Fitting the Multinomial Probit Model via Markov chain Monte Carlo

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Description

mnp is used to fit (Bayesian) multinomial probit model via Markov chain Monte Carlo. mnp can also fit the model with different choice sets for each observation, and complete or partial ordering of all the available alternatives. The computation uses the efficient marginal data augmentation algorithm that is developed by Imai and van Dyk (2005a).

Usage

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mnp(formula, data = parent.frame(), choiceX = NULL, cXnames = NULL,
    base = NULL, latent = FALSE, invcdf = FALSE, trace = TRUE, n.draws = 5000,
    p.var = "Inf", p.df = n.dim+1, p.scale = 1, coef.start = 0,
    cov.start = 1, burnin = 0, thin = 0, verbose = FALSE) 

Arguments

formula

A symbolic description of the model to be fit specifying the response variable and covariates. The formula should not include the choice-specific covariates. Details and specific examples are given below.

data

An optional data frame in which to interpret the variables in formula and choiceX. The default is the environment in which mnp is called.

choiceX

An optional list containing a matrix of choice-specific covariates for each category. Details and examples are provided below.

cXnames

A vector of the names for the choice-specific covariates specified in choiceX. The details and examples are provided below.

base

The name of the base category. For the standard multinomial probit model, the default is the lowest level of the response variable. For the multinomial probit model with ordered preferences, the default base category is the last column in the matrix of response variables.

latent

logical. If TRUE, then the latent variable W will be returned. See Imai and van Dyk (2005) for the notation. The default is FALSE.

invcdf

logical. If TRUE, then the inverse cdf method is used for truncated normal sampling. If FALSE, then the rejection sampling method is used. The default is FALSE.

trace

logical. If TRUE, then the trace of the variance covariance matrix is set to a constant (here, it is equal to n.dim) instead of setting its first diagonal element to 1. The former avoids the arbitrariness of fixing one particular diagonal element in order to achieve identification (see Burgette and Nordheim, 2009).

n.draws

A positive integer. The number of MCMC draws. The default is 5000.

p.var

A positive definite matrix. The prior variance of the coefficients. A scalar input can set the prior variance to the diagonal matrix whose diagonal element is equal to that value. The default is "Inf", which represents an improper noninformative prior distribution on the coefficients.

p.df

A positive integer greater than n.dim-1. The prior degrees of freedom parameter for the covariance matrix. The default is n.dim+1, which is equal to the total number of alternatives.

p.scale

A positive definite matrix. When trace = FALSE, its first diagonal element is set to 1 if it is not equal to 1 already. The prior scale matrix for the covariance matrix. A scalar input can be used to set the scale matrix to a diagonal matrix with diagonal elements equal to the scalar input value. The default is 1.

coef.start

A vector. The starting values for the coefficients. A scalar input sets the starting values for all the coefficients equal to that value. The default is 0.

cov.start

A positive definite matrix. When trace = FALSE, its first diagonal element is set to 1 if it is not equal to 1 already. The starting values for the covariance matrix. A scalar input can be used to set the starting value to a diagonal matrix with diagonal elements equal to the scalar input value. The default is 1.

burnin

A positive integer. The burnin interval for the Markov chain; i.e., the number of initial Gibbs draws that should not be stored. The default is 0.

thin

A positive integer. The thinning interval for the Markov chain; i.e., the number of Gibbs draws between the recorded values that are skipped. The default is 0.

verbose

logical. If TRUE, helpful messages along with a progress report of the Gibbs sampling are printed on the screen. The default is FALSE.

Details

To fit the multinomial probit model when only the most preferred choice is observed, use the syntax for the formula, y ~ x1 + x2, where y is a factor variable indicating the most preferred choice and x1 and x2 are individual-specific covariates. The interactions of individual-specific variables with each of the choice indicator variables will be fit.

To specify choice-specific covariates, use the syntax, choiceX=list(A=cbind(z1, z2), B=cbind(z3, z4), C=cbind(z5, z6)), where A, B, and C represent the choice names of the response variable, and z1 and z2 are each vectors of length n that record the values of the two choice-specific covariates for each individual for choice A, likewise for z3, , z6. The corresponding variable names via cXnames=c("price", "quantity") need to be specified, where price refers to the coefficient name for z1, z3, and z5, and quantity refers to that for z2, z4, and z6.

