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A Likelihood-Based Biostatistical Model for Analyzing Consumer
Movement in Simultaneous Choice Experiments
Author(s): Adam R. Zeilinger, Dawn M. Olson and David A. Andow
Source: Environmental Entomology, 43(4):977-988. 2014.
Published By: Entomological Society of America
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A Likelihood-Based Biostatistical Model for Analyzing Consumer
Movement in Simultaneous Choice Experiments
ADAM R. ZEILINGER,
DAWN M. OLSON,
AND DAVID A. ANDOW
Environ. Entomol. 43(4): 977Ð988 (2014); DOI: http://dx.doi.org/10.1603/EN13287
ABSTRACT Consumer feeding preference among resource choices has critical implications for basic
ecological and evolutionary processes, and can be highly relevant to applied problems such as
ecological risk assessment and invasion biology. Within consumer choice experiments, also known as
feeding preference or cafeteria experiments, measures of relative consumption and measures of
consumer movement can provide distinct and complementary insights into the strength, causes, and
consequences of preference. Despite the distinct value of inferring preference from measures of
consumer movement, rigorous and biologically relevant analytical methods are lacking. We describe
a simple, likelihood-based, biostatistical model for analyzing the transient dynamics of consumer
movement in a paired-choice experiment. With experimental data consisting of repeated discrete
measures of consumer location, the model can be used to estimate constant consumer attraction and
leaving rates for two food choices, and differences in choice-speciÞc attraction and leaving rates can
be tested using model selection. The model enables calculation of transient and equilibrial proba-
bilities of consumer-resource association, which could be incorporated into larger scale movement
models. We explore the effect of experimental design on parameter estimation through stochastic
simulation and describe methods to check that data meet model assumptions. Using a dataset of modest
sample size, we illustrate the use of the model to draw inferences on consumer preference as well as
underlying behavioral mechanisms. Finally, we include a userÕs guide and computer code scripts in
R to facilitate use of the model by other researchers.
KEY WORDS attraction rate, host selection, leaving rate, movement ecology, transient dynamics
Consumer feeding preference among resources has
critical implications for larger ecological and evolu-
tionary patterns and processes. Feeding preference
can be a signiÞcant driver of ecological specialization,
assortative mating, and thus speciation (Linn et al.
2003). Feeding preference of a disease vector for dif-
ferent hosts of a pathogen has important and complex
implications for infectious disease spread in both
plant and animal host populations (Kingsolver 1987,
Zeilinger and Daugherty 2014). In invasion biology,
feeding preference has been used to support the biotic
resistance hypothesis (Morrison and Hay 2011). In
ecological risk assessment, feeding preference is often
used to help assess potential nontarget effects of in-
troduced biological control agents (Babendreier et al.
2005) and genetically engineered organisms (Prager
et al. 2014).
Consumer preference is also an important process
in optimal foraging theory. In optimal patch foraging
models, Þtness is a function of residence time of the
consumer within a patch (Stephens and Krebs 1986).
Patch residence time should be a function of both
attractiveness and consumption rate of the resource;
both of which are fundamental components of feeding
preference (Nicotri 1980, Schoonhoven et al. 2005).
Optimal patch foraging models are essentially models
of consumer movement (Nathan et al. 2008). Describ-
ing and understanding the movement of consumers
among multiple resourcesÑin other words, relating
preference to movementÑcould help inform optimal
patch foraging models.
Within consumer choice experiments or assaysÑ
where two or more choices are provided simultane-
ously to a consumerÑfeeding preference is inferred
from either measures of relative consumption among
resource choices (Larrinaga 2010, Morrison and Hay
2011), or measures of initial consumer movement or
consumer location (Nicotri 1980, Rovenska´et al. 2005,
Bakonyi et al. 2006, Zirbes et al. 2011). Measures of
consumption and measures of movement provide dis-
tinct and complementary information about feeding
preference. Measures of consumption may be more
relevant for investigating the potential for a consumer to
suppress a resource, such as in a test of a potential bio-
Conservation Biology Program, Department of Entomology, Uni-
versity of Minnesota, 1980 Folwell Ave., St. Paul, MN 55108.
Present Address: Adam Zeilinger, Berkeley Initiative for Global
Change Biology, University of California Berkeley, 3101 Valley Life
Sciences Bldg., Berkeley, CA 94720.
Corresponding author, e-mail: email@example.com.
Crop Protection and Research Management Unit, USDAÐARS,
2747 Davis Rd., Tifton, GA 31793.
Department of Entomology and Center for Community Genetics,
University of Minnesota, 1980 Folwell Ave., St. Paul, MN 55108.
logical control agent (Babendreier et al. 2005). Move-
ment-based measures of feeding preference, however,
may incorporate elements of habitat preference (Un-
derwood et al. 2004). As a result, compared with mea-
sures of consumption, measures of movement may be
more relevant for investigating questions relating to the
degree of association of a consumer with different re-
sources, such as in optimal patch foraging models and in
investigations into the evolutionary and ecological re-
sponses of consumers to novel resources (Linn et al.
2003). Movement can be quantiÞed for any mobile con-
sumer, whereas consumption can be difÞcult to measure
for consumers with haustellate or sucking mouthparts.
Moreover, the transient dynamics and equilibria associ-
ated with consumer movement behavior can, in some
cases, be more revealing about consumer preference
than consumption experiments, especially for mobile
species. For example, leaving rates are probably related
to assessment, handling, and consumption times, and
provide a broader perspective than consumption exper-
iments by themselves.
