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Statistical modeling and inferences on dynamic decision-making mechanism underlying longitudinal distribution data with resource heterogeneity
Journal of the Korean Statistical Society 39 (2010) 409–416
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Journal of the Korean Statistical Society journal homepage: www.elsevier.com/locate/jkss
Statistical modeling and inferences on dynamic decision-making mechanism underlying longitudinal distribution data with resource heterogeneity Fushing Hsieh a , Rou-Ling Yang b,∗ a
Department of Statistics, University of California, Davis, CA 95616, USA
b
Department of Entomology, National Taiwan University, Taipei 100, Taiwan
article
info
Article history: Received 21 January 2009 Accepted 28 August 2009 Available online 25 September 2009 AMS 2000 subject classifications: primary 62H05 secondary 91B06 Keywords: Fitness-gain manifold Longitudinal distribution data Minimum Sum of Chi-squared approach Resource heterogeneity
abstract We propose a new way of constructing dynamic decision-making model and a nonlikelihood based statistical approach for analyzing a new data type: longitudinal distribution data. This longitudinal data records a trajectory of an animal’s dynamic decision-making when continuously exploiting a relative large, but close environment. The ensemble of all hosts contained in the environment is postulated to constitute a manifold of species-specific fitness-gains at any time point, and traverses through two major distinct phases: abundance vs. scarcity of pristine hosts. As such a manifold provides the relative potentials to all possible hosts available for selection, we construct a phase-dependent dynamic decision-making mechanism in a form of a self-adaptive conditional probabilistic model. We devise a Minimum Sum of Chi-squared approach to simultaneously evaluate individual cognitive capability within the two distinct phases and address the validity of the manifold based dynamic decision-making model on the longitudinal distribution data. We analyze three real data sets of seed beetle Callosobruchus maculatus collected from three experimental designs with different degrees of resource heterogeneity. Our statistical inferences are shown to successfully resolve the behavioral ecology issue of whether animal adaptively employs a dynamic decision-making mechanism in response to gradual environmental changes. © 2009 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.
1. Introduction Animal intelligent strategies of ecological relevance are reported in many behavioral studies as attributes to the evolutionary success leading to species’ persistent existence (Shettleworth, 1998). Especially many supporting evidences on the population level are derived from analyses relying on stationary assumptions (Goubault, Plantegenest, Poinsot, & Cortesero, 2003; Heimpel, Rosenheim, & Mangel, 1996; McGregor & Roitberg, 2000; Morris & Fellowes, 2002). Recently many researchers have realized that the applicability of such stationary results is rather limited. Many state-space type of mathematical modeling techniques are proposed to take potential nonlinear dynamics into behavioral process analysis. Important successes have been achieved along this line as reported and collected in two recent books by Clark and Mangel (2000) and Houston and McNamara (1999). However these techniques are commonly based on assumptions that certain optimality is achieved by all individual animals. These assumptions embed the philosophy of evolution into the behavior study on one hand, while its mathematical
∗ Corresponding address: Department of Conservation and Preservation, National Palace Museum, No. 221, Sec. 2, Zhishan Road., Taipei 11143, Taiwan. Tel.: +886 2 2881 2021x2120; fax: +886 2 2882 1440. E-mail address: [email protected] (R.-L. Yang). 1226-3192/$ – see front matter © 2009 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jkss.2009.08.006
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optimizations restrict many aspects of information that could be possibly extracted from behavioral data on the other. This is why validations of such state-space models via rigorous statistical goodness-of-fit testing in general are missing in the literature of animal behavior. In this paper we propose a new and biologically realistic construction for a dynamic decisionmaking model that is capable of adapting varying environmental conditions, and devise an non-likelihood based statistical approach for simultaneously analyzing longitudinal behavioral data and testing validity of the proposed dynamic model. The particular behavioral data considered here is a new type of longitudinal distribution data observed when an animal individually exploits a relative large, but close environment. This longitudinal data records the quality distribution of an ensemble of hosts at any observational time point without replacement. Since any change on successive distributions is caused by the individual animal’s decisions made between the two time points. Therefore we can perceive this longitudinal distribution data as a trajectory of dynamic decision-making exclusively made by the animal. In order to evaluate individual decision-making mechanism, we construct a geometry at any time point on the set of all decisions available for the animal. Within a chosen geometry, decisions are not only ranked, but also quantified with a ‘‘potential of preference’’ that prescribes their relative positions. Such a geometry is simply called a manifold of the set of all potential decisions. Hence, based on the landscape of ‘‘potential of preference’’ at any time point, a self-adaptive conditional probabilistic distribution of decision on host selection can be specified according to such a manifold. In this fashion the dynamics of such a manifold is transformed into a dynamic decision-making mechanism that statistically models an individual animal’s longitudinal distributions along the observational temporal axis. This dynamic decision-making mechanism is exclusively illustrated via one species of female seed beetles’, Callosobruchus maculatus (F.), oviposition (egglaying) behavioral process throughout this paper, which indeed motives this study. For statistical inferences on this dynamic decision-making model regulated by a specific manifold, we propose the Minimum Sum of Chi-squared approach to simultaneously estimate the parameters of the dynamic decision-making model, which represents individual’s cognitive capability of discerning the manifold structure, and evaluate the goodness-of-fit testing of the model. The ability of making simultaneous inferences is not the only reason for our proposal. The other important reason is that computational loading of the maximum likelihood approach is too high. For instance if there are 20 decisions which have been made between two successive observations. The set of possible segmental trajectories of length 20 is very large. This largeness of possible trajectories will cost a rather too large an amount of computing to be reasonably handled by a personal computer. Thus the task of performing optimizing the likelihood function becomes too costly to do, so is the task of goodness-of-fit testing. Further testing dynamic decision-making models are theoretically involved with a martingale structure (Hsieh, Horng, Lin, & Lan, 2007). Given the heavy computational loading needed for maximum likelihood estimation, the martingale residuals require at least a similar amount of computation. In contrast the Minimum sum of Chi-squared computations employed here can dynamically summarize information contents within a feasible computational loading for both statistical inferences. This paper is organized as follows. The motivating example with regard to the subject animal’s biology and the existing knowledge of decision-making mechanisms are introduced in Section 2. The designs of three different degrees of resource heterogeneity are given in Section 3. The self-adapted manifold structure and the dynamic decision-making model are derived and constructed in Section 4. The minimum sum of the Chi-squared approach is proposed in Section 5. Applications on three real data sets are performed in Section 6. Other statistical and relevant biological issues are discussed in Section 7. 2. Motivating example Our technique and computations are exclusively illustrated through longitudinal distribution data of a cosmopolitan pest on legumes, the female seed beetle, C. maculatus (F.), which is also called the bean weevil. Seed beetles lay their eggs singly on the surface of legume species. After hatching, the larvae will penetrate into the seeds and grow up within them. They will not leave the seed before emerging as an adult (Southgate, 1979). Thus, the seed quality has a direct influence on a seed beetles’ progeny fitness. Therefore it is thought that seed beetles have evolved with a discriminating ability for host quality. To precisely identify the effect of different host traits has attracted much attention in the literature of behavioral ecology. For instance, the number of eggs on hosts, host species, host size and seed texture, etc., could be discriminated by seed beetles were reported (Mitchell, 1990; Thanthianga & Mitchell, 1990; Wasserman, 1986). Among these discrimination abilities, egg discrimination (i.e., discriminating egg-load among beans) was demonstrated to have a relatively higher ranking than others in the oviposition process of seed beetles (Thanthianga & Mitchell, 1990). Beetles can detect the presence of eggs either by physical contact or a chemical cue (Messina & Renwick, 1985). Thus, here we considered the egg discrimination ability as a primary cognitive capability of female beetles. Yang, Fushing, and Horng (2006) have shown that egg and size discrimination play different important roles in different oviposition stages. For further exploration of the oviposition decision-making process of female beetles, here we consider size discrimination as a secondary cognitive capability and examine its interaction with the primary cognitive capability of distinguishing egg-load. Also it has been found that female beetles would more likely reject smaller beans while facing a drastic bean-size change. Beyond all aforementioned findings, in this paper we attempt to answer the dynamic question of whether an individual female insect becomes choosier when facing greater resource heterogeneity, while constantly keeping track of the road map of host preference prescribed via the fitness manifold. The latter innate capability is hypothesized based on whether a host quality at any given time point to a female insect is its expected fitness gain, or the number of potential adult offspring.
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a
b
411
c
Fig. 1. Four sizes of adzuki beans were arranged into three designs with increasing heterogeneity from (a) 2 × 2, (b) 4 × 4, to (c) 8 × 8. The size of beans was increased with increasing Arabic numerals.
