NominalLogisticBiplot (version 0.2)

polylogist: Multinomial logistic regression with ridge penalization

Description

This function does a logistic regression between a dependent variable y and some independent variables x, and solves the separation problem in this type of regression using ridge regression and penalization.

Usage

polylogist(y, x, penalization = 0.2, cte = TRUE, tol = 1e-04, maxiter = 200, show = FALSE)

Arguments

y

Dependent variable.

x

A matrix with the independent variables.

penalization

Penalization used in the diagonal matrix to avoid singularities.

cte

Should the model have a constant?

tol

Tolerance for the iterations.

maxiter

Maximum number of iterations.

show

Should the iteration history be printed?.

Value

An object of class "polylogist". This has components

fitted

Matrix with the fitted probabilities

cov

Covariance matrix among the estimates

Y

Indicator matrix for the dependent variable

beta

Estimated coefficients for the multinomial logistic regression

stderr

Standard error of the estimates

logLik

Logarithm of the likelihood

Deviance

Deviance of the model

AIC

Akaike information criterion indicator

BIC

Bayesian information criterion indicator

Details

The problem of the existence of the estimators in logistic regression can be seen in Albert (1984), a solution for the binary case, based on the Firth's method, Firth (1993) is proposed by Heinze(2002). The extension to nominal logistic model was made by Bull (2002). All the procedures were initially developed to remove the bias but work well to avoid the problem of separation. Here we have chosen a simpler solution based on ridge estimators for logistic regression Cessie(1992).

Rather than maximizing $L_j (G | b_j0 , B_j)$ we maximize

$${{L_j}(\left. {\bf{G}} \right|{{\bf{b}}_{j0}},{{\bf{B}}_j})} - \lambda \left( {\left\| {{{\bf{b}}_{j0}}} \right\| + \left\| {{{\bf{B}}_j}} \right\|} \right)$$ Changing the values of $\lambda$ we obtain slightly different solutions not affected by the separation problem.

References

Albert,A. & Anderson,J.A. (1984),On the existence of maximum likelihood estimates in logistic regression models, Biometrika 71(1), 1--10. Bull, S.B., Mak, C. & Greenwood, C.M. (2002), A modified score function for multinomial logistic regression, Computational Statistics and dada Analysis 39, 57--74. Firth, D.(1993), Bias reduction of maximum likelihood estimates, Biometrika 80(1), 27--38 Heinze, G. & Schemper, M. (2002), A solution to the problem of separation in logistic regression, Statistics in Medicine 21, 2109--2419 Le Cessie, S. & Van Houwelingen, J. (1992), Ridge estimators in logistic regression, Applied Statistics 41(1), 191--201.

Examples

  
  data(HairColor)
  data = data.matrix(HairColor)
  G = NominalMatrix2Binary(data)
  mca=afc(G,dim=2)
  depVar = data[,1]
  nomreg = polylogist(depVar,mca$RowCoordinates[,1:2],penalization=0.1)
  nomreg