For at least four decades, biologists have debated
and reÞned the design and analysis of consumer
choice experiments based on measures of relative con-
sumption, resulting in a rigorous set of methodologies
(Manly 1974, 1993; Roa 1992; Horton 1995; Lockwood
III 1998; Prince et al. 2004; Underwood et al. 2004;
Taplin 2007; Larrinaga 2010). However, design and
analysis of choice experiments based on measures of
movement have lagged behind. At the same time,
recent analyses of organismal movement have made
signiÞcant advances using stateÐspace models, which
link probabilistic statistical models of observations of
organismal location to biologically relevant stochastic
models of movement (Jonsen et al. 2003, Patterson et
Rigorous analysis of repeated measures of move-
ment within consumer choice experiments have been
lacking because frequentist goodness-of-Þt tests for
repeated-measures multinomial data do not exist.
While measures of movement have been used exten-
sively in choice experiments, preference is often in-
ferred from a single movement event, such as a con-
sumerÕs Þrst choice or location at the end of a trial
(Nicotri 1980, Bakonyi et al. 2006, Zirbes et al. 2011).
In such cases, the transient behavior of the consumer
is ignored, the parameters estimated often have little
biological relevance, and the results are difÞcult to
relate in a quantitative manner to larger scale models
of movement (e.g., Nathan et al. 2008).
In the present article, we develop a simple, likeli-
hood-based, biostatistical model to analyze repeated
measures of consumer location between paired re-
source choices. The model incorporates all available
information on consumer location to and from choices
during the trial, but does not require constant obser-
vation. Estimated choice-speciÞc attraction and leav-
ing rates are then used to make inferences on prefer-
ence: higher attraction rates, lower leaving rates, or
both, associated with one food choice indicate a
greater preference. The model also allows for calcu-
lation of transient and equilibrial probabilities of as-
sociation between consumer and resource choices.
Using stochastic simulation, we explore elements of
the model relevant to experimental design. Finally,
using an empirical dataset on herbivorous stink bug
preference, we illustrate that the model can be used
to draw inferences on consumer preference.
Materials and Methods
Theory and Model Development. In general, move-
ment among potential resource choices is a stochastic
process comprising probabilities of movement to (at-
traction) and from (leaving) a resource item or patch.
Following the work of Haccou and Meelis (1992) on
statistical analysis of continuous and discrete obser-
vations of animal behavior, we modeled the move-
ment of a consumer between two choices as a con-
tinuous time, stochastic Markov chain model. In
contrast to Haccou and Meelis (1992), we focus only
on movements relevant to food choice with the goal
of estimating attraction and leaving rates for each
choice in a paired-choice experiment. Let P
probability that a consumer is feeding on or associated
with state jat observation time t
nfor an experiment with n1 food choices and i
1,2,... , mfor mtotal number of discrete-time ob-
servations of consumer location per trial. The proba-
bility that a consumer is in the neutral space, i.e., not
at a food choice, deÞned as P
), at time t
The probability that the consumer is associated with
choice j, where jn, P
), is a function of the
probability that the consumer is feeding on food
choice jat time t
t, deÞned as P
t), and the
probability that the consumer is in the neutral space
at time t
t, deÞned as P
t). We assume that
consumers leave food choice jat a choice-speciÞc
constant leaving probability
tand move toward
food choice jat a choice-speciÞc attraction probability
t. We further assume that consumers can only
move to a food choice from the neutral space; in other
words, a switching event from one choice to the other
can be decomposed into two independent eventsÑ
leaving the Þrst choice and subsequent attraction to-
ward the second choice (Fig. 1). Note that the model
assumes that some amount of physical distance exists
between choices that could reasonably considered
neutral space; the boundary between neutral space
and a choice can at times be vague and researchers
should consider this in their experimental design.
The probability that the consumer is at food choice
where o(t) are higher order terms of t. As tgoes
to zero, equation 2 becomes
978 ENVIRONMENTAL ENTOMOLOGY Vol. 43, no. 4
Combining equations 1 and 3 and considering the
special case of an experiment with only two choices,
j1, 2, the system can be modeled as
where the parameters p
constants (Fig. 1), and P
is a special case of equation
1 for a system with only two choices. We substituted
the equation for P
into the differential equations to
produce the following two-equation model of linear,
nonhomogeneous differential equations:
We solved system (5) analytically with respect to t
using KolmogoroffÕs forward differential equations
method (Tijms 2003) to produce a set of three dy-
namic probability functions P
), and P
are arbitrary constants and
are eigenvalues of the system (Supp Material 1 [online
The system of equation 6 describes the probability
that the consumer is associated with choice 1 or choice
2 at time tprojected from time 0. In an experi-
mental context, a researcher will often want to mea-
sure the location of each consumer multiple times over
the course of the trials. Given mtotal observations at
where i1,2,....,m, then the conditional
probabilities of consumer association, P
,can be cal-
culated for the interval
are determined by the observed dis-
tribution of consumers at time t
. The system (7)
describe the conditional probabilities of Þnding con-
sumers in states 1, 2, and 3, given that an observed
number of them were found in each state at the be-
ginning of the time interval.