3. Designs of heterogeneous resource and its information content A heterogeneous resource was provided by offering a seed beetle with four sizes of adzuki beans (i.e., heterogeneous quality) arranged in three designs. The sizes were classified according to beans’ major axes and weights: ‘‘1’’ for 5.0–5.5 mm(87.22 ± 0.96 mg), ‘‘2’’ for 5.5–6.0 mm(114.66 ± 0.74 mg), ‘‘3’’ for 6.0–6.5 mm(146.4 ± 1.83 mg), and ‘‘4’’ for 6.5–7.0 mm(180.6 ± 2.65 mg), and three lattice designs were: (a) 2 × 2, (b) 4 × 4 and (c) 8 × 8 (respectively shown in Fig. 1 a, b and c). All designs consist of 64 un-attacked beans, i.e., 16 beans for each size, placed on a Styrofoam board in a 9-cm Petri dish. The idea behind these three designs is that the marginal probability of encountering any one of the four sizes of host bean is equal, while the conditional probability of experiencing various host sizes given in locality are drastically different. Therefore a female beetle is expected to more likely experience constant sized hosts locally in 2 × 2 layout, and to encounter increasing resource heterogeneity in 4 × 4 and 8 × 8 layouts. If a female seed beetle could discriminate between different sizes of beans, then related patterns should be more evidently observed in the 8 × 8 than in the 4 × 4 or 2 × 2 designs. This heuristic idea is tested under different phases, i.e., pristine and parasitized phases, of environmental conditions. The pristine phase is defined when the overall bean resources reach the point that the average number of eggs per bean is below 50 . When a constantly exploited environmental condition 64 deteriorates beyond the pristine phase it is termed the parasitized phase. In the pristine phase, female beetles in general only lay on clean beans, while in the parasitized phase oviposition on beans with 1 or 2 eggs, thereafter termed 1-egg or 2-egg beans, is not uncommon even though there might be clean beans scattered sparsely among the 8 × 8 lattice. The Seed beetle C. maculatus (stock 4C6-4) used here was collected from and kept on commercial adzuki beans (Vigna angularis) for more than 100 generations in a dark growth chamber at 28 ± 1 ◦ C, 60%–70% RH (Horng, 1997). Its life cycle takes about 30 days, and adult female weighs about 7.68 ± 0.10 mg. It can lay 83 ± 5 eggs and live 12.8 ± 0.7 days without any food or water supply after eclosion (Yang, 2004). After mating, female beetles were assigned separately and randomly to each design arena, and their oviposition process as distribution of eggs among seeds was counted 4 times daily from 8 am to 8 pm. There are 13 replications for each design. 4. Fitness manifold and dynamic decision-making models Here we proceed to construct the fitness manifold first and then derive an individual animal’s dynamic decision-making models pertaining to longitudinal distribution data. Let M (Ω ) denote a manifold of a potential fitness value defined on the domain Ω as an ensemble of all available hosts with time-dependent qualities. In the aforementioned bean beetle example, the quality of a host at one point of time is determined by its size and the number of eggs having been laid onto it. One natural candidate of a manifold is the one representing the fitness-gain of host qualities, as shown in the Fig. 2. Here fitness-gain is defined as the probability of success that an egg laid upon a bean of specific quality successfully goes through several developmental stages into the final adult stage. The fitness-gain reported in Fig. 2 was previously determined in an independent experiment. Further we specifically consider two distinct phases of environmental conditions: pristine and parasitized conditions, to objectively separate an individual longitudinal choice data into two segments. The pristine phase (Ω0 ) is postulated for the setting when the environment is abundant with ‘‘good quality’’ hosts, while the parasitized phase (Ω1 ) is for the setting when a ‘‘good quality’’ host is hard to find. In our motivating example, from the starting point of the pristine phase with 64 clean adzuki beans, it is known that a female beetle can effectively carry out her decision-making by rejecting almost all host beans with eggs. At this stage, the weight is the most essential characteristic of host bean quality. Denote the average weights of the four sizes of bean as w1 , w2 , w3 , w4 , in increasing order. Further we know qualitatively that a female beetle hardly accepts a host bean that has been laid with one or more eggs, when she still encounters clean bean from time to time. In contrast, only when a female has been frustrated by encountering a long enough series of hosts with positive egg load, does she then switch to accepting a host loaded with one egg. To summarize above established knowledge, we propose the following two manifolds for pristine and parasitized phases, respectively. At any longitudinal time point t, the Ω can be represented by the distribution data of number of hosts with
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Fig. 2. The surface of larva-to-adult survivorship of the seed beetle from seeds with specific weights (x) and contain y larvae in them.
an egg-load being larger than or equal to 1, O(t ) = (O1 (t ), O2 (t ), O3 (t ), O4 (t )), and equivalently by the distribution of the remaining number of clean hosts N (t ) = (N1 (t ), N2 (t ), N3 (t ), N4 (t )) with Ni (t ) = 16 − Oi (t ). [Manifold Specifications]: 1. When an environment is in its pristine phase, the manifold M (Ω0 ) is accordingly defined via host quality, that is, wi for a clean bean in i-th weight category and −∞ for any host with a positive egg load; 2. When an environment is in its parasitized phase, the manifold M (Ω1 ) is defined via the fitness-gain given in Fig. 2. It is seen in Fig. 2 that there is a large gap separating fitness-gains of 1-egg-beans from 2- and 3-egg-beans. This gap heuristically implies that there would be no significant change in the decision-making mechanism, separately, if we substitute M (Ω0 ) by M (Ω1 ). The reason that these two manifolds are discussed separately is to illustrate one important aspect of dynamic decision-making modeling: different choices of manifold could lead to employ different perspectives of longitudinal distribution data. Next the dynamic decision-making models are derived via the following conditional transition probability models based on manifolds M (Ωi ), i = 0, 1, respectively. [Conditional transition probability specifications] 1. [Under the pristine phase with M (Ω0 ) manifold:] Then conditional probabilities of adding one more egg on one host size i at time tk+1 given the previously observed egg-distribution O(tk ) is modeled by: Pi [β; O(tk )] = Ni (tk )eβwi
− 4
Nj (tk )eβwj
for i = 1, 2, 3, 4.
(1)
j =1
2. [Under the parasitized phase with M (Ω1 ) manifold:] A female beetle’s decision-making of host selection adaptively discriminates the total fitness value pertaining to the category, instead of only the number of clean host beans. Here the total fitness value of category i at time tk is the linear sum of fitness-gains values in manifold M (Ω1 ) of all 16 beans in the category, and denotes this value by Fi (tk ). Then correspondingly the conditional probability of observing one more egg on host size i given egg-dispersion O(tk ) is modeled by: Pi [β; O(tk )] = Fi (tk )e ∗
βwi
− 4
Fj (tk )eβwj
for i = 1, 2, 3, 4.
(2)
j =1
The probabilities given in Eq. (1) are understood in the following fashion. First, when β = 0, then Pi [0; O(tk )] = ∑4 Ni (tk )/ j=1 Nj (tk ), that is, the probability of observing one egg laid on a particular host size depends only on the remaining number of clean beans of that size. This stands for the extreme case where the female beetle has no capability of discriminating among host weights. Second, when β is a very large positive number, then Pi [β; O(tk )] = Ni (tk )eβ(wi −w4 )
− 4
Nj (tk )eβ(wj −w4 )
j =1
is close to 0 for i = 1, 2, 3, since wi − w4 < 0, and is nearly equal to 1 with i = 4 if N4 (tk ) > 0.. That is another extreme case that her decision-making criterion is to lay egg only on the largest host bean as if the female beetle ‘‘knows’’ the complete environmental conditions. Third, if β is negative and far away from 0, then the probability given in (1) prescribes a decisionmaking criteria of only laying an egg on the smallest clean host bean. This is an unrealistic case. Hence we expect β to be positive in this seed beetle example. It is also worth emphasizing again that the dynamic decision-making mechanism
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specified by manifold M (Ω0 ) and the conditional transition probability in (1) has the built-in criteria of rejecting any host bean with eggs. Within the parasitized phase, the female beetle might search for long enough period of time without encountering any clean host bean. This experience would force her to change her decision-making criteria, including accepting a host loaded with an egg for oviposition. Thus it is biologically reasonable to model the dynamic decision-making mechanism via manifold M (Ω1 ) and conditional transitional probability in (2) under the parasitized phase. Behaviorally speaking the decision-making mechanism proposed with manifold M (Ω0 ) takes 1-egg-bean as an inhibitory stimulus, and only the 0-egg-bean is excitatory within the pristine phase. In contrast, all host beans either with or without an egg load are excitatory stimuli under the framework of manifold M (Ω1 ) within the parasitized phase. Therefore probabilities (1) and (2) are indeed coherent and the parameter β in both equations should be comparable to a certain degree. 5. Minimum sum of Chi-squared statistics In this section we construct the minimum sum of Chi-squared statistics within the pristine and parasitized phases. Usually there are several ovipositions involved with a transition from O(tk ) to O(tk+1 ). Denote the vector of increment in the four categories as 1O(tk+1 ) = O(tk+1 ) − O(tk ). We can estimate β via the minimum Chi-squared estimation by finding the minimizer of Chi-squared statistics at time tk+1 :
χ 2 (β|k + 1) =
4 − (1Oj (tk+1 ) − Nj (tk )Pj [β; O(tk )])2 j=1
Nj (tk )Pj [β; O(tk )]
.
