Robust Multinomial Regression
Multinomial Robust Estimation
multinomRob
fits the overdispersed multinomial regression model
for grouped count data using the hyperbolic tangent (tanh) and least quartile
difference (LQD) robust estimators.
 Keywords
 robust, models, regression
Usage
multinomRob(model, data, starting.values=NULL, equality=NULL, genoud.parms=NULL, print.level=0, iter = FALSE, maxiter = 10, multinom.t=1, multinom.t.df=NA, MLEonly=FALSE)
Arguments
 model

The regression model specification. This is a list of formulas, with one
formula for each category of outcomes for which counts have been measured
for each observation. For example, in the following,
model=list(y1 ~ x1, y2 ~ x2, y3 ~ 0)
the outcome variables containing counts are
y1
,y2
andy3
, and the linear predictor fory1
is a coefficient timesx1
plus a constant, the linear predictor fory2
is a coefficient timesx2
plus a constant, and the linear predictor fory3
is zero. Each formula has the formatcountvar ~ RHS
, wherecountvar
is the name of a vector, in the dataframe referenced by thedata
argument, that gives the counts for all observations for one category.RHS
denotes the righthand side of a formula using the usual syntax for formulas, where each variable in the formula is the name of a vector in the dataframe referenced by thedata
argument. For example, aRHS
specification ofvar1 + var2*var3
would specify that the regressors are to bevar1
,var2
,var3
, the terms generated by the interactionvar2:var3
, and the constant.The set of outcome alternatives may be specified to vary over observations, by putting in a negative value for alternatives that do not exist for particular observations. If the value of an outcome variable is negative for an observation, then that outcome is considered not available for that observation. The predicted counts for that observation are defined only for the available observations and are based on the linear predictors for the available observations. The same set of coefficient parameter values are used for all observations. Any observation for which fewer than two outcomes are available is omitted.
Observations with missing data (
NA
) in any outcome variable or regressor are omitted (listwise deletion).In a model that has the same regressors for every category, except for one category for which there are no regressors in order to identify the model (the reference category), the
RHS
specification must be given for all the categories except the reference category. The formula for the reference category must include aRHS
specification that explicitly omits the constant, e.g.,countvar ~ 1
orcountvar ~ 0
. The number of coefficient parameters to be estimated equals the number of terms generated by all the formulas, subject to equality constraints that may be specified using theequality
argument.  data

The dataframe that contains all the variables referenced in the
model
argument, which are the data to be analyzed.  starting.values

Starting values for the regression coefficient parameters, as a vector.
The parameter ordering matches the ordering of the formulas in the
model
argument: parameters for the terms in the first formula appear first, then come parameters for the terms in the second formula, etc. In practice it will usually be better to start by letting multinomRob find starting values by using themultinom.t
option, then using the results from one run as starting values for a subsequent run done with, perhaps, a larger population of operators for rgenoud.  equality

List of equality constraints. This is a list of lists of
formulas. Each formula has the same format as in the model specification,
and must include only a subset of the outcomes and regressors used in the
model specification formulas. All the coefficients specified by the
formulas in each list will be constrained to have the same value during
estimation. For example, in the following,
multinomRob(model=list(y1 ~ x1, y2 ~ x2, y3 ~ 0), data=dtf, equality=list(list(y1 ~ x1 + 0, y2 ~ x2 + 0)) );
the model to be estimated is
list(y1 ~ x1, y2 ~ x2, y3 ~ 0)
and the coefficients of x1 and x2 are constrained equal by
equality=list(list(y1 ~ x1 + 0, y2 ~ x2 + 0))
In the equality formulas it is necessary to say
+ 0
so the intercepts are not involved in the constraints. If a parameter occurs in two different lists in theequality=
argument, then all the parameters in the two lists are constrained to be equal to one another. In the output this is described as consolidating the lists.  genoud.parms
 List of named arguments used to control the rgenoud optimizer, which is used to compute the LQD estimator.
 print.level
 Specify 0 for minimal printing, 1 to print more detailed information about LQD and other intermediate computations, 2 to print details about the tanh computations, or 3 to print details about starting values computations.
 iter

TRUE
means to iterate between LQD and tanh estimation steps until either the algorithm converges, the number of iterations specified by themaxiter
argument is reached, or if an LQD step occurs that produces a larger value than the previous step did for the overdispersion scale parameter. This option is often improves the fit of the model.  maxiter
 The maximum number of iterations to be done between LQD and tanh estimation steps.
 multinom.t

1
means use the multinomial multivariatet model to compute starting values for the coefficient parameters. But if the MNL results are better (as judged by the LQD fit), MNL values will be used instead.0
means use nonrobust maximum likelihood estimates for a multinomial regression model.2
forces the use of the multivariatet model for starting values even if the MNL estimates provide better starting values for the LQD. Note that withmultinom.t=1
ormultinom.t=2
, multivariatet starting values will not be used if the model cannot generate valid standard errors. To force the use of multivariatet estimates even in this circumstance, see themultinom.t.df
argument.If the
starting.values
argument is notNULL
, the starting values given in that argument are used and themultinom.t
argument is ignored. Multinomial multivariatet starting values are not available when the number of outcome alternatives varies over the observations.  multinom.t.df

NA
means that the degrees of freedom (DF) for the multivariatet model (when used) should be estimated. Ifmultinom.t.df
is a number, that number will be used for the degrees of freedom and the DF will not be estimated. Only a positive number should be used. Settingmultinom.t.df
to a number also implies that, ifmultinom.t=1
ormultinom.t=2
, the multivariatet starting values will be used (depending on the comparison with the MNL estimates ifmultinom.t=1
is set) even if the standard errors are not defined.  MLEonly

