logitr: Fast Estimation of Multinomial and Mixed Logit Models with Preference Space and Willingness to Pay Space Utility Parameterizations

Abstract

This paper introduces the logitr R package for fast maximum likelihood estimation of multinomial logit and mixed logit models with unobserved heterogeneity across individuals, which is modeled by allowing parameters to vary randomly over individuals according to a chosen distribution. The package is faster than other similar packages such as mlogit, gmnl, mixl, and apollo, and it supports utility models specified with "preference space" or "willingness to pay (WTP) space" parameterizations, allowing for the direct estimation of marginal WTP. The typical procedure of computing WTP post-estimation using a preference space model can lead to unreasonable distributions of WTP across the population in mixed logit models. The paper provides a discussion of some of the implications of each utility parameterization for WTP estimates. It also highlights some of the design features that enable logitr's performant estimation speed and includes a benchmarking exercise with similar packages. Finally, the paper highlights additional features that are designed specifically for WTP space models, including a consistent user interface for specifying models in either space and a parallelized multi-start optimization loop, which is particularly useful for searching the solution space for different local minima when estimating models with non-convex log-likelihood functions.

Full text

JSS Journal of Statistical Software
MMMMMM YYYY, Volume VV, Issue II. doi: 10.18637/jss.v000.i00
logitr: Fast Estimation of Multinomial and Mixed Logit
Models with Preference Space and Willingness to Pay
Space Utility Parameterizations
John Paul Helveston
George Washington University
Abstract
This paper introduces the logitr Rpackage for fast maximum likelihood estimation of multi-
nomial logit and mixed logit models with unobserved heterogeneity across individuals, which
is modeled by allowing parameters to vary randomly over individuals according to a chosen
distribution. The package is faster than other similar packages such as mlogit,gmnl,mixl,
and apollo, and it supports utility models specified with “preference space” or “willingness to
pay (WTP) space” parameterizations, allowing for the direct estimation of marginal WTP. The
typical procedure of computing WTP post-estimation using a preference space model can lead
to unreasonable distributions of WTP across the population in mixed logit models. The paper
provides a discussion of some of the implications of each utility parameterization for WTP esti-
mates. It also highlights some of the design features that enable logitr’s performant estimation
speed and includes a benchmarking exercise with similar packages. Finally, the paper highlights
additional features that are designed specifically for WTP space models, including a consistent
user interface for specifying models in either space and a parallelized multi-start optimization
loop, which is particularly useful for searching the solution space for different local minima when
estimating models with non-convex log-likelihood functions.
Keywords: logit, utility, preference, willingness to pay, discrete choice models, R, maximum likeli-
hood estimation.
1. Introduction
Choice modeling is a well-established statistical method for assessing consumer preferences across
a wide variety of fields. One of the most common approaches for modeling choice is the maximum
likelihood estimation of multinomial logit models (McFadden 1974), which is rooted in the theory
of random utility models (Louviere, Hensher, and Swait 2000;Train 2009). The central assumption
of these models is that individual consumers make choices that maximize an underlying random
arXiv:2210.10875v1 [stat.ME] 19 Oct 2022

2logitr: Preference and WTP Space Logit Models in R
utility model, which can be parameterized as a function of a product’s observed attributes and
a random variable representing the portion of utility unobservable to the modeler. These models
produce estimates of the marginal utility for changes in each attribute relative to one another.
In many applications, modelers are interested in estimating marginal “willingness to pay” (WTP)
for changes in product attributes. The typical procedure to obtain these estimates is to divide the
estimated parameters of a “preference space” utility model by the negative of the price parameter.
Despite this common practice, it can yield unreasonable distributions of WTP across the population
in heterogeneous random parameter (or “mixed logit”) models (Train and Weeks 2005;Sonnier,
Ainslie, and Otter 2007;Helveston, Feit, and Michalek 2018). For example, if the parameters for
the price attribute and another non-price attribute are both assumed to be normally distributed
across the population, then the resulting WTP estimate follows a Cauchy distribution, implying
that WTP has an infinite variance across the population.
An alternative approach is to re-parameterize the utility model into the “WTP space” prior to
estimation. Estimating a WTP space model allows the modeler to directly specify assumptions
of how WTP is distributed, which has been found to yield more reasonable estimates of WTP
(Train and Weeks 2005;Sonnier et al. 2007;Daly, Hess, and Train 2012). WTP space models have
also been found to be more consistent with respondent’s true underlying preferences (Crastesa,
Beaumaisb, Mahieud, Martinez-Camblore, and Scarpa 2014). Finally, since WTP estimates are
independent of error scaling, they can be conveniently compared across different models estimated
on different data.
Several statistical packages support the estimation of multinomial and mixed logit models with
WTP space utility parameterizations. One of the most common approaches involves an adaptation
of the generalized multinomial logit (GMNL) model (Fiebig, Keane, Louviere, and Wasi 2010) to fit
WTP space models via an implementation of the scaled multinomial logit (SMNL) model, though
this requires that the price parameter estimate and standard error be calculated post-estimation.
Estimation of WTP space models via GMNL has been implemented in R with the gmnl package
(Sarrias, Daziano et al. 2017) and in STATA with the gmnl package (Gu, Hole, and Knox 2013).
WTP space models can also be estimated using the apollo (Hess and Palma 2019) and mixl (Molloy,
Becker, Schmid, and Axhausen 2021) R packages as they allow the user to hand-specify any valid
utility model. Finally, Professor Arne Rise Hole developed two STATA packages that share a
common syntax for estimating mixed logit models in the preference space (mixlogit) and WTP
space (mixlogitwtp) (Hole 2007). Many other packages exist for estimating a wider variety of logit
models, but they are limited to preference space models. Of these, package mlogit (Croissant 2020)
is perhaps the most complete and widely used for estimating multinomial logit and mixed logit
models in R via maximum likelihood estimation.
The logitr package is designed specifically to support the estimation of multinomial logit and mixed
logit models models with either preference space or WTP space utility parameterizations. While
logitr is less general in scope compared to more flexible packages like mixl and apollo, it offers
other functionality that is particularly useful for estimating WTP space models and conveniently
switching between preference and WTP space models. For example, given their non-linear utility
specification, WTP space models often diverge during estimation and can be sensitive to starting
parameters. To address this, the package includes a parallelized multi-start optimization loop
to search for different local minima from different random starting points when minimizing the
negative log-likelihood. The user interface is also more streamlined and simplified for estimating
models in either space.
Package logitr is also computationally efficient and faster than other similar packages, including the

Journal of Statistical Software 3
mixl package which uses high performance C++ code to compile the log-likelihood function (Molloy
et al. 2021) (though mixl can be accelerated considerably via multi-core processing). The perfor-
mance gains are the result of a combination of design features, including how the choice probabilities
are specified, avoiding redundant computation by pre-computing constant intermediate variables,
and the use of analytic gradients that are optimized for efficiency.
The rest of the article is organized as follows: Section 2 provides an overview of the models sup-
ported by logitr, including multinomial and mixed logit models with preference space and WTP
space utility parameterizations. Section 3 discusses several important implications of preference
versus WTP space utility parameterizations on WTP estimates. Section 4 describes the software
architecture and performance. Section 5 then introduces the logitr package, including examples
of estimating multinomial and mixed logit models in both preference and WTP spaces as well as
additional functionality for estimating weighted models and making predictions. Section 6 explains
some limitations of WTP space models. Finally, Section 7 concludes the paper.
2. Models
2.1. The random utility model in two spaces
Random utility models assume that consumers choose the alternative jfrom a set of alternatives
that has the greatest utility uj. Utility is a random variable that is modeled as uj=vj+εj, where
vjis the “observed utility” (a function of the observed attributes such that vj=f(xj)) and εjis a
random variable representing the portion of utility unobservable to the modeler.
Adopting the same notation as in Helveston et al. (2018), consider the following utility model:
u∗
j=β∗>xj+α∗pj+ε∗
j, ε∗
j∼Gumbel 0, σ2π2
6!(1)
where β∗is the vector of coefficients for non-price attributes xj,α∗is the coefficient for price pj,
and the error term, ε∗
j, is an IID random variable with a Gumbel extreme value distribution of
mean zero and variance σ2(π2/6).
This model is not identified since there exists an infinite set of combinations of values for β∗,α∗,
and σthat will produce the same choice probabilities. In order to specify an identifiable model,
Equation 1must be normalized. One approach is to normalize the scale of the error term by
dividing Equation 1by σ, producing the “preference space” utility specification (Train and Weeks
2005):
u∗
j
σ!=β∗
σ>
xj+α∗
σpj+ ε∗
j
σ!, ε∗
j
σ!∼Gumbel 0,π2
6!(2)
The typical preference space parameterization of the multinomial logit model can then be written
by rewriting Equation 2with uj= (u∗
j/σ),β= (β∗/σ),α= (α∗/σ), and εj= (ε∗
j/σ):
uj=β>xj+αpj+εjεj∼Gumbel 0,π2
6!(3)
The vector βin Equation 3represents the marginal utility for changes in each non-price attribute
(relative to the standardized scale of the error term), and αrepresents the marginal utility obtained

4logitr: Preference and WTP Space Logit Models in R
from changes in price (relative to the standardized scale of the error term). The coefficients βand
αare only relative values rather than absolute and do not have units. Using this model, estimates
of the marginal WTP for changes in each non-price attribute could be computed by dividing ˆ
βby
−ˆα, where the “hat” symbol indicates a parameter estimate.
An alternative approach to normalizing Equation 1is to divide by −α∗instead of σ, resulting in
the “WTP space” utility parameterization:
u∗
j
−α∗!=β∗
−α∗>
xj+α∗
−α∗pj+ ε∗
j
−α∗!, ε∗
j
−α∗!∼Gumbel 0,σ2
(−α∗)2
π2
6!(4)
Since the error term in Equation 4is scaled by λ2=σ2/(−α∗)2, it can be rewritten by multiplying
both sides by λ= (−α∗/σ) and renaming uj= (λu∗
j/−α∗),ω= (β∗/−α∗), and εj= (λε∗
j/−α∗):
uj=λω>xj−pj+εjεj∼Gumbel 0,π2
6!(5)
The vector ωin Equation 5represents the marginal WTP for changes in each non-price attribute,
and λrepresents the scale of the deterministic portion of utility relative to the standardized scale
of the random error term (also called the “scale parameter”). In contrast to the βcoefficients
from the preference space model in Equation 3, the ωcoefficients have absolute value with units
of currency.
The logitr package can fit logit models with either utility parameterization, and it contains functions
that facilitate the comparison of WTP estimates between models from the two model spaces.
2.2. Multinomial and mixed logit probabilities
By assuming that the error term in Equations 3and 5follows a Gumbel extreme value distribution,
the probability that a consumer will choose alternative jin choice situation nfollows a convenient,
closed form expression, cf. Train (2009):
Pnj =exp (vnj )
PJ
kexp (vnk),(6)
where vnj is the deterministic portion of the utility model and Jis the number of alternatives
in choice situation n. The multinomial logit model assumes homogeneous preferences across the
population and possess the independence of irrelevant alternatives (IIA) property, which means that
the ratio of any two probabilities is independent of the functions determining any other outcome
since
Pnj
Pnk
=exp (vnj )
exp (vnk),(7)
To relax this assumption and allow for heterogeneity of preferences across the population, the
multinomial logit model can be extended to the random coefficients “mixed” logit model (McFadden
and Train 2000) where probabilities are the integrals of standard logit probabilities over a density
of parameters across people:
Pnj =Z exp (vnj )
PJ
kexp (vnk)!f(θ)dθ,(8)