(3)
Denote the estimate by βˆ (k+1) and minimum Chi-square value by χ 2 (βˆ k+1 |k + 1). The advantage of this statistical inference is that we can simultaneously perform the estimation for the β parameter, and the testing for goodness-of-fit of the decisionmaking model within the pristine phase via χ 2 (βˆ k+1 |k + 1). We illustrate the above minimum Chi-square computations on a hypothetical case within the pristine phase, let O(tk ) = (0, 2, 9, 13) and O(tk+1 ) = (3, 5, 12, 16), so 1O(tk+1 ) = (3, 3, 3, 3). Intuitively the minimum Chi-square statistics will give rise to a large positive estimate βˆ (k+1) , since there are only three 0-egg-beans left in the size category 4 after time tk and the female does indeed pick them up in the pool of clean beans. That is, when N1 (tk ) + N2 (tk ) are about 10 times N4 (tk ) and in order to compensate these differences by adjusting factors in the Eq. (1) and observing 3 new oviposition on category 4, the βˆ (k+1) (=9.04) value must be positive and relatively large, and so the minimum Chi-square value is χ 2 (βˆ k+1 |k + 1) = 5.174, which has its p-value greater than 0.1 according to Chi-squared distribution with degree of freedom 3. The above example with a p-value nearly rejecting the dynamic decision-making model assumption heuristically indicates that a Chi-squared statistic based on one single transition of distribution data with too few ovipositions (much less than 12) would only provide a small amount of information and hinder the sensitivity of goodness-of-fit testing due to the increment of degrees of freedom. On the other hand, one single Chi-squared statistic based on all pooled transitions with too many ovipositions (much more than 12) would be neither efficient on parameter estimation, nor goodness-of-fit testing. Therefore we propose to accumulate successive transitions up to having around 12 oviposition for one Chi-squared statistic and then sum them up. We term this the sum of the Chi-square statistic. An explicit construction on the real data is given in the next section. In summary our construction of minimum sum of the Chi-squared statistic is to simultaneously balance the power of goodness-of-fit testing and efficiency of estimating the β parameter. Specifically a minimum sum of the Chi-squared statistic consisting of more separate Chi-squared statistics would have a higher efficiency in β estimation, while the power of the computed minimum Chi-squared statistic for testing the model assumption becomes less because of the larger degree of freedom, see Hsieh (2001) for more detail discussion. Further we can extend our dynamic decision-making model by accommodating the distribution of 0-, 1- and 2-egg beans in each size category of the longitudinal distribution data into the conditional transition probabilities given in (1) and (2). This kind of model extension would better describe an individual decision-making. 6. Real data analysis In this section we perform an individual based minimum sum of Chi-squared analysis of a dynamic decision-making model first on the pristine phase and then on the parasitized phase, separately. The duration of the pristine phase is further divided into three sub-periods, each of which contains around 12 ovipositions. Its duration is roughly equal to time spans of 8 or 12 h. Upon each sub-period a Chi-square statistic is constructed based on the probability in Eq. (3), and denoted by χ12 (β), χ22 (β) & χ32 (β). And then we derive the βˆ for the pristine phase by computing the minimizer of the sum of the Chi-square statistic χT2 (β) = χ12 (β) + χ22 (β) + χ32 (β). This minimum sum of Chi-squared computation is performed for each of the 13 individuals in the three designs. As the larger positive βˆ value indicates that the individual is choosier, we would like to check whether female beetles facing higher resource heterogeneity tend to be significantly choosier or not. Comparisons via the Receiver Operating Characteristic (ROC) curve based on the three collections of βˆ estimates are proposed to shed light on the above biological problem. An introductory discussion and asymptotical theory with connection to nonparametric ROC statistics are given
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Table 1 Minimum Chi-squares values and associated beta values (in parentheses) arrived at by fitting model to sequential decisions made by seed beetles in different environments (* means the Chi-square value is larger than critical point of 0.05 significant; d.f. = 9).
χ 2 (βˆ )
Pristine phase
Parasitized phase
Design
1
2
3
4
5
6
7
8
9
10
11
12
13
2×2
2.69 (2.59)
6.37 (7.2)
4.25 (3.36)
14.28 (5.06)
17.97* (−0.3)
6.12 (−0.7)
3.22 (−3.04)
17.87* (2.07)
7.73 (2.4)
10.28 (4.09)
6.73 (0.81)
7.29 (2.05)
7.02 (2.00)
4×4
4.91 (6.16)
8.75 (6.08)
4.71 (2.14)
3.13 (1.67)
2.93 (4.17)
5.90 (3)
1.53 (3.18)
0.49 (2.58)
1.26 (−0.34)
8.02 (4.66)
1.51 (4.27)
4.53 (4.38)
5.31 (3.47)
8×8
4.95 7.92 (13.61) (5.69)
5.12 (3.20)
3.85 (3.70)
9.02 (2.99)
9.53 (−2.25)
2.34 (3.19)
4.50 (1.04)
3.14 (3.22)
4.13 (4.90)
6.60 (3.19)
2.54 (3.17)
6.60 (4.89)
2×2
4.05 (2.18)
7.66 (9.57)
6.10 (7.57)
4.17 (1.80)
0.33 (18.14)
3.67 (0.12)
– –
1.88 (1.84)
6.52 (4.63)
11.97* (4.52)
1.17 7.69 (11.97) (4.64)
8.47 (3.73)
4×4
4.86 (8.09)
8.33 (5.02)
5.31 (7.70)
– –
5.41 (1.96)
3.29 (2.36)
6.37 (2.27)
– –
5.96 (4.83)
1.69 15.68* (10.08) (4.27)
0.85 5.19 (11.57) (4.73)
8×8
0.83 (5.36)
5.99 (−0.01)
7.13 (7.47)
4.42 (5.98)
4.07 (6.90)
– –
4.34 (9.28)
5.07 (2.17)
4.46 (4.24)
5.38 (3.06)
– –
– –
2.14 (4.83)
in Hsieh and Turnbull (1996). The construction of an ROC curve for comparing two empirical distributions is carried out in the following fashion. We rank the estimated values in increasing order, denoting them as {βˆ 2j }, {βˆ 4j }, {βˆ 8j }, j = 1, 2, . . . 13 for 2 × 2, 4 × 4 and 8 × 8 designs, respectively. Correspondingly the minimum sum of Chi-square values are calculated and denoted by χT2 (βˆ 2j ), χT2 (βˆ 4j ) & χT2 (βˆ 8j ), j = 1, 2, . . . 13 (Table 1). The ROC curve of a pair of distributions, for example #of {βˆ ≤βˆ
,j=1,2,...,13}
2j 4k k 1 3 {βˆ 2j }vs.{βˆ 4j }, is constructed as a curve of ( 13 , ), k = 1, 2, . . . , 13. For example, ( 13 , 13 ) means that 13 2 5 ˆ ˆ there are three β2j estimates smaller than or equal to the smallest value among the 13 β4j estimates, and ( 13 , 13 ) indicates that there are five βˆ 2j values smaller than or equal to the second smallest βˆ 4j estimates. In this comparison the distribution corresponding to {βˆ 4j } is taken as the baseline. This choice is based on that values of {βˆ 2j } tend to be stochastically smaller than values of {βˆ 4j }, the ROC curve is concave and above the diagonal line, i.e., female beetles are choosier by being able to discriminate host size much better in 4 × 4 than in 2 × 2 designs. In contrast, values of {βˆ 8j } are stochastically larger than values of {βˆ 4j }, and the ROC curve is convex and below the diagonal, see the two ROC curves {βˆ 2j }vs.{βˆ 4j } and {βˆ 8j }vs.{βˆ 4j }
shown in Fig. 3a. For the parasitized phase, the number of ovipositions varies more from individual to individual than in the pristine phase. Therefore there might be only one or two sub-periods in which the cumulative increment of oviposition is above 12. Likewise we calculate minimum sum of Chi-squared estimate of parameter β and its sum of the Chi-squared value χT2 (β) based on the ∗ probability given in Eq. (2), and denoted {βˆ 2j }, {βˆ 4j∗ }{βˆ 8j∗ }, and χT2∗ (βˆ 2j ), χT2∗ (βˆ 4j ) & χT2∗ (βˆ 8j ), j = 1, 2, . . . 13 (Table 1). The
∗ two ROC curves for the {βˆ 2j }vs.{βˆ 4j∗ } and {βˆ 8j∗ }vs.{βˆ 4j∗ } in the parasitized phase are seen being intersecting with the diagonal in Fig. 3b. Thus we can conclude that in general the resource heterogeneity evidently leads individuals to become choosier when making a decision in the pristine phase, but not in the parasitized phase. This condition is reasonable because host quality is more critically determined by egg-load than by host size in this phase. In comparison of β estimates of the pristine phase with that of parasitized phases, as summarized in Fig. 4, we see that ∗ the {βˆ 2j } is significantly stochastically smaller than {βˆ 2j }, so is {βˆ 4j }vs.{βˆ 4j∗ }. And it is less evident in the case of {βˆ 8j }vs.{βˆ 8j∗ }. Overall these comparisons coincide with the biological understanding that an animal in general invests more foraging efforts on host quality discrimination when the environmental condition deteriorates into a phase where hosts of ideal quality become significantly harder to find. From the goodness-of-fit perspective, the model (1) in the pristine phase and model (2) in the parasitized phase work reasonably well based on results reported in Table 1. Hence we can conclude that the female’s decision-making criteria are reasonably prescribed by the fitness manifold throughout her decision-making process going from pristine to parasitized phases.
