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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

URL: http://www.bioone.org/doi/full/10.1603/EN13287

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BEHAVIOR

A Likelihood-Based Biostatistical Model for Analyzing Consumer

Movement in Simultaneous Choice Experiments

ADAM R. ZEILINGER,

1,2,3

DAWN M. OLSON,

4

AND DAVID A. ANDOW

5

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-

1

Conservation Biology Program, Department of Entomology, Uni-

versity of Minnesota, 1980 Folwell Ave., St. Paul, MN 55108.

2

Present Address: Adam Zeilinger, Berkeley Initiative for Global

Change Biology, University of California Berkeley, 3101 Valley Life

Sciences Bldg., Berkeley, CA 94720.

3

Corresponding author, e-mail: arz@berkeley.edu.

4

Crop Protection and Research Management Unit, USDAÐARS,

2747 Davis Rd., Tifton, GA 31793.

5

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

al. 2008).

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

j

(t

i

)bethe

probability that a consumer is feeding on or associated

with state jat observation time t

i

,where j1,2,...,

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

n

(t

i

), at time t

i

is

Pnti⫽1⫺

冘

j1

n1

Pjti. [1]

The probability that the consumer is associated with

choice j, where jn, P

j

(t

i

), is a function of the

probability that the consumer is feeding on food

choice jat time t

i

t, deÞned as P

j

(t

i

t), and the

probability that the consumer is in the neutral space

at time t

i

t, deÞned as P

n

(t

i

t). We assume that

consumers leave food choice jat a choice-speciÞc

constant leaving probability

j

tand move toward

food choice jat a choice-speciÞc attraction probability

p

j

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

jis

Pjti⫽1⫺

jtPjti⫺t⫹pjtPnti⫺t

⫹ot[2]

where o(t) are higher order terms of t. As tgoes

to zero, equation 2 becomes

978 ENVIRONMENTAL ENTOMOLOGY Vol. 43, no. 4

dPj

dt ⫽

jPj⫹pjPn. [3]

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

dP1

dt ⫽

1P1⫹p1P3

dP2

dt ⫽

2P2⫹p2P3

P3⫽1⫺P1⫺P2[4]

where the parameters p

1

,p

2

,

1,

and

2

are Þtted

constants (Fig. 1), and P

3

is a special case of equation

1 for a system with only two choices. We substituted

the equation for P

3

into the differential equations to

produce the following two-equation model of linear,

nonhomogeneous differential equations:

dP1

dt ⫽

1⫹p1P1⫺p1P2⫹p1

dP2

dt ⫽

2⫹p2P2⫺p2P1⫹p2. [5]

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

1

(t

i

), P

2

(t

i

), and P

3

(t

i

)

P1ti⫽c1p1e

1ti⫺c2p1e

2ti⫹p1

2

1p2⫹

2p1⫹

1

2

P2ti⫽c1

1⫹p1⫹

1e

1ti⫹c2

1⫹p1⫹

2e

2ti

⫹p2

1

1p2⫹

2p1⫹

1

2

P3ti⫽1⫺P1⫺P2[6]

where c

1

and c

2

are arbitrary constants and

1

and

2

are eigenvalues of the system (Supp Material 1 [online

only]).

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

times t

i

where i1,2,....,m, then the conditional

probabilities of consumer association, P

j

,can be cal-

culated for the interval

i

t

i

t

i1

:

P1

i⫽c1p1e

1

i⫺c2p1e

2

i⫹p1

2

1p2⫹

2p1⫹

1

2

P2

i⫽c1

1⫹p1⫹

1e

1

i⫹c2

1⫹p1⫹

2e

2

i

⫹p2

1

1p2⫹

2p1⫹

1

2

P3

i⫽1⫺P1⫺P2[7]

where c

1

and c

2

are determined by the observed dis-

tribution of consumers at time t

i1

. 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

i1

is (n

1

(t

i1

), n

2

(t

i1

), n

3

(t

i1

)), then c

1

and

c

2

can be found by solving the initial value problem,

where P

1

(t

i

)n

1

(t

i1

)/N, P

2

(t

i

)n

2

(t

i1

)/N, and

t0:

n1ti1/N⫽c1p1⫺c2p1⫹p1

2

1p2⫹

2p1⫹

1

2

n2ti1/N⫽c1

1⫹p1⫹

1⫹c2

1⫹p1⫹

2

⫹p2

1

1p2⫹

2p1⫹

1

2

. [8]

Once system (8) is solved for c

1

and c

2

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

1

and c

2

have been calculated explicitly in the appendix. Be-

cause of the Markov properties of the model, these

sequential conditional probabilities are independent

(Tijms 2003).

To derive a likelihood function, let n

1

(t

i

), n

2

(t

i

), and

n

3

(t

i

) be the observed number of consumers at each

location at observation time t

i

. Then P

1

(t

i

), P

2

(t

i

), and

P

3

(t

i

) can be modeled as the parameters of a multi-

nomial distribution, so (n

1

(t

i

), n

2

(t

i

), n

3

(t

i

)) Multi-

nom(N, [P

1

(t

i

), P

2

(t

i

), P

3

(t

i

)]), for Nsample size and

i1 ... mnumber of observation times per trial.

