bayesian methods in plant conservation biology
TRANSCRIPT
Bayesian methods in plant conservation biology
Juan M. Marın*, Raquel Montes Diez, David Rıos Insua
Statistics and Decision Sciences Group, Department of Experimental Sciences and Engineering, Universidad Rey Juan Carlos,
Tulipan s/n. 28933 Mostoles, Madrid, Spain
Abstract
We provide an introduction to Bayesian methods in conservation biology, illustrating inferences, prediction and decision makingissues. After presenting the basic framework with a recovery plan evaluation problem, we illustrate more complex issues related toforecasting trends of structured populations using matrix population models and, finally, describe relevant topics in spatial and
logistic regression problems. Computational and other implementation difficulties are also discussed.# 2003 Published by Elsevier Science Ltd.
Keywords: Conservation biology; Bayesian analysis; Matrix population models; Recovery plans; Viability; Mixtures
1. Introduction
The Bayesian framework provides a unified andcoherent approach to solving inference and decisionmaking problems under conditions of uncertainty. Assuch, we would expect Bayesian methods to have a greatimpact on conservation biology. Conservation scientistsgather and analyse data to improve resource manage-ment in environmental problems. These data areusually affected by uncertainty and therefore require astatistical analysis. In environmental and ecologicalproblems, the end use of such analysis is often to reach adecision, hence requiring uncertainty to be formallyaccounted for in the decision making process. Bayesiananalysis seems, therefore, the appropriate approach tobioconservation statistical analysis, as it provides acoherent framework that integrates multiple sources ofuncertain information (data, expert opinion) within deci-sion making. French and Rıos Insua (2000) provide acomplete description.
Many other advantages support the use of Baye-sian methods in biological conservation studies. Forexample, prior information is frequently available,possibly from previous related studies from whichour analysis might benefit, if appropriately mod-elled. Also, we sometimes need to combine severalstudies, see Osenberg et al. (1999) for a review on
metanalysis issues in ecology, and the Bayesianapproach provides a natural way of tackling those pro-blems through hierarchical analysis. Spatial issues arealso becoming increasingly important in ecology andrecent Bayesian models provide useful tools to handlethem.
However powerful we might think the Bayesianapproach is, we should admit that most statisticalanalyses in this field are still performed within theclassical paradigm, although the situation seems to begradually changing. As in other areas, controversy haspermeated the eventual application of Bayesian meth-ods in ecology—see Dennis (1996) for an example. Inspite of its foundational strength, we believe that thebest way to introduce and further advance these meth-ods in conservation biology is through practice. This isprecisely the goal of this paper, to provide a motivatedintroduction to Bayesian methods. Other reviews inthe field, such as Reckhow (1990), Ellison (1996), orWade (2000), have remained at a more conceptual level.We shall emphasise issues related to plant conservationbiology.
After introducing the basic Bayesian framework, wefirst illustrate the ideas with a simple example relatingto the assessment of alternative actions in recoveryplans. We then describe an alternative approach tomatrix population models based on dynamic linearmodels. In Section 5, we illustrate a logistic model toforecast germination, comparing Bayesian and stan-dard methods. Finally, we introduce a novel model tohandle spatial issues, when spatial clustering of indivi-duals takes place.
0006-3207/03/$ - see front matter # 2003 Published by Elsevier Science Ltd.
doi:10.1016/S0006-3207(03)00124-1
Biological Conservation 113 (2003) 379–387
www.elsevier.com/locate/biocon
* Corresponding author. Tel.: +34-916647475; fax: +34-
916647434.
E-mail address: [email protected] (J.M. Marın).
2. An overview of Bayesian methods
The Bayesian framework for inference and decisionmaking problems can be easily described. Indeed, wefeel that one of its strengths is, precisely, the ease withwhich basic concepts are put into place.
Frequently one of the goals in conservation biology isto learn about one or more parameters (e.g. plant mor-tality rates, expected lifetimes or dispersion) whichdescribe an ecological phenomenon of interest. � Shalldesignate such parameters. To learn about them, weshall observe the phenomenon, collect data (e.g. countsof plants at various ages or their locations) and form thelikelihood p xj�ð Þ which describes how data x depends(probabilistically) on the parameters �, that is the dis-tribution of data x when the parameter values are fixedat �. The first novel ingredient in the Bayesian approachis to take into account another source of informationabout the parameters: very frequently, the conservationbiologist will have access to expert information, basedon past experience or related studies, which we shallmodel through the prior distribution p �ð Þ.