Journal of Statistical Software 5
where f(θ)is a density function and θcontains the parameters in the deterministic portion of the
utility model, which are βand αfor preference space models (Equation 3) and ωand λfor WTP
space models (Equation 5). The mixed logit probability can be interpreted as a weighted average
of the multinomial logit probability with weights given by the density f(θ).
Modelers often specify different mixing distributions for parameters in θdepending on assumptions
of how preferences might be distributed across the population. For example, modelers may assume
αfollows a log-normal or zero-censored normal distribution to force the price coefficient to remain
positive—an assumption based on the logic that most people prefer price decreases rather than
increases. Likewise, parameters in βare often assumed to follow a normal distribution if it is
unclear whether the utility parameters for attributes xjshould be positive or negative.
2.3. Maximum likelihood estimation
Parameters in the preference or WTP space utility models can be estimated by maximizing the
log-likelihood function. For the multinomial logit model, the log-likelihood is given by:
L=
N
X
n
J
X
j
ynj ln Pnj (9)
where ynj = 1 if alternative jis chosen in situation nand 0otherwise, Nis the number of choice
situations, Jis the number of alternatives in choice situation n, and the probabilities Pnj are given
by Equation 6.
For mixed logit models, the log-likelihood can be estimated using simulation to obtain estimates
of Pnj in Equation 8(Train 2009). Over a series of iterations, parameters are drawn from f(θ)
and used to compute the logit probability in Equation 6. The average probabilities over all of
the iterations, ˆ
Pnj , are then used in place of Pnj in Equation 9to compute the simulated log-
likelihood. Should the data contain a panel structure where multiple observations come from the
same individual, the product of the logit probabilities in Equation 6over all trials for each individual
must first computed and then averaged over the draws of each parameter drawn from f(θ)(Train
2009).
McFadden (1974) shows that the log-likelihood function is globally concave for linear-in-parameters
utility models with fixed parameters. This implies that optimization algorithms should always arrive
at a global solution when minimizing the negative log-likelihood for preference space models with
fixed parameters. In contrast, WTP space utility models (as well as mixed logit models with either
utility parameterization) have non-convex log-likelihood functions and thus are not guaranteed
to arrive at a global solution. For these models, different optimization strategies should be used
to minimize the negative log-likelihood, such as using a multi-start loop where the optimization
algorithm is run multiple times from different random starting points to search for multiple local
minima.
3. Implications of preference versus WTP space utility parameterizations
WTP estimates can be obtained from both preference and WTP space utility parameterizations. In
the preference space utility model given by Equation 3, WTP is estimated as ˆ
β/−ˆα; in the WTP
space model given by Equation 5, WTP is simply ˆ
ω. The choice of which approach to use can
have important implications for estimates of WTP, and modelers should consider which outcomes

6logitr: Preference and WTP Space Logit Models in R
and measures are most relevant to any one particular study when making a choice between the two
parameterizations.
3.1. Distribution of WTP estimates across the population
Depending on the utility parameterization used, the distribution of WTP in mixed logit models
can be sensitive to distributional assumptions of model parameters (Train and Weeks 2005;Sonnier
et al. 2007). For example, in a preference space model, if αand βwere each assumed to be normally
distributed, then the WTP for marginal changes in xjwould follow a Cauchy distribution, implying
that WTP has an infinite variance across the population. This WTP distribution is not likely what
the modeler had in mind when making individual distributional assumptions on αand β, but it
is the implied result. In contrast, in a WTP space model the distribution of WTP for marginal
changes in xjcan be directly specified.
Several prior studies have also identified this issue, and all find that WTP space utility parame-
terizations yield more reasonable estimates of WTP. In a study on preferences for alternative-fuel
vehicles, Train and Weeks (2005) found that while a preference space model with a log-normally dis-
tributed price coefficient fit the data better, it resulted in unreasonably large estimates of WTP; in
contrast, a WTP space model produced much more reasonable estimates of WTP. Using a Bayesian
approach, Sonnier et al. (2007) similarly found that a preference space model with heterogeneity
distributions for attribute and price coefficients resulted in poorly behaved posterior WTP distri-
butions and that the problem was particularly bad in small sample settings. Finally, Daly et al.
(2012) show that when the price coefficient is modeled with a variety of popular distributions, in-
cluding the normal, truncated normal, uniform, and triangular, the resulting distribution of WTP
has infinite moments.
To illustrate this issue, consider an example of three preference space models with different assump-
tions on how the price parameter is distributed. This example uses the yogurt data set included
in package logitr. In each model, coefficients for the yogurt brand (βin the preference space and
ωin the WTP space) are modeled as normally distributed. However, the price parameter, α,
in the preference space model (and likewise the scale parameter, λ, in the WTP space model) is
modeled three different ways: 1) as a fixed coefficient, 2) normally distributed, and 3) log-normally
distributed.
Figure 1compares the WTP distribution for the Yoplait brand across the population from each
preference space and WTP space model. In the case where αand λare modeled as fixed coefficients
(panel A), the WTP distributions from each model are identical. But when αand λare modeled
as normally distributed (panel B), the WTP distribution from the preference space model changes
dramatically. Since WTP in this case is the ratio of two normal distributions, the variance increases
by two orders of magnitude and the mean shifts upward. A similar outcome occurs when αand λ
are modeled as log-normally distributed1(panel C). Note that the WTP distribution in the WTP
space model remains nearly the same regardless of how λwas assumed to be distributed.
This example illustrates the sensitivity of the WTP distribution to modeling choices made in
preference space utility models. In general, WTP space models yield more reasonable estimates of
WTP distributions across the population, a consistent finding across multiple prior studies (Train
and Weeks 2005;Sonnier et al. 2007;Das, Anderson, and Swallow 2009;Helveston et al. 2018).
3.2. Prediction
1Log-normal distributions are often used to force parameter positivity.

Journal of Statistical Software 7
Figure 1: Comparison of WTP distribution for the Yoplait brand from mixed logit
models with preference space (red) and WTP space (gray) utility parameterizations.
In each panel, the price parameter is modeled as fixed, normally distributed, or log-
normally distributed. The “SD” in the labels means “Standard Deviation”.
Whether preference space or WTP space utility models predict better is an empirical question that
has not been definitively addressed in prior studies. Of the few studies that have estimated models
in the WTP space, none have provided conclusive evidence that one parameterization systematically
predicts better than the other. Both Sonnier et al. (2007) and Train and Weeks (2005) found that
a preference space parameterization fit the data better, but they also both found that the resulting
estimates of WTP had unreasonably large tails. Das et al. (2009) also estimated both preference
and WTP space models using data on preferences for landfill site attributes in Rhode Island, and
they found nearly identical model fit on out-of-sample predictions with each model specification,
though the WTP space model yielded more reasonable estimates of WTP.
3.3. Practical considerations
In addition to considerations of WTP estimates, model fit, and prediction, there are other practical
implications to consider when choosing to estimate a preference versus WTP space model. Perhaps
the simplest and most obvious distinction is that a WTP space model yields estimates of WTP
without needing additional post-estimation calculations. And because WTP space coefficients have
units of currency, they have a concrete meaning that can be immediately interpreted. In contrast,
preference space model coefficients only have relative meaning along an abstract scale of utility,
and modelers often compute WTP from preference space coefficients to help make results more
interpretable.
Perhaps less obvious is the fact that WTP estimates can be directly compared with those from
models estimated on other datasets since WTP coefficients are independent of error scaling. This
is particularly convenient for comparing WTP estimates from different subsets of a dataset. In
a preference space model, parameters are proportional to error scaling, and thus due to potential
scale differences coefficients estimated from different data sets cannot be directly compared (Swait
and Andrews 2003;Helveston et al. 2018). This also poses a challenge for comparative studies or
literature reviews that seek to compare outcomes on similar topics across multiple studies.
Another perhaps less obvious implication of the preference space parameterization is the assumption

8logitr: Preference and WTP Space Logit Models in R
that distributions of marginal utilities are independent across attributes, which induces a strong
correlation structure among WTP values (Train and Weeks 2005). This can make it difficult to
evaluate alternatives with different attribute levels since WTP cannot be added across attributes.
In a WTP space model, this problem can be avoided by directly incorporating the correlation
structure among WTP coefficients, and as a result dollar values can be summed to yield the total
WTP for an alternative.
Finally, there is no theoretical basis for believing that marginal utilities versus marginal WTPs
should follow standard distributions (e.g., normal and log-normal). In the absence of any theoret-
ical basis for these assumptions, the modeler is left to consider differences in empirical outcomes,
which as previously noted, there has not been much definitive evidence that models in one space
systematically out-perform the other along all measures of significance.
4. Software architecture and performance
4.1. Design features for increased estimation speed
In maximum likelihood estimation (and simulated MLE for mixed logit models), the log-likelihood
function is computed many times as the algorithm searches for parameters that minimize the
negative of the log-likelihood via gradient descent. The logitr package uses several strategies to
accelerate this process.
First, minimization of the negative log-likelihood is handled via the nloptr package, which is an
Rinterface to NLopt—an open source program for nonlinear optimization started by Steven G.
Johnson (Ypma and Johnson 2020). One benefit of using nloptr is that both the log-likelihood
function and its gradient can be computed within the same function. This reduces redundant com-
putations as many intermediate calculations are shared between the log-likelihood and its gradient.
Furthermore, analytic gradients are implemented for both preference space and WTP space models
and for multinomial and mixed logit models.
Another important feature is that the choice probabilities are reformulated to reduce the number of
calculations needed to compute the log-likelihood function. Instead of using Equation 6to compute
the probability of each alternative in a choice set, the choice probability for the chosen alternative,
Pc, can be calculated as:
Pc=1
1 + PJ
j6=cexp(vj−vc),(10)
This results in a more stable and computationally faster calculation of the log-likelihood, which is
simplified from Equation 9as
L=
N
X
n
J
X
j
ln Pnc (11)
In addition, logitr takes advantage of the fact that, except for the parameters, the data used
in computing the log-likelihood function and its gradient do not change, enabling a considerable
amount of memory reduction by pre-computing several intermediate computations that remain
constant throughout the estimation process. For example, the gradient with respect to parameters

