R: Fit a Logistic Surface Model to Data

cusp.logist {cusp}

R Documentation

Fit a Logistic Surface Model to Data

Description

This function fits a logistic curve model to data using maximum likelihood under the assumption of normal errors (i.e., nonlinear least squares). Both the response variable may be modelled by a linear combination of variables and design factors, as well as the normal/asymmetry factor alpha and bifurction/splitting factor beta.

Usage

cusp.logist(formula, alpha, beta, data, ..., model = TRUE, x =
                 FALSE, y = TRUE)

Arguments

`formula`, `alpha`, `beta`	`formula`s for the response variable and the regression variables (see below)
`data`	`data.frame` containing `n` observations of all the variables named in the formulas
`...`	named arguments that are passed to `nlm`
`model`, `x`, `y`	logicals. If `TRUE` the corresponding components of the fit (the model frame, the model matrix, and the response are returned.

Details

A nonlinear regression is carried out of the model

y_i = \frac{1}{1+\exp(-\alpha_i/\beta_i^2)} + \epsilon_i

for i = 1, 2, \ldots, n, where

y_i = w_0 + w_1 Y_{i1} + \cdots + w_p Y_{ip}

\alpha_i = a_0 + a_1 X_{i1} + \cdots + a_p X_{ip}

\beta_i = b_0 + b_1 X_{i1} + \cdots + b_q X_{iq}

in which the a_j's, and b_j's, are estimated. The Y_{ij}'s are variables in the data set and specified by formula; the X_{ij}'s are variables in the data set and are specified in alpha and beta. Variables in alpha and beta need not be the same. The w_j's are estimated implicitely using concentrated likelihood methods, and are not returned explicitely.

Value

List with components

`minimum`	Objective function value at minimum
`estimate`	Coordinates of objective function minimum
`gradient`	Gradient of objective function at minimum.
`code`	Convergence `code` returned by `optim`
`iterations`	Number of iterations used by `optim`
`coefficients`	A named vector of estimates of `a_j, b_j`'s
`linear.predictors`	Estimates of `\alpha_i`'s and `\beta_i`'s.
`fitted.values`	Predicted values of `y_i`'s as determined from the `linear.predictors`
`residuals`	Residuals
`rank`	Numerical rank of matrix of predictors for `\alpha_i`'s plus rank of matrix of predictors for `\beta_i`'s plus rank of matrix of predictors for the `y_i`'s.
`deviance`	Residual sum of squares.
`logLik`	Log of the likelihood at the minimum.
`aic`	Akaike's information criterion
`rsq`	R Squared (proportion of explained variance)
`df.residual`	Degrees of freedom for the residual
`df.null`	Degrees of freedom for the Null residual
`rss`	Residual sum of squares
`hessian`	Hessian matrix of objective function at the minimum if `hessian=TRUE`.
`Hessian`	Hessian matrix of log-likelihood function at the minimum (currently unavailable)
`qr`	QR decomposition of the `hessian` matrix
`converged`	Boolean indicating if optimization convergence is proper (based on exit code `optim`, gradient, and, if `hessian=TRUE` eigen values of the hessian).
`weights`	`weights` (currently unused)
`call`	the matched call
`y`	If requested (the default), the matrix of response variables used.
`x`	If requested, the model matrix used.
`null.deviance`	The sum of squared deviations from the mean of the estimated `y_i`'s.

Author(s)

Raoul Grasman

References

Hartelman PAI (1997). Stochastic Catastrophe Theory. Amsterdam: University of Amsterdam, PhDthesis.