We now need a way to combine both sources ofinformation. This is done through Bayes’ formula. Itprovides the posterior distribution p xj�ð Þ, which is thedistribution of the parameter � given the data x:
p �jxð Þ ¼p �ð Þp xj�ð ÞÐp �ð Þp xj�ð Þd�
:
Note that the posterior will be proportional ( /) tothe numerator, which we shall represent
p �jxð Þ / p �ð Þp xj�ð Þ:
The posterior distribution summarises all availableinformation about the parameters and may be used tosolve all inference problems of interest in conservationbiology, such as point and interval estimation, hypoth-esis testing or prediction. For example, for point esti-mation, we could summarise the posterior through, forexample, its mean, that is
E �jxð Þ ¼
ð�p �jxð Þd�:
To predict further y values of the phenomenon, wecan use the predictive distribution. If we knew the valueof �, we would use the distribution p yj�ð Þ. Since there isuncertainty about �, modelled through the posteriorp �jxð Þ, we integrate it out to get the predictive
p yjxð Þ ¼
ðp yj�ð Þp �jxð Þd�:
The ultimate aim of statistical research in conserva-tion biology is usually to support decision making, forinstance, to decide between several alternative actions ina recovery plan. For each action a and each future result
y, we would associate a consequence c a; yð Þ. For exam-ple, if action a succeeds (y, the future result in this case),we would have incurred certain expenses, created somejobs and recovered a certain piece of land for conserva-tion purposes. Such consequence will be evaluated withits utility u a; yð Þ and we will choose the action max-imising the expected utilityð
u a; yð Þp yjxð Þdy:
Our aim now is to illustrate these ideas with severalapplications in plant conservation biology, showingtheir potentiality and their computational and model-ling subtleties.
3. A simple example: evaluating recovery plans
To illustrate the basic concepts, we provide examplesconcerning the evaluation of actions in recovery plans.We assume that a certain protocol establishes action Afor a certain plant. Our aim is to evaluate its success.The parameter in this case is the probability �1 of asuccessful intervention through action A. Let us sup-pose that 9 out of 12 cases ended in success. If X1 is thenumber of successes in 12 trials, we have a binomialmodel Bin 12; �1ð Þ and, accordingly, we have that theprobability of obtaining nine successful interventions in12 trials given �1 is:
Pr X1 ¼ 9j�1ð Þ ¼129
� ��91 1 � �1ð Þ
3:
As we are novel to the problem, let us view all valuesof �1 as being equally likely and use the uniform priorp �1ð Þ ¼ 1; �1 2 0; 1½ . Hence, applying Bayes’ formula,we get the posterior
p �1jX1 ¼ 9ð Þ / p �1ð Þ Pr X1 ¼ 9j�1ð Þ
/ 1 �91 1 � �1ð Þ3; �1 2 0; 1½
which corresponds to a beta distribution with para-meters � ¼ 10 and � ¼ 4 that we represent as Be 10; 4ð Þ.Remember that a random variable � has a Be �; �ð Þ dis-tribution, �; � > 0, if p �ð Þ ¼
G �þ�ð Þ
G �ð ÞG �ð Þ���1 1 � �ð Þ
��1, 8� 20; 1½ its mean being �
�þ�.Given the sequential nature of Bayes’ formula, this
posterior would serve as prior for further studies. Forexample, suppose that in a subsequent study, three outof five implementations of our action A were successful.The new posterior would be
p �1jX1 ¼ 9;X2 ¼ 3ð Þ / p �1jX1 ¼ 9ð ÞPr X2 ¼ 3j�1ð Þ
/ �91 1��1ð Þ3�31 1��1ð Þ
2
/ �121 1��1ð Þ5; �1 2 0; 1½
corresponding to a Be 13; 6ð Þ distribution.
380 J.M. Marın et al. / Biological Conservation 113 (2003) 379–387
As mentioned, we may sometimes have expert infor-mation available about parameters. For example, let ussuppose that we believe that higher values of �1 aremore likely and we expect that around 80% of the timesthat this action is applied in a similar environment it willbe successful. We could model this with a Be 4; 1ð Þ priordistribution. Note that, as required, its mean is 0:8 andits mode is 1. With such prior, the posteriors would beBe 13; 4ð Þ, after the first sample, and Be 16; 6ð Þ, after thesecond one.