7. Discussion In this paper we propose a new way of constructing dynamic decision-making models that can adapt to ever changing environmental conditions. The novel idea is to build dynamic decision-making models based on a self-adaptive fitness manifold, which has an empirical basis derived from experimental and relevant biological knowledge. We also propose a new minimum sum of Chi-squared statistics for analyzing individual longitudinal distribution data, which is a new data type in longitudinal analysis. And the ROC analyses synthesizing three minimum sum of Chi-squared analyses on three real data sets provide a resolution to an interesting and important animal behavioral issue regarding host selection under different environmental phases. Statistical significances of our manifold based dynamic decision-making modeling are that they provide a flexible platform for constructing a more refined model for analyzing behavioral and cognitive processing on an individual level.
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Fig. 3. The number of βˆ estimates in 2 × 2 or 8 × 8 designs are smaller or equal to the value of {βˆ 4j } at the (a) pristine and (b) parasitized phases of seed beetles’ oviposition process.
8 pristine phase
Mean beta value
7
parasitized phase
6 5 4 3 2 1 0
2x2
4x4 Designs
8x8
Fig. 4. The level of choosiness in seed size (means of βˆ ) by seed beetles in different heterogeneous designs at different phases of the oviposition process (Mean ± SE).
Though the computational simplicity of minimum sum of Chi-squared statistics can be seen as a pilot analysis of the more refined martingale based analysis (Hsieh et al., 2007), it is necessary to further comment this from information content and computational perspectives. The martingale approach requires us to summarize the associated high dimensional
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martingale difference into one dimensional martingale statistics and perform the conditional variation calculations at every observational time point. First, any dimension reduction transformation requires the resultant summarizing statistics to retain the martingale structure. Thus most often a linear transformation is employed. However a linear transformation could hardly capture the dynamic features of decision-making simultaneously involved with evaluating and balancing two or more biological characteristics. Second, the involving computational loading of the conditional variations could be also heavy as well. Finally and critically we need to justify the theoretical approximation via a martingale central theorem applied to the ratio statistic of the sum of transformed martingale differences and square-root of the sum of conditional variations, and consider its approximation error. In contrast, the minimum sum of Chi-square computations has an essential computational advantage. And the required theoretical approximation seems more evident and natural for most of real world applications. Biological significances of our modeling methodology proposed here are the following. First our design of resource heterogeneity ranging from 2 × 2 to 8 × 8 resolution on 8 × 8 lattices could be useful prototypes in designing experiments to induce stochastically distinct input-experiences onto individual study subjects. Second we can explicitly evaluate individual female seed beetles’ size-discrimination capability (secondary) with simultaneously adjustment upon eggload discrimination capability (primary) under distinct phases of environmental conditions. Hence behavioral scientists need not redirect their research efforts to pursue certain marginal stimulus–response (S–R) relationships. These S–R relationships often are not flexible enough to effectively predict individual behavioral processing under ever changing environmental conditions. And their corresponding analysis could be unreliable because of difficulties stemming from individual heterogeneity. Finally we conclude our modeling methodology and the statistical analyzing approach proposed here might greatly widen the spectrum of the study of animal cognitive tactics of ecological relevance without the burden of experimental and analytic limitations involved in the S–R relationship. Our constructions of dynamic decision-making models are not limited to animal behavioral study per se. They can be indispensable in the study of genetic and physiological mechanisms underpinning a target behavioral strategy. By selection a strain of a capable performer could be established against a strain of a poor one. And then genetic comparison can be reliably used to identify potential regulating genes and their pathways leading to the intelligent strategy of interest (Fox, Stillwell, Amarillo-s, Czesak, & Messina, 2004; Kuo, 2004; Messina & Slade, 1997). Acknowledgements This work was partly supported by Academia Sinica of Taiwan and the National Science Council of Taiwan through the grant (NSC 93-2118-M-001-002). References Clark, C. W., & Mangel, M. (2000). Dynamic state variable models in ecology: Methods and applications. Oxford: Oxford University Press. Fox, C. W., Stillwell, R. C., Amarillo-s, A. R., Czesak, M. E., & Messina, F. J. (2004). Genetic architecture of population differences in oviposition behaviour of the seed beetle Callosobruchus maculatus. Journal of Evolutionary Biology, 17, 1141–1151. Goubault, M., Plantegenest, M., Poinsot, D., & Cortesero, A. M. (2003). Effects of expected offspring survival probability on host selection in a solitary parasitoid. Entomological Experimentalis et Applicata, 109, 123–131. Heimpel, G. E., Rosenheim, J. A., & Mangel, M. (1996). Egg limitation, host quality, and dynamic behavior by a parasitoid in the field. Ecology, 77, 2410–2420. Horng, S.-B. (1997). Larval competition and egg-laying decisions by the bean weevil, Callosobruchus maculatus. Animal Behaviour, 53, 1–12. Houston, A. I., & McNamara, J. M. (1999). Models of adaptive behaviour: An approach based on state. Cambridge: Cambridge University Press. Hsieh, F. (2001). On heteroscedastic hazards regression models: Theory and application. Journal of the Royal Statistical Society: Series B, 63, 63–79. Hsieh, F., & Turnbull, B. W. (1996). Non- and semi-parametric estimation of the receiver operating characteristics (ROC) curve. Annals of Statistics, 24, 25–40. Hsieh, F., Horng, S.-B., Lin, H.-Y., & Lan, Y.-C. (2007). Testing dynamic rules of animal cognitive processing with longitudinal distribution data. Statistica Sinica, 17, 735–748. Kuo, T.-H. (2004). Behavioral and genetic studies of incipient speciation in Drosophila melanogaster. Master’s thesis. National Tsing Hua University, Taiwan. McGregor, R. R., & Roitberg, B. D. (2000). Size-selective oviposition by parasitoids and the evolution of life-history timing in hosts: Fixed preferences vs. frequency-dependent host selection. Oikos, 89, 305–312. Messina, F. J., & Renwick, J. A. A. (1985). Mechanism of egg recognition by the cowpea weevil Callosobruchus maculatus. Entomological Experimentalis et Applicata, 37, 241–245. Messina, F. J., & Slade, A. F. (1997). Inheritance of host-plant choice in the seed beetle Callosobruchus maculatus (Coleoptera: Bruchidae). Annals of the Entomological Society of America, 90, 848–855. Mitchell, R. (1990). Behavioral ecology of Callosobruchus maculatus. In K. Fujii, A. M. R. Gatehouse, C. D. Johnson, R. Mitchel, & T. Yoshida (Eds.), Bruchids and Legumes: Economics, ecology, and coevolution (pp. 317–330). New York: Springer Publishing Company. Morris, R. J., & Fellowes, M. D. E. (2002). Learning and natal host influence host preference, handling time and sex allocation behaviour in a pupal parasitoid. Behavioral Ecology & Sociobiology, 51, 386–393. Shettleworth, S. J. (1998). Cognition, evolution, and behavior. Oxford: Oxford University press. Southgate, B. J. (1979). Biology of Bruchidae. Annual Review of Entomology, 24, 449–473. Thanthianga, C., & Mitchell, R. (1990). The fecundity and oviposition behavior of South Indian strain of Callosobruchus maculatus. Entomological Experimentalis et Applicata, 57, 133–142. Wasserman, S. S. (1986). Genetic variation in adaptation to foodplants among populations of the southern cowpea weevil, Callosobruchus maculatus: Evolution of oviposition preference. Entomological Experimentalis et Applicata, 42, 201–212. Yang, R.-L. 2004. Host size discrimination behavior and its mechanism of Callosobruchus maculatus (F.) in heterogeneous environment. Ph.D. thesis. National Taiwan University, Taiwan. Yang, R.-L., Fushing, H., & Horng, S.-B. (2006). Effects of search experience in a resource-heterogeneous environment on the oviposition decision of the seed beetle, Callosobruchus maculatus (F.). Ecological Entomology, 31, 285–293.
