# Multinomial Regression Tanh Estimator Gauss-Newton Optimization

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##### Multinomial Regression Hyperbolic Tangent (Tanh) Estimator Gauss-Newton Optimization

mGNtanh uses Gauss-Newton 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 Gauss-Newton stages. Each stage does at most 100 Gauss-Newton 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 M-estimator. 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 Gauss-Newton iterations.
error
Error code: 0, no errors; 2, sum(w) < nobs*(ncats-1)/2 (weights are too small); 32, Hessian not positive definite in the final Newton step.
GNlist
List reporting final results of the Gauss-Newton optimization. Elements: coefficients, vector of coefficient parameters (same as coeffvec value in list returned by mGNtanh); tvec, matrix of coefficient parameters (same as coefficients 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) log-likelihood values; convflag, TRUE/FALSE convergence flag; iters, number of iterations done in final Gauss-Newton 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 done Y[!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): 391--410. http://sekhon.berkeley.edu/multinom.pdf