Given Ntotal consumers and that their distribution
at time t
)), then c
can be found by solving the initial value problem,
Once system (8) is solved for c
and the solu-
tions are substituted into equation 7, the conditional
probability for any time interval and any initial ob-
servation can be calculated in terms of the parameters
of interest. For the initial conditions of the experiment
(i.e., all consumers start in the neutral state), c
have been calculated explicitly in the appendix. Be-
cause of the Markov properties of the model, these
sequential conditional probabilities are independent
To derive a likelihood function, let n
) be the observed number of consumers at each
location at observation time t
. Then P
) can be modeled as the parameters of a multi-
nomial distribution, so (n
)]), for Nsample size and
i1 ... mnumber of observation times per trial.
Accordingly, the log-likelihood function at each time
), is the log of the probability mass func-
tion for the multinomial distribution (Boos and Ste-
fanski 2013). Summing over all observations produces
the following likelihood function:
Fig. 1. Conceptual diagram of host plant choice and
movement between three locations within a paired-choice
experimental arena. The three circles represent the three
possible states or locations: the two food choices (choices 1
and 2) and the neutral space between the choices. The
arrows represent possible movements between the three
probability that consumers are on
choices 1 and 2, respectively; P
probability that consumers
are in neutral space; p
attraction rates to choices
1 and 2, respectively;
leaving rates from choices
1 and 2, respectively.
August 2014 ZEILINGER ET AL.: ANALYZING CONSUMER CHOICE EXPERIMENTS 979
is a vector of model parameters p
. Note that, whereas p
tin equation 2
were deÞned as attraction and leaving probabilities for
choice j, respectively, here in system (7) and likeli-
hood function (9), p
are deÞned as attraction
and leaving rates, respectively.
Statistical Inference. From this model, consumer
feeding preference can be inferred from differences
between the choice-speciÞc attraction and leaving
rates. To test for differences in preference between
choices, we compared four variants of the likelihood
function (9). First, we set both the attraction rates and
the leaving rates equal to each other, which we call the
Fixed model (p
). In the Fixed model,
the optimization algorithm is forced to Þt one attrac-
tion rate and one leaving rate for the data from both
choices; the Fixed model represents a null model of no
preference. Second, in the Free Leaving model, we set
attraction rates, p
,equal to each other but
allowed the leaving rates to vary (p
Third, in the Free Attraction model, we allowed the
attraction rates to vary but set the leaving rates,
, equal to each other (p
). Finally, in
the Free model, we allowed all four parameters to be
Þt independently (p
among the maximum likelihood estimates (MLEs) of
these four model variants can be tested with AkaikeÕs
information criterion (AIC). Models do not need to be
nested for AIC (Bolker 2008, Burnham et al. 2011), so
all variants can be tested together. Inference can be
made either based on the best modelÑthe one with
the lowest AIC valueÑor through model averaging
(Burnham and Anderson 2002).
Variances for parameter estimates can be estimated
using either the proÞle method or the normal approx-
imation method, if the MLE is at or near the global
minimum (Bolker 2008, Millar 2011). In the normal
approximation method, variances are extracted from a
varianceÐcovariance matrix that is calculated by in-
verting the Hessian matrix of the MLE (Bolker 2008),
which is often an output of derivative-based MLE
algorithms. However, if a parameter estimate is on the
boundary of the inequality constraint, then the MLE
is unlikely to be at the global minimum and the proÞle
and normal approximation methods for estimating
variance are no longer valid. In this case, variances and
SEs can be estimated using jackknife methods. Im-
portantly, CIs should not be calculated from jackknife
estimates of SE because, in general, their probability
distributions are unknown (Efron and Tibshirani
Finally, parameter estimates can be used to calcu-
late the probabilities, at equilibrium, that a consumer
will be associated with the two choices using equations
(A12) in Supp Material 1 (online only).
Testing Model Assumptions. The present model as-
sumes that attraction and leaving rates are constant for
the duration of the trials, although it is possible to
develop more general models with time-varying pa-
rameters. The assumption of time-constancy can be
interrogated using graphical inspection of ln(t
)]}from KaplanÐMeier survival func-
tions of the attraction rates and leaving rates, where t
is the time of an attraction or leaving event, and S(t
is the proportion of individuals remaining at time t
the data follow the line of best Þt, then the attraction
or leaving rates are constant (Machin et al. 2006). Also,
Machin et al. (2006) note that the y-intercept of the
line of best Þt provides an estimate of the natural log
of the constant hazard rate, ln(
). This estimate of the
, can be used as an initial parameter value
in the MLE algorithm.
The model also assumes that consecutive consumer
choices are independent from previous choices. This
assumption can be interrogated with contingency ta-
ble analyses in which consecutive choices are the
factors: Þrst choice versus second choice, second
choice versus third choice, etc. (Andow and Kiritani
1984). Note that such an analysis requires following
individuals through time. Alternatively, indepen-
dence between choices can be assessed by examining
correlations between model parameter estimates at
the MLE. As described in the Statistical Inference
section, the varianceÐcovariance matrix can be cal-
culated from the Hessian matrix. When the varianceÐ
covariance matrix is scaled to the variances, then the
off-diagonal elements of the matrix provide correla-
tions between parameter estimates (Bolker 2008).
Contingency table analysis will be invalid when move-
ment event frequencies are too small; namely, when
one or more table cells are 0. Estimating correlations
between parameter estimates will be invalid when
parameter estimates are on a constraint boundary.
