RSS: Residual Sum of Squares
In jeksterslabds/jeksterslabRlinreg: Jekster's Lab Linear Regression Utilities

Description Usage Arguments Details Value Author(s) References See Also Examples

Calculates the residual sum of squares ≤ft( \mathrm{RSS} \right) using

\mathrm{RSS} = ∑_{i = 1}^{n} ≤ft( Y_i - \hat{Y}_i \right)^2 \\ = ∑_{i = 1}^{n} ≤ft( Y_i - ≤ft[ \hat{β}_{1} + \hat{β}_{2} X_{2i} + \hat{β}_{3} X_{3i} + … + \hat{β}_{k} X_{ki} \right] \right)^2 \\ = ∑_{i = 1}^{n} ≤ft( Y_i - \hat{β}_{1} - \hat{β}_{2} X_{2i} - \hat{β}_{3} X_{3i} - … - \hat{β}_{k} X_{ki} \right)^2 .

In matrix form

\mathrm{RSS} = ∑_{i = 1}^{n} ≤ft( \mathbf{y} - \mathbf{\hat{y}} \right)^{2} \\ = ∑_{i = 1}^{n} ≤ft( \mathbf{y} - \mathbf{X} \boldsymbol{\hat{β}} \right)^{2} \\ = ≤ft( \mathbf{y} - \mathbf{X} \boldsymbol{\hat{β}} \right)^{\prime} ≤ft( \mathbf{y} - \mathbf{X} \boldsymbol{\hat{β}} \right) .

Or simply

\mathrm{RSS} = ∑_{i = 1}^{n} \boldsymbol{\hat{\varepsilon}}_{i}^{2} = \boldsymbol{\hat{\varepsilon}}^{\prime} \boldsymbol{\hat{\varepsilon}}

where \boldsymbol{\hat{\varepsilon}} is an n \times 1 vector of residuals, that is, the difference between the observed and predicted value of \mathbf{y} ≤ft( \boldsymbol{\hat{\varepsilon}} = \mathbf{y} - \mathbf{\hat{y}} \right). Equivalent computational matrix formula

\mathrm{RSS} = \mathbf{y}^{\prime} \mathbf{y} - 2 \boldsymbol{\hat{β}} \mathbf{X}^{\prime} \mathbf{y} + \boldsymbol{\hat{β}}^{\prime} \mathbf{X}^{\prime} \mathbf{X} \boldsymbol{\hat{β}}.

Note that

\mathrm{TSS} = \mathrm{ESS} + \mathrm{RSS}.

RSS(X, y)

`X`	`n` by `k` numeric matrix. The data matrix \mathbf{X} (also known as design matrix, model matrix or regressor matrix) is an n \times k matrix of n observations of k regressors, which includes a regressor whose value is 1 for each observation on the first column.
`y`	Numeric vector of length `n` or `n` by `1` matrix. The vector \mathbf{y} is an n \times 1 vector of observations on the regressand variable.

If betahat = NULL, betahat computed using betahat().

Returns residual sum of squares ≤ft( \mathrm{RSS} \right).

Ivan Jacob Agaloos Pesigan

Wikipedia: Residual Sum of Squares

Wikipedia: Explained Sum of Squares

Wikipedia: Total Sum of Squares

Wikipedia: Coefficient of Determination

Other sum of squares functions: .ESS(), .RSS(), ESS(), TSS()

# Simple regression------------------------------------------------
X <- jeksterslabRdatarepo::wages.matrix[["X"]]
X <- X[, c(1, ncol(X))]
y <- jeksterslabRdatarepo::wages.matrix[["y"]]
RSS(X = X, y = y)

# Multiple regression----------------------------------------------
X <- jeksterslabRdatarepo::wages.matrix[["X"]]
# age is removed
X <- X[, -ncol(X)]
RSS(X = X, y = y)