Journal of Statistical Software 9
θof the log-likelihood in equation 11 for multinomial logit models can be written as follows:
∂L
∂θ=
N
X
n=1
−Pnc 

J
X
j6=c
exp(vnj −vnc)∂
∂θ(vnj −vnc)
(12)
In preference space models where vnj =β>xnj +αpnj , the partial derivative ∂/∂θin equation 12
is:
∂
∂α (vnj −vnc) = pnj −pnc ,∂
∂β(vnj −vnc) = xnj −xnc (13)
In WTP space models where vnj =λ(ω>xnj −pnj ), the partial derivatives ∂/∂θin equation 12
are:
∂
∂λ (vnj −vnc) = 1
λ(vnj −vnc),∂
∂ω(vnj −vnc) = λ(xnj −xnc )(14)
The values of pnj −pnc and xnj −xnc in Equations 13 and 14 are constant and can be computed
prior to starting the optimization loop. Furthermore, since Pnc,(vnj −vnc ), and exp(vnj −vnc )
are already computed when calculating the log-likelihood, they can be used to quickly compute the
analytic gradient with only a few additional calculations in each iteration of the algorithm.
Finally, the parallel package is also used to simultaneously estimate multiple models from dif-
ferent starting points when estimating a multi-start loop. For machines with multiple cores, this
can dramatically increase the size of the solution space searched without substantially increasing
estimation time.
4.2. Performance benchmarking
The design features implemented in logitr result in impressive gains in overall efficiency compared to
similar packages. To compare its performance, a preference space mixed logit model was estimated
using logitr,mlogit,mixl,gmnl, and apollo. Figure 2shows the estimation time for each package
plotted against the number of random draws used in the mixed logit model. The benchmark
was carried out in a Google Colab notebook at https://colab.research.google.com/drive/
1vYlBdJd4xCV43UwJ33XXpO3Ys8xWkuxx?usp=sharing.
With just 50 random draws, logitr is particularly fast, clocking in at 4.3 times faster than mixl with
one core, 3.6 times faster than mixl with two cores, 5.1 times faster than mlogit, 4.5 times faster
than gmnl, 8.2 times faster than apollo with one core, and 10.6 times faster than apollo with two
cores. The comparative difference decreases with higher numbers of draws, though even at 1,000
draws logitr is still over twice as fast as mlogit, the next-fastest package in the benchmark (see
Table 1). Speeds could potentially be further improved by parallelizing elements of the gradient
calculation. In addition, it is important to note that mixl in particular may perform better at very
large numbers of draws (e.g. 10,000 or more) if used on a machine with a large number of cores.

10 logitr: Preference and WTP Space Logit Models in R
Figure 2: Comparison of package estimation times for a preference space mixed logit
model with four normally-distributed random parameters. The x-axis shows the num-
ber of random draws used in simulating the log-likelihood function.
Table 1: Comparison of package estimation times for a preference space mixed logit model with
four normally-distributed random parameters.
Estimation time (s) Times slower than logitr
50 200 400 600 800 1000 50 200 400 600 800 1000
logitr 3 9 14 24 33 39 1.0 1.0 1.0 1.0 1.0 1.0
mixl (1 core) 11 50 80 158 229 271 3.9 5.6 5.8 6.6 6.8 6.9
mixl (2 cores) 9 42 66 130 185 231 3.2 4.7 4.8 5.4 5.5 5.9
mlogit 12 20 88 60 101 98 4.3 2.2 6.4 2.5 3.0 2.5
gmnl 11 31 70 122 99 141 3.8 3.5 5.1 5.1 3.0 3.6
apollo (1 core) 17 44 84 129 164 198 6.3 4.9 6.1 5.4 4.9 5.1
apollo (2 cores) 22 53 83 120 164 197 8.0 6.0 6.0 5.0 4.9 5.0

Journal of Statistical Software 11
5. Using the logitr package
5.1. Installation
The logitr package can be installed from CRAN:
R> install.packages("logitr")
The development version can be installed from GitHub using the remotes package:
R> remotes::install_github("jhelvy/logitr")
The package is loaded in Rwith:
R> library("logitr")
5.2. Data format
The logitr package requires that data be structured in a data.frame and arranged in a “long”
format (Wickham et al. 2014) where each row contains data on a single alternative from a choice
observation. The choice observations do not have to be symmetric, meaning they can have a
“ragged” structure where different choice observations have different numbers of alternatives. The
data must include variables for each of the following:
•Outcome: a dummy-coded variable that identifies which alternative was chosen (1is chosen,
0is not chosen). Only one alternative should have a 1per choice observation.
•Observation ID: a sequence of repeated numbers that identifies each unique choice observation.
For example, if the first three choice observations had 2 alternatives each, then the first 6
rows of the obsID variable would be 1, 1, 2, 2, 3, 3.
•Covariates: other variables that will be used as model covariates.
The logitr package contains several example data sets that illustrate this structure. The yogurt
data set will be used as a running example throughout this paper to illustrate the key features of
the package. The data set contains observations of yogurt purchases by a panel of 100 households
(Jain, Vilcassim, and Chintagunta 1994). Choice is identified by the choice column, each choice
observation is identified by the obsID column, and the columns price,feat, and brand can be
used as model covariates:
R> head(yogurt)
# A tibble: 6 x 7
id obsID alt choice price feat brand
<dbl> <int> <int> <dbl> <dbl> <dbl> <chr>
1 1 1 1 0 8.1 0 dannon
2 1 1 2 0 6.10 0 hiland
3 1 1 3 1 7.90 0 weight
4 1 1 4 0 10.8 0 yoplait
5 1 2 1 1 9.80 0 dannon
6 1 2 2 0 6.40 0 hiland

12 logitr: Preference and WTP Space Logit Models in R
This data set also includes an alt variable that determines the alternatives included in the choice
set of each observation and an id variable that determines the individual as the data have a panel
structure containing multiple choice observations from each individual.
5.3. Model specification interface
Models are specified and estimated using the logitr() function. The data argument should be set
to the data frame containing the data, and the outcome and obsID arguments should be set to the
column names in the data frame that correspond to the dummy-coded outcome (choice) variable
and the observation ID variable, respectively. All variables to be used as model covariates should
be provided as a vector of column names to the pars argument. Each variable in the vector is
additively included as a covariate in the utility model, with the interpretation that they represent
utilities in preference space models and WTPs in a WTP space model.
For example, consider a preference space model where the utility for yogurt is given by the following
utility model:
uj=αpj+β1xj1+β2xj2+β3xj3+β4xj4+εj,(15)
where pjis price,xj1is feat, and xj2−4are dummy-coded variables for each brand (with the
fourth brand representing the reference level). This model can be estimated using the logitr()
function as follows:
R> mnl_pref <- logitr(
+ data = yogurt,
+ outcome = "choice",
+ obsID = "obsID",
+ pars = c("price", "feat", "brand")
+ )
The equivalent model in the WTP space is given by the following utility model:
uj=λ(ω1xj1+ω1xj2+ω1xj3+ω2xj4−pj) + εj,(16)
To specify this model, simply move "price" from the pars argument to the scalePar argument:
R> mnl_wtp <- logitr(
+ data = yogurt,
+ outcome = "choice",
+ obsID = "obsID",
+ pars = c("feat", "brand"),
+ scalePar = "price"
+ )
In the above model, the variables in pars are marginal WTPs, whereas in the preference space model
they are marginal utilities. Price is separately specified with the scalePar argument because it
acts as a scaling term in WTP space models. While price is the most typical scaling variable, other
continuous variables can also be used; for example, time could be used to obtain marginal estimates
of “willingness to wait.”

Journal of Statistical Software 13
Interactions between covariates can be entered in the pars vector separated by the *symbol.
For example, an interaction between price with feat in the above preference space model could
be included by specifying pars = c("price", "feat", "brand", "price*feat"), or even more
concisely just pars = c("price*feat", "brand") as the interaction between price and feat will
produce individual parameters for price and feat in addition to the interaction parameter.
Although the logitr model specification interface is a departure from the popular formula interface
used in other similar packages such as mlogit, it was designed to be more uniform and streamlined
for estimating either preference or WTP space models. For WTP space models in particular, using
the formula interface can be confusing as it requires that covariates be additively specified (e.g.,
choice ~ price + feat + brand), which is inconsistent with the underlying WTP space utility
model parameterization in which the price parameter (λ) scales the WTP parameters.
For example, consider how the formula interface is used in the gmnl package. Using gmnl, WTP
space models are specified by 1) modifying the price attribute in the data to be the negative of
price prior to estimation, 2) specifying the model argument as "smnl", 3) additively including all
parameters in the formula (including price) along with appropriate 0s and 1s in the additional
formula components to properly specify a scaled multinomial logit model, 4) specifying a vector of
TRUE and FALSE values to the fixed argument for every model parameter, and 5) providing starting
values where the price / scale parameter is set to 1for stability (otherwise the default value of 0
will be used, which usually results in a convergence failure). To estimate the previous example
WTP space model using gmnl, the data would first need to be formatted using the mlogit.data()
function:
R> data_gmnl <- mlogit.data(
+ data = yogurt,
+ shape = "long",
+ choice = "choice",
+ id.var = 'id',
+ alt.var = 'alt',
+ chid.var = 'obsID',
+ opposite = 'price'
+ )
The model would then be estimated using the gmnl() function:
R> mnl_wtp <- gmnl(
+ data = data_gmnl,
+ formula = choice ~ price + feat + brand | 0 | 0 | 0 | 1,
+ fixed = c(TRUE, FALSE, FALSE, FALSE, FALSE, TRUE, FALSE),
+ model = "smnl", method = "bhhh",
+ start = c(1, 0, 0, 0, 0, 0, 0)
+ )
Compared to the logitr interface, the above syntax is considerably more complex, and it is also not
obvious that this specification will produce WTP estimates. The additive inclusion of price in
formula is particularly confusing for a model that produces WTP estimates as it is inconsistent with
the WTP space utility model. Finally, if the user fails to remember the specific set of preparation
steps prior to estimation (such as taking the negative of price), results could be confusing.