Stochastic Simulation. To explore the behavior of
the model and maximum likelihood estimation of pa-
rameters, we used stochastic simulation to investigate
how various dimensions of consumer choice experi-
mental design inßuence parameter estimation. We
simulated consumer location data, n
), using a Markov stochastic process (Pielou 1969)
of model (7). The simulated data were used with the
likelihood function (8) to estimate the parameters
using the Free model variant.
First, we explored how sample size inßuences pa-
rameter estimation by simulating data at low, medium,
and very large sample sizes: N10, 20, and 1,000,
respectively. We also explored how the rate of move-
ment by experimental consumers may inßuence pa-
rameter estimation. SpeciÞcally, we hypothesized
that, for accurate parameter estimation, the distribu-
tion of observation times should match the rate of
movement by consumers such that both the transient
dynamics and equilibrium are observed. In the slow
consumer scenario, we simulated data using each com-
bination of low and high parameter values in which
the observation times covered only the transient dy-
namics; the low and high values were 0.02 and 0.1,
980 ENVIRONMENTAL ENTOMOLOGY Vol. 43, no. 4
respectively (Table 1). This generated 16 different
combinations of true parameter values. In the fast
consumer scenario, we repeated this process with low
and high parameter values that allowed for observa-
tion of both transient dynamics and the equilibrium. In
this case, the low and high parameter values were 0.2
Second, we explored how the number of and inter-
vals between per trial observations inßuenced param-
eter estimation. We simulated data at low, medium,
and high numbers of per trial observation times: m
10, 20, and 40, respectively. For each level of m, we
compared two different interval schemes: when those
observations were evenly spaced at a constant interval
of 1 time step and when the observations were
weighted toward the beginning of the experiment
(i.e., front-loaded observations).
To assess the performance of the MLE process, we
calculated expected proportional bias as E((d
ˆis the maximum likelihood parameter esti-
mate and dis the true parameter value, from 4,000
simulation runs for each true parameter combination
(PC). The MLE converged in 100% of the simulations.
Consumer Choice Experiment. To illustrate the
practical use of the model, we used data from an
experiment conducted to test for feeding preference
of nymphs of the herbivorous stink bug species Eu-
schistus servus Say and Nezara viridula L. (Heterop-
tera: Pentatomidae) between a cotton plant (Gos-
sypium hirsutum L.) that had been damaged by a larval
Helicoverpa zea (Boddie) (Lepidoptera: Noctuidae)
and an undamaged cotton plant. The study was de-
signed to test the hypothesis that induced plant re-
sponses to H. zea herbivory inßuenced stink bug feed-
ing preference (Zeilinger et al. 2011). Brießy, three
possible locations of the stink bug were recorded in
each trial: on the undamaged plant, on the damaged
plant, or in neutral space between plants. We moni-
tored the location of the stink bug at 10, 30 min, 1, 12,
18, 24, and 36 h from the start of the experiment. We
focused our observations in the beginning of the ex-
periment, i.e., front-loaded observations because stink
bug movement was most likely to occur during this
period (A.R.Z, unpublished data). We obtained sam-
ple sizes of 15 and 19 for trials with E. servus and N.
viridula, respectively. For model selection, we used
AIC corrected for small sample size (AIC
and 95% CIs were calculated using the normal approx-
imation method (Bolker 2008, Millar 2011). Parameter
estimates and variances were averaged for all models
7 following Burnham et al. (2011). For
more detail on the experimental design see Zeilinger
Table 1. True parameter values (p
) for fast consumers (0.2 and 0.6) and slow consumers (0.02 and 0.1), equilibrium
*, and P
*), and the time to equilibrium of the model for each PC
Fast consumer true parameter values
1 0.2 0.2 0.2 0.2 0.33 0.33 0.33 18
2 0.6 0.2 0.2 0.2 0.60 0.20 0.20 12
3 0.2 0.6 0.2 0.2 0.20 0.60 0.20 12
4 0.6 0.6 0.2 0.2 0.43 0.43 0.14 8
5 0.2 0.2 0.6 0.2 0.14 0.43 0.43 35
6 0.6 0.2 0.6 0.2 0.33 0.33 0.33 43
7 0.2 0.6 0.6 0.2 0.08 0.69 0.23 26
8 0.6 0.6 0.6 0.2 0.20 0.60 0.20 30
9 0.2 0.2 0.2 0.6 0.43 0.14 0.43 35
10 0.6 0.2 0.2 0.6 0.69 0.08 0.23 26
11 0.2 0.6 0.2 0.6 0.33 0.33 0.33 43
12 0.6 0.6 0.2 0.6 0.60 0.20 0.20 30
13 0.2 0.2 0.6 0.6 0.20 0.20 0.60 11
14 0.6 0.2 0.6 0.6 0.43 0.14 0.43 8
15 0.2 0.6 0.6 0.6 0.14 0.43 0.43 8
16 0.6 0.6 0.6 0.6 0.33 0.33 0.33 6
Slow consumer true parameter values
1 0.02 0.02 0.02 0.02 0.33 0.33 0.33 177
2 0.1 0.02 0.02 0.02 0.71 0.14 0.14 99
3 0.02 0.1 0.02 0.02 0.14 0.71 0.14 99
4 0.1 0.1 0.02 0.02 0.45 0.45 0.09 63
5 0.02 0.02 0.1 0.02 0.09 0.45 0.45 389
6 0.1 0.02 0.1 0.02 0.33 0.33 0.33 405
7 0.02 0.1 0.1 0.02 0.03 0.81 0.16 146
8 0.1 0.1 0.1 0.02 0.14 0.71 0.14 255
9 0.02 0.02 0.02 0.1 0.45 0.09 0.45 389
10 0.1 0.02 0.02 0.1 0.81 0.03 0.16 146
11 0.02 0.1 0.02 0.1 0.33 0.33 0.33 405
12 0.1 0.1 0.02 0.1 0.71 0.14 0.14 255
13 0.02 0.02 0.1 0.1 0.14 0.14 0.71 80
14 0.1 0.02 0.1 0.1 0.45 0.09 0.45 63
15 0.02 0.1 0.1 0.1 0.09 0.45 0.45 63
16 0.1 0.1 0.1 0.1 0.33 0.33 0.33 36
Time to equilibrium indicates the min. time step where P
*to a precision of 5 decimal places.