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Journal of the Korean Statistical Society journal homepage: www.elsevier.com/locate/jkss
Statistical modeling and inferences on dynamic decision-making mechanism underlying longitudinal distribution data with resource heterogeneity Fushing Hsieh a , Rou-Ling Yang b,∗ a
Department of Statistics, University of California, Davis, CA 95616, USA
b
Department of Entomology, National Taiwan University, Taipei 100, Taiwan
article
info
Article history: Received 21 January 2009 Accepted 28 August 2009 Available online 25 September 2009 AMS 2000 subject classifications: primary 62H05 secondary 91B06 Keywords: Fitness-gain manifold Longitudinal distribution data Minimum Sum of Chi-squared approach Resource heterogeneity
abstract We propose a new way of constructing dynamic decision-making model and a nonlikelihood based statistical approach for analyzing a new data type: longitudinal distribution data. This longitudinal data records a trajectory of an animal’s dynamic decision-making when continuously exploiting a relative large, but close environment. The ensemble of all hosts contained in the environment is postulated to constitute a manifold of species-specific fitness-gains at any time point, and traverses through two major distinct phases: abundance vs. scarcity of pristine hosts. As such a manifold provides the relative potentials to all possible hosts available for selection, we construct a phase-dependent dynamic decision-making mechanism in a form of a self-adaptive conditional probabilistic model. We devise a Minimum Sum of Chi-squared approach to simultaneously evaluate individual cognitive capability within the two distinct phases and address the validity of the manifold based dynamic decision-making model on the longitudinal distribution data. We analyze three real data sets of seed beetle Callosobruchus maculatus collected from three experimental designs with different degrees of resource heterogeneity. Our statistical inferences are shown to successfully resolve the behavioral ecology issue of whether animal adaptively employs a dynamic decision-making mechanism in response to gradual environmental changes. © 2009 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.
1. Introduction Animal intelligent strategies of ecological relevance are reported in many behavioral studies as attributes to the evolutionary success leading to species’ persistent existence (Shettleworth, 1998). Especially many supporting evidences on the population level are derived from analyses relying on stationary assumptions (Goubault, Plantegenest, Poinsot, & Cortesero, 2003; Heimpel, Rosenheim, & Mangel, 1996; McGregor & Roitberg, 2000; Morris & Fellowes, 2002). Recently many researchers have realized that the applicability of such stationary results is rather limited. Many state-space type of mathematical modeling techniques are proposed to take potential nonlinear dynamics into behavioral process analysis. Important successes have been achieved along this line as reported and collected in two recent books by Clark and Mangel (2000) and Houston and McNamara (1999). However these techniques are commonly based on assumptions that certain optimality is achieved by all individual animals. These assumptions embed the philosophy of evolution into the behavior study on one hand, while its mathematical
∗ Corresponding address: Department of Conservation and Preservation, National Palace Museum, No. 221, Sec. 2, Zhishan Road., Taipei 11143, Taiwan. Tel.: +886 2 2881 2021x2120; fax: +886 2 2882 1440. E-mail address: [email protected] (R.-L. Yang). 1226-3192/$ – see front matter © 2009 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jkss.2009.08.006
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optimizations restrict many aspects of information that could be possibly extracted from behavioral data on the other. This is why validations of such state-space models via rigorous statistical goodness-of-fit testing in general are missing in the literature of animal behavior. In this paper we propose a new and biologically realistic construction for a dynamic decisionmaking model that is capable of adapting varying environmental conditions, and devise an non-likelihood based statistical approach for simultaneously analyzing longitudinal behavioral data and testing validity of the proposed dynamic model. The particular behavioral data considered here is a new type of longitudinal distribution data observed when an animal individually exploits a relative large, but close environment. This longitudinal data records the quality distribution of an ensemble of hosts at any observational time point without replacement. Since any change on successive distributions is caused by the individual animal’s decisions made between the two time points. Therefore we can perceive this longitudinal distribution data as a trajectory of dynamic decision-making exclusively made by the animal. In order to evaluate individual decision-making mechanism, we construct a geometry at any time point on the set of all decisions available for the animal. Within a chosen geometry, decisions are not only ranked, but also quantified with a ‘‘potential of preference’’ that prescribes their relative positions. Such a geometry is simply called a manifold of the set of all potential decisions. Hence, based on the landscape of ‘‘potential of preference’’ at any time point, a self-adaptive conditional probabilistic distribution of decision on host selection can be specified according to such a manifold. In this fashion the dynamics of such a manifold is transformed into a dynamic decision-making mechanism that statistically models an individual animal’s longitudinal distributions along the observational temporal axis. This dynamic decision-making mechanism is exclusively illustrated via one species of female seed beetles’, Callosobruchus maculatus (F.), oviposition (egglaying) behavioral process throughout this paper, which indeed motives this study. For statistical inferences on this dynamic decision-making model regulated by a specific manifold, we propose the Minimum Sum of Chi-squared approach to simultaneously estimate the parameters of the dynamic decision-making model, which represents individual’s cognitive capability of discerning the manifold structure, and evaluate the goodness-of-fit testing of the model. The ability of making simultaneous inferences is not the only reason for our proposal. The other important reason is that computational loading of the maximum likelihood approach is too high. For instance if there are 20 decisions which have been made between two successive observations. The set of possible segmental trajectories of length 20 is very large. This largeness of possible trajectories will cost a rather too large an amount of computing to be reasonably handled by a personal computer. Thus the task of performing optimizing the likelihood function becomes too costly to do, so is the task of goodness-of-fit testing. Further testing dynamic decision-making models are theoretically involved with a martingale structure (Hsieh, Horng, Lin, & Lan, 2007). Given the heavy computational loading needed for maximum likelihood estimation, the martingale residuals require at least a similar amount of computation. In contrast the Minimum sum of Chi-squared computations employed here can dynamically summarize information contents within a feasible computational loading for both statistical inferences. This paper is organized as follows. The motivating example with regard to the subject animal’s biology and the existing knowledge of decision-making mechanisms are introduced in Section 2. The designs of three different degrees of resource heterogeneity are given in Section 3. The self-adapted manifold structure and the dynamic decision-making model are derived and constructed in Section 4. The minimum sum of the Chi-squared approach is proposed in Section 5. Applications on three real data sets are performed in Section 6. Other statistical and relevant biological issues are discussed in Section 7. 2. Motivating example Our technique and computations are exclusively illustrated through longitudinal distribution data of a cosmopolitan pest on legumes, the female seed beetle, C. maculatus (F.), which is also called the bean weevil. Seed beetles lay their eggs singly on the surface of legume species. After hatching, the larvae will penetrate into the seeds and grow up within them. They will not leave the seed before emerging as an adult (Southgate, 1979). Thus, the seed quality has a direct influence on a seed beetles’ progeny fitness. Therefore it is thought that seed beetles have evolved with a discriminating ability for host quality. To precisely identify the effect of different host traits has attracted much attention in the literature of behavioral ecology. For instance, the number of eggs on hosts, host species, host size and seed texture, etc., could be discriminated by seed beetles were reported (Mitchell, 1990; Thanthianga & Mitchell, 1990; Wasserman, 1986). Among these discrimination abilities, egg discrimination (i.e., discriminating egg-load among beans) was demonstrated to have a relatively higher ranking than others in the oviposition process of seed beetles (Thanthianga & Mitchell, 1990). Beetles can detect the presence of eggs either by physical contact or a chemical cue (Messina & Renwick, 1985). Thus, here we considered the egg discrimination ability as a primary cognitive capability of female beetles. Yang, Fushing, and Horng (2006) have shown that egg and size discrimination play different important roles in different oviposition stages. For further exploration of the oviposition decision-making process of female beetles, here we consider size discrimination as a secondary cognitive capability and examine its interaction with the primary cognitive capability of distinguishing egg-load. Also it has been found that female beetles would more likely reject smaller beans while facing a drastic bean-size change. Beyond all aforementioned findings, in this paper we attempt to answer the dynamic question of whether an individual female insect becomes choosier when facing greater resource heterogeneity, while constantly keeping track of the road map of host preference prescribed via the fitness manifold. The latter innate capability is hypothesized based on whether a host quality at any given time point to a female insect is its expected fitness gain, or the number of potential adult offspring.
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a
b
411
c
Fig. 1. Four sizes of adzuki beans were arranged into three designs with increasing heterogeneity from (a) 2 × 2, (b) 4 × 4, to (c) 8 × 8. The size of beans was increased with increasing Arabic numerals.