14 logitr: Preference and WTP Space Logit Models in R
The apollo and mixl packages can also be used to estimate WTP space models, but they require
that the user hand-specify the utility model either as a function or string. While this provides
greater flexibility in the types of models that can be estimated, it also requires much more effort by
the user to carefully specify every model. Simple modifications, such as adding in one more variable,
require that the user modify multiple settings, including modifying the starting parameter vector
as well as re-defining the utility model function or string (among other potential required changes).
Even with helpful guides and examples provided by the developers of these packages, more effort is
required by the user to appropriately use these packages to estimate even the simplest of models.
These issues motivated the use of an alternative model specification interface for package logitr,
with the goal of developing a syntax that is at least as intuitive as the formula interface but more
uniform for estimating models in either the preference or WTP space.
5.4. Continuous and discrete variable coding
Variables are modeled in logitr as either continuous or discrete based on their data type. Numeric
variables are modeled with a single “slope” coefficient, and character or factor type variables
are modeled as categorical variables with dummy-coded coefficients for all but the first level, which
serves as the reference level. For example, since the price variable in the yogurt data frame is a
numeric variable, it will be modeled with a single coefficient representing the change in utility for
marginal changes in price. In contrast, since brand is a character type with the levels "dannon",
"hiland","weight", and "yoplait", it will be modeled with three dummy-coded coefficients with
the "dannon" brand set as the reference level as it is alphabetically first.
To change the reference level for discrete variables, modify the factor levels for that variable prior
to model estimation with the factor() function. For example, the following code will set "weight"
instead of "dannon" as the reference level for the brand variable:
R> yogurt2 <- logitr::yogurt
R> yogurt2$brand <- factor(
+ x = yogurt2$brand,
+ levels = c("weight", "hiland", "yoplait", "dannon")
+ )
R> levels(yogurt2$brand)
[1] "weight" "hiland" "yoplait" "dannon"
Any variable can be made discrete by either converting it to a character or factor type prior
to model estimation or by creating new dummy-coded variables for each level and including all
but the reference level as model covariates. A recommended approach is to use the dummy_cols()
function from the fastDummies package, which generates dummy-coded variables for each unique
level of a discrete variable (Kaplan 2020).
5.5. Estimating multinomial logit models
The logitr() function estimates preference space models by default. Once a model is estimated,
the summary() function can be used to print a summary of the estimated model results to the
console. For example, consider again the preference space model with price,feat, and brand as
model covariates:

Journal of Statistical Software 15
R> mnl_pref <- logitr(
+ data = yogurt,
+ outcome = "choice",
+ obsID = "obsID",
+ pars = c("price", "feat", "brand")
+ )
R> summary(mnl_pref)
=================================================
Model estimated on: Tue Oct 04 11:58:55 2022
Using logitr version: 0.8.0
Call:
logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = c("price",
"feat", "brand"))
Frequencies of alternatives:
1234
0.402156 0.029436 0.229270 0.339138
Exit Status: 3, Optimization stopped because ftol_rel or ftol_abs was reached.
Model Type: Multinomial Logit
Model Space: Preference
Model Run: 1 of 1
Iterations: 21
Elapsed Time: 0h:0m:0.02s
Algorithm: NLOPT_LD_LBFGS
Weights Used?: FALSE
Robust? FALSE
Model Coefficients:
Estimate Std. Error z-value Pr(>|z|)
price -0.366555 0.024365 -15.0441 < 2.2e-16 ***
feat 0.491439 0.120062 4.0932 4.254e-05 ***
brandhiland -3.715477 0.145417 -25.5506 < 2.2e-16 ***
brandweight -0.641138 0.054498 -11.7645 < 2.2e-16 ***
brandyoplait 0.734519 0.080642 9.1084 < 2.2e-16 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Log-Likelihood: -2656.8878790
Null Log-Likelihood: -3343.7419990
AIC: 5323.7757580
BIC: 5352.7168000

16 logitr: Preference and WTP Space Logit Models in R
McFadden R2: 0.2054148
Adj McFadden R2: 0.2039195
Number of Observations: 2412.0000000
The summary includes information about the logitr() function call, the frequency of chosen
alternatives, the optimization exit status, the estimated coefficients, the log-likelihood value at the
solution, and several measures of model fit. In this example, three coefficients were estimated for
the “brand” attribute, with "dannon" set as the default reference level.
These results indicate that, all else being equal, people in this sample on average preferred the
“Yoplait” brand the most, followed by “Dannon” (the reference level, which would be 0 on the utility
scale), “Weight Watchers”, and finally “Hiland”. The results also indicate that utility decreased
by a value of -0.367 for every dollar increase in price, which is logically consistent with people
preferring lower rather than higher prices, all else being equal.
Values from an estimated model can be extracted using methods designed for objects of class
logitr, such as the following:
The estimated coefficients:
R> coef(mnl_pref)
price feat brandhiland brandweight brandyoplait
-0.3665546 0.4914392 -3.7154773 -0.6411384 0.7345195
The coefficient standard errors:
R> se(mnl_pref)
price feat brandhiland brandweight brandyoplait
0.02436526 0.12006175 0.14541671 0.05449794 0.08064229
The log-likelihood:
R> logLik(mnl_pref)
'log Lik.'-2656.888 (df=5)
The variance-covariance matrix:
R> vcov(mnl_pref)
price feat brandhiland brandweight
price 0.0005936657 5.729619e-04 0.001851795 1.249988e-04
feat 0.0005729619 1.441482e-02 0.000855011 5.092011e-06
brandhiland 0.0018517954 8.550110e-04 0.021146019 1.490080e-03
brandweight 0.0001249988 5.092011e-06 0.001490080 2.970026e-03
brandyoplait -0.0015377721 -1.821331e-03 -0.003681036 7.779428e-04
brandyoplait

Journal of Statistical Software 17
price -0.0015377721
feat -0.0018213311
brandhiland -0.0036810363
brandweight 0.0007779427
brandyoplait 0.0065031782
5.6. Estimating willingness to pay
Coefficients in preference space models reflect marginal changes in utility, which only have relative
value. To make these coefficients more interpretable, modelers often divide the non-price parameters
by the negative of the price parameter to obtain estimates of WTP (the negative of the price
parameter is used so that marginal WTPs have a positive interpretation). This can be computed
using the wtp() function:
R> wtp(mnl_pref, scalePar = "price")
Estimate Std. Error z-value Pr(>|z|)
scalePar 0.366555 0.024378 15.0361 < 2.2e-16 ***
feat 1.340699 0.360539 3.7186 0.0002003 ***
brandhiland -10.136219 0.583206 -17.3802 < 2.2e-16 ***
brandweight -1.749094 0.181960 -9.6125 < 2.2e-16 ***
brandyoplait 2.003848 0.143323 13.9813 < 2.2e-16 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
The wtp() function returns a data frame of the WTP estimates and standard errors. The coefficient
labeled scalePar is the negative of the price coefficient from the preference space model, which
can also be interpreted as the scale coefficient in the WTP space model as it scales all the WTP
coefficients. Standard errors are estimated using the Krinsky and Robb parametric bootstrapping
method (Krinsky and Robb 1986).
In contrast to the preference space model coefficients, the WTP estimates above have units of
currency (in this case $US dollars) and can be interpreted as how much the average person in
the sample would be willing to pay for each feature, all else being equal. For example, the brand
coefficients suggest that, relative to the “Dannon” brand, consumers are on average willing to pay
an additional $2.00 for the “Yoplait” brand, -$1.75 for the “Weight Watchers” brand, and -$10.14
for the “Hiland” brand (negative WTPs indicate a relative preference for “Dannon”).
WTPs can also be directly estimated using a WTP space model. In this case, the pars argument
should contain only attributes for which WTPs are to be estimate. The variable for “price” should
be provided separately using the scalePar argument. For example, consider again the WTP space
model with feat and brand WTP covariates:

18 logitr: Preference and WTP Space Logit Models in R
R> set.seed(123)
R>
R> mnl_wtp <- logitr(
+ data = yogurt,
+ outcome = "choice",
+ obsID = "obsID",
+ pars = c("feat", "brand"),
+ scalePar = "price",
+ numMultiStarts = 10,
+ numCores = 1
+ )
In the above example, a 10-iteration multi-start optimization loop was implemented by setting
numMultiStarts = 10. This runs the minimization of the negative log-likelihood function 10 times
from 10 different sets of random starting points (the first iteration uses all 0s except for the price
parameter which starts at 1). This is recommended as WTP space models have a non-convex log-
likelihood function and thus could have multiple local minimia. Note also that the multi-start loop
can be parallelized by setting numCores to an integer greater than 1. The default value is one less
than the total number of available cores, but numCores = 1 is used here to ensure reproducibility.
The WTP estimates have the same interpretation as those computed from the preference space
model. In the summary output, a short summary of each iteration of the multi-start loop is provided
first followed by the same summary information about the preference space model. Because a multi-
start loop was used, only the summary of the “best” estimated model is returned (determined by
the iteration with the largest log-likelihood value):
R> summary(mnl_wtp)
=================================================
Model estimated on: Tue Oct 04 11:58:55 2022
Using logitr version: 0.8.0
Call:
logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = c("feat",
"brand"), scalePar = "price", numMultiStarts = 10, numCores = 1)
Frequencies of alternatives:
1234
0.402156 0.029436 0.229270 0.339138
Summary Of Multistart Runs:
Log Likelihood Iterations Exit Status
1 -2656.888 38 3
2 -2656.888 52 3
3 -2656.888 59 3
4 -2656.888 40 3

