In charlotte-ngs/asmss2022: Applied Statistical Methods in Animal Science - Spring Semester 2022

knitr::opts_chunk$set(echo = FALSE)
knitr::knit_hooks$set(hook_convert_odg = rmdhelp::hook_convert_odg)

Obtain Parameter Estimates in R

• Computations are tedious
• Use R builtin functions
• Assuming data is available in dataframe tbl_reg with columns Body Weight and Breast Circumference

\vspace{3ex}

lm_bw_bc <- lm(`Body Weight` ~ `Breast Circumference`, 
               data = tbl_reg)
summary(lm_bw_bc)

The General Case

• Not only one $x$-variable, but many of them
• Parameter estimates can be derived the same way, but very cumbersome
• Use matrix-vector notation, for an example with two $x$-variables
• Define

mat_X <- matrix(c("x_{10}", "x_{11}", "x_{12}",
                  "x_{20}", "x_{21}", "x_{22}",
                  ".", ".", ".",
                  ".", ".", ".",
                  ".", ".", ".",
                  "x_{N0}", "x_{N1}", "x_{N2}"), ncol = 3, byrow = TRUE)
vec_y <- c("y_1", "y_2", ".", ".", ".", "y_N")
vec_e <- c("e_1", "e_2", ".", ".", ".", "e_N")
vec_b <- c("b_0", "b_1", "b_2")

cat("$$\n")
cat(paste0(rmdhelp::bmatrix(pmat = mat_X, ps_name = "\\mathbf{X}"), collapse = ""), "\n")
cat(", \\ \n")
cat(paste0(rmdhelp::bcolumn_vector(pvec = vec_y, ps_name = "\\mathbf{y}")))
cat(", \\ \n")
cat(paste0(rmdhelp::bcolumn_vector(pvec = vec_e, ps_name = "\\mathbf{e}")))
cat("\\text{ and }\n")
cat(paste0(rmdhelp::bcolumn_vector(pvec = vec_b, ps_name = "\\mathbf{b}")))
cat("$$\n")

Linear Regression Model

\begin{equation} \mathbf{y} = \mathbf{X}\mathbf{b} + \mathbf{e} \text{, with } E(\mathbf{y}) = \mathbf{X}\mathbf{b} \notag \end{equation}

• General case with $k$ $x$-variables

mat_X_gen <- matrix(c("x_{10}", "x_{11}", ".", "x_{1k}",
                  "x_{20}", "x_{21}", ".", "x_{2k}",
                  ".", ".", ".", ".",
                  ".", ".", ".", ".",
                  ".", ".", ".", ".",
                  "x_{N0}", "x_{N1}", ".", "x_{Nk}"), ncol = 4, byrow = TRUE)
vec_b_gen <- c("b_0", "b_1", ".", ".", "b_k")

cat("$$\n")
cat(paste0(rmdhelp::bmatrix(pmat = mat_X_gen, ps_name = "\\mathbf{X}"), collapse = ""), "\n")
cat(", \\ \n")
cat(paste0(rmdhelp::bcolumn_vector(pvec = vec_b_gen, ps_name = "\\mathbf{b}")))
cat("$$\n")

Random Error Terms

• Properties of random error terms in vector $\mathbf{e}$

\begin{equation} E(\mathbf{e}) = \mathbf{0} \notag \end{equation}

\begin{equation} var(\mathbf{e}) = E\left[\mathbf{e} - E(\mathbf{e}) \right]\left[\mathbf{e} - E(\mathbf{e}) \right]^T = E(\mathbf{e}\mathbf{e}^T) = \sigma^2 \mathbf{I}_N \notag \end{equation}

Least Squares Estimates

\begin{align} \mathbf{e}^T\mathbf{e} &= \left[\mathbf{y} - E(\mathbf{y}) \right]^T\left[\mathbf{y} - E(\mathbf{y}) \right] \notag \ &= \left[\mathbf{y} - \mathbf{Xb} \right]^T\left[\mathbf{y} - \mathbf{Xb} \right]\notag \ &= \mathbf{y}^T\mathbf{y} - 2 \mathbf{b}^T\mathbf{X}^T\mathbf{y} + \mathbf{b}^T\mathbf{X}^T\mathbf{X}\mathbf{b} \notag \end{align}

• Setting

$$\frac{\partial \mathbf{e}^T\mathbf{e}}{\partial \mathbf{b}} = \mathbf{0}$$

• yields least squares normal equations

\begin{equation} \mathbf{X}^T\mathbf{X}\hat{\mathbf{b}} = \mathbf{X}^T\mathbf{y} \notag \end{equation}

Solution for Least Squares Estimators

\begin{equation} \hat{\mathbf{b}} = (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{y} \notag \end{equation}

charlotte-ngs/asmss2022 documentation built on June 7, 2022, 1:33 p.m.