Fig. 1 provides the six distributions involved in thisexample. Those in the left column refer to the evolutionof beliefs about �1, the probability of successful imple-mentation of action A stemming from the uniformprior, whereas those on the right stem from the Be 4; 1ð Þ
prior. The left column shows a remarkable change inbeliefs, from total uncertainty to a distribution fairlyconcentrated in the region [0.5,0.9]. Also note that theposterior becomes increasingly concentrated as moredata become available. Similar comments hold for theright column, though we model a lot of prior informa-tion, and, therefore, the evolution of beliefs is not thatremarkable. Finally, note that the bottom figures arefairly similar, showing the usual fact that, as dataincrease, prior relevance mitigates.
The posterior distribution contains all relevant infor-mation for further inferences and predictions. The
following comments refer to the case that stems fromuniform prior beliefs and we have observed nine suc-cesses and three failures, i.e. the current beliefs about �1follow a Be 10; 4ð Þ distribution. Let us suppose we need apoint estimate for the probability of success of action A.We could use the posterior mean, which is10= 10 þ 4ð Þ ¼ 0:72. Should we need an interval esti-mate, we could provide a region which covers �1 withposterior probability 0:9; for example, we have thatp �1 2 0:505; 0:887½ jX1 ¼ 9ð Þ 0:9; where we have cho-sen the interval so that p �1 4 0:505jX1 ¼ 9ð Þ 0:05 andp �1 5 0:887jX1 ¼ 9ð Þ 0:05. From an applied point ofview, we may be more interested in computing pre-dictive probabilities. For example, suppose that we areabout to implement action A in seven more locationsand we are interested in computing the probability ofsucceeding five or more times. We have, for eachk; k ¼ 0; . . . ; 7, that
Pr Y ¼ kjX1 ¼ 9ð Þ ¼
ðPr Y ¼ kj�1ð Þp �1jX1 ¼ 9ð Þd�1
¼
ð7k
� ��k1 1 � �1ð Þ
7-k G 14ð Þ
G 10ð ÞG 4ð Þ�91 1 � �1ð Þ
3d�1
¼7k
� �G 14ð Þ
G 10ð ÞG 4ð Þ
ð�9þk1 1 � �1ð Þ
10�kd�1
¼7k
� �13! 9 þ kð Þ! 10 � kð Þ!
9!3!20!
Fig. 1. Distributions involved in the recovery plan evaluation problem. �1: probability of a successful intervention through action A.
J.M. Marın et al. / Biological Conservation 113 (2003) 379–387 381
Therefore,
Pr Y5 5jX1 ¼ 9ð Þ ¼X7
k¼5
Pr Y ¼ kjX1 ¼ 9ð Þ; 0:6641:
The comparison of alternative actions follows a simi-lar path. Suppose, for example, that another action Bhas been used in a similar context, resulting in 10applications with Z ¼ 6 successes. We compare �1, theprobability of success with action A, with �2, the prob-ability of success with action B. The model we consideris
Xj�1 � Bin 12; �1ð Þ
Zj�2 � Bin 10; �2ð Þ
�1; �2 � Unif 0; 1½ ; independent:
For comparison purposes, we computer ¼ Pr �1 5 �2jX1 ¼ 9;Z ¼ 6ð Þ. To do so, we see that, aposteriori, �1jX1 ¼ 9 � Be 10; 4ð Þ, �2jZ ¼ 6 � Be 7; 5ð Þ
and that they are independent. The posterior distribu-tion of �1��2 cannot be obtained in closed form, but wemay easily simulate it. For example, we may draw 1000values from the posterior distributions of �1 and �2,obtain their differences �1��2, count the number of suchdifferences which are non-negative and divide by 1000to approximate r. In our example, we undertook such aprocedure to obtain r 0:772. This result suggests thataction A seems better than action B, as we have a betterchance of success with the former.