Journal of Statistical Software 19
5 -2656.888 44 3
6 -2656.888 44 3
7 -2656.888 63 3
8 -2656.888 36 3
9 -2656.888 40 3
10 -2656.888 45 3
Use statusCodes() to view the meaning of each status code
Exit Status: 3, Optimization stopped because ftol_rel or ftol_abs was reached.
Model Type: Multinomial Logit
Model Space: Willingness-to-Pay
Model Run: 1 of 10
Iterations: 38
Elapsed Time: 0h:0m:0.04s
Algorithm: NLOPT_LD_LBFGS
Weights Used?: FALSE
Robust? FALSE
Model Coefficients:
Estimate Std. Error z-value Pr(>|z|)
scalePar 0.366583 0.024366 15.0448 < 2.2e-16 ***
feat 1.340593 0.355867 3.7671 0.0001651 ***
brandhiland -10.135764 0.576089 -17.5941 < 2.2e-16 ***
brandweight -1.749083 0.179898 -9.7226 < 2.2e-16 ***
brandyoplait 2.003821 0.142377 14.0740 < 2.2e-16 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Log-Likelihood: -2656.8878779
Null Log-Likelihood: -3343.7419990
AIC: 5323.7757559
BIC: 5352.7168000
McFadden R2: 0.2054148
Adj McFadden R2: 0.2039195
Number of Observations: 2412.0000000
In the above summary, iteration 1 converged to a solution with a log-likelihood value of -2656.888.
While all of the iterations arrived at the same solution in this particular example, this is not always
the case nor is it guaranteed. Because the previous examples are both fixed parameter models, the
WTP estimates from the WTP space model are identical to those computed from the preference
space model. The WTPs from each model can be quickly compared using the wtpCompare()
function:

20 logitr: Preference and WTP Space Logit Models in R
R> wtpCompare(
+ model_pref = mnl_pref,
+ model_wtp = mnl_wtp,
+ scalePar = "price"
+ )
pref wtp difference
scalePar 0.3665546 0.3665832 0.00002867
feat 1.3406987 1.3405926 -0.00010605
brandhiland -10.1362190 -10.1357635 0.00045548
brandweight -1.7490940 -1.7490826 0.00001133
brandyoplait 2.0038476 2.0038208 -0.00002686
logLik -2656.8878790 -2656.8878779 0.00000106
In the above summary, the pref column contains the computed WTPs from the preference space
model, and the wtp column contains the directly estimated WTPs from the WTP space model. The
difference column is the computed difference between them. This is a helpful tool for assessing
whether the two models converged to the same solution.
5.7. Estimating mixed logit models
The mixed logit model is a popular approach for modeling unobserved heterogeneity across in-
dividuals, which is implemented by assuming that parameters vary randomly across individuals
according to a chosen distribution (McFadden and Train 2000). A mixed logit model is specified
by setting the randPars argument in the logitr() function equal to a named vector defining
parameter distributions. The current package version (0.8.0) supports the following distributions:
•Normal: "n"
•Log-normal: "ln"
•Zero-censored normal: "cn"
Mixed logit models will estimate a mean and standard deviation of the underlying normal distri-
bution for each random coefficient. Note that log-normal or zero-censored normal parameters force
positivity, so when using these it is often necessary to use the negative of a value (e.g. for “price”,
which typically has a negative coefficient). Mixed logit models in logitr assume uncorrelated hetero-
geneity covariances by default, though full covariances can be estimated using the correlation =
TRUE argument. For WTP space models, the scalePar parameter can also be modeled as following
a random distribution by setting the randScale argument equal to "n","ln", or "cn".
The code below is an example of a mixed logit model where the observed utility for yogurts is
vj=αpj+β1xj1+β2xj2+β3xj3+β4xj4, where pjis price,xj1is feat, and xj2−4are dummy-
coded variables for brand. To model feat as well as each of the brands as normally-distributed,
set randPars = c(feat = "n", brand = "n"). Since mixed logit models have a non-convex log-
likelihood function, it is recommended to use a multi-start search to run the optimization multiple
times from different random starting points. Mixed logit models typically take longer to estimate
than fixed parameter models, so setting a larger number for numMultiStarts could take several
minutes to complete.
Note that since the yogurt data has a panel structure (i.e. multiple choice observations for each
respondent), it is necessary to set the panelID argument to the id variable, which identifies the

Journal of Statistical Software 21
individual. This will use the panel version of the log-likelihood (see (Train 2009) section 6.7 for
details).
R> set.seed(456)
R>
R> mxl_pref <- logitr(
+ data = yogurt,
+ outcome = 'choice',
+ obsID = 'obsID',
+ panelID = 'id',
+ pars = c('price','feat','brand'),
+ randPars = c(feat = 'n', brand = 'n'),
+ numMultiStarts = 10,
+ numCores = 1
+ )
R> summary(mxl_pref)
=================================================
Model estimated on: Tue Oct 04 11:58:57 2022
Using logitr version: 0.8.0
Call:
logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = c("price",
"feat", "brand"), randPars = c(feat = "n", brand = "n"),
panelID = "id", numMultiStarts = 10, numCores = 1)
Frequencies of alternatives:
1234
0.402156 0.029436 0.229270 0.339138
Summary Of Multistart Runs:
Log Likelihood Iterations Exit Status
1 -1266.550 34 3
2 -1300.751 64 3
3 -1260.216 35 3
4 -1261.216 43 3
5 -1269.066 40 3
6 -1239.294 56 3
7 -1343.221 59 3
8 -1260.006 55 3
9 -1273.143 52 3
10 -1304.384 59 3
Use statusCodes() to view the meaning of each status code

22 logitr: Preference and WTP Space Logit Models in R
Exit Status: 3, Optimization stopped because ftol_rel or ftol_abs was reached.
Model Type: Mixed Logit
Model Space: Preference
Model Run: 6 of 10
Iterations: 56
Elapsed Time: 0h:0m:2s
Algorithm: NLOPT_LD_LBFGS
Weights Used?: FALSE
Panel ID: id
Robust? FALSE
Model Coefficients:
Estimate Std. Error z-value Pr(>|z|)
price -0.448338 0.039987 -11.2120 < 2.2e-16 ***
feat 0.776990 0.193521 4.0150 5.944e-05 ***
brandhiland -6.367360 0.520828 -12.2255 < 2.2e-16 ***
brandweight -3.668683 0.307207 -11.9421 < 2.2e-16 ***
brandyoplait 1.122492 0.203483 5.5164 3.460e-08 ***
sd_feat 0.567495 0.225004 2.5222 0.01166 *
sd_brandhiland -3.181844 0.371697 -8.5603 < 2.2e-16 ***
sd_brandweight 4.097130 0.232495 17.6225 < 2.2e-16 ***
sd_brandyoplait 3.261281 0.219902 14.8306 < 2.2e-16 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Log-Likelihood: -1239.2944250
Null Log-Likelihood: -3343.7419990
AIC: 2496.5888500
BIC: 2548.6828000
McFadden R2: 0.6293690
Adj McFadden R2: 0.6266774
Number of Observations: 2412.0000000
Summary of 10k Draws for Random Coefficients:
Min. 1st Qu. Median Mean 3rd Qu. Max.
feat -Inf 0.3938347 0.7765564 0.7761956 1.1591475 Inf
brandhiland -Inf -8.5118797 -6.3663393 -6.3644102 -4.2201174 Inf
brandweight -Inf -6.4342648 -3.6720435 -3.6750045 -0.9090452 Inf
brandyoplait -Inf -1.0817673 1.1169084 1.1141118 3.3162383 Inf
Since the feat and brand attributes were modeled as normally distributed across the population,
each of these covariates have two parameters that describe the mean and standard deviation of a
normal distribution. For example, the coefficients brandyoplait and sd_brandyoplait indicate
that, relative to the “Dannon” brand, the marginal utility for the “Yoplait” brand follows a normal
distribution across the population where β1∼ N (µ= 1.122, σ = 3.261). For mixed logit models, a
summary of all random parameter distributions is printed at the bottom of the summary output. In
this example, there appears to be less heterogeneity in preferences for the “Yoplait” and “Hiland”

Journal of Statistical Software 23
brands compared to the “Weight Watchers” brand, which has a larger standard deviation parameter.
Note that mixed logit models can sometimes produce negative values for the standard deviation
parameters; in these cases, the parameters should be interpreted as positive. The negative values are
an artifact of how the simulated MLE algorithm works. Since the normal distribution is symmetric,
it does not matter if draws are generated with µ+σZ or µ−σZ , where Zis a standard normal
and µand σare the mean and standard deviation parameters. By allowing the standard deviation
parameter to be negative, the optimization is unconstrained, making it a much easier problem to
solve.
5.8. Estimating WTP space mixed logit models
WTP space mixed logit models have the advantage of being able to directly specify the assumed
distribution of WTP across the population. As with fixed parameter models, estimating a mixed
logit WTP space model requires that price be separately specified with the scalePar argument.
The randPars argument is used to specify random parameters. For example, the following can be
used to estimate a model where the WTP distributions for feat and each brand is assumed to be
normally distributed:
R> set.seed(6789)
R>
R> mxl_wtp <- logitr(
+ data = yogurt,
+ outcome = "choice",
+ obsID = "obsID",
+ panelID = 'id',
+ pars = c("feat", "brand"),
+ scalePar = "price",
+ randPars = c(feat = "n", brand = "n"),
+ numMultiStarts = 10,
+ numCores = 1
+ )
R> summary(mxl_wtp)
=================================================
Model estimated on: Tue Oct 04 11:59:16 2022
Using logitr version: 0.8.0
Call:
logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = c("feat",
"brand"), scalePar = "price", randPars = c(feat = "n", brand = "n"),
panelID = "id", numMultiStarts = 10, numCores = 1)
Frequencies of alternatives:
1234
0.402156 0.029436 0.229270 0.339138

24 logitr: Preference and WTP Space Logit Models in R
Summary Of Multistart Runs:
Log Likelihood Iterations Exit Status
1 -1256.886 110 3
2 -1252.536 76 3
3 -1258.974 87 3
4 -1342.036 111 4
5 -1250.922 111 3
6 -1266.990 66 3
7 -1268.352 81 3
8 -1239.294 77 3
9 -1258.974 60 3
10 -1239.294 51 3
Use statusCodes() to view the meaning of each status code
Exit Status: 3, Optimization stopped because ftol_rel or ftol_abs was reached.
Model Type: Mixed Logit
Model Space: Willingness-to-Pay
Model Run: 8 of 10
Iterations: 77
Elapsed Time: 0h:0m:3s
Algorithm: NLOPT_LD_LBFGS
Weights Used?: FALSE
Panel ID: id
Robust? FALSE
Model Coefficients:
Estimate Std. Error z-value Pr(>|z|)
scalePar 0.448563 0.039982 11.2191 < 2.2e-16 ***
feat 1.731133 0.491792 3.5201 0.0004315 ***
brandhiland -14.223308 1.365310 -10.4176 < 2.2e-16 ***
brandweight -8.172665 0.955928 -8.5495 < 2.2e-16 ***
brandyoplait 2.503597 0.407192 6.1484 7.825e-10 ***
sd_feat 1.266802 0.497472 2.5465 0.0108816 *
sd_brandhiland -7.114726 0.944233 -7.5349 4.885e-14 ***
sd_brandweight 9.130682 0.923411 9.8880 < 2.2e-16 ***
sd_brandyoplait 7.270250 0.752617 9.6600 < 2.2e-16 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Log-Likelihood: -1239.2939746
Null Log-Likelihood: -3343.7419990
AIC: 2496.5879491
BIC: 2548.6819000
McFadden R2: 0.6293691
Adj McFadden R2: 0.6266775