3. Designs of heterogeneous resource and its information content A heterogeneous resource was provided by offering a seed beetle with four sizes of adzuki beans (i.e., heterogeneous quality) arranged in three designs. The sizes were classified according to beans’ major axes and weights: ‘‘1’’ for 5.0–5.5 mm(87.22 ± 0.96 mg), ‘‘2’’ for 5.5–6.0 mm(114.66 ± 0.74 mg), ‘‘3’’ for 6.0–6.5 mm(146.4 ± 1.83 mg), and ‘‘4’’ for 6.5–7.0 mm(180.6 ± 2.65 mg), and three lattice designs were: (a) 2 × 2, (b) 4 × 4 and (c) 8 × 8 (respectively shown in Fig. 1 a, b and c). All designs consist of 64 un-attacked beans, i.e., 16 beans for each size, placed on a Styrofoam board in a 9-cm Petri dish. The idea behind these three designs is that the marginal probability of encountering any one of the four sizes of host bean is equal, while the conditional probability of experiencing various host sizes given in locality are drastically different. Therefore a female beetle is expected to more likely experience constant sized hosts locally in 2 × 2 layout, and to encounter increasing resource heterogeneity in 4 × 4 and 8 × 8 layouts. If a female seed beetle could discriminate between different sizes of beans, then related patterns should be more evidently observed in the 8 × 8 than in the 4 × 4 or 2 × 2 designs. This heuristic idea is tested under different phases, i.e., pristine and parasitized phases, of environmental conditions. The pristine phase is defined when the overall bean resources reach the point that the average number of eggs per bean is below 50 . When a constantly exploited environmental condition 64 deteriorates beyond the pristine phase it is termed the parasitized phase. In the pristine phase, female beetles in general only lay on clean beans, while in the parasitized phase oviposition on beans with 1 or 2 eggs, thereafter termed 1-egg or 2-egg beans, is not uncommon even though there might be clean beans scattered sparsely among the 8 × 8 lattice. The Seed beetle C. maculatus (stock 4C6-4) used here was collected from and kept on commercial adzuki beans (Vigna angularis) for more than 100 generations in a dark growth chamber at 28 ± 1 ◦ C, 60%–70% RH (Horng, 1997). Its life cycle takes about 30 days, and adult female weighs about 7.68 ± 0.10 mg. It can lay 83 ± 5 eggs and live 12.8 ± 0.7 days without any food or water supply after eclosion (Yang, 2004). After mating, female beetles were assigned separately and randomly to each design arena, and their oviposition process as distribution of eggs among seeds was counted 4 times daily from 8 am to 8 pm. There are 13 replications for each design. 4. Fitness manifold and dynamic decision-making models Here we proceed to construct the fitness manifold first and then derive an individual animal’s dynamic decision-making models pertaining to longitudinal distribution data. Let M (Ω ) denote a manifold of a potential fitness value defined on the domain Ω as an ensemble of all available hosts with time-dependent qualities. In the aforementioned bean beetle example, the quality of a host at one point of time is determined by its size and the number of eggs having been laid onto it. One natural candidate of a manifold is the one representing the fitness-gain of host qualities, as shown in the Fig. 2. Here fitness-gain is defined as the probability of success that an egg laid upon a bean of specific quality successfully goes through several developmental stages into the final adult stage. The fitness-gain reported in Fig. 2 was previously determined in an independent experiment. Further we specifically consider two distinct phases of environmental conditions: pristine and parasitized conditions, to objectively separate an individual longitudinal choice data into two segments. The pristine phase (Ω0 ) is postulated for the setting when the environment is abundant with ‘‘good quality’’ hosts, while the parasitized phase (Ω1 ) is for the setting when a ‘‘good quality’’ host is hard to find. In our motivating example, from the starting point of the pristine phase with 64 clean adzuki beans, it is known that a female beetle can effectively carry out her decision-making by rejecting almost all host beans with eggs. At this stage, the weight is the most essential characteristic of host bean quality. Denote the average weights of the four sizes of bean as w1 , w2 , w3 , w4 , in increasing order. Further we know qualitatively that a female beetle hardly accepts a host bean that has been laid with one or more eggs, when she still encounters clean bean from time to time. In contrast, only when a female has been frustrated by encountering a long enough series of hosts with positive egg load, does she then switch to accepting a host loaded with one egg. To summarize above established knowledge, we propose the following two manifolds for pristine and parasitized phases, respectively. At any longitudinal time point t, the Ω can be represented by the distribution data of number of hosts with
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Fig. 2. The surface of larva-to-adult survivorship of the seed beetle from seeds with specific weights (x) and contain y larvae in them.
an egg-load being larger than or equal to 1, O(t ) = (O1 (t ), O2 (t ), O3 (t ), O4 (t )), and equivalently by the distribution of the remaining number of clean hosts N (t ) = (N1 (t ), N2 (t ), N3 (t ), N4 (t )) with Ni (t ) = 16 − Oi (t ). [Manifold Specifications]: 1. When an environment is in its pristine phase, the manifold M (Ω0 ) is accordingly defined via host quality, that is, wi for a clean bean in i-th weight category and −∞ for any host with a positive egg load; 2. When an environment is in its parasitized phase, the manifold M (Ω1 ) is defined via the fitness-gain given in Fig. 2. It is seen in Fig. 2 that there is a large gap separating fitness-gains of 1-egg-beans from 2- and 3-egg-beans. This gap heuristically implies that there would be no significant change in the decision-making mechanism, separately, if we substitute M (Ω0 ) by M (Ω1 ). The reason that these two manifolds are discussed separately is to illustrate one important aspect of dynamic decision-making modeling: different choices of manifold could lead to employ different perspectives of longitudinal distribution data. Next the dynamic decision-making models are derived via the following conditional transition probability models based on manifolds M (Ωi ), i = 0, 1, respectively. [Conditional transition probability specifications] 1. [Under the pristine phase with M (Ω0 ) manifold:] Then conditional probabilities of adding one more egg on one host size i at time tk+1 given the previously observed egg-distribution O(tk ) is modeled by: Pi [β; O(tk )] = Ni (tk )eβwi
− 4
Nj (tk )eβwj
for i = 1, 2, 3, 4.
(1)
j =1
2. [Under the parasitized phase with M (Ω1 ) manifold:] A female beetle’s decision-making of host selection adaptively discriminates the total fitness value pertaining to the category, instead of only the number of clean host beans. Here the total fitness value of category i at time tk is the linear sum of fitness-gains values in manifold M (Ω1 ) of all 16 beans in the category, and denotes this value by Fi (tk ). Then correspondingly the conditional probability of observing one more egg on host size i given egg-dispersion O(tk ) is modeled by: Pi [β; O(tk )] = Fi (tk )e ∗
βwi
− 4
Fj (tk )eβwj
for i = 1, 2, 3, 4.
(2)
j =1
The probabilities given in Eq. (1) are understood in the following fashion. First, when β = 0, then Pi [0; O(tk )] = ∑4 Ni (tk )/ j=1 Nj (tk ), that is, the probability of observing one egg laid on a particular host size depends only on the remaining number of clean beans of that size. This stands for the extreme case where the female beetle has no capability of discriminating among host weights. Second, when β is a very large positive number, then Pi [β; O(tk )] = Ni (tk )eβ(wi −w4 )
− 4
Nj (tk )eβ(wj −w4 )
j =1
is close to 0 for i = 1, 2, 3, since wi − w4 < 0, and is nearly equal to 1 with i = 4 if N4 (tk ) > 0.. That is another extreme case that her decision-making criterion is to lay egg only on the largest host bean as if the female beetle ‘‘knows’’ the complete environmental conditions. Third, if β is negative and far away from 0, then the probability given in (1) prescribes a decisionmaking criteria of only laying an egg on the smallest clean host bean. This is an unrealistic case. Hence we expect β to be positive in this seed beetle example. It is also worth emphasizing again that the dynamic decision-making mechanism
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specified by manifold M (Ω0 ) and the conditional transition probability in (1) has the built-in criteria of rejecting any host bean with eggs. Within the parasitized phase, the female beetle might search for long enough period of time without encountering any clean host bean. This experience would force her to change her decision-making criteria, including accepting a host loaded with an egg for oviposition. Thus it is biologically reasonable to model the dynamic decision-making mechanism via manifold M (Ω1 ) and conditional transitional probability in (2) under the parasitized phase. Behaviorally speaking the decision-making mechanism proposed with manifold M (Ω0 ) takes 1-egg-bean as an inhibitory stimulus, and only the 0-egg-bean is excitatory within the pristine phase. In contrast, all host beans either with or without an egg load are excitatory stimuli under the framework of manifold M (Ω1 ) within the parasitized phase. Therefore probabilities (1) and (2) are indeed coherent and the parameter β in both equations should be comparable to a certain degree. 5. Minimum sum of Chi-squared statistics In this section we construct the minimum sum of Chi-squared statistics within the pristine and parasitized phases. Usually there are several ovipositions involved with a transition from O(tk ) to O(tk+1 ). Denote the vector of increment in the four categories as 1O(tk+1 ) = O(tk+1 ) − O(tk ). We can estimate β via the minimum Chi-squared estimation by finding the minimizer of Chi-squared statistics at time tk+1 :
χ 2 (β|k + 1) =
4 − (1Oj (tk+1 ) − Nj (tk )Pj [β; O(tk )])2 j=1
Nj (tk )Pj [β; O(tk )]
.