If
TRUE
, then only the standard maximumlikelihood MNL model is estimated. No robust estimation model and no overdispersion parameter is estimated.
Details
The tanh estimator is a redescending Mestimator, and the LQD estimator is a generalized Sestimator. The LQD is used to estimate the scale of the overdispersion. Given that scale estimate, the tanh estimator is used to estimate the coefficient parameters of the linear predictors of the multinomial regression model.
If starting values are not supplied, they are computed using a multinomial multivariatet model. The program also computes and reports nonrobust maximum likelihood estimates for the multinomial regression model, reporting sandwich estimates for the standard errors that are adjusted for a nonrobust estimate of the error dispersion.
Value

multinomRob returns a list of 15 objects. The returned objects are:
 coefficients

The tanh coefficient estimates in matrix format. The matrix has one
column for each formula specified in the
model
argument. The name of each column is the name used for the count variable in the corresponding formula. 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 (/), andNA
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 amodel
formula regressor.  se

The tanh coefficient estimate standard errors in matrix format. The
format and labelling used for the matrix is the same as is used for the
coefficients
. The standard errors are derived from the estimated asymptotic sandwich covariance estimate.  LQDsigma2
 The LQD dispersion (variance) parameter estimate. This is the LQD estimate of the scale value, squared.
 TANHsigma2
 The tanh dispersion parameter estimate.
 weights

The matrix of tanh weights for the orthogonalized residuals. The matrix
has one row for each observation in the data and as many columns as
there are formulas specified in the
model
argument. The first column of the matrix has names for the observations, and the remaining columns contain the weights. Each of the latter columns has a name derived from the name of one of the count variables named in themodel
argument. Ifcount1
is the name of the count variable used in the first formula, then the second column in the matrix is namedweights:count1
, etc.If an observation has negative values specified for some outcome variables, indicating that those outcome alternatives are not available for that observation, then values ofNA
appear in the weights matrix for that observation, as manyNA
values as there are unavailable alternatives. TheNA
values will be the last values in the affected row of the weights matrix, regardless of which outcome alternatives were unavailable for the observation.  Hdiag

Weights used to fully studentize the orthogonalized residuals. The matrix
has one row for each observation in the data and as many columns as
there are formulas specified in the
model
argument. The first column of the matrix has names for the observations, and the remaining columns contain the weights. Each of the latter columns has a name derived from the name of one of the count variables named in themodel
argument. Ifcount1
is the name of the count variable used in the first formula, then the second column in the matrix is namedHdiag:count1
, etc.If an observation has negative values specified for some outcome variables, indicating that those outcome alternatives are not available for that observation, then values of 0 appear in the weights matrix for that observation, as many 0 values as there are unavailable alternatives. Values of 0 that are created for this reason will be the last values in the affected row of the weights matrix, regardless of which outcome alternatives were unavailable for the observation.  prob
 The matrix of predicted probabilities for each category for each observation based on the tanh coefficient estimates.
 residuals.rotate
 Matrix of studentized residuals which have been made comparable by rotating each choice category to the first position. These residuals, unlike the student and standard residuals below, are no longer orthogonalized because of the rotation. These are the residuals displayed in Table 6 of the reference article.
 residuals.student
 Matrix of fully studentized orthogonalized residuals.
 residuals.standard
 Matrix of orthogonalized residuals, standardized by dividing by the overdispersion scale.
 mnl

List of nonrobust maximum likelihood estimation results from function
multinomMLE
.  multinomT

List of multinomial multivariatet estimation results from function
multinomT
.  genoud

List of LQD estimation results obtained by rgenoud optimization, from
function
genoudRob
.  mtanh

List of tanh estimation results from function
mGNtanh
.  error

Exit error code, usually from function
mGNtanh
.  iter
 Number of LQDtanh iterations.
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/.
Examples
# make some multinomial data
x1 < rnorm(50);
x2 < rnorm(50);
p1 < exp(x1)/(1+exp(x1)+exp(x2));
p2 < exp(x2)/(1+exp(x1)+exp(x2));
p3 < 1  (p1 + p2);
y < matrix(0, 50, 3);
for (i in 1:50) {
y[i,] < rmultinomial(1000, c(p1[i], p2[i], p3[i]));
}
# perturb the first 5 observations
y[1:5,c(1,2,3)] < y[1:5,c(3,1,2)];
y1 < y[,1];
y2 < y[,2];
y3 < y[,3];
# put data into a dataframe
dtf < data.frame(x1, x2, y1, y2, y3);
## Set parameters for Genoud
## Not run:
# ## For production, use these kinds of parameters
# zz.genoud.parms < list( pop.size = 1000,
# wait.generations = 10,
# max.generations = 100,
# scale.domains = 5,
# print.level = 0
# )
# ## End(Not run)
## For testing, we are setting the parmeters to run quickly. Don't use these for production
zz.genoud.parms < list( pop.size = 10,
wait.generations = 1,
max.generations = 1,
scale.domains = 5,
print.level = 0
)
# estimate a model, with "y3" being the reference category
# true coefficient values are: (Intercept) = 0, x = 1
# impose an equality constraint
# equality constraint: coefficients of x1 and x2 are equal
mulrobE < multinomRob(list(y1 ~ x1, y2 ~ x2, y3 ~ 0),
dtf,
equality = list(list(y1 ~ x1 + 0, y2 ~ x2 + 0)),
genoud.parms = zz.genoud.parms,
print.level = 3, iter=FALSE);
summary(mulrobE, weights=TRUE);
#Do only MLE estimation. The following model is NOT identified if we
#try to estimate the overdispersed MNL.
dtf < data.frame(y1=c(1,1),y2=c(2,1),y3=c(1,2),x=c(0,1))
summary(multinomRob(list(y1 ~ 0, y2 ~ x, y3 ~ x), data=dtf, MLEonly=TRUE))