Journal of Statistical Software 25
Number of Observations: 2412.0000000
Summary of 10k Draws for Random Coefficients:
Min. 1st Qu. Median Mean 3rd Qu. Max.
feat -Inf 0.8758265 1.730165 1.729359 2.584211 Inf
brandhiland -Inf -19.0185364 -14.221025 -14.216712 -9.421990 Inf
brandweight -Inf -14.3359189 -8.180155 -8.186754 -2.022659 Inf
brandyoplait -Inf -2.4102743 2.491150 2.484915 7.394032 Inf
The summary shows the solution for iteration 8 out of the 10 multi-starts—the one with the
largest log-likelihood value. Since the feat and brand WTPs were both modeled as normally
distributed across the population, each of these covariates have two parameters that describe the
mean and standard deviation of a normal distribution. The results again suggest that there is
greater heterogeneity for the “Weight Watchers” brand compared to “Yoplait” and “Hiland”, which
can be seen with its larger standard deviation coefficient and wider WTP range in the distribution
summary at the bottom of the summary output. In fact, these results indicate that although the
mean WTP for the “Weight Watchers” brand is still higher than that of the “Hiland” brand, the
heterogeneity in WTP spans a much wider range.
In this case, since the price parameter was modeled as a fixed parameter, the WTP estimates from
the preference space and those from the WTP space model are nearly identical:
R> wtpCompare(
+ model_pref = mxl_pref,
+ model_wtp = mxl_wtp,
+ scalePar = "price"
+ )
pref wtp difference
scalePar 0.4483378 0.4485634 0.00022557
feat 1.7330459 1.7311326 -0.00191335
brandhiland -14.2021478 -14.2233082 -0.02116033
brandweight -8.1828533 -8.1726652 0.01018805
brandyoplait 2.5036744 2.5035970 -0.00007740
sd_feat 1.2657757 1.2668022 0.00102646
sd_brandhiland -7.0969787 -7.1147261 -0.01774740
sd_brandweight 9.1384873 9.1306816 -0.00780576
sd_brandyoplait 7.2741604 7.2702504 -0.00391001
logLik -1239.2944250 -1239.2939746 0.00045043
If, however, the price parameter in either model were modeled as a random parameter (which
is controlled via the randScale argument), the resulting WTP estimates could be substantially
different.
5.9. Weighted models
Sometimes the modeler may wish to differentially weight individual choice observations in model
estimation. For example, if a particular group was over- or under-represented in a sample relative
to that of a target population, the choice observations of that group could be weighted such that

26 logitr: Preference and WTP Space Logit Models in R
they have a stronger or weaker contribution to the log-likelihood in an attempt to balance the
sample to match the proportions of the target population.
The cars_us data set that comes with the package includes a weights column and is useful for
illustrating how to estimate weighted models. This data set contains 384 stated choice observations
from a conjoint survey of U.S. car buyers fielded online using Amazon Mechanical Turk in 2012 and
in person at the 2013 Pittsburgh Auto show (Helveston, Liu, Feit, Fuchs, Klampfl, and Michalek
2015). Participants were asked to select a vehicle from a set of three alternatives, and each partic-
ipant answered 15 choice questions. The data set contains variables for different types of vehicles
("hev","phev10","phev20","phev40","bev75","bev100","bev150"), different brands repre-
sented by the country of origin ("american","japanese","chinese","skorean"), fast charging
options ("phevFastcharge" and "bevFastcharge"), price ("price"), operating cost ("opCost"),
and 0-60 mph acceleration time ("accelTime").
To compare the impact of the weights on the estimated parameters, an unweighted and weighted
model are estimated, both in the WTP space to replicate the models estimated in Helveston et
al. (2015). In both models, the argument robust = TRUE clusters the standard errors using the
obsID variable for clustering, which should be done for weighted models. Standard errors can also
be clustered at other levels by specifying a clusterID variable. For example, a common desired
clustering is to cluster at the individual level in conjoint studies where survey respondents answer
multiple sequential choice questions to account for potential correlations among these questions.
The unweighted model is estimated with the following code:
R> set.seed(5678)
R>
R> mnl_wtp_unweighted <- logitr(
+ data = cars_us,
+ outcome = "choice",
+ obsID = "obsnum",
+ pars = c(
+ "hev", "phev10", "phev20", "phev40", "bev75", "bev100", "bev150",
+ "american", "japanese", "chinese", "skorean", "phevFastcharge",
+ "bevFastcharge","opCost", "accelTime"),
+ scalePar = "price",
+ robust = TRUE,
+ numMultiStarts = 10,
+ numCores = 1
+ )
R> summary(mnl_wtp_unweighted)
=================================================
Model estimated on: Tue Oct 04 11:59:52 2022
Using logitr version: 0.8.0
Call:
logitr(data = cars_us, outcome = "choice", obsID = "obsnum",

Journal of Statistical Software 27
pars = c("hev", "phev10", "phev20", "phev40", "bev75", "bev100",
"bev150", "american", "japanese", "chinese", "skorean",
"phevFastcharge", "bevFastcharge", "opCost", "accelTime"),
scalePar = "price", robust = TRUE, numMultiStarts = 10, numCores = 1)
Frequencies of alternatives:
123
0.34323 0.33507 0.32170
Summary Of Multistart Runs:
Log Likelihood Iterations Exit Status
1 -4616.952 26 3
2 -4616.955 31 3
3 -4616.952 45 3
4 -4616.952 35 3
5 -4616.952 34 3
6 -4616.952 36 3
7 -4616.952 34 3
8 -4616.952 33 3
9 -4616.952 34 3
10 -4616.952 32 3
Use statusCodes() to view the meaning of each status code
Exit Status: 3, Optimization stopped because ftol_rel or ftol_abs was reached.
Model Type: Multinomial Logit
Model Space: Willingness-to-Pay
Model Run: 8 of 10
Iterations: 33
Elapsed Time: 0h:0m:0.1s
Algorithm: NLOPT_LD_LBFGS
Weights Used?: FALSE
Cluster ID: obsnum
Robust? TRUE
Model Coefficients:
Estimate Std. Error z-value Pr(>|z|)
scalePar 0.0738787 0.0021929 33.6900 < 2.2e-16 ***
hev 0.8072448 0.9990581 0.8080 0.4190872
phev10 1.1658652 1.0614987 1.0983 0.2720648
phev20 1.6478081 1.0617443 1.5520 0.1206665
phev40 2.5794026 1.0499274 2.4567 0.0140203 *
bev75 -16.0458795 1.2541265 -12.7945 < 2.2e-16 ***
bev100 -13.0031631 1.2388544 -10.4961 < 2.2e-16 ***
bev150 -9.5733561 1.1641772 -8.2233 2.220e-16 ***
american 2.3442854 0.7979689 2.9378 0.0033053 **
japanese -0.3747714 0.7998315 -0.4686 0.6393821

28 logitr: Preference and WTP Space Logit Models in R
chinese -10.2685448 0.8859347 -11.5906 < 2.2e-16 ***
skorean -6.0311955 0.8514340 -7.0836 1.405e-12 ***
phevFastcharge 2.8793913 0.8028804 3.5863 0.0003354 ***
bevFastcharge 2.9184681 0.9181323 3.1787 0.0014794 **
opCost -1.6360487 0.0686313 -23.8382 < 2.2e-16 ***
accelTime -1.6970364 0.1638091 -10.3598 < 2.2e-16 ***
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Log-Likelihood: -4616.9517800
Null Log-Likelihood: -6328.0067827
AIC: 9265.9035600
BIC: 9372.4426000
McFadden R2: 0.2703940
Adj McFadden R2: 0.2678655
Number of Observations: 5760.0000000
Number of Clusters 5760.0000000
The estimated WTP coefficients have units of $1,000. The results indicate large negative WTP
values for the three full electric vehicle types: -$16,000 for bev75, -$13,000 for bev100, and -$9,600
for bev150 relative to conventional gasoline vehicles (the numbers in each bev type indicate driving
ranges in miles on a full charge). There also appear to be strong brand preferences, with WTP
values ranging as much as -$10,300 for Chinese brands to $2,300 for American brands relative to
German brands.
To estimate a weighted model, the argument weights = "weights" is added in the logitr()
function call. This sets the weights column in the cars_us data frame to be used to weight
each choice observation. The weights are specific to each individual survey respondent and were
calculated to account for over-sampling of younger and less-wealthy car buyers (Helveston et al.
2015). In this example, the weights have values ranging from 0.2 to 5, meaning some choice
observations could have as much as 25 times the weight of others in contributing to the log-
likelihood.
R> set.seed(5678)
R>
R> mnl_wtp_weighted <- logitr(
+ data = cars_us,
+ outcome = "choice",
+ obsID = "obsnum",
+ pars = c(
+ "hev", "phev10", "phev20", "phev40", "bev75", "bev100", "bev150",
+ "american", "japanese", "chinese", "skorean", "phevFastcharge",
+ "bevFastcharge","opCost", "accelTime"),
+ scalePar = "price",
+ weights = "weights",
+ robust = TRUE,
+ numMultiStarts = 10,
+ numCores = 1
+ )

Journal of Statistical Software 29
R> summary(mnl_wtp_weighted)
=================================================
Model estimated on: Tue Oct 04 11:59:55 2022
Using logitr version: 0.8.0
Call:
logitr(data = cars_us, outcome = "choice", obsID = "obsnum",
pars = c("hev", "phev10", "phev20", "phev40", "bev75", "bev100",
"bev150", "american", "japanese", "chinese", "skorean",
"phevFastcharge", "bevFastcharge", "opCost", "accelTime"),
scalePar = "price", weights = "weights", robust = TRUE, numMultiStarts = 10,
numCores = 1)
Frequencies of alternatives:
123
0.34323 0.33507 0.32170
Summary Of Multistart Runs:
Log Likelihood Iterations Exit Status
1 -3425.633 19 3
2 -3425.630 33 3
3 -3425.631 37 3
4 -3425.630 30 3
5 -3425.633 36 3
6 -3425.631 34 3
7 -3425.630 31 3
8 -3425.630 29 3
9 -3425.631 31 3
10 -3425.630 29 3
Use statusCodes() to view the meaning of each status code
Exit Status: 3, Optimization stopped because ftol_rel or ftol_abs was reached.
Model Type: Multinomial Logit
Model Space: Willingness-to-Pay
Model Run: 10 of 10
Iterations: 29
Elapsed Time: 0h:0m:0.09s
Algorithm: NLOPT_LD_LBFGS
Weights Used?: TRUE
Cluster ID: obsnum
Robust? TRUE
Model Coefficients:

