Methods and formulas for parameter estimates in Nonlinear Regression

Enforce parameter constraints by transforming the parameters.¹

If	Then
a < θ	θ = a + exp( φ )
θ < b	θ = b - exp( φ )
a < θ < b	θ = a +((b - a) / (1 + exp( -φ )))

Term	Description
a and b	numeric constants
θ's	parameters
φ	transformed parameters

Minitab performs these transforms, and displays the results in terms of the original parameters.

1. Bates and Watts (1988). Nonlinear Regression Analysis and Its Applications. John Wiley & Sons, Inc.

The approximate standard error of the estimate of θ_p is S times the square root of diagonal element p of , which is written as:

where e_p is a P by 1 vector with element p equal to 1 and all other elements equal to 0. Minitab computes:

by back-solving:

Notation

Term	Description
n	nth observation
N	total number of observations
p	number of free (unlocked) parameters
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 variance-covariance matrix of the parameter estimates is:

The approximate correlation between the estimates of θ_p and θ_q is:

Because R is triangular, Minitab can obtain its inverse by back-solving rather than by a general-purpose inversion algorithm.

Notation

Term	Description
R	the (upper triangular) R matrix from the QR decomposition of Vi for the final iteration
P	number of free (unlocked) parameters
v0	gradient matrix = ( ∂f(xn, θ) / ∂θ p), the P by 1 vector of partial derivatives of f( x0, θ), evaluated at θ*
θ's	parameters

Let θ = (θ₁, . . . . θ_p) * with θ* being the final iteration for θ.

The likelihood-based 100 (1 - α) % confidence limits satisfy:

where S( θ_p ) is the SSE obtained when holding θ_p fixed and minimizing over the other parameters.¹ This is equivalent to solving:

S(θ_p) = S(θ*) + (t_α/2)² MSE

Notation

Term	Description
θ's	parameters
n	nth observation
N	total number of observations
P	number of free (unlocked) parameters
tα/2	upper α/2 point of the t distribution with N - P degrees of freedom
S(θ)	Sum of the squared error
MSE	mean squared error