bws.response: Generating artificial responses to Case 1 best-worst scaling questions

Description

This function synthesizes responses to Case 1 best-worst scaling (BWS) questions on the basis of a maximum difference model.

Usage

bws.response(design, item.names = NULL, b, n, detail = FALSE,
             seed = NULL)

Value

The function bws.response returns a data frame that contains synthesized responses to Case 1 BWS questions, in either a detailed or a simple format. The detailed format dataset contains the following variables, as well as independent variables according to the argument item.names.

id: An identification number of artificial respondents.
Q: A serial number of questions.
PAIR: A serial number of possible pairs of the best and worst items for each question.
BEST: An item number treated as the best in the possible pairs of the best and worst items for each question.
WORST: An item number treated as the worst in the possible pairs of the best and worst items for each question.
RES: Responses to BWS questions, taking the value of 1 if a possible pair of the best and worst items is selected by the synthesized respondents and 0 otherwise.
STR: A stratification variable used to identify each combination of respondent and question.

The simple format dataset contains the following variables.

id: An identification number of artificial respondents.
Bi: A variable describing the row number of the item that is selected as the best in the \(i\)-th BWS question (see the help for bws.dataset for a row number format). The serial number of questions is appended to the tail of the variable name (e.g., B1 for the first question, B2 for the second question, and B3 for the third question).
Wi: A variable describing the row number of the item that is selected as the worst in the \(i\)-th BWS question (see the help for bws.dataset for a row number format). The serial number of questions is appended to the tail of the variable name (e.g., W1 for the first question, W2 for the second question, and W3 for the third question).

The detailed format dataset includes a dependent variable and independent variables for the analysis, and thus is available for discrete choice analysis functions such as the function clogit in the survival package. On the other hand, the simple format dataset only contains variables that correspond to responses to BWS questions, as well as id variable. It must be converted using the function bws.dataset in the package for the analysis. For details, see the Example section.

Arguments

design: A matrix or data frame containing a balanced incomplete block design (BIBD).
item.names: A character vector containing the names of items: if NULL (default), default names (i.e., ITEM1, ITEM2, ...) are used in the resultant dataset. The argument is valid for the dataset in a detailed format.
b: A vector containing parameters of independent variables in the model. The vector is used to calculate utilities for alternatives.
n: An integer value showing the number of respondents in the resultant dataset.
detail: A logical variable: if TRUE, the dataset is returned in a detailed format; and if FALSE (default), the dataset is returned in a simple format.
seed: Seed for a random number generator.

Details

This function synthesizes responses to Case 1 BWS questions on the basis of a maximum difference model. The model assumes that there are \(m\) items to be evaluated, and that \(k\) (\(< m\)) items exist in a choice set (a question). The number of possible pairs where item \(i\) is selected as the best and item \(j\) is selected as the worst (\(i \neq j\)) from \(k\) items is given by \(k \times (k - 1)\). The model also assumes that the respondents select item \(i\) as the best and item \(j\) as the worst because the difference in utility between items \(i\) and \(j\) is the highest among all of the \(k \times (k - 1)\) differences in utility. The systematic component of the utility is assumed to be a linear additive function of the item variables. If the error component of the utility is assumed to be an independently, identically distributed type I extreme value, the probability of selecting item \(i\) as the best and item \(j\) as the worst is expressed as a conditional logit model.

Given the parameter values assigned to the argument b, and the choice sets assigned to the argument design, the function bws.response calculates the utility for the items. The parameter values assigned to the argument b are set as a numerical vector where the elements correspond to the parameters of item variables. For example, assume that seven items A, B, C, D, E, F, and G exist in the list for evaluation, and their corresponding dummy-coded item variables \(D_{A}\), \(D_{B}\), \(D_{C}\), \(D_{D}\), \(D_{E}\), and \(D_{G}\) (item F is assumed to be the reference item) have parameter values of 0.5, 1.2, 1.6, 1.8, 2.1, and 0.9, respectively. A vector assigned to the argument b is c(0.5, 1.2, 1.6, 1.8, 2.1, 0, 0.9), where the sixth element (i.e., item F) corresponds to the reference level, and thus has a value of 0. After calculating the utility values (by adding the calculated values of the systematic component of the utility and random numbers generated from a type I extreme value distribution), the function bws.response finds the pair with the highest difference in utility from the \(k \times (k - 1)\) differences in utility.

Examples

Run this code

# The following lines of code synthesize responses to Case 1 BWS questions,
# return them in detailed and simple format, and then fit the models using
# the function clogit in the survival package. The questions evaluate seven
# items. The choice sets consist of seven questions with four items each.
# The function find.BIB in the crossdes package creates the corresponding
# BIBD with seven treatments, seven blocks, and size four. The systematic
# component of the utility for items is the same as that explained in the
# Details section.

if (FALSE) {
# Load packages
library(survival)
library(crossdes)

# Generate BIBD
set.seed(123)
bibd <- find.BIB(trt = 7, b = 7, k = 4)
isGYD(bibd)
bibd

# Synthesize responses to Case 1 BWS questions
b <- c(0.5, 1.2, 1.6, 1.8, 2.1, 0, 0.9)
items = c("A", "B", "C", "D", "E", "F", "G")
dat.detail <- bws.response(
  design = bibd, item.names = items,
  b = b, n = 100,
  detail = TRUE, seed = 123)
str(dat.detail)
dat.simple <- bws.response(
  design = bibd,
  b = b, n = 100,
  detail = FALSE, seed = 123)
str(dat.simple)

# Convert dat.simple into dataset for the analysis
response.vars <- colnames(dat.simple)[-1]
dat.simple.md <- bws.dataset(
  respondent.dataset = dat.simple,
  response.type = 1,
  choice.sets = bibd,
  design.type = 2,
  item.names = items,
  id = "id",
  response = response.vars,
  model = "maxdiff")

# Fit conditional logit models
mf <- RES ~ A + B + C + D + E + G + strata(STR)
out.detail <- clogit(mf, dat.detail)
out.simple <- clogit(mf, dat.simple.md)
out.simple
all.equal(coef(out.detail), coef(out.simple))
}