30 logitr: Preference and WTP Space Logit Models in R
Estimate Std. Error z-value Pr(>|z|)
scalePar 0.0522802 0.0040688 12.8489 < 2.2e-16 ***
hev -1.1745214 2.9133014 -0.4032 0.6868318
phev10 0.0275518 3.1280284 0.0088 0.9929723
phev20 1.6949071 3.0997221 0.5468 0.5845208
phev40 2.6494989 2.9851858 0.8875 0.3747834
bev75 -20.1362768 3.6671641 -5.4910 3.997e-08 ***
bev100 -19.4967470 3.6256286 -5.3775 7.554e-08 ***
bev150 -13.6909374 3.4926845 -3.9199 8.859e-05 ***
american 8.1877347 2.4052979 3.4040 0.0006640 ***
japanese 0.9337835 2.3603628 0.3956 0.6923927
chinese -19.0068520 2.8539795 -6.6598 2.743e-11 ***
skorean -9.5109238 2.5234809 -3.7690 0.0001639 ***
phevFastcharge 3.9438186 2.3624185 1.6694 0.0950384 .
bevFastcharge 3.3428976 2.8087011 1.1902 0.2339704
opCost -1.5975429 0.1948476 -8.1989 2.220e-16 ***
accelTime -1.1719313 0.4834735 -2.4240 0.0153513 *
---
Signif. codes: 0 '***'0.001 '**'0.01 '*'0.05 '.'0.1 ' ' 1
Log-Likelihood: -3425.6302862
Null Log-Likelihood: -4360.5909275
AIC: 6883.2605723
BIC: 6989.7997000
McFadden R2: 0.2144115
Adj McFadden R2: 0.2107422
Number of Observations: 5760.0000000
Number of Clusters 5760.0000000
With both models estimated, it is helpful to directly compare the estimated coefficients side-by-side:
R> data.frame(
+ Unweighted = coef(mnl_wtp_unweighted),
+ Weighted = coef(mnl_wtp_weighted)
+ )
Unweighted Weighted
scalePar 0.07387865 0.05228019
hev 0.80724480 -1.17452143
phev10 1.16586524 0.02755184
phev20 1.64780809 1.69490706
phev40 2.57940264 2.64949894
bev75 -16.04587947 -20.13627677
bev100 -13.00316310 -19.49674699
bev150 -9.57335615 -13.69093743
american 2.34428544 8.18773467
japanese -0.37477137 0.93378346
chinese -10.26854481 -19.00685205

Journal of Statistical Software 31
skorean -6.03119552 -9.51092383
phevFastcharge 2.87939127 3.94381855
bevFastcharge 2.91846813 3.34289759
opCost -1.63604869 -1.59754290
accelTime -1.69703637 -1.17193131
From this comparison, it is clear that in the weighted model the negative WTP for full electric
vehicles is slightly larger than that in the unweighted model, and the range of WTP for each brand
also increased in the weighted model. Nonetheless, all of the statistically significant coefficients
maintained the same sign and significance.
5.10. Predicting probabilities
Once a model has been estimated, it can be used to predict probabilities, outcomes, or both for a
set of alternatives using the predict() method. Predictions can be made for any set of alternatives
so long as the columns in the alternatives correspond to estimated coefficients in the model. By
default, if no new data are provided via the newdata argument, then predictions will be made for
the original data used to estimate the model.
Predictions can be made using both preference space and WTP space models, as well as multinomial
logit and mixed logit models. For mixed logit models, heterogeneity is modeled by simulating draws
from the population estimates of the estimated model. In the example below, the preference space
MNL model from Section 5.5 (mnl_pref) is used to predict probabilities for the data used to
estimate the model:
R> probs <- predict(mnl_pref)
R> head(probs)
obsID predicted_prob
1 1 0.41802407
2 1 0.02118240
3 1 0.23691737
4 1 0.32387615
5 2 0.26643822
6 2 0.02255486
The predict() method returns a data frame containing the observation ID as well as the predicted
probabilities. The original data can also be returned in the data frame by setting returnData =
TRUE:
R> probs <- predict(mnl_pref, returnData = TRUE)
R> head(probs)
obsID predicted_prob price feat brandhiland brandweight brandyoplait choice
1 1 0.41802407 8.1 0 0 0 0 0
2 1 0.02118240 6.1 0 1 0 0 0
3 1 0.23691737 7.9 0 0 1 0 1
4 1 0.32387615 10.8 0 0 0 1 0
5 2 0.26643822 9.8 0 0 0 0 1
6 2 0.02255486 6.4 0 1 0 0 0

32 logitr: Preference and WTP Space Logit Models in R
To make predictions for a new set of alternatives, use the newdata argument. The example below
makes predictions for just two of the choice observations from the yogurt dataset:
R> data <- subset(
+ yogurt, obsID %in% c(42, 13),
+ select = c('obsID','alt','price','feat','brand'))
R>
R> probs_mnl_pref <- predict(
+ mnl_pref,
+ newdata = data,
+ obsID = "obsID",
+ )
R>
R> probs_mnl_pref
obsID predicted_prob
1 13 0.43685145
2 13 0.03312986
3 13 0.19155548
4 13 0.33846321
5 42 0.60764778
6 42 0.02602007
7 42 0.17803313
8 42 0.18829902
The ci argument can be used to obtain upper and lower bounds of a confidence interval for predicted
probabilities, which are estimated using the Krinsky and Robb parametric bootstrapping method
(Krinsky and Robb 1986). For example, a 95% CI is obtained with ci = 0.95:
R> set.seed(5678)
R>
R> probs_mnl_pref <- predict(
+ mnl_pref,
+ newdata = data,
+ obsID = "obsID",
+ ci = 0.95
+ )
R>
R> probs_mnl_pref
obsID predicted_prob predicted_prob_lower predicted_prob_upper
1 13 0.43685145 0.41629325 0.45825284
2 13 0.03312986 0.02608194 0.04111778
3 13 0.19155548 0.17572618 0.20815699
4 13 0.33846321 0.31873136 0.35843950
5 42 0.60764778 0.57346649 0.64162051
6 42 0.02602007 0.01872744 0.03598088
7 42 0.17803313 0.16138477 0.19504384
8 42 0.18829902 0.16787563 0.20791429

Journal of Statistical Software 33
WTP space models can also be used to predict probabilities. In the example below, the WTP space
MNL model from Section 5.6 (mnl_wtp) is used to predict probabilities for the data object defined
above:
R> set.seed(5678)
R>
R> probs_mnl_wtp <- predict(
+ mnl_wtp,
+ newdata = data,
+ obsID = "obsID",
+ ci = 0.95
+ )
R>
R> probs_mnl_wtp
obsID predicted_prob predicted_prob_lower predicted_prob_upper
1 13 0.43686141 0.41508522 0.45701654
2 13 0.03312947 0.02681696 0.04280579
3 13 0.19154829 0.17721984 0.20777595
4 13 0.33846083 0.31813769 0.35837404
5 42 0.60767120 0.57441503 0.64107061
6 42 0.02601800 0.01840558 0.03636293
7 42 0.17802363 0.16340562 0.19459951
8 42 0.18828717 0.16671672 0.20890971
5.11. Predicting outcomes
The predict() method can also be used to predict outcomes by setting type = "outcome" (the
default is "prob" for predicting probabilities). In the examples below, outcomes are predicted using
the same preference space and WTP space models as in the previous examples. The returnData
argument is also set to TRUE so that the predicted outcomes can be compared to the actual choices
made:
R> set.seed(5678)
R>
R> outcomes_pref <- predict(
+ mnl_pref,
+ type = "outcome",
+ returnData = TRUE
+ )
R> head(outcomes_pref)
obsID predicted_outcome price feat brandhiland brandweight brandyoplait
1 1 1 8.1 0 0 0 0
2 1 0 6.1 0 1 0 0
3 1 0 7.9 0 0 1 0

34 logitr: Preference and WTP Space Logit Models in R
4 1 0 10.8 0 0 0 1
5 2 0 9.8 0 0 0 0
6 2 0 6.4 0 1 0 0
choice
1 0
2 0
3 1
4 0
5 1
6 0
R> set.seed(5678)
R>
R> outcomes_wtp <- predict(
+ mnl_wtp,
+ type = "outcome",
+ returnData = TRUE
+ )
R> head(outcomes_wtp)
obsID predicted_outcome feat brandhiland brandweight brandyoplait scalePar
1 1 1 0 0 0 0 8.1
2 1 0 0 1 0 0 6.1
3 1 0 0 0 1 0 7.9
4 1 0 0 0 0 1 10.8
5 2 0 0 0 0 0 9.8
6 2 0 0 1 0 0 6.4
choice
1 0
2 0
3 1
4 0
5 1
6 0
The accuracy of each model can be computed by dividing the number of correctly predicted choices
by the total number of choices:
R> chosen_pref <- subset(outcomes_pref, choice == 1)
R> chosen_pref$correct <- chosen_pref$choice == chosen_pref$predicted_outcome
R> accuracy_pref <- sum(chosen_pref$correct) / nrow(chosen_pref)
R> accuracy_pref
[1] 0.3706468
R> chosen_wtp <- subset(outcomes_wtp, choice == 1)
R> chosen_wtp$correct <- chosen_wtp$choice == chosen_wtp$predicted_outcome