However, to formally make a decision, we shouldconsider the utility of various consequences. We shalluse a simplified utility function here; for further detailssee French and Rıos Insua (2000). Suppose that actionA implies a considerable cost, but action B implies vir-tually no cost. Let us also suppose that the utility func-tion is as described in Table 1 where, for example, if weimplement action A successfully, the utility obtained is0.8. Then, conditional on �1 and �2, the expected utilitiesof the actions are, respectively, 0:8�1 þ 0 1 � �1ð Þ ¼ 0:8�1and �2 þ 0:2 1 � �2ð Þ ¼ 0:2 þ 0:8�2, whereas their poster-ior expected utilities are, respectively,
0:8E �1jX1 ¼ 9ð Þ ¼ 0:8 10
14¼
4
7;
and
382 J.M. Marın et al. / Biological
0:2 þ 0:8E �2jZ ¼ 6ð Þ ¼ 0:2 þ 0:8 7
12¼
2
3:
Therefore, we would actually recommend plan B,since its expected utility is higher.
4. A complex example: forecasting population viability
One of the most important issues in conservationbiology is the evaluation of a population’s stability interms of its size at each stage of its biological cycle. Thefinal goal is usually to predict whether a population isdecreasing, increasing or is stabilised, hence assessing itsviability. To do this, a demographic analysis and amethodology to study population growth trajectoriesand identify life history stages are required.
Many authors, for example, Schemske et al. (1994)and Caswell (2001), have advocated the use of matrixpopulation models in the context of conservation biol-ogy. Individuals are classified into several categoriesthat reflect natural stages of the life cycle. Fig. 2 shows ahypothetical graph of a plant population model withfive states: seed, seedling, juvenile, subadult and adult,with the corresponding valid transitions, pij prob-abilities and Fj per capita fertilities at each stage. Forexample, p12 describes the probability of a seed becom-ing a seedling after one year, and F5 is the number ofseeds produced by an adult per year. As usual in con-servation biology, we do not show a death stage eitherin the life cycle or in the transition matrix in Table 2where there are implicit transitions to the death stagewith the corresponding probability. For example, theprobability of seed mortality would be 1 � p11 � p12. AMarkov chain model is implicitly assumed, transitionprobabilities are estimated and the evolution of thepopulation size can be studied, for instance, to comparedifferent management strategies. Thus, this procedurecan be used to estimate seeding needs or the expectednumber of adults in a period of time.
To characterise the process, we shall consider theprobabilities given in the associated transition matrix inTable 2. The parameters defining our matrix populationmodel process are, therefore, y=(p11,p12,p22,p33,p34,p35,p44,p45,p53,p54,p55,F5). We assume they are, a priori,independent. Inference procedures for such parametersfollow.
For the p parameters corresponding to each column,that is, the fates of a particular life stage after a timeperiod, we shall assume a Dirichlet-multinomial model(see O’Hagan, 1994). This model, in which more thantwo classes are used, is a straightforward generalisationof the b-binomial model applied in Section 3. Let ussuppose that there are three classes and let X ¼
X1;X2;X3ð Þ designate the number of occurrences inclasses 1, 2 and 3 after n trials, with X1 þ X2 þ X3 ¼ n.Then, we assume Xj� � Mult n; �1; �2ð Þ, where �i is the
Table 1
Utilities for actions A and B in a recovery plan
Succeeds
Does not succeedAction A
0.8 0Action B
1 0.2Conservation 113 (2003) 379–387
probability of an observation in class i, i ¼ 1; 2 and �3 ¼1 � �1 � �2 is the probability of an observation in class3. A priori, � � Dir �1; �2; �3ð Þ, a Dirichlet distributionwith parameters �1; �2 and �3. Remember that theDirichlet distribution is a multivariate generalisation ofthe Beta distribution. For example, the Dir �1; �2; �3ð Þ
density is p �1; �2; �3ð Þ / ��1�11 ��2�1
2 ��3�13 where 0 < �i < 1
andP
i�i ¼ 1. A posteriori, we see that
�j x1; x2; x3ð Þ � Dir �1 þ x1; �2 þ x2; �3 þ x3ð Þ;
with posterior means
E �ijxð Þ ¼�i þ xiPj
�j þ n; i ¼ 1; 2; 3:
If there is little information from experts, we usenoninformative priors. We start with 1000 seeds, out of
which 800 germinate, 100 die and 100 remain dormant.From the 800 seedlings, 600 pass to juveniles, 180 dieand the rest remain in that stage, and so on until wecomplete a whole cycle.