August 2014 ZEILINGER ET AL.: ANALYZING CONSUMER CHOICE EXPERIMENTS 981
All programming was done in R 3.1.0 (R Core Team
2014, Vienna, Austria). To maximize the negative log-
likelihood function, we used the optimx function
(Nash and Varadhan 2011) with the BarzilaiÐBorwein
spectral projected gradient (“spg”) optimization al-
gorithm (Varadhan and Gilbert 2009). The spg
method was used because preliminary simulations
showed that other constrained optimization algo-
rithms, namely L-BFGS-B, did not consistently con-
verge on an MLE (results not shown). For MLE of
simulated and empirical data, convergence tolerance
was set at 10
and the number of maximum itera-
tions was set at 10,000. To improve MLE convergence,
we used inequality constraints of 10 ⱖ
supplied exact gradient functions. Gradient functions
were derived in Mathematica 9 (Wolfram Research,
Inc. 2012, Champaign, IL) and veriÞed by calculat-
ing numerical derivatives with the grad function in
R (Gilbert 2012). To facilitate the future use of the
model, we have developed a userÕs guide (Supp
Material 2 [online only]) and supplied R script for
maximum likelihood estimation with the four model
variants, model selection, variance estimation using
the normal approximation method, and jackknife
method, and testing assumptions (Supp Material 3
[online only]). Current R scripts and future revi-
sions and extensions to the model will also be avail-
able at: https://github.com/arzeilinger/Consumer-
Stochastic Simulation of Sample Size. Some of the
16 PCs were reciprocals, in which the true parameter
values were switched between the two choices, and in
these cases, bias estimates were switched and equiv-
alent (Supp Fig. 1 [online only]). For example, PC 6
0.02) is reciprocal
to PC 11 (p
was overestimated in PC 6, whereas p
overestimated in PC 11 (Supp Fig. 1 [online only]).
That estimated bias was equivalent between recipro-
cal PCs indicates that accuracy in parameter estimates
between choice 1 and choice 2 were equivalent. Fol-
lowing this, we show only results from the 10 unique
PCs (Figs. 2 and 3).
Estimated proportional bias was generally greatest
at low sample size (N10), decreased at intermediate
sample size (N20), and was negligible at very large
sample size (N1,000; Fig. 2). These results suggest
that bias was a sampling problem, and not intrinsic to
the model and estimation method. Bias estimates were
generally positive, indicating that the parameter esti-
mates were greater than the true values (Fig. 2).
Estimated proportional bias also depended on the
overall movement rates (Fig. 2). The patterns of bias
indicate that low accuracy (high bias) may be because
of poor estimation of either the transient dynamics
(when the model moves quickly to equilibrium) or
equilibrium values (when the model moves slowly to
Fig. 2. Sample Size Simulation. Expected proportional bias estimates, E((d
ˆd)/d) for rate parameters over a range
of sample sizes, N10, 20, and 1,000, for 10 unique PCs from 4,000 simulation runs. Left-side panels (“Fast movement”)
represent combinations of greater true parameter values: 0.2 and 0.6 for high and low values, respectively (see Table 1).
Right-side panels (“Slow movement”) represent combinations of smaller true parameter values: 0.02 and 0.1. Number of
observation times m40 for each simulation and spaced at 1 time step. These values were within the range of initial parameter
estimates from empirical data from trials with herbivorous stink bugs on cotton plants (Zeilinger et al. in review).
982 ENVIRONMENTAL ENTOMOLOGY Vol. 43, no. 4
equilibrium). The model requires four degrees of free-
dom (dfs) to estimate the four parameter values. The
equilibria of the three state variables provide two dfs,
so the remaining information is in the transient dy-
namics of the system. We evaluated the time to equi-
librium for each PC using equations (A12). Consistent
with our hypothesis, the greatest bias estimates oc-
curred with PCs that caused the system to move to
equilibrium quicklyÑPCs 2 and 16Ñand PCs that
caused the system to move to equilibrium slowlyÑ
PCs 1, 5, and 6 (Fig. 2; Table 1) relative to the time
step and duration of observations. SpeciÞcally, all
instances of high bias are associated with PCs in
which there are insufÞcient observations of both the
transient period and the equilibrium. For instance,
under the slow consumer scenario, PCs 1, 2, 5, and
6 take ⱖ99 time steps to reach equilibrium (Table 1)
while observations were made up to 40 time steps,
excluding any observations of the equilibria. How-
ever, when rates of movement were increasedÑ
under the fast consumer scenarioÑthe time to equi-
librium approached 40 and bias estimates were
greatly reduced (Fig. 2; Table 1).