(3)
Denote the estimate by βˆ (k+1) and minimum Chi-square value by χ 2 (βˆ k+1 |k + 1). The advantage of this statistical inference is that we can simultaneously perform the estimation for the β parameter, and the testing for goodness-of-fit of the decisionmaking model within the pristine phase via χ 2 (βˆ k+1 |k + 1). We illustrate the above minimum Chi-square computations on a hypothetical case within the pristine phase, let O(tk ) = (0, 2, 9, 13) and O(tk+1 ) = (3, 5, 12, 16), so 1O(tk+1 ) = (3, 3, 3, 3). Intuitively the minimum Chi-square statistics will give rise to a large positive estimate βˆ (k+1) , since there are only three 0-egg-beans left in the size category 4 after time tk and the female does indeed pick them up in the pool of clean beans. That is, when N1 (tk ) + N2 (tk ) are about 10 times N4 (tk ) and in order to compensate these differences by adjusting factors in the Eq. (1) and observing 3 new oviposition on category 4, the βˆ (k+1) (=9.04) value must be positive and relatively large, and so the minimum Chi-square value is χ 2 (βˆ k+1 |k + 1) = 5.174, which has its p-value greater than 0.1 according to Chi-squared distribution with degree of freedom 3. The above example with a p-value nearly rejecting the dynamic decision-making model assumption heuristically indicates that a Chi-squared statistic based on one single transition of distribution data with too few ovipositions (much less than 12) would only provide a small amount of information and hinder the sensitivity of goodness-of-fit testing due to the increment of degrees of freedom. On the other hand, one single Chi-squared statistic based on all pooled transitions with too many ovipositions (much more than 12) would be neither efficient on parameter estimation, nor goodness-of-fit testing. Therefore we propose to accumulate successive transitions up to having around 12 oviposition for one Chi-squared statistic and then sum them up. We term this the sum of the Chi-square statistic. An explicit construction on the real data is given in the next section. In summary our construction of minimum sum of the Chi-squared statistic is to simultaneously balance the power of goodness-of-fit testing and efficiency of estimating the β parameter. Specifically a minimum sum of the Chi-squared statistic consisting of more separate Chi-squared statistics would have a higher efficiency in β estimation, while the power of the computed minimum Chi-squared statistic for testing the model assumption becomes less because of the larger degree of freedom, see Hsieh (2001) for more detail discussion. Further we can extend our dynamic decision-making model by accommodating the distribution of 0-, 1- and 2-egg beans in each size category of the longitudinal distribution data into the conditional transition probabilities given in (1) and (2). This kind of model extension would better describe an individual decision-making. 6. Real data analysis In this section we perform an individual based minimum sum of Chi-squared analysis of a dynamic decision-making model first on the pristine phase and then on the parasitized phase, separately. The duration of the pristine phase is further divided into three sub-periods, each of which contains around 12 ovipositions. Its duration is roughly equal to time spans of 8 or 12 h. Upon each sub-period a Chi-square statistic is constructed based on the probability in Eq. (3), and denoted by χ12 (β), χ22 (β) & χ32 (β). And then we derive the βˆ for the pristine phase by computing the minimizer of the sum of the Chi-square statistic χT2 (β) = χ12 (β) + χ22 (β) + χ32 (β). This minimum sum of Chi-squared computation is performed for each of the 13 individuals in the three designs. As the larger positive βˆ value indicates that the individual is choosier, we would like to check whether female beetles facing higher resource heterogeneity tend to be significantly choosier or not. Comparisons via the Receiver Operating Characteristic (ROC) curve based on the three collections of βˆ estimates are proposed to shed light on the above biological problem. An introductory discussion and asymptotical theory with connection to nonparametric ROC statistics are given
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Table 1 Minimum Chi-squares values and associated beta values (in parentheses) arrived at by fitting model to sequential decisions made by seed beetles in different environments (* means the Chi-square value is larger than critical point of 0.05 significant; d.f. = 9).
χ 2 (βˆ )
Pristine phase
Parasitized phase
Design
1
2
3
4
5
6
7
8
9
10
11
12
13
2×2
2.69 (2.59)
6.37 (7.2)
4.25 (3.36)
14.28 (5.06)
17.97* (−0.3)
6.12 (−0.7)
3.22 (−3.04)
17.87* (2.07)
7.73 (2.4)
10.28 (4.09)
6.73 (0.81)
7.29 (2.05)
7.02 (2.00)
4×4
4.91 (6.16)
8.75 (6.08)
4.71 (2.14)
3.13 (1.67)
2.93 (4.17)
5.90 (3)
1.53 (3.18)
0.49 (2.58)
1.26 (−0.34)
8.02 (4.66)
1.51 (4.27)
4.53 (4.38)
5.31 (3.47)
8×8
4.95 7.92 (13.61) (5.69)
5.12 (3.20)
3.85 (3.70)
9.02 (2.99)
9.53 (−2.25)
2.34 (3.19)
4.50 (1.04)
3.14 (3.22)
4.13 (4.90)
6.60 (3.19)
2.54 (3.17)
6.60 (4.89)
2×2
4.05 (2.18)
7.66 (9.57)
6.10 (7.57)
4.17 (1.80)
0.33 (18.14)
3.67 (0.12)
– –
1.88 (1.84)
6.52 (4.63)
11.97* (4.52)
1.17 7.69 (11.97) (4.64)
8.47 (3.73)
4×4
4.86 (8.09)
8.33 (5.02)
5.31 (7.70)
– –
5.41 (1.96)
3.29 (2.36)
6.37 (2.27)
– –
5.96 (4.83)
1.69 15.68* (10.08) (4.27)
0.85 5.19 (11.57) (4.73)
8×8
0.83 (5.36)
5.99 (−0.01)
7.13 (7.47)
4.42 (5.98)
4.07 (6.90)
– –
4.34 (9.28)
5.07 (2.17)
4.46 (4.24)
5.38 (3.06)
– –
– –
2.14 (4.83)
in Hsieh and Turnbull (1996). The construction of an ROC curve for comparing two empirical distributions is carried out in the following fashion. We rank the estimated values in increasing order, denoting them as {βˆ 2j }, {βˆ 4j }, {βˆ 8j }, j = 1, 2, . . . 13 for 2 × 2, 4 × 4 and 8 × 8 designs, respectively. Correspondingly the minimum sum of Chi-square values are calculated and denoted by χT2 (βˆ 2j ), χT2 (βˆ 4j ) & χT2 (βˆ 8j ), j = 1, 2, . . . 13 (Table 1). The ROC curve of a pair of distributions, for example #of {βˆ ≤βˆ
,j=1,2,...,13}
2j 4k k 1 3 {βˆ 2j }vs.{βˆ 4j }, is constructed as a curve of ( 13 , ), k = 1, 2, . . . , 13. For example, ( 13 , 13 ) means that 13 2 5 ˆ ˆ there are three β2j estimates smaller than or equal to the smallest value among the 13 β4j estimates, and ( 13 , 13 ) indicates that there are five βˆ 2j values smaller than or equal to the second smallest βˆ 4j estimates. In this comparison the distribution corresponding to {βˆ 4j } is taken as the baseline. This choice is based on that values of {βˆ 2j } tend to be stochastically smaller than values of {βˆ 4j }, the ROC curve is concave and above the diagonal line, i.e., female beetles are choosier by being able to discriminate host size much better in 4 × 4 than in 2 × 2 designs. In contrast, values of {βˆ 8j } are stochastically larger than values of {βˆ 4j }, and the ROC curve is convex and below the diagonal, see the two ROC curves {βˆ 2j }vs.{βˆ 4j } and {βˆ 8j }vs.{βˆ 4j }
shown in Fig. 3a. For the parasitized phase, the number of ovipositions varies more from individual to individual than in the pristine phase. Therefore there might be only one or two sub-periods in which the cumulative increment of oviposition is above 12. Likewise we calculate minimum sum of Chi-squared estimate of parameter β and its sum of the Chi-squared value χT2 (β) based on the ∗ probability given in Eq. (2), and denoted {βˆ 2j }, {βˆ 4j∗ }{βˆ 8j∗ }, and χT2∗ (βˆ 2j ), χT2∗ (βˆ 4j ) & χT2∗ (βˆ 8j ), j = 1, 2, . . . 13 (Table 1). The
∗ two ROC curves for the {βˆ 2j }vs.{βˆ 4j∗ } and {βˆ 8j∗ }vs.{βˆ 4j∗ } in the parasitized phase are seen being intersecting with the diagonal in Fig. 3b. Thus we can conclude that in general the resource heterogeneity evidently leads individuals to become choosier when making a decision in the pristine phase, but not in the parasitized phase. This condition is reasonable because host quality is more critically determined by egg-load than by host size in this phase. In comparison of β estimates of the pristine phase with that of parasitized phases, as summarized in Fig. 4, we see that ∗ the {βˆ 2j } is significantly stochastically smaller than {βˆ 2j }, so is {βˆ 4j }vs.{βˆ 4j∗ }. And it is less evident in the case of {βˆ 8j }vs.{βˆ 8j∗ }. Overall these comparisons coincide with the biological understanding that an animal in general invests more foraging efforts on host quality discrimination when the environmental condition deteriorates into a phase where hosts of ideal quality become significantly harder to find. From the goodness-of-fit perspective, the model (1) in the pristine phase and model (2) in the parasitized phase work reasonably well based on results reported in Table 1. Hence we can conclude that the female’s decision-making criteria are reasonably prescribed by the fitness manifold throughout her decision-making process going from pristine to parasitized phases.
