ESS: Explained Sum of Squares
In jeksterslabds/jeksterslabRlinreg: Jekster's Lab Linear Regression Utilities

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

Calculates the explained sum of squares ≤ft( \mathrm{ESS} \right) using

\mathrm{ESS} = ∑_{i = 1}^{n} ≤ft( \hat{Y}_{i} - \bar{Y} \right)^2 \\ = ∑_{i = 1}^{n} ≤ft( \hat{β}_{1} + \hat{β}_{2} X_{2i} + \hat{β}_{3} X_{3i} + … + \hat{β}_{k} X_{ki} - \bar{Y} \right)^2

In matrix form

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

where \mathbf{\hat{y}} ≤ft( \mathbf{X} \boldsymbol{\hat{β}} \right) is an n \times 1 matrix of predicted values of \mathbf{y}, and \mathbf{\bar{Y}} is the mean of \mathbf{y}. Equivalent computational matrix formula

\mathrm{ESS} = \boldsymbol{\hat{β}}^{\prime} \mathbf{X}^{\prime} \mathbf{X} \boldsymbol{\hat{β}} - n \mathbf{\bar{Y}}^{2}.

Note that

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

ESS(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.

Returns the explained sum of squares ≤ft( \mathrm{ESS} \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(), RSS(), TSS()

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

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