Stochastic Simulation of Per Trial Observation
Times. Using the fast consumer scenario, we explored
the effects of varying the per trial observation timesÑ
both total number and intervals between observa-
tionsÑon parameter estimation. As with the sample
size simulation, estimated proportional bias was great-
est at a low number of observation times (m10) and
decreased substantially at intermediate and large
numbers of observation times (m20 and m40,
respectively; Fig. 3).
We also found that the interval schemeÑconstant
intervals of 1 time step or front-loaded observationsÑ
affected bias estimates (Fig. 3). Bias estimates tended
to be greater when observations were evenly spaced;
front-loading observations resulted in consistently
small bias estimates (proportional bias 1) across PCs,
particularly for mⱖ20.
Consumer Choice Experiment. For the assumption
of independent consecutive choices, we tested for
independence between Þrst and second choices for N.
viridula using contingency table analysis; the frequen-
cies of E. servus movement were too small for such
analysis (Table 2). Consecutive choices made by N.
viridula nymphs were independent (odds ratio 0.06;
95% CI [0.0007, 1.34]; P0.07). Independence is
also supported by correlations between parameter es-
timates, calculated from the Hessian matrix of the Free
Choice model; correlation between
was 0.09 and correlation between
was 0.14. For E. servus, parameter correlations sug-
gested that choices were independent as well; r
0.009 and r
0.039. For the assumption of con-
stant attraction and leaving rates, the data available
followed the line of best Þt, indicating that the attrac-
tion and leaving rates were constant during the ex-
periment (Fig. 4).
Fig. 3. Number of Per trial Observation Times Simulation. Expected proportional bias estimates, E((d
ˆd)/d), for rate
parameters over a range of number of per trial observations and intervals between observation times for 10 unique PCs from 4,000
simulation runs. Range of number of per trial observations included m10, 20, and 40. Left-side panels (“Constant intervals”)
represent simulations with constant intervals between observations, with an observation every 1 time step. Right-side panels
(“Front-loaded observations”) represent simulations with a greater concentration of observations, with shorter intervals, at the
beginning of trials. PC numbers correspond to those in Table 1. Sample size, N, for each simulation 20.
August 2014 ZEILINGER ET AL.: ANALYZING CONSUMER CHOICE EXPERIMENTS 983
For E. servus trials, the Free Attraction model Þt the
data best, but all models were good with AIC
(Table 3). Averaged parameter estimates and CIs from
these models showed that E. servus was signiÞcantly
more attracted to undamaged plants than to H. zea-
damaged plants. Differences in attraction rates had a
strong effect in determining preference, whereas leav-
ing rates between choices were indistinguishable (Fig.
5). Based on the model-averaged parameter estimates
(Fig. 5), the probability that E. servus is associated
with undamaged plants is predicted to be much
greater than the probability of association with H.
zea-damaged plants (Fig. 6).
For N. viridula, the best model was the Fixed model
but once again all four model variants were good with
7 (Table 3). Using all four models to estimate
model-averaged parameter values, we found that N.
viridula attraction rates and leaving rates were equiv-
alent between undamaged and damaged plants (Fig.
5). At equilibrium, we predict that N. viridula will be
Table 2. Contingency tables of the outcomes of consecutive
choices made by stink bugs between H. zea-damaged and undam-
aged cotton plants
Species First choice Second choice
E. servus Damaged 2
Undamaged 1 1
N. viridula Damaged 1 4
Undamaged 6 1
E. servus Damaged 0 1
Undamaged 0 0
N. viridula Damaged 0 3
Undamaged 2 0
Number of stink bug nymphs on the damaged plant for their Þrst
choice and damaged plant for their second choice, meaning that the
stink bug was observed to have left the damaged plant and to have
returned to the damaged plant. Only the Þrst contingency table,
between Þrst and second choices, was analyzed.
Fig. 4. Graphical inspection of model assumptions: con-
stant attraction rates (AÐD) and leaving rates (EÐH) for each
stink bug species, E. servus (A, C, E, and G) and N. viridula (B,
D, F, and H), on undamaged cotton plants (A, B, E, and F) and
H. zea-damaged plants (C, D, G, and H). The variable “time”on
the x-axis indicates the time (in hours) when an attraction or
leaving event occurred (i.e., when one or more stink bugs
moved to a plant or left a plant). The variable “S(t)”on the y-axis
indicates the proportion of individuals remaining, i.e., “surviv-
ing,”in neutral space or remaining on the plant after the event
at time t. Each datum represents the proportion of stink bugs
that moved from a choice jat observation time t
out of the total
number of stink bugs at location jat observation time t
that each panel is used to assess one rate parameter; thus each
must be assessed using different plots. If the data points
follow the line of best Þt, then the rate is constant.
Table 3. Degrees of freedom, information criterion corrected
for small sample size (AIC
) values, and change in AIC
each model variant in the stink bug choice experiment
species Model variant df
E. servus Free attraction model 3 56.66 0
Free leaving model 3 60.19 3.53
Free model 4 60.46 3.80
Fixed model 2 62.50 5.84
N. viridula Fixed model 2 50.40 0
Free attraction model 3 52.65 2.25
Free leaving model 3 53.02 2.62
Free model 4 55.50 5.10
Degrees of freedom associated with the model variant.
indicates the change in AIC
relative to the minimum AIC
value among model variants. Model variants are ordered according to
Models with AIC
7 were considered good models and se-
lected for averaging following Burnham et al (2011).