7. Discussion In this paper we propose a new way of constructing dynamic decision-making models that can adapt to ever changing environmental conditions. The novel idea is to build dynamic decision-making models based on a self-adaptive fitness manifold, which has an empirical basis derived from experimental and relevant biological knowledge. We also propose a new minimum sum of Chi-squared statistics for analyzing individual longitudinal distribution data, which is a new data type in longitudinal analysis. And the ROC analyses synthesizing three minimum sum of Chi-squared analyses on three real data sets provide a resolution to an interesting and important animal behavioral issue regarding host selection under different environmental phases. Statistical significances of our manifold based dynamic decision-making modeling are that they provide a flexible platform for constructing a more refined model for analyzing behavioral and cognitive processing on an individual level.
F. Hsieh, R.-L. Yang / Journal of the Korean Statistical Society 39 (2010) 409–416
415
Fig. 3. The number of βˆ estimates in 2 × 2 or 8 × 8 designs are smaller or equal to the value of {βˆ 4j } at the (a) pristine and (b) parasitized phases of seed beetles’ oviposition process.
8 pristine phase
Mean beta value
7
parasitized phase
6 5 4 3 2 1 0
2x2
4x4 Designs
8x8
Fig. 4. The level of choosiness in seed size (means of βˆ ) by seed beetles in different heterogeneous designs at different phases of the oviposition process (Mean ± SE).
Though the computational simplicity of minimum sum of Chi-squared statistics can be seen as a pilot analysis of the more refined martingale based analysis (Hsieh et al., 2007), it is necessary to further comment this from information content and computational perspectives. The martingale approach requires us to summarize the associated high dimensional
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martingale difference into one dimensional martingale statistics and perform the conditional variation calculations at every observational time point. First, any dimension reduction transformation requires the resultant summarizing statistics to retain the martingale structure. Thus most often a linear transformation is employed. However a linear transformation could hardly capture the dynamic features of decision-making simultaneously involved with evaluating and balancing two or more biological characteristics. Second, the involving computational loading of the conditional variations could be also heavy as well. Finally and critically we need to justify the theoretical approximation via a martingale central theorem applied to the ratio statistic of the sum of transformed martingale differences and square-root of the sum of conditional variations, and consider its approximation error. In contrast, the minimum sum of Chi-square computations has an essential computational advantage. And the required theoretical approximation seems more evident and natural for most of real world applications. Biological significances of our modeling methodology proposed here are the following. First our design of resource heterogeneity ranging from 2 × 2 to 8 × 8 resolution on 8 × 8 lattices could be useful prototypes in designing experiments to induce stochastically distinct input-experiences onto individual study subjects. Second we can explicitly evaluate individual female seed beetles’ size-discrimination capability (secondary) with simultaneously adjustment upon eggload discrimination capability (primary) under distinct phases of environmental conditions. Hence behavioral scientists need not redirect their research efforts to pursue certain marginal stimulus–response (S–R) relationships. These S–R relationships often are not flexible enough to effectively predict individual behavioral processing under ever changing environmental conditions. And their corresponding analysis could be unreliable because of difficulties stemming from individual heterogeneity. Finally we conclude our modeling methodology and the statistical analyzing approach proposed here might greatly widen the spectrum of the study of animal cognitive tactics of ecological relevance without the burden of experimental and analytic limitations involved in the S–R relationship. Our constructions of dynamic decision-making models are not limited to animal behavioral study per se. They can be indispensable in the study of genetic and physiological mechanisms underpinning a target behavioral strategy. By selection a strain of a capable performer could be established against a strain of a poor one. And then genetic comparison can be reliably used to identify potential regulating genes and their pathways leading to the intelligent strategy of interest (Fox, Stillwell, Amarillo-s, Czesak, & Messina, 2004; Kuo, 2004; Messina & Slade, 1997). Acknowledgements This work was partly supported by Academia Sinica of Taiwan and the National Science Council of Taiwan through the grant (NSC 93-2118-M-001-002). References Clark, C. W., & Mangel, M. (2000). Dynamic state variable models in ecology: Methods and applications. Oxford: Oxford University Press. Fox, C. W., Stillwell, R. C., Amarillo-s, A. R., Czesak, M. E., & Messina, F. J. (2004). Genetic architecture of population differences in oviposition behaviour of the seed beetle Callosobruchus maculatus. Journal of Evolutionary Biology, 17, 1141–1151. Goubault, M., Plantegenest, M., Poinsot, D., & Cortesero, A. M. (2003). Effects of expected offspring survival probability on host selection in a solitary parasitoid. Entomological Experimentalis et Applicata, 109, 123–131. Heimpel, G. E., Rosenheim, J. A., & Mangel, M. (1996). Egg limitation, host quality, and dynamic behavior by a parasitoid in the field. Ecology, 77, 2410–2420. Horng, S.-B. (1997). Larval competition and egg-laying decisions by the bean weevil, Callosobruchus maculatus. Animal Behaviour, 53, 1–12. Houston, A. I., & McNamara, J. M. (1999). Models of adaptive behaviour: An approach based on state. Cambridge: Cambridge University Press. Hsieh, F. (2001). On heteroscedastic hazards regression models: Theory and application. Journal of the Royal Statistical Society: Series B, 63, 63–79. Hsieh, F., & Turnbull, B. W. (1996). Non- and semi-parametric estimation of the receiver operating characteristics (ROC) curve. Annals of Statistics, 24, 25–40. Hsieh, F., Horng, S.-B., Lin, H.-Y., & Lan, Y.-C. (2007). Testing dynamic rules of animal cognitive processing with longitudinal distribution data. Statistica Sinica, 17, 735–748. Kuo, T.-H. (2004). Behavioral and genetic studies of incipient speciation in Drosophila melanogaster. Master’s thesis. National Tsing Hua University, Taiwan. McGregor, R. R., & Roitberg, B. D. (2000). Size-selective oviposition by parasitoids and the evolution of life-history timing in hosts: Fixed preferences vs. frequency-dependent host selection. Oikos, 89, 305–312. Messina, F. J., & Renwick, J. A. A. (1985). Mechanism of egg recognition by the cowpea weevil Callosobruchus maculatus. Entomological Experimentalis et Applicata, 37, 241–245. Messina, F. J., & Slade, A. F. (1997). Inheritance of host-plant choice in the seed beetle Callosobruchus maculatus (Coleoptera: Bruchidae). Annals of the Entomological Society of America, 90, 848–855. Mitchell, R. (1990). Behavioral ecology of Callosobruchus maculatus. In K. Fujii, A. M. R. Gatehouse, C. D. Johnson, R. Mitchel, & T. Yoshida (Eds.), Bruchids and Legumes: Economics, ecology, and coevolution (pp. 317–330). New York: Springer Publishing Company. Morris, R. J., & Fellowes, M. D. E. (2002). Learning and natal host influence host preference, handling time and sex allocation behaviour in a pupal parasitoid. Behavioral Ecology & Sociobiology, 51, 386–393. Shettleworth, S. J. (1998). Cognition, evolution, and behavior. Oxford: Oxford University press. Southgate, B. J. (1979). Biology of Bruchidae. Annual Review of Entomology, 24, 449–473. Thanthianga, C., & Mitchell, R. (1990). The fecundity and oviposition behavior of South Indian strain of Callosobruchus maculatus. Entomological Experimentalis et Applicata, 57, 133–142. Wasserman, S. S. (1986). Genetic variation in adaptation to foodplants among populations of the southern cowpea weevil, Callosobruchus maculatus: Evolution of oviposition preference. Entomological Experimentalis et Applicata, 42, 201–212. Yang, R.-L. 2004. Host size discrimination behavior and its mechanism of Callosobruchus maculatus (F.) in heterogeneous environment. Ph.D. thesis. National Taiwan University, Taiwan. Yang, R.-L., Fushing, H., & Horng, S.-B. (2006). Effects of search experience in a resource-heterogeneous environment on the oviposition decision of the seed beetle, Callosobruchus maculatus (F.). Ecological Entomology, 31, 285–293.