This repository has all the necessary functions to replicate the results from the article “Modelling preference heterogeneity using model-based decision trees” (Gutiérrez-Vargas, Vandebroek & Meulders, 2023) forthcoming at the Journal of Choice Modelling. You can find the In Press pre-print here.
The source code is contained in a small R package I developed to store
all the necessary functions which is called MobMixlogit and can be
found in the
\src
folder and can be installed locally by running the R script
\src\install_MobMixlogit.R
The different parts of the article are organized in different folders:
\simulations_replication: contains the replications from sections 5 and 6.\empirical_application: contains the replications from section 7.
For citations please use:
@article{gutierrez_vargas_2023,
title = {Modeling preference heterogeneity using model-based decision trees},
journal = {Journal of Choice Modelling},
volume = {46},
pages = {100393},
year = {2023},
issn = {1755-5345},
doi = {https://doi.org/10.1016/j.jocm.2022.100393},
url = {https://www.sciencedirect.com/science/article/pii/S1755534522000501},
author={Guti{\'e}rrez-Vargas, {\'A}lvaro A and Meulders, Michel and Vandebroek, Martina}
}
Here is a brief illustration on how to implement a Mixed Logit at the end leaves using the MOB algorithm. We start by creating some data that contains a partition.
library(MobMixlogit)
library(partykit)
library(mlogit)
# Simulate some data with a partition (see Figure 2a on the article)
df <- MobMixlogit::dgp_tree(N = 500, # 500 individuals
t = 10, # 10 choice sets each
xi = 0.5, # xi = 0.5 implies balanced groups (see Section 5.1)
delta = 1 # delta regulates how different are the parameters of each partition
)
# Create the choice variable for the wide format
df$choice <- with(df,ave(x = altern * chosen,
group = id,
FUN = max))
# Delete chosen variable
df$chosen <- NULL ; df$id_choice_situation_of_individual_n <- NULL
# Reshape from long to wide.
df_wide <- stats::reshape(data = df,
idvar = c("id","id_ind"),
timevar = "altern",
v.names = c("x1", "x2"),
direction = "wide",
sep = "_")Now we have to create an auxiliary function that will be useful to run
the Mixed Logit model using mlogit. This way of constructing the
auxiliary function was greatly inspired by an answer given by professor
Achim Zeileis on
StackOverflow.
# Create a auxiliar function to make mlogit work together with partykit.
mixl_for_partykit <- function(y,
x = NULL,
start = NULL,
weights = NULL,
offset = NULL, ...) {
# Create an artificial dataset binding the x's and the choice variable (y)
yx <- cbind(y,x)
# Declare dfidx data
d <- dfidx::dfidx(data = yx,
shape = "wide",
choice = "choice",
varying = 2:7, ## first if "y", then "x's" and finally "identifiers".
sep = "_",
idx = list(c("id", "id_ind")), #obs= choice situation; id = individual
idnames = c(NA, "altern"))
# Fit a Mixed Logit using mlogit to the "d" data.
clogit_dfidx <- mlogit::mlogit(formula = choice ~ x1 + x2 |0 , #+0 to drop intercepts
data = d, # dataset
rpar = c(x1 = 'n', # Both random parameters are assumed normal
x2 = 'n'),
R = 500, # Number of Halton draws
panel = TRUE, # Activate Panel structure (more than one response per individual)
method = 'bhhh') # Optimization technique.
return(clogit_dfidx)
}Then we call the
partykit::mob() function
and provide our auxiliary function mixl_for_partykit as our model fit
to it.
(mob_mixl <- partykit::mob(formula = choice ~ # Choice variable
x1_1 + x1_2 + x1_3 + # Variables for the first attribute
x2_1 + x2_2 + x2_3 + # Variables of the second attribute
id + id_ind + 0 | # Identifiers + drop the intercept.
z1 +z2 + z3 + z4 + z5 ,
data = df_wide, # data set to be used (wide format)
fit = mixl_for_partykit, # auxiliary function just defined above
cluster = id_ind, # cluster standard errors by individual
control = partykit::mob_control(
ytype = "data.frame", # Declare the choice variable as data.frame()
xtype = "data.frame", # Declare the attribute variables as data.frame()
minsize = 100, # Minimum number of choice sets to create a partition
alpha = 0.05 ))) # Level of confidence to reject null hypothesis## Model-based recursive partitioning (mixl_for_partykit)
##
## Model formula:
## choice ~ x1_1 + x1_2 + x1_3 + x2_1 + x2_2 + x2_3 + id + id_ind +
## 0 | z1 + z2 + z3 + z4 + z5
##
## Fitted party:
## [1] root
## | [2] z1 <= 0.5: n = 2690
## | x1 x2 sd.x1 sd.x2
## | 0.9772931 1.0125683 0.4678080 0.4293633
## | [3] z1 > 0.5
## | | [4] z2 <= 0.5: n = 1120
## | | x1 x2 sd.x1 sd.x2
## | | 1.9899722 1.9976571 0.5546160 0.3708133
## | | [5] z2 > 0.5: n = 1190
## | | x1 x2 sd.x1 sd.x2
## | | 3.1129389 2.0251336 0.2657006 0.4117270
##
## Number of inner nodes: 2
## Number of terminal nodes: 3
## Number of parameters per node: 4
## Objective function: 3084.055
Finally, we can also provide a graphical illustration of the estimated tree. The figure shows that a first partition was made using variable Z_1, and then a subsequent partition on Z_2. We observe also that both tests rejected the hypothesis of parameter stability with p-values smaller than 0.001. Some other ways of presenting the end nodes can be found here and here.
## Summary function inspired by:
## https://stackoverflow.com/questions/65495322/partykit-modify-terminal-node-to-include-standard-deviation-and-significance-of/65500344#65500344
fn_summary_for_partykit_plots <- function(info, digits = 2) {
n <- info$nobs
individuals <- length(unique(info$object$model$idx$id))
c(paste("T =", n),
paste("N =", individuals))
}
plot(mob_mixl,
terminal_panel = node_terminal,
tp_args = list(FUN = fn_summary_for_partykit_plots,
fill = "white"))
In our article we also provided an illustration of one possible way to
use the MOB algorithm as a variable selection step for the allocation
model of Latent Class models. You can find this code in the script
\simulations_replication\MOB_MNL_on_LCMIXL_data.R.