Methods and formulas for parameter estimates in Nonlinear Regression

Select the method or formula of your choice.

Parameter constraints

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 )))
TermDescription
a and bnumeric constants
θ'sparameters
φtransformed 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.

Standard error of the parameter estimate

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

TermDescription
nnth observation
Ntotal number of observations
pnumber of free (unlocked) parameters
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

Correlation matrix of the parameter estimates

The approximate variance-covariance matrix of the parameter estimates is:
The approximate correlation between the estimates of θp and θq is:
Because R is triangular, Minitab can obtain its inverse by back-solving rather than by a general-purpose inversion algorithm.

Notation

TermDescription
Rthe (upper triangular) R matrix from the QR decomposition of Vi for the final iteration
Pnumber of free (unlocked) parameters
v0gradient matrix = ( ∂f(xn, θ) / ∂θ p), the P by 1 vector of partial derivatives of f( x0, θ), evaluated at θ*
θ'sparameters

Profile likelihood confidence intervals for parameters

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

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

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

S(θp) = S(θ*) + (tα/2)2 MSE

Notation

TermDescription
θ'sparameters
nnth observation
Ntotal number of observations
Pnumber of free (unlocked) parameters
tα/2upper α/2 point of the t distribution with N - P degrees of freedom
S(θ)Sum of the squared error
MSEmean squared error
  1. Bates and Watts (1988). Nonlinear Regression Analysis and Its Applications. John Wiley & Sons, Inc.
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