Journal of Statistical Software 35
R> accuracy_wtp <- sum(chosen_wtp$correct) / nrow(chosen_wtp)
R> accuracy_wtp
[1] 0.3706468
These results show that both models correctly predicted choice for approximately 37% of the
observations in the yogurt data frame, which is significantly better than random (25%).
5.12. Additional options
The logitr() function contains many other arguments for controlling different aspects of the model
specification and estimation procedure. For example, the estimated solution for mixed logit models
can sometimes be sensitive to the resolution of the simulated random parameter distributions. By
default, 50 Halton draws are used, but this can be increased using the numDraws argument. The
user can also use Sobol draws by specifying drawType = "sobol" (defaults to "halton"), which is
recommended in models with a larger number of random parameters (Czajkowski and Budziński
2019).
Details of the optimization procedure can be controlled via the options argument, which must
be a named list of control options. For example, the optimization tolerance levels can be con-
trolled by changing the values for xtol_rel,xtol_abs,ftol_rel, and ftol_abs. The function
nloptr::nloptr.print.options() prints details on these control settings to the console.
Finally, it can be helpful to provide a custom set of starting values for models that have trouble
converging. For WTP space models in particular, one strategy is to first compute the WTP from
a preference space model and then use those results as the starting values. For example, using the
mnl_pref model estimated in Section 5.5, we can first compute the corresponding WTP using the
wtp() function:
R> wtp_est <- wtp(mnl_pref, scalePar = "price")$Estimate
The computed wtp_est vector can then be passed to the startVals argument for the logitr()
function. If a multi-start is used, the user-provided starting values will only be used for the first
iteration.
R> set.seed(5678)
R>
R> mnl_wtp2 <- logitr(
+ data = yogurt,
+ outcome = 'choice',
+ obsID = 'obsID',
+ pars = c('feat','brand'),
+ scalePar = 'price',
+ startVals = wtp_est,
+ numMultiStarts = 10,
+ numCores = 1
+ )

36 logitr: Preference and WTP Space Logit Models in R
6. Limitations of WTP space models
Although package logitr was designed to simplify the estimation of WTP space models, these
models do have several important limitations. First, since only one scale parameter is estimated
in a WTP space model, the true WTP could be over- or under-estimated if there are multiple
latent classes in the sample that each have different sensitivities to price. In contrast, in preference
space models interactions can be used to estimate these differences in price sensitivities for different
groups in a given sample, which could then be used to compute WTP for each group. While one
could estimate separate WTP space models on each latent class to account for differences in price
sensitivities by group, this requires that every model parameter be separately estimated across each
group, which is a stricter assumption. A latent class implementation of WTP space models has not
yet been explored.
In addition, given the non-linear utility specification of WTP space models, these models can often
diverge during estimation and can be highly sensitive to starting parameters. Models in which the
scale parameter is modeled as a random parameter in particular tend to diverge more often during
estimation. For example, if the scale parameter is assumed to be log-normally distributed to force
positivity (which can be done by setting randScale = "ln" in the logitr() function), the model
may not consistently converge on a solution as the draws from this distribution can sometimes have
extremely large values that could have dramatic effects on the optimization search.
Even so, {logitr} tends to perform better and converge more often compared to many other packages
that support WTP space models. By using the parallelized multi-start optimization loop, the
package can efficiently search for different local minima from different random starting points when
minimizing the negative log-likelihood, improving the chances of converging to a solution for at
least some of the multi-start iterations. For more details, see the package vignette titled “WTP
space convergence issues in other packages”.
Finally, WTP space models can be computationally expensive. Since WTP space models have a
non-convex log-likelihood function, there is no guaranteed that any one iteration of the optimization
algorithm will reach a global solution. As a result, it is recommended that a multi-start optimization
loop always be used for WTP space models, which increases computation time. In practice, it may
be helpful to initially use a relatively small number of multi-start iterations (e.g., 10) to ensure
that the optimization is converging in most iterations before using a larger number of iterations to
conduct a broader search. The startValBounds argument, which is set to c(-1, 1) by default,
can be used to specify the lower and upper boundaries of the random starting parameters used in
each iteration of a multi-start loop.
7. Conclusions
Package logitr implements the maximum likelihood estimation of multinomial logit and mixed logit
models with unobserved heterogeneity across individuals, which is modeled by allowing parameters
to vary randomly over individuals according to a chosen distribution. Distinguishing features
include fast estimation speeds and support for utility models that can be specified using either a
“preference space” or “WTP space” parameterization, allowing for the direct estimation of marginal
WTP. This offers several advantages over the typical procedure of computing WTP using the
estimated parameters of a preference space model, including greater control over how WTP is
assumed to be distributed across the population.
While logitr is less general in scope than other similar packages in terms of the variety of supported
models, it is considerably faster and offers other functionality that is particularly useful for esti-

Journal of Statistical Software 37
mating WTP space and mixed logit models. For example, a parallelized multi-start optimization
loop offers a convenient interface for searching the solution space for different local minima when
estimating models with non-convex log-likelihood functions (i.e., WTP space and mixed logit mod-
els). In addition, although the user interface departs from the popular formula input, it is more
uniform and streamlined for estimating models with preference or WTP utility parameterizations.
The package could be further improved by adding support for other features, such as individual-level
parameter estimates, and other random parameter distributions, such as the triangle distribution
and the exponential distribution. The package source code and documentation can be found at
https://github.com/jhelvy/logitr.
Acknowledgements
I would like to express my gratitude to Connor Forsythe for contributing to the package by
adding support for clustering errors, Elea Feit for providing critical feedback in early drafts of
this manuscript, Jeremy Michalek for introducing me to choice modeling, Kenneth Train for his
incredibly well-written and accessible texts, and the JSS reviewers whose comments and suggestions
substantially improved the package and this manuscript.
References
Crastesa R, Beaumaisb O, Mahieud PA, Martinez-Camblore P, Scarpa R (2014). “External validity
of WTP estimates: comparing preference and WTP-space model results.” In 2nd Workshop on
Non-market Valuation.
Croissant Y (2020). “mlogit: Random Utility Models in R.” Journal of Statistical Software,95(11).
doi:10.18637/jss.v095.i11.
Czajkowski M, Budziński W (2019). “Simulation error in maximum likelihood estimation of discrete
choice models.” Journal of choice modelling,31, 73–85.
Daly A, Hess S, Train K (2012). “Assuring Finite Moments For Willingness To Pay In Random
Coefficient Models.” Transportation,39(1), 19–31.
Das C, Anderson CM, Swallow SK (2009). “Estimating Distributions of Willingness to Pay for
Heterogeneous Populations.” Southern Economic Journal, pp. 593–610.
Fiebig DG, Keane MP, Louviere J, Wasi N (2010). “The Generalized Multinomial Logit Model:
Accounting For Scale and Coefficient Heterogeneity.” Marketing Science,29(3), 393–421.
Gu Y, Hole AR, Knox S (2013). “Fitting the Generalized Multinomial Logit Model in STATA.”
The Stata Journal,13(2), 382–397.
Helveston JP, Feit EM, Michalek JJ (2018). “Pooling Stated and Revealed Preference Data in the
Presence of RP Endogeneity.” Transportation Research Part B: Methodological,109, 70–89.
Helveston JP, Liu Y, Feit EM, Fuchs ER, Klampfl E, Michalek JJ (2015). “Will Subsidies
Drive Electric Vehicle Adoption? Measuring Consumer Preferences in the U.S. and China.”
Transportation Research Part A: Policy and Practice,73, 96–112. ISSN 09658564. doi:
10.1016/j.tra.2015.01.002.

38 logitr: Preference and WTP Space Logit Models in R
Hess S, Palma D (2019). “Apollo: a flexible, powerful and customisable freeware package for choice
model estimation and application.” Journal of choice model ling,32, 100170.
Hole AR (2007). “Fitting Mixed Logit Models by Using Maximum Simulated Likelihood.” The
Stata Journal,7(3), 388–401.
Jain DC, Vilcassim NJ, Chintagunta PK (1994). “A Random-Coefficients Logit Brand-Choice
Model Applied to Panel Data.” Journal of Business & Economic Statistics,12(3), 317–328.
Kaplan J (2020). fastDummies: Fast Creation of Dummy (Binary) Columns and Rows from
Categorical Variables.Rpackage version 1.6.3, URL https://CRAN.R-project.org/package=
fastDummies.
Krinsky I, Robb AL (1986). “On Approximating the Statistical Properties of Elasticities.” The
Review of Economics and Statistics, pp. 715–719.
Louviere JJ, Hensher DA, Swait J (2000). Stated Choice Methods: Analysis and Applications.
Cambridge University Press, Cambridge, Massachusetts.
McFadden D (1974). “Conditional Logit Analysis of Qualitative Choice Behaviour.” In Frontiers
in Econometrics, chapter 4, pp. 105–142.
McFadden D, Train KE (2000). “Mixed MNL Models for Discrete Response.” J. Appl.
Econom.,15(5), 447–470. ISSN 0883-7252. doi:10.1002/1099-1255(200009/10)15:5<447::
AID-JAE570>3.0.CO;2-1.
Molloy J, Becker F, Schmid B, Axhausen KW (2021). “mixl: An open-source R package for
estimating complex choice models on large datasets.” Journal of choice modelling,39, 100284.
Sarrias M, Daziano R, et al. (2017). “Multinomial Logit Models with Continuous and Discrete
Individual Heterogeneity in R: The gmnl Package.” Journal of Statistical Software,79(2), 1–46.
doi:10.18637/jss.v079.i02.
Sonnier G, Ainslie A, Otter T (2007). “Heterogeneity Distributions of Willingness-To-Pay in Choice
Models.” Quant. Mark. Econ.,5(3), 313–331. doi:10.1007/s11129-007-9024-6.
Swait J, Andrews RL (2003). “Enriching Scanner Panel Models with Choice Experiments.” Mar-
keting Science,22(4), 442–460.
Train KE (2009). Discrete Choice Methods with Simulation. 2nd edition. Cambridge University
Press.
Train KE, Weeks M (2005). “Discrete Choice Models in Preference and Willingness-to-Pay Space.”
In Appl. Simul. Methods Environ. Resour. Econ., chapter 1, pp. 1–16.
Wickham H, et al. (2014). “Tidy Data.” Journal of statistical software,59(10), 1–23.
Ypma J, Johnson SG (2020). nloptr:RInterface to NLopt.Rpackage version 1.2.2.2, URL
https://CRAN.R-project.org/package=nloptr.

Journal of Statistical Software 39
Affiliation:
John Paul Helveston
George Washington University
Department of Engineering Management and Systems Engineering,
School of Engineering and Applied Sciences,
800 22nd St NW
Washington, DC 20052
E-mail: jph@gwu.edu
URL: https://www.jhelvy.com
Journal of Statistical Software http://www.jstatsoft.org/
published by the Foundation for Open Access Statistics http://www.foastat.org/
MMMMMM YYYY, Volume VV, Issue II Submitted: yyyy-mm-dd
doi:10.18637/jss.v000.i00 Accepted: yyyy-mm-dd