For the first column, we would have that, a posteriori,
1 � p11 � p12; p11; p12ð Þ � Dir 100 þ 1; 100 þ 1; 800 þ 1ð Þ;
with, respectively, posterior means
100þ11000þ3 ¼ 0:1007; 100þ1
1000þ3 ¼ 0:1007; 800þ11000þ3 ¼ 0:7986
� �
and very small posterior variances, namely,
9:0197 10�5; 9:0197 10�5; 1:6019 10�4
:
The whole process is completed resulting in very con-centrated distributions around the posterior means,with matrix
P ¼
p11 0 0 0 0p12 p22 0 0 00 p23 p33 0 p53
0 0 p34 p44 p54
0 0 p35 p45 p55
0BBBB@
1CCCCA
Fig. 2. Transitions from different states of the life cycle of a plant.
Table 2
Associated population transition matrix
Seed
Seedling Juvenile Subadult AdultSeed
p11 0 0 0 F5Seedling
p12 p22 0 0 0Juvenile
0 p23 p33 0 p53Subadult
0 0 p34 p44 p54Adult
0 0 p35 p45 p55J.M. Marın et al. / Biological Conservation 113 (2003) 379–387 383
The above transition matrix P is conditional on theparameter values, that is, the entries are actuallypijj� ¼ Pr Xnþ1 ¼ jjXn ¼ i; �ð Þ. From a Bayesian point ofview, the transition matrix that we should consider isthe corresponding marginal distribution. However, asour posteriors are very concentrated (see Marın etal., 2002b), we may approximate pijjdata byPr Xnþ1 ¼ jjXn ¼ i; �
� �¼ pijj�, where � are our posterior
point estimates. F5 may be similarly estimated bycounting the number of seeds produced by severaladults. We could then use the methods in Caswell (2001)to undertake forecasting tasks. Other cases with moreuncertainty in the point estimates require simulation, aswe shall illustrate in a forthcoming paper.
A shortcoming of such models is that the involvedparameters are fixed through time. We describe here aDynamic Linear Model that permits dynamic variationon such parameters. Our aim is to forecast the popula-tion structure several periods (years) ahead, by using theavailable population structure history.
Let us suppose that nst; nsl
t ; njt; nsa
t and nat are, respec-
tively, the number of seeds, seedlings, juveniles, sub-adults and adults at a given year t. We model theunderlying population evolution in terms of an (auto-regressive) multivariate dynamic linear model, withobservation equations
nstþ1 ¼ pt
11nst þ Ft
5nat þ "t
s
nsltþ1 ¼ pt
12nst þ pt
22nslt þ "t
sl
njtþ1 ¼ pt
23nst þ pt
33njt þ pt
53nat þ "t
j
nsatþ1 ¼ pt
34njt þ pt
44nsat þ pt
54nat þ "t
sa
natþ1 ¼ pt
35njt þ pt
45nsat þ pt
55nat þ "t
a
where ptij; i; j ¼ 1; . . . ; 5 are the probabilities of changing
(surviving, growing,. . .) from stage i to stage j and Ft;5 isthe fertility per capita parameter, within period t. Forexample, the first equation describes that the number ofseeds ns;tþ1 in year t þ 1 depends on the number of seedspt11n
st , remaining at such stage from the previous year,
and seeds Ft5n
at coming from adults in the previous year.
Similarly, the last equation suggests that the number ofadults na
tþ1 in year t þ 1 depends on the numbers ofadults na
t , subadults nsat and juveniles nj
t in the previousyear. The error terms are assumed to be jointly indepen-dent and normal with zero mean and unknown variance.Note that if we drop these terms, we recover a standardmatrix population model with time dependent para-meters. We now assume that the parameters will varyrandomly with time according to the system equations
ptþ1ij ¼ pt
ij þ tij
F tþ15 ¼ Ft
5 þ t5
where the pij’s satisfy the constraints ptij 5 0 andP5
j¼1pij ¼ 1, 8i, and the t terms are also independentand normally distributed.
This model, in fact, generalises classical matrix popu-lation models by considering random evolving para-meters. As the parameters of the model may evolve withtime, the model includes a system of equations toupgrade their values. Following the Bayesian paradigm,we would assign a prior distribution to parameters pij
and F5 at time 0. The posterior distribution of para-meters is updated at each stage, also giving the pre-dictive distributions of the variables of interest severalstages ahead.
