Robust Multinomial Regression {multinomRob} R Documentation

## Multinomial Robust Estimation

### Description

`multinomRob` fits the overdispersed multinomial regression model for grouped count data using the hyperbolic tangent (tanh) and least quartile difference (LQD) robust estimators.

### 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` and `y3`, and the linear predictor for `y1` is a coefficient times `x1` plus a constant, the linear predictor for `y2` is a coefficient times `x2` plus a constant, and the linear predictor for `y3` is zero. Each formula has the format `countvar ~ RHS`, where `countvar` is the name of a vector, in the dataframe referenced by the `data` 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 the `data` argument. For example, a `RHS` specification of `var1 + var2*var3` would specify that the regressors are to be `var1`, `var2`, `var3`, the terms generated by the interaction `var2: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 a `RHS` specification that explicitly omits the constant, e.g., `countvar ~ -1` or `countvar ~ 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 the `equality` 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 the `multinom.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 the `equality=` 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 the `maxiter` 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 multivariate-t 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 multivariate-t model for starting values even if the MNL estimates provide better starting values for the LQD. Note that with `multinom.t=1` or `multinom.t=2`, multivariate-t starting values will not be used if the model cannot generate valid standard errors. To force the use of multivariate-t estimates even in this circumstance, see the `multinom.t.df` argument. If the `starting.values` argument is not `NULL`, the starting values given in that argument are used and the `multinom.t` argument is ignored. Multinomial multivariate-t 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 multivariate-t model (when used) should be estimated. If `multinom.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. Setting `multinom.t.df` to a number also implies that, if `multinom.t=1` or `multinom.t=2`, the multivariate-t starting values will be used (depending on the comparison with the MNL estimates if `multinom.t=1` is set) even if the standard errors are not defined. `MLEonly` If `TRUE`, then only the standard maximum-likelihood MNL model is estimated. No robust estimation model and no overdispersion parameter is estimated.

### Details

The tanh estimator is a redescending M-estimator, and the LQD estimator is a generalized S-estimator. 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 multivariate-t 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 (/), 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 `model` 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 the `model` argument. If `count1` is the name of the count variable used in the first formula, then the second column in the matrix is named `weights: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 `NA` appear in the weights matrix for that observation, as many `NA` values as there are unavailable alternatives. The `NA` 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 the `model` argument. If `count1` is the name of the count variable used in the first formula, then the second column in the matrix is named `Hdiag: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 multivariate-t 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 LQD-tanh iterations.

### Author(s)

Walter R. Mebane, Jr., Cornell University, wrm1@cornell.edu, http://macht.arts.cornell.edu/wrm1/

Jasjeet S. Sekhon, UC Berkeley, sekhon@berkeley.edu, http://sekhon.berkeley.edu/

### 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

### 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
zz.genoud.parms <- list( pop.size             = 1000,
wait.generations      = 10,
max.generations       = 100,
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))
```

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