Methods and formulas for predictions in Nonlinear Regression

Select the method or formula of your choice.

Fitted value

The expected response for the nth observation at θ*:

Notation

TermDescription
θ*the final iteration
xnvector of values for the predictors at the nth observation
v0gradient matrix = ( ∂f(xn, θ) / ∂θp ), the P by 1 vector of partial derivatives of f(x0, θ), evaluated at θ*

Confidence interval of the prediction

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

TermDescription
tα/2upper α/2 point of the t distribution with N – P degrees of freedom
se fitstandard error of the fit
nnth observation
Ntotal number of observations
Pnumber of free (unlocked) parameters
fitted value
b(R')-1v0
Rthe (upper triangular) R matrix from the QR decomposition of Vi for the final iteration
v0gradient matrix = ( ∂f(xn, θ) / ∂θp), the P by 1 vector of partial derivatives of f(x0, θ), evaluated at θ*
S

Prediction interval

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

TermDescription
tα/2upper α/2 point of the t distribution with N – P degrees of freedom
se fitstandard error of the fit
nnth observation
Ntotal number of observations
Pnumber of free (unlocked) parameters
fitted value
b(R')-1v0
Rthe (upper triangular) R matrix from the QR decomposition of Vi for the final iteration
v0gradient matrix = ( ∂f(xn, θ) / ∂θp), the P by 1 vector of partial derivatives of f(x0, θ), evaluated at θ*
S

Standard error of the fitted value

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

TermDescription
nnth observation
Ntotal number of observations
Pnumber of free (unlocked) parameters
x0vector of values for the predictors
f(x0, θ*)
v0gradient matrix = ( ∂f(xn, θ) / ∂θp), the P by 1 vector of partial derivatives of f(x0, θ), evaluated at θ*
S
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