This class of models has been used in many otherfields within engineering and applied sciences. Its mainadvantage is its flexibility to adapt to many situations,where the calculations are automatically performed byusing the Kalman Filter procedure, a set of embeddedlinear equations that resume the main stages of estima-tions of the model. A full description of this technique isoutside the scope of this paper, but may be seen, forexample, in West and Harrison (1997).
5. Logistic regression
Logistic regression is a popular technique in life sci-ences. It can be regarded as an extension of standardregression models. Its main aim is to predict the prob-ability of success within a given trial in relation to agroup of predictive variables. For example, in a simplelogistic (binomial) regression, we may have independentobservations from I populations. We assume that eachone is distributed according to a binomial law:yi � Bin ni; �ið Þ, for i ¼ 1; . . . ; I, where E yið Þ ¼ ni�i, andthat the explanatory variable x;i is available. Odds aredefined as �i= 1 � �ið Þ. The logistic regression specifies alinear structure for the logarithm of the odds (log odds):
logit �ið Þ � log�i
1-�i
� �¼ �0 þ �1xi; i ¼ 1; . . . ; I:
Note that by varying xi, the term on the right ofequation can take any real value, but �i is alwaysrestricted between 0 and 1.
Standard methods for analysing binomial regressiondata rely on asymptotic inference, requiring a greatamount of data to provide accurate estimates. Thisproblem is straightforwardly worked out by Bayesianmethodology, allowing us to include prior information,use diagnostic studies, undertake model selectionamong different models and perform sensitivity analy-sis. This has been facilitated by the recent introductionof Markov chain Monte Carlo methods, which in ourmodel permit computations for prediction and makinginferences on regression coefficients.
384 J.M. Marın et al. / Biological Conservation 113 (2003) 379–387
To fix ideas, suppose we have data yi; x0i
; i ¼ 1; . . . n,
available where xis are known vectors of k covariatesand the yis are independent binomial random variableswith ni trials. The success probability � for any singletrial y with covariate x is F x0�ð Þ, i.e.F x0�ð Þ � � � Pr y ¼ 1jx; �ð Þ, where � is an unknownk-dimensional vector of regression coefficients, and F is
F x0�ð Þ ¼exp x0�ð Þ
1 þ exp x0�ð Þ:
The likelihood function for the complete data Y ¼
y1; . . . ynð Þ is
L �jYð Þ �Yn
i¼1
ni
yi
� �F x0
i� � �yi 1 � F x0
i� � �ni�yi
For a given prior distribution on �, namely � �ð Þ, toobtain posterior and predictive distributions we need tocompute the posterior distribution of �:
� �jYð Þ ¼L �jYð Þ� �ð ÞÐL �jYð Þ� �ð Þd�
:
As many important quantities are obtained throughcumbersome and intractable integrals involving � �jYð Þ,approximations must be undertaken.
Monte Carlo methods provide discrete approxima-tions to the posterior distribution. Given a functionh �ð Þ, the posterior expectation E h �ð ÞjY½ may beapproximated by
E h �ð ÞjY½ ¼
ðh �ð Þ� �jYð Þd�
1
t
Xt
i¼1
h �i
;
for a sample �i� �t
i¼1of size t, from the posterior dis-
tribution � �jYð Þ. The Strong Law of Large Numbersensures that the error in the approximation converges tozero as the simulation sample size t increases (seeFrench and Rıos Insua, 2000 for details). A number ofstrategies have been devised to obtain samples from theposterior, as e.g. Gibbs and Metropolis Hastings sam-plers. Interestingly, the whole process may be auto-mated for relatively complex models, as in BUGS, a freesoftware for Bayesian analysis using the Gibbs sampler(see Spiegelhalter et al., 1999).
We illustrate the previous concepts with an exampletaken from Crowder (1978). It concerns the proportionof seeds that germinated on each of 21 plates, arrangedaccording to a 22 factorial layout by seed and type ofroot extract. The data are shown in Table 3, where yi
and ni are the number of germinations and the totalnumber of seeds on the ith plate i ¼ 1; . . . ; n, respec-tively. The model is essentially a random effects logisticregression model, allowing for over-dispersion. If �i isthe probability of germination on the ith plate, weassume
yi � Bin ni; �ið Þ
logit �ið Þ ¼ �0 þ �1x1i þ �2x2i þ �12x1ix2i þ "i
"i � N 0; �2
where x1i, x2i are the seed type and root extract of theith plate, and an interaction term �12x1ix2i is included.�0, �1, �2, �12, �
2 are given independent noninformativepriors.
