Package 'MNP'

Extract Multinomial Probit Model Coefficients

Description

coef.mnp is a function which extracts multinomial probit model coefficients from ojbects returned by mnp. coefficients.mnp is an alias for it. coef method for class mnp.

Usage

## S3 method for class 'mnp'
coef(object, subset = NULL, ...)

Arguments

object	An output object from mnp.
subset	A scalar or a numerical vector specifying the row number(s) of param in the output object from mnp. If specified, the posterior draws of coefficients for those rows are extracted. The default is NULL where all the posterior draws are extracted.
...	further arguments passed to or from other methods.

Value

coef.mnp returns a matrix (when a numerical vector or NULL is specified for subset argument) or a vector (when a scalar is specified for subset arugment) of multinomila probit model coefficients.

Author(s)

Kosuke Imai, Department of Government and Department of Statistics, Harvard University [email protected]

See Also

mnp, vcov.mnp;

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Detergent Brand Choice

Description

This dataset gives the laundry detergent brand choice by households and the price of each brand.

Format

A data frame containing the following 7 variables and 2657 observations.

choice	factor	a brand chosen by each household
TidePrice	numeric	log price of Tide
WiskPrice	numeric	log price of Wisk
EraPlusPrice	numeric	log price of EraPlus
SurfPrice	numeric	log price of Surf
SoloPrice	numeric	log price of Solo
AllPrice	numeric	log price of All

References

Chintagunta, P. K. and Prasad, A. R. (1998) “An Empirical Investigation of the ‘Dynamic McFadden’ Model of Purchase Timing and Brand Choice: Implications for Market Structure”. Journal of Business and Economic Statistics vol. 16 no. 1 pp.2-12.

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Voters' Preferences of Political Parties in Japan (1995)

Description

This dataset gives voters' preferences of political parties in Japan on the 0 (least preferred) - 100 (most preferred) scale. It is based on the 1995 survey data of 418 individual voters. The data also include the sex, education level, and age of the voters. The survey allowed voters to choose among four parties: Liberal Democratic Party (LDP), New Frontier Party (NFP), Sakigake (SKG), and Japanese Communist Party (JCP).

Format

A data frame containing the following 7 variables for 418 observations.

LDP	numeric	preference for Liberal Democratic Party	0 - 100
NFP	numeric	preference for New Frontier Party	0 - 100
SKG	numeric	preference for Sakigake	0 - 100
JCP	numeric	preference for Japanese Communist Party	0 - 100
gender	factor	gender of each voter	male or female
education	numeric	levels of education for each voter
age	numeric	age of each voter

---

Fitting the Multinomial Probit Model via Markov chain Monte Carlo

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

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, $\ldots$, 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 Government and Department of Statistics, Harvard University [email protected], https://imai.fas.harvard.edu; David A. van Dyk, Statistics Section, Department of Mathematics, Imperial College London.

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, vcov.mnp, predict.mnp, summary.mnp;

Examples

###
### 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 = 100, burnin = 10,
            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)

---

Posterior Prediction under the Bayesian Multinomial Probit Models

Description

Obtains posterior predictions under a fitted (Bayesian) multinomial probit model. predict method for class mnp.

Usage

## S3 method for class 'mnp'
predict(
  object,
  newdata = NULL,
  newdraw = NULL,
  n.draws = 1,
  type = c("prob", "choice", "order"),
  verbose = FALSE,
  ...
)

Arguments

object	An output object from mnp.
newdata	An optional data frame containing the values of the predictor variables. Predictions for multiple values of the predictor variables can be made simultaneously if newdata has multiple rows. The default is the original data frame used for fitting the model.
newdraw	An optional matrix of MCMC draws to be used for posterior predictions. The default is the original MCMC draws stored in object.
n.draws	The number of additional Monte Carlo draws given each MCMC draw of coefficients and covariance matrix. The specified number of latent variables will be sampled from the multivariate normal distribution, and the quantities of interest will be calculated by averaging over these draws. This will be particularly useful calculating the uncertainty of predicted probabilities. The default is 1.
type	The type of posterior predictions required. There are four options: type = "prob" returns the predictive probabilities of being the most preferred choice among the choice set. type = "choice" returns the Monte Carlo sample of the most preferred choice, and type = "order" returns the Monte Carlo sample of the ordered preferences,
verbose	logical. If TRUE, helpful messages along with a progress report on the Monte Carlo sampling from the posterior predictive distributions are printed on the screen. The default is FALSE.
...	additional arguments passed to other methods.