Accordingly, the log-likelihood function at each time

point, ᐉ(

,t

i

), 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

locations. P

1

and P

2

probability that consumers are on

choices 1 and 2, respectively; P

3

probability that consumers

are in neutral space; p

1

and p

2

attraction rates to choices

1 and 2, respectively;

1

and

2

leaving rates from choices

1 and 2, respectively.

August 2014 ZEILINGER ET AL.: ANALYZING CONSUMER CHOICE EXPERIMENTS 979

ᐉ

⫽

冘

i1

m

冋

ln

冉

N!

n1

i!n2

i!n3

i!

冊

⫹lnP1

in1

i

⫹ln(P2

in2

i⫹ln(P3

in3

i

册

[9]

where

is a vector of model parameters p

1

,p

2

,

1

, and

2

. Note that, whereas p

j

tand

j

tin equation 2

were deÞned as attraction and leaving probabilities for

choice j, respectively, here in system (7) and likeli-

hood function (9), p

j

and

j

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

1

p

2

,

1

2

). 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

1

and p

2

,equal to each other but

allowed the leaving rates to vary (p

1

p

2

,

1

2

).

Third, in the Free Attraction model, we allowed the

attraction rates to vary but set the leaving rates,

1

, and

2

, equal to each other (p

1

p

2

,

1

2

). Finally, in

the Free model, we allowed all four parameters to be

Þt independently (p

1

p

2

,

1

2

). Differences

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

1993).

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

i

) ver-

sus ln{ln[S(t

i

)]}from KaplanÐMeier survival func-

tions of the attraction rates and leaving rates, where t

i

is the time of an attraction or leaving event, and S(t

i

)

is the proportion of individuals remaining at time t

i

.If

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

hazard rate,

, 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

1

(t

i

), n

2

(t

i

),and

n

3

(t

i

), 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

and 0.6.

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

ˆd)/d),

where 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

c

). Variances

and 95% CIs were calculated using the normal approx-

imation method (Bolker 2008, Millar 2011). Parameter

estimates and variances were averaged for all models

with AIC

c

7 following Burnham et al. (2011). For

more detail on the experimental design see Zeilinger

(2011).

Table 1. True parameter values (p

1

,p

2

,

1

, and

2

) for fast consumers (0.2 and 0.6) and slow consumers (0.02 and 0.1), equilibrium

values (P

1

*,P

2

*, and P

3

*), and the time to equilibrium of the model for each PC

PC p

1

p

2

1

2

P

1

*P

2

*P

3

*Time to

equilibrium

a

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

a

Time to equilibrium indicates the min. time step where P

j

(t) P

j

*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

20

and the number of maximum itera-

tions was set at 10,000. To improve MLE convergence,

we used inequality constraints of 10 ⱖ

ⱖ0.0001 and

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-

Choice-model.

Results

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

(p

1

0.6, p

2

0.2,

1

0.6,

2

0.02) is reciprocal

to PC 11 (p

1

0.2, p

2

0.6,

1

0.2,

2

0.6);

likewise, p

1

was overestimated in PC 6, whereas p

2

was

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

2

and p

1

(r

2,p1

)

was 0.09 and correlation between

1

and p

2

(r

1,p2

)

was 0.14. For E. servus, parameter correlations sug-

gested that choices were independent as well; r

2,p1

0.009 and r

1,p2

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

c

7

(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

AIC

c

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

Damaged Undamaged

E. servus Damaged 2

a

0

Undamaged 1 1

N. viridula Damaged 1 4

Undamaged 6 1

Second choice

Third choice

Damaged Undamaged

E. servus Damaged 0 1

Undamaged 0 0

N. viridula Damaged 0 3

Undamaged 2 0

a

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

i

out of the total

number of stink bugs at location jat observation time t

i1

.Note

that each panel is used to assess one rate parameter; thus each

p

j

and

j

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

c

) values, and change in AIC

c

(⌬AIC

c

) for

each model variant in the stink bug choice experiment

Stink bug

species Model variant df

a

AIC

c

AIC

cb

E. servus Free attraction model 3 56.66 0

c

Free leaving model 3 60.19 3.53

c

Free model 4 60.46 3.80

c

Fixed model 2 62.50 5.84

c

N. viridula Fixed model 2 50.40 0

c

Free attraction model 3 52.65 2.25

c

Free leaving model 3 53.02 2.62

c

Free model 4 55.50 5.10

c

a

Degrees of freedom associated with the model variant.

b

AIC

c

indicates the change in AIC

c

relative to the minimum AIC

c

value among model variants. Model variants are ordered according to

AIC

c

values.

c

Models with AIC

c

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).

Discussion

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)

under study.

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

3

0 and

1

p

2

and

2

p

1

.

If parameter

␥

12

is the rate of switching from choice 1

to choice 2 and

␥

21

is the rate of switching from choice

2 to choice 1, then system (4) reduces to

dP1

dt ⫽

␥

12P1⫹

␥

21P2

dP2

dt ⫽

␥

21P2⫹

␥

12P1[10]

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-

ther research.

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

preference experiments.

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

per arena.

986 ENVIRONMENTAL ENTOMOLOGY Vol. 43, no. 4

Acknowledgments

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,

J. Sarhad, A. Fahimipour, S. Hayes, H. Hulton, P. Rueda-

Cedil, and R. Swab for helpful comments on earlier drafts.

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Received 11 October 2013; accepted 9 June 2014.

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