `solve()`. Occurs when $\mathbf{X}'\mathbf{X}$ is not full rank (Week 12). | | **Object not found** | `Error: object 'variable_name' not found` | Add `data=` argument to `lm()`: `lm(y ~ x, data=mydata)`. Ensures variables are found in the data frame. | | **Wrong SS type** | Type I SS gives different results than expected in unbalanced data | Use `car::Anova(model, type=3)` instead of base `anova()` for unbalanced designs (Weeks 9-10). | | **Non-estimable contrasts** | Different g-inverses give different parameter estimates | Verify contrast is estimable: check $\sum c_i = 0$ for treatment contrasts. Only interpret estimable functions (Week 8, 12-13). | | **Uncentered covariate** | Intercept not interpretable as group mean at $\bar{x}$ | Center covariate: `lm(y ~ group + I(x - mean(x)))` (Week 10). | | **Rank deficiency warnings** | `Coefficients: (1 not defined because of singularities)` | Either use cell means model (`~ factor - 1`) or apply explicit constraints. Check design matrix rank: `qr(X)$rank` (Week 12). | | **Factor vs numeric** | `contrasts can be applied only to factors with 2 or more levels` | Convert numeric variable to factor: `factor(variable)` or include in formula as `factor(variable)`. | | **Dimension mismatch** | `non-conformable arrays` or `arguments imply differing number of rows` | Check dimensions with `dim(X)`, `length(y)`. Matrix multiplication requires matching inner dimensions (Week 1-2). | :::{.callout-important} ## Debugging Strategy When encountering errors, follow this systematic approach: 1. **Check your data**: `str(data)`, `summary(data)`, `head(data)` 2. **Check your formula**: `formula(model)` 3. **Check design matrix**: `X <- model.matrix(model)`, then `dim(X)`, `qr(X)$rank` 4. **Verify dimensions**: Ensure $n \times p$ design matrix matches $n \times 1$ response 5. **Check for rank deficiency**: Compare `qr(X)$rank` to `ncol(X)` 6. **Compare with lm()**: Verify manual calculations match `lm()` output 7. **Test estimability**: For contrasts, verify $\mathbf{c}'\boldsymbol{\beta}$ is estimable See Week 11 (Diagnostics) and Week 12 (Non-Full Rank) for detailed troubleshooting. ::: --- ## Quick Troubleshooting Guide {#sec-rfunc-troubleshoot} ### Common Error Messages and Solutions **"Error in solve.default(): system is computationally singular"** - **Cause**: Matrix $\mathbf{X}'\mathbf{X}$ is not full rank (determinant ≈ 0) - **Solution**: ```r library(MASS) b <- ginv(t(X) %*% X) %*% t(X) %*% y # Use generalized inverse ``` - **Related**: Week 2 (Linear Algebra), Week 12 (Non-Full Rank Models) **"Error: object 'variable_name' not found"** - **Cause**: Variable not in current environment or data frame not specified - **Solution**: ```r model <- lm(y ~ x, data=mydata) # Always use data= argument ``` - **Related**: Week 4 (Simple Regression) **"Coefficients: (1 not defined because of singularities)"** - **Cause**: Design matrix is rank deficient (overparameterized model) - **Solution**: ```r # Option 1: Cell means model model <- lm(y ~ breed - 1, data=mydata) # Option 2: Check which parameters are aliased alias(model) ``` - **Related**: Week 12 (Non-Full Rank Models) **"contrasts can be applied only to factors with 2 or more levels"** - **Cause**: Categorical variable coded as numeric instead of factor - **Solution**: ```r mydata$breed <- factor(mydata$breed) # Or in formula: model <- lm(y ~ factor(breed), data=mydata) ``` - **Related**: Week 3 (Design Matrices), Week 7 (ANOVA) **"non-conformable arrays"** - **Cause**: Matrix dimensions don't match for the operation - **Solution**: ```r dim(X) # Check dimensions dim(y) # For matrix multiplication A %*% B, need ncol(A) == nrow(B) ``` - **Related**: Week 1-2 (Matrix Operations) --- ### Getting Help in R **Function documentation**: ```r ?lm # Quick help help(lm) # Same as ? ??regression # Search all help files ``` **Examples**: ```r example(lm) # Run examples from help file demo("graphics") # Interactive demonstrations ``` **Package information**: ```r help(package="MASS") # Package overview and function list ``` **Vignettes** (detailed tutorials): ```r vignette() # List all available vignettes vignette("emmeans") # View specific vignette browseVignettes("emmeans") # Open in browser ``` **Online resources**: - R Documentation: https://www.rdocumentation.org/ - Stack Overflow: https://stackoverflow.com/questions/tagged/r - R-bloggers: https://www.r-bloggers.com/ - Quick-R: https://www.statmethods.net/ --- ### Verification Strategy When building your own least squares solvers, always verify against `lm()`: ```r # 1. Manual calculation X <- model.matrix(~ breed + sex, data=mydata) y <- mydata$weight b_manual <- solve(t(X) %*% X) %*% t(X) %*% y # 2. Using lm() model <- lm(weight ~ breed + sex, data=mydata) b_lm <- coef(model) # 3. Compare (should be near-zero) max(abs(b_manual - b_lm)) # 4. Use all.equal() for floating-point comparisons all.equal(as.vector(b_manual), b_lm) # Should return TRUE (or very small tolerance message) # 5. Verify fitted values y_hat_manual <- X %*% b_manual y_hat_lm <- fitted(model) all.equal(as.vector(y_hat_manual), y_hat_lm) # 6. Verify residuals e_manual <- y - y_hat_manual e_lm <- residuals(model) all.equal(as.vector(e_manual), e_lm) # 7. Verify SSE SSE_manual <- sum(e_manual^2) SSE_lm <- deviance(model) all.equal(SSE_manual, SSE_lm) ``` :::{.callout-tip} ## Floating-Point Comparison Never use `==` to compare floating-point numbers! Use `all.equal()` instead: ```r # BAD b_manual == b_lm # May fail due to rounding errors # GOOD all.equal(b_manual, b_lm) # Allows small tolerance (default: 1.5e-8) ``` ::: --- ## Week-by-Week Function Index {#sec-rfunc-by-week} Quick reference showing which functions are introduced or used heavily in each week. | Week | Topic | Key Functions | |------|-------|---------------| | **1** | Overview & Foundations | `matrix()`, `t()`, `%*%`, `c()`, `mean()`, `sum()`, `length()` | | **2** | Linear Algebra | `solve()`, `ginv()`, `qr()`, `det()`, `diag()`, `eigen()`, `svd()`, `rankMatrix()` | | **3** | Design Matrices | `model.matrix()`, `factor()`, `cbind()`, `rbind()`, `formula()`, `contr.treatment()`, `contr.sum()` | | **4** | Simple Regression | `lm()`, `coef()`, `fitted()`, `residuals()`, `predict()`, `summary()` | | **5** | Least Squares Theory | `crossprod()`, `tcrossprod()`, `vcov()`, `sigma()`, `deviance()`, `confint()` | | **6** | Multiple Regression | `anova()`, `drop1()`, `add1()`, `vif()`, `cor()`, `AIC()`, `update()` | | **7** | One-Way ANOVA | `anova()`, `tapply()`, `aggregate()`, `aov()`, `TukeyHSD()` | | **8** | Contrasts | `emmeans()`, `contrast()`, `pairs()`, `contr.sum()`, `contr.treatment()`, `contrasts()`, `relevel()` | | **9** | Two-Way ANOVA | `Anova()` (Type III), `interaction.plot()`, `model.tables()`, `emmeans()` with interactions | | **10** | ANCOVA | `emmeans()` (adjusted means), `anova()` (sequential tests), `Anova()` (partial tests), centering covariates | | **11** | Diagnostics | `plot()`, `hatvalues()`, `cooks.distance()`, `rstandard()`, `rstudent()`, `influence.measures()`, `dffits()`, `dfbetas()`, `qqnorm()` | | **12** | Non-Full Rank | `ginv()`, `rankMatrix()`, `qr()`, `all.equal()`, `alias()`, checking estimability | | **13** | Special Topics I | `options(contrasts)`, `relevel()`, `is.estimable()`, different constraint systems | | **14** | Special Topics II | `poly()`, `lm(weights=)`, `lmer()` (preview), `boxcox()`, `logLik()` | | **15** | Capstone | **Integration of all functions above** - comprehensive livestock data analysis | :::{.callout-note} ## Progressive Learning - **Weeks 1-3**: Foundations (matrix operations, design matrices) - **Weeks 4-6**: Core regression (simple, theory, multiple) - **Weeks 7-10**: ANOVA family (one-way, contrasts, two-way, ANCOVA) - **Week 11**: Diagnostics (checking all assumptions) - **Weeks 12-14**: Advanced topics (rank deficiency, special cases) - **Week 15**: Integration (real-world application) Each week builds on previous concepts - review earlier weeks as needed. ::: --- ## Additional Resources ### R Cheat Sheets - **Base R**: https://iqss.github.io/dss-workshops/R/Rintro/base-r-cheat-sheet.pdf - **RStudio**: https://www.rstudio.com/resources/cheatsheets/ ### Textbooks with R Code - *Linear Models with R* by Julian Faraway - *The R Book* by Michael Crawley - *Applied Linear Statistical Models* by Kutner et al. (has R companion) ### Online Courses - DataCamp: "Introduction to Linear Modeling in R" - Coursera: "Regression Models" (Johns Hopkins) - R-exercises: https://www.r-exercises.com/ ### Package Documentation - **MASS**: https://cran.r-project.org/web/packages/MASS/MASS.pdf - **car**: https://cran.r-project.org/web/packages/car/car.pdf - **emmeans**: https://cran.r-project.org/web/packages/emmeans/vignettes/ --- This completes Appendix C. For mathematical notation, see [Appendix A](#sec-appendix-a). For in-depth matrix algebra, see [Appendix B](#sec-appendix-b).