cusp.logist: Fit a Logistic Surface Model to Data
In cusp: Cusp-Catastrophe Model Fitting Using Maximum Likelihood

Description Usage Arguments Details Value Author(s) References See Also

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.

1 2	cusp.logist(formula, alpha, beta, data, ..., model = TRUE, x = FALSE, y = TRUE)

`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.

A nonlinear regression is carried out of the model

y[i] = 1/(1+exp(-α[i]/β[i]^2)) + ε[i]

for i = 1, 2, …, n, where

y[i] = w[0] + w[1] * Y[i,1] + \cdots + w[p] * Y[i,p],

α[i] = a[0] + a[1] * X[i,1] + ... + a[p] * X[i,p],

β[i] = b[0] + b[1] * X[i,1] + ... + b[p] * X[i,p],

in which the a[j]'s, and b[j]'s, are estimated. The Y[i,j]'s are variables in the data set and specified by formula; the X[i,j]'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.

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 α[i]'s and β[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 α[i]'s plus rank of matrix of predictors for β[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.