dot-ESS: Explained Sum of Squares (from \mathbf{\hat{y}} and...
In jeksterslabds/jeksterslabRlinreg: Jekster's Lab Linear Regression Utilities

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

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} .

1	.ESS(yhat = NULL, ybar = NULL, X, y, betahat = NULL)

`yhat`	Numeric vector of length `n` or `n` by `1` numeric matrix. n \times 1 vector of predicted values of \mathbf{y} ≤ft( \mathbf{\hat{y}} \right).
`ybar`	Numeric. Mean of `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.
`betahat`	Numeric vector of length `k` or `k` by `1` matrix. The vector \boldsymbol{\hat{β}} is a k \times 1 vector of estimates of k unknown regression coefficients.