Details

The posterior predictive values are computed using the Monte Carlo sample stored in the mnp output (or other sample if newdraw is specified). Given each Monte Carlo sample of the parameters and each vector of predictor variables, we sample the vector-valued latent variable from the appropriate multivariate Normal distribution. Then, using the sampled predictive values of the latent variable, we construct the most preferred choice as well as the ordered preferences. Averaging over the Monte Carlo sample of the preferred choice, we obtain the predictive probabilities of each choice being most preferred given the values of the predictor variables. Since the predictive values are computed via Monte Carlo simulations, each run may produce somewhat different values. The computation may be slow if predictions with many values of the predictor variables are required and/or if a large Monte Carlo sample of the model parameters is used. In either case, setting verbose = TRUE may be helpful in monitoring the progress of the code.

Value

predict.mnp yields a list of class predict.mnp containing at least one of the following elements:

o	A three dimensional array of the Monte Carlo sample from the posterior predictive distribution of the ordered preferences. The first dimension corresponds to the rows of newdata (or the original data set if newdata is left unspecified), the second dimension corresponds to the alternatives in the choice set, and the third dimension indexes the Monte Carlo sample. If n.draws is greater than 1, then each entry will be an average over these additional draws.
p	A two or three dimensional array of the posterior predictive probabilities for each alternative in the choice set being most preferred. The first demension corresponds to the rows of newdata (or the original data set if newdata is left unspecified), the second dimension corresponds to the alternatives in the choice set, and the third dimension (if it exists) indexes the Monte Carlo sample. If n.draws is greater than 1, then the third diemsion exists and indexes the Monte Carlo sample.
y	A matrix of the Monte Carlo sample from the posterior predictive distribution of the most preferred choice. The first dimension correspond to the rows of newdata (or the original data set if newdata is left unspecified), and the second dimension indexes the Monte Carlo sample. n.draws will be set to 1 when computing this quantity of interest.
x	A matrix of covariates used for prediction

Author(s)

Kosuke Imai, Department of Government and Department of Statistics, Harvard University [email protected]

See Also

mnp

---

Print the summary of the results for the Multinomial Probit Models

Description

summary print method for class mnp.

Usage

## S3 method for class 'summary.mnp'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

x	An object of class summary.mnp.
digits	the number of significant digits to use when printing.
...	further arguments passed to or from other methods.

Author(s)

Kosuke Imai, Department of Politics, Princeton University [email protected]

See Also

mnp

---

Summarizing the results for the Multinomial Probit Models

Description

summary method for class mnp.

Usage

## S3 method for class 'mnp'
summary(object, CI = c(2.5, 97.5), ...)

Arguments

object	An output object from mnp.
CI	A 2 dimensional vector of lower and upper bounds for the credible intervals used to summarize the results. The default is the equal tail 95 percent credible interval.
...	further arguments passed to or from other methods.

Value

summary.mnp yields an object of class summary.mnp containing the following elements:

call	The call from mnp.
n.alt	The total number of alternatives.
base	The base category used for fitting.
n.obs	The number of observations.
n.param	The number of estimated parameters.
n.draws	The number of Gibbs draws used for the summary.
coef.table	The summary of the posterior distribution of the coefficients.
cov.table	The summary of the posterior distribution of the covariance matrix.

This object can be printed by print.summary.mnp

Author(s)

Kosuke Imai, Department of Government and Department of Statistics, Harvard University [email protected]

See Also

mnp

---

Extract Multinomial Probit Model Covariance Matrix

Description

vcov.mnp is a function which extracts the posterior draws of covariance matrix from objects returned by mnp.

Usage

## S3 method for class 'mnp'
vcov(object, subset = NULL, ...)

Arguments

object	An output object from mnp.
subset	A scalar or a numerical vector specifying the row number(s) of param in the output object from mnp. If specified, the posterior draws of covariance matrix for those rows are extracted. The default is NULL where all the posterior draws are extracted.
...	further arguments passed to or from other methods.

Value

When a numerical vector or NULL is specified for subset argument, vcov.mnp returns a three dimensional array where the third dimension indexes posterior draws. When a scalar is specified for subset arugment, vcov.mnp returns a matrix.

Author(s)

Kosuke Imai, Department of Government and Department of Statistics, Harvard University [email protected]

See Also

mnp, coef.mnp;

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