Fig. 5. Model-averaged parameter estimates 95% CIs for
attraction rates (A) and leaving rates (B) for E. servus (ÔEsÕ,
closed circles) and N. viridula (ÔNvÕ, open circles) trials for
undamaged and H. zea-damaged cotton plants. Parameter esti-
mates are averaged from good models identiÞed in Table 3.
984 ENVIRONMENTAL ENTOMOLOGY Vol. 43, no. 4
equally distributed between undamaged and H. zea-
damaged plants (Fig. 6).
Methods for the design and analysis of consumer
choice assays using measures of consumption have
been debated and reÞned for at least four decades
(Manly 1974, 1993; Roa 1992; Horton 1995; Prince et
al. 2004; Underwood et al. 2004; Taplin 2007; Larrinaga
2010). In contrast, similar attention has been lacking
for choice assays using measures of movement. Mea-
sures of movement provide distinct and complimen-
tary insight into feeding preference compared with
measures of consumption; movement-based infer-
ences on preference may be more directly related
than measures of consumption to optimal patch for-
aging models and other classes of movement models
(Stephens and Krebs 1986, Patterson et al. 2008). Sim-
ilar to some state-space models described by (Patter-
son et al. 2008), we modeled the probability of a
mobile consumer being associated with two resource
choices as a function of choice-speciÞc attraction and
leaving rates. Using repeated measures of consumer
location with choice trials, attraction and leaving rates
were estimated using maximum likelihood estimation
and inferences on the differences of these rates de-
termined by model selection methods. Finally, tran-
sient and equilibrial probabilities of association be-
tween the consumer and the resource choices can be
calculated from the model.
We simulated data to explore the effects on param-
eter estimation from variation in movement rates, sam-
ple size, the number of per trial observation times, and
the intervals between observation times. Increasing
sample size and increasing the number of observations
generally improved the accuracy of parameter esti-
mates. Greater sample sizes should enhance “valleys”
and “ridges”in the likelihood surface, making it easier
to Þnd the MLE (Bolker 2008). Increasing the number
of observation times and changing the spacing of ob-
servations improve the accuracy of parameter estima-
tion because they allow information to be gathered on
both transient dynamics and the equilibrium. Ac-
curate parameter estimation depends on multiple
observations covering both transient and equilibrial
periods of consumer movement. Overall, bias esti-
mates tended to be positive, indicating that param-
eter estimates tended to be greater than the true
values. Importantly, parameter bias was symmetri-
cal between choices, indicating that difference be-
tween the choices in parameter estimates relate to
real differences in consumer choice rather than
artifacts from the model or the MLE procedure.
Our simulations suggest that capturing both tran-
sient dynamics and the equilibrium of consumer lo-
cation are important. We were able to improve accu-
racy by increasing sample size, increasing the per trial
number of observation times, or changing the spacing
of observations to better estimate transients and equi-
libria. From an experimental perspective, increasing
the number of observation times and changing their
temporal spacing would be more efÞcient than in-
creasing sample size. In practice, the number of per
trial observations and their spacing must be deter-
mined by the movement behavior of the consumer(s)
In the analysis of stink bug feeding preference data,
we found greater attraction rates toward undamaged
plants for E. servus relative to H. zea-damaged plants
and equivalent movement rates between choices for
N. viridula. The E. servus results correspond to PC 2 in
the stochastic simulation (Table 1). The simulation
results predict that such a PC at a modest sample size,
modest number of observation times, and front-loaded
observations should result in moderate overestimation
of the leaving rate of the less-preferred choice (Figs.
2 and 3). While the leaving rate from the H. zea-
damaged plant may be overestimated, the implications
from the estimated parameters were not affected. In-
deed, if it is overestimated, the true effect is even
greater than the estimated effect. We expect little bias
in the parameter estimates for N. viridula and all es-
timates to be biased equally. In both cases, the pre-
dicted biases in the parameter estimates do not alter
the interpretation of the results.
Data on the feeding preference of stink bug nymphs
between H. zea-damaged and undamaged cotton
plants largely conformed to the assumptions of the
model; stink bug attraction and leaving rates were
constant over the duration of the trials and consecu-
tive choices were independent. The number of data
points produced from KaplanÐMeier survival analysis
will depend on the number of observations and the
intervals between observations in relation to mobility
of the consumer. In our data on E. servus, the number
of observations was too few to rigorously test the
assumption of constant movement rates. Again, the
Fig. 6. Predicted dynamics of E. servus (A) and N.
viridula (B) selecting H. zea-damaged cotton plants (solid
line) and undamaged plants (dashed line), calculated using
model-averaged parameter estimates shown in Fig. 5.
August 2014 ZEILINGER ET AL.: ANALYZING CONSUMER CHOICE EXPERIMENTS 985
number of per trial observations and their spacing
should be determined by the behavior of the con-
sumer under study.
The model also assumes that movement choices are
sequentially independent, violation of which will not
necessarily invalidate parameter estimates and model
selection. Rather, positive correlation between con-
secutive choices may inßate attraction rates. For
highly mobile consumers, it may be difÞcult to observe
the consumer in neutral space, possibly resulting in
more switching events being recorded than leaving-
and-returning events and a violation of the indepen-
dent choice assumption. This could be resolved by
increasing the frequency of per trial observations.
