Multinomial Regression Tanh Estimator GaussNewton Optimization
Multinomial Regression Hyperbolic Tangent (Tanh) Estimator GaussNewton Optimization
mGNtanh
uses GaussNewton optimization to compute the
hyperbolic tangent (tanh) estimator for the overdispersed multinomial
regression model for grouped count data. This function is not meant
to be called directly by the user. It is called by
multinomRob
, which constructs the various arguments.
 Keywords
 robust, models, regression, optimize
Usage
mGNtanh(bstart, sigma2, resstart, Y, Ypos, Xarray, xvec, tvec, jacstack, itmax = 100, print.level = 0)
Arguments
 bstart
 Vector of starting values for the coefficient parameters.
 sigma2
 Value of the dispersion parameter (variance). The estimator does not update this value.
 resstart
 Array of initial orthogonalized (but not standardized) residuals.
 Y

Matrix (observations by alternatives) of outcome counts.
Values must be nonnegative. Missing data (
NA
values) are not allowed.  Ypos
 Matrix indicating which elements of Y are counts to be analyzed (TRUE) and which are values to be skipped (FALSE). This allows the set of outcome alternatives to vary over observations.
 Xarray
 Array of regressors. dim(Xarray) = c(observations, parameters, alternatives).
 xvec
 Matrix (parameters by alternatives) that represents the model structure. It has a 1 for an estimated parameter, an integer greater than 1 for an estimated parameter constrained equal to another estimated parameter (all parameters constrained to be equal to one another have the same integer value in xvec) and a 0 otherwise.
 tvec
 Starting values for the regression coefficient parameters, as a matrix (parameters by alternatives). Parameters that are involved in equality constraints are repeated in tvec.
 jacstack
 Array of regressors used to facilitate computing the gradient and the Hessian matrix. dim(jacstack) = c(observations, unique parameters, alternatives).
 itmax
 Maximum number of GaussNewton stages. Each stage does at most 100 GaussNewton steps.
 print.level
 Specify 0 for minimal printing (error messages only) or 2 to print details about the tanh computations.
Details
The tanh estimator is a redescending Mestimator. Given an estimate of the scale of the overdispersion, the tanh estimator estimates the coefficient parameters of the linear predictors of the multinomial regression model.
Value

mGNtanh returns a list of 16 objects. The returned objects are:
 coefficients

The tanh coefficient estimates in matrix format. The matrix has one
column for each outcome alternative. The label for each row of the matrix
gives the names of the regressors to which the coefficient values in the row
apply. The regressor names in each label are separated by a forward
slash (/), and
NA
is used to denote that no regressor is associated with the corresponding value in the matrix. The value 0 is used in the matrix to fill in for values that do not correspond to a regressor.  coeffvec
 A vector containing the tanh coefficient estimates.
 dispersion

Value of the dispersion parameter (variance). This is the value specified
in the argument
sigma2
in the call to the function.  w

Vector of weights based on the tanh estimator's
psi
function for each observation.  psi

Vector of values of the tanh estimator's
psi
function for each observation.  A
 The outer product of the gradient (expected information) divided by the moment estimate of the dispersion.
 B
 The inverse of the Hessian matrix (observed formation).
 covmat
 Sandwich estimate of the asymptotic covariance of the tanh coefficient estimates.
 iters
 Number of GaussNewton iterations.
 error

Error code:
0, no errors;
2,
sum(w) < nobs*(ncats1)/2
(weights are too small); 32, Hessian not positive definite in the final Newton step.  GNlist

List reporting final results of the GaussNewton optimization. Elements:
coefficients
, vector of coefficient parameters (same ascoeffvec
value in list returned by mGNtanh);tvec
, matrix of coefficient parameters (same ascoefficients
value in list returned by mGNtanh);formation
, inverse Hessian matrix;score
, score (or gradient element) matrix;LLvals
, list containing weighted (LLvals$LL
) and unweighted (LLvals$LLu
) loglikelihood values;convflag
, TRUE/FALSE convergence flag;iters
, number of iterations done in final GaussNewton stage;posdef
, TRUE if Hessian is positive definite.  tanhsigma2
 The tanh overdispersion parameter estimate, which is a weighted moment estimate of the dispersion: weighted mean sum of squared orthogonalized residuals (adjusted for effective sample size after weighting and degrees of freedom lost to estimated coefficients).
 Y

The same
Y
matrix that was supplied as input, except modified by having doneY[!Ypos] < 0
.  Ypos

The same
Ypos
matrix that was supplied as input.  probmat
 The matrix of predicted probabilities for each category for each observation based on the coefficient estimates.
 jacstack

The same
jacstack
that was supplied as an input argument.  Xarray

The same
Xarray
that was supplied as an input argument.
References
Walter R. Mebane, Jr. and Jasjeet Singh Sekhon. 2004. ``Robust Estimation and Outlier Detection for Overdispersed Multinomial Models of Count Data.'' American Journal of Political Science 48 (April): 391410. http://sekhon.berkeley.edu/multinom.pdf
For additional documentation please visit http://sekhon.berkeley.edu/robust/.