Methods and formulas for predictions in Nonlinear Regression

The expected response for the n^th observation at θ*:

Notation

Term	Description
θ*	the final iteration
xn	vector of values for the predictors at the nth observation
v0	gradient matrix = ( ∂f(xn, θ) / ∂θp ), the P by 1 vector of partial derivatives of f(x0, θ), evaluated at θ*

The range in which the mean response is expected to fall given specified settings of the predictors. An approximate 100(1 - α)% confidence interval for the prediction is:

Notation

Term	Description
tα/2	upper α/2 point of the t distribution with N – P degrees of freedom
se fit	standard error of the fit
n	nth observation
N	total number of observations
P	number of free (unlocked) parameters
	fitted value
b	(R')-1v0
R	the (upper triangular) R matrix from the QR decomposition of Vi for the final iteration
v0	gradient matrix = ( ∂f(xn, θ) / ∂θp), the P by 1 vector of partial derivatives of f(x0, θ), evaluated at θ*
S

The range in which the predicted response for a single new observation is expected to fall. A new observation has an approximate 100(1 - α)% prediction interval of:

Notation

Term	Description
tα/2	upper α/2 point of the t distribution with N – P degrees of freedom
se fit	standard error of the fit
n	nth observation
N	total number of observations
P	number of free (unlocked) parameters
	fitted value
b	(R')-1v0
R	the (upper triangular) R matrix from the QR decomposition of Vi for the final iteration
v0	gradient matrix = ( ∂f(xn, θ) / ∂θp), the P by 1 vector of partial derivatives of f(x0, θ), evaluated at θ*
S

The approximate standard error of the fitted value is:

where R is the (upper triangular) R matrix from the QR decomposition of Vⁱ for the final iteration. Minitab computes:

by back-solving:

Notation

Term	Description
n	nth observation
N	total number of observations
P	number of free (unlocked) parameters
x0	vector of values for the predictors
	f(x0, θ*)
v0	gradient matrix = ( ∂f(xn, θ) / ∂θp), the P by 1 vector of partial derivatives of f(x0, θ), evaluated at θ*
S