Alternatively, consider a scenario of extreme choice
dependence, where no observations are made of the
consumer in neutral space and each attraction rate is
exactly equal to the leaving rate from the opposite
choice. In this case, P
is the rate of switching from choice 1
to choice 2 and
is the rate of switching from choice
2 to choice 1, then system (4) reduces to
and the assumption of independent choice is relaxed.
However, using model (10) does not allow one to
distinguish between preferences through greater at-
traction versus lower leaving rates.
The present model deÞnes preference as the bal-
ance between choice-speciÞc attraction and leaving
rates. A difference in attraction rates between choices
indicates that feeding preference is likely inßuenced
by consumer selection behavior (Vinson 1976, Ber-
nays and Chapman 1994, Schoonhoven et al. 2005) and
that cues detected from a distance may be important
determinants of preference. However, a difference in
leaving rates indicates that preference is likely inßu-
enced by consumer acceptance behavior and patch
giving up times. A wide variety of cues are known to
affect acceptance behavior, including visual cues, ol-
factory cues in the hostÐfood headspace, or surface or
internal chemistry (Vinson 1976, Bernays and Chap-
man 1994). Information on whether preference is de-
termined by attraction rates or leaving rates would
facilitate developing hypotheses on the particular
mechanisms underlying preference for testing in fur-
The distinction between preference owing to at-
traction rates and leaving rates can be ecologically
valuable. The feeding preference of vectors of plant
and animal pathogens can greatly inßuence pathogen
spread (Kingsolver 1987). Theory predicts that the
epidemiological importance of preference will de-
pend on disease prevalence but only when preference
is determined when selecting a host, i.e., by attraction
rates (Sisterson 2008). Preference determined after
vector feedingÑrelating to differences in leaving
ratesÑwill have a relatively minor inßuence on dis-
ease spread. Further, the host manipulation hypoth-
esis predicts that spread of nonpersistent vector-borne
pathogens (pathogens that do not enter the circula-
tory system of the vector) will be greatest if vectors are
preferentially attracted to infected hosts but also leave
them quickly (Mescher 2012). Thus, our model pro-
vides an efÞcient way to estimate the inßuence of
vector feeding preference on disease spread and test
the host manipulation hypothesis using simple feeding
Leaving rates are used widely in optimal patch for-
aging models, but the assumed processes underlying
patch leaving in optimal foraging models differ from
those assumed in our model. Here, we assume that
leavingÐinducing cues are constant over the duration
of the experiment. If, however, leaving rates are de-
termined by resource depletionÑwhich is central to
optimal patch foraging models (Stephens and Krebs
1986)Ñthen leaving rates will not be constant, but will
increase over time, and this will be apparent from tests
of model assumptions (e.g., Fig. 4). Indeed, tests of
timeÐconstancy in parameters could potentially de-
tect any changes in the resource choices during the
experiment that are relevant to preference, including
autogenic changes (Manly 1993).
From estimated attraction and leaving rates, the
model can be used to calculate transient and equilib-
rial probabilities that the consumer will be associated
with the resource choices tested. Such probabilities
could be incorporated into movement and foraging
models, or to test the inßuence of preference in as-
sociations of consumer and resource. For example, do
innate preferences for different resources or the rel-
ative abundance of those resources explain consumerÐ
resource associations in the environment (Spotswood
et al. 2013)?
Our consumer movement model described here
could also be used to analyze data from repeated
measures of consumer colonization of habitat patches.
In particular, the method would be well suited for
markÐreleaseÐrecapture data, where individuals are
followed to habitat patches (Kuussaari et al. 1996).
The model described here provides all of the advan-
tages of state-space models recently developed for
movement ecology data: it enables the estimation of
states (probabilities of location), biologically mean-
ingful model parameters (attraction and leaving
rates), observation error (variance of parameter esti-
mates), and enables statistical inferences from model
selection among biologically relevant models (Patter-
son et al. 2008). The technique can be used for paired-
choice experiments of any duration, with any number
of repeated observations of consumer location, and
any interval scheme between observation times as
long as observation times are allocated in both the
transient and equilibrium periods. Possible future ex-
tensions of the model include time-varying parame-
ters and adaptations to experiments using more than
two simultaneous choices or more than one consumer
986 ENVIRONMENTAL ENTOMOLOGY Vol. 43, no. 4
This study was partially supported by a grant from the
National Research Initiative of the U.S. Department of Ag-
riculture, National Institute of Food and Agriculture (grant
2008-02409) to D.A.A. and D.M.O., an IGERT grant from U.S.
National Science Foundation to the University of Minnesota,
and a Thesis Research Grant, Doctoral Dissertation Fellow-
ship, and grants from the DaytonÐWilkie Fund from the
Graduate School and Bell Museum of Natural History, Uni-
versity of Minnesota, to A.R.Z. We thank the R Help online
community for assistance with R programming, and M. Ga-
nesh and R. Almeida for access to the Lawrence Berkeley
LaboratoryÕs Computation Genomics Research Laboratory
computing cluster to run the stochastic simulation. We also
thank C. Neuhauser, M. Daugherty, K. Anderson, H. Regan,
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Received 11 October 2013; accepted 9 June 2014.
988 ENVIRONMENTAL ENTOMOLOGY Vol. 43, no. 4