If the choice set varies from one observation to another, use the syntax, cbind(y1, y2, y3) ~ x1 + x2, in the case of a three choice problem, and indicate unavailable alternatives by NA. If only the most preferred choice is observed, y1, y2, and y3 are indicator variables that take on the value one for individuals who prefer that choice and zero otherwise. The last column of the response matrix, y3 in this particular example syntax, is used as the base category.

To fit the multinomial probit model when the complete or partial ordering of the available alternatives is recorded, use the same syntax as when the choice set varies (i.e., cbind(y1, y2, y3, y4) ~ x1 + x2). For each observation, all the available alternatives in the response variables should be numerically ordered in terms of preferences such as 1 2 2 3. Ties are allowed. The missing values in the response variable should be denoted by NA. The software will impute these missing values using the specified covariates. The resulting uncertainty estimates of the parameters will properly reflect the amount of missing data. For example, we expect the standard errors to be larger when there is more missing data.

Value

An object of class mnp containing the following elements:

param

A matrix of the Gibbs draws for each parameter; i.e., the coefficients and covariance matrix. For the covariance matrix, the elements on or above the diagonal are returned.

call

The matched call.

x

The matrix of covariates.

y

The vector or matrix of the response variable.

w

The three dimensional array of the latent variable, W. The first dimension represents the alternatives, and the second dimension indexes the observations. The third dimension represents the Gibbs draws. Note that the latent variable for the base category is set to 0, and therefore omitted from the output.

alt

The names of alternatives.

n.alt

The total number of alternatives.

base

The base category used for fitting.

invcdf

The value of invcdf.

p.var

The prior variance for the coefficients.

p.df

The prior degrees of freedom parameter for the covariance matrix.

p.scale

The prior scale matrix for the covariance matrix.

burnin

The number of initial burnin draws.

thin

The thinning interval.

Author(s)

Kosuke Imai, Department of Politics, Princeton University kimai@Princeton.Edu, http://imai.princeton.edu; David A. van Dyk, Department of Statistics, University of California, Irvine dvd@uci.edu, http://www.ics.uci.edu/~dvd.

References

Imai, Kosuke and David A. van Dyk. (2005a) “A Bayesian Analysis of the Multinomial Probit Model Using the Marginal Data Augmentation,” Journal of Econometrics, Vol. 124, No. 2 (February), pp.311-334.

Imai, Kosuke and David A. van Dyk. (2005b) “MNP: R Package for Fitting the Multinomial Probit Models,” Journal of Statistical Software, Vol. 14, No. 3 (May), pp.1-32.

Burgette, L.F. and E.V. Nordheim. (2009). “An alternate identifying restriction for the Bayesian multinomial probit model,” Technical report, Department of Statistics, University of Wisconsin, Madison.

See Also

coef.mnp, cov.mnp, predict.mnp, summary.mnp; MNP home page at http://imai.princeton.edu/research/MNP.html

Examples

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###
### NOTE: this example is not fully analyzed. In particular, the
### convergence has not been assessed. A full analysis of these data
### sets appear in Imai and van Dyk (2005b).
###

## load the detergent data
data(detergent)
## run the standard multinomial probit model with intercepts and the price
res1 <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice,
                                       Wisk=WiskPrice, EraPlus=EraPlusPrice,
                                       Solo=SoloPrice, All=AllPrice),
            cXnames = "price", data = detergent, n.draws = 500, burnin = 100,
            thin = 3, verbose = TRUE)
## summarize the results
summary(res1)
## calculate the quantities of interest for the first 3 observations
pre1 <- predict(res1, newdata = detergent[1:3,])

## load the Japanese election data
data(japan)
## run the multinomial probit model with ordered preferences
res2 <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan,
            verbose = TRUE)
## summarize the results
summary(res2)
## calculate the predicted probabilities for the 10th observation
## averaging over 100 additional Monte Carlo draws given each of MCMC draw.
pre2 <- predict(res2, newdata = japan[10,], type = "prob", n.draws = 100,
                verbose = TRUE)