Custom Linear Regression Estimation

Provides a custom implementation of R's stats::lm() linear regression function that uses the S3 system.

Usage

my_lm(x, ...)

# Default S3 method
my_lm(x, ...)

# S3 method for class 'formula'
my_lm(formula, data = list(), ...)

# S3 method for class 'matrix'
my_lm(x, y, ...)

Arguments

x: Either a design matrix with dimensions $n \times p$ or formula with the data parameter specified
...: Not used.
formula: An object of class formula() which provides a symbolic description of the model to be fitted.
data: An optional data frame, list or environment.
y: A vector of responses with dimensions $n \times 1$.

Value

An object of class my_lm that contains:

coefficients: Estimated parameter values of $\hat{\beta}$ with dimensions $p \times 1$
cov_mat: Covariance matrix of estimated parameter values with dimensions $p \times p$.
sigma: Standard deviation of residuals
df: Degrees of Freedom given by $df = N - p$
fitted.values: Fitted Values given by $\hat{y} = X\hat{\beta}$
residuals: Residuals given by $e = y - \hat{y}$
call: Information on how the my_lm() function was called.

Details

Given a response vector $y$ with dimensions $n \times 1$, a design matrix $X$ with dimensions $n \times p$, a vector of parameters $\beta$ with dimensions $p \times 1$, and an error vector $\epsilon$ with dimensions $n \times 1$ from $\epsilon \sim N\left( {0,{\sigma ^2}} \right)$, the standard linear regression model can be stated as:

$$y = {X'}\beta + \epsilon$$

The least ordinary squares (OLS) solutions are then:

$$\hat \beta = {\left( {{X'}X} \right)^{ - 1}}{X'}y$$ $$cov\left( {\hat \beta } \right) = {\sigma ^2}{\left( {{X^T}X} \right)^{ - 1}}$$

Examples

## Matrix interface

# Create a design matrix
x = cbind(1, mtcars$disp)

# Extract response
y = mtcars$mpg

# Calculate outcome
my_model = my_lm(x, y)

## Formula interface

# Calculate 
my_model = my_lm(mpg ~ disp, data = mtcars)