Table 4 shows the different estimations for the logisticregression model obtained with maximum likelihood andthe Gibbs sampler results using BUGS. Note, however,that in this case classical methods based on asymptoticproperties are not appropriate because of the small samplesizes; thus confidence intervals are unrealistic. The resultsbased on posterior inference show larger intervals, there-fore acknowledging more uncertainty.
6. Spatial issues
Spatial issues are of extreme importance in conserva-tion biology as we try to relate biological phenomena tolocation-related variables (soil, climate, presence ofhuman populations. . .) (e.g. Rubio and Escudero,2000). They have also been of recent interest in Bayesianstatistics (see Besag and Green, 1993, for general refer-ences and Gregoire et al., 1997, for references related toecology). Here we outline a model which, we feel, may berelevant in conservation biology. The underlying theme ismodelling with mixtures which, again, we feel is animportant issue in this field for classification and dis-crimination purposes, as we may model population het-erogeneity in a natural way. Diebolt and Robert (1994)provide a general framework for mixture modelling. Forfurther details and extensions, see Marın et al. (2002a).
Our goal is to model the spatial structure of a plantpopulation. Our basic data are the locations of theseplants. We shall consider an ecologically distributed
Table 3
Proportion of germinated seeds of two varieties of Orobanche aegyp-
tiaca in presence of bean and cucumber root extracts. (Crowder, 1978)
O. aegyptiaca 75
O. aegyptiaca 73Bean
Cucumber Bean Cucumbery
n y/n y n y/n y n y/n y n y/n10
39 0.26 5 6 0.83 8 16 0.50 3 12 0.2523
62 0.37 53 74 0.72 10 30 0.33 22 41 0.5423
81 0.28 55 72 0.76 8 28 0.29 15 30 0.5026
51 0.51 32 51 0.63 23 45 0.51 32 51 0.6317
39 0.44 46 79 0.58 0 4 0.0 3 7 0.43–
– – 10 13 0.77 – – – – – –y: number of germinated seeds in each plate; n: total number of seeds
in each plate.
J.M. Marın et al. / Biological Conservation 113 (2003) 379–387 385
environment, for example regulated by wind. Underusual circumstances, normal winds will facilitate seeddispersal and establishment of individuals in nearbylocations. Note that we shall have uncertainty about thenumber of clusters in the population.
At the level of observed data, we induce clustering byassuming a mixture of normal models with an unknownnumber M of terms. Therefore, we assume that plantcharacteristics yi are distributed as
XMj¼1
qjN �j j;Sj
; i ¼ 1; . . . ;N:
By this, we understand that plants are clustered (nor-mally) within one of M groups, each cluster centre being j and each cluster dispersion matrix being �j,j ¼ 1; . . . ;M. The proportion of plants within the jthgroup would be qj, j ¼ 1; . . . ;M, with
PMj¼1qj ¼ 1. The
model is completed with priors on the parameters,including the size M of the mixture, and the weightsq ¼ q1; . . . ; qMð Þ, as described in Marın et al. (2002a).
We have already mentioned the computational diffi-culties associated with performing posterior inference.Section 3 illustrated a direct Monte Carlo approach,whereas in Section 4 we described how to undertake anMCMC based approach using BUGS. In other cases,however, such automated approaches are not feasibleand we must perform specific analysis for the problemat hand.
As another example, let us suppose we have 48 mixedmeasurements on the time of seed germination of agiven species collected in a population (Table 5). Wewish to know if the collected seeds come from plantsoccurring in two different microhabitats. Thus, we fit amixture of two normal distributions with the same var-iance, so that each observation yi is assumed to bedrawn from one (but we do not know which) of the twomicrohabitats (1 or 2).
We denote Ti ¼ 1; 2 as the true microhabitat of the ithobservation, (i ¼ 1; . . . ; 48), where microhabitat j(j ¼ 1; 2) has a normal distribution with mean j andvariance �2. We assume that an unknown fraction p ofobservations belongs to microhabitat 1 and, hence, thefraction 1 � pð Þ belongs to microhabitat 2. The modelcan be written as
yi � N Ti; �2
where Ti, i ¼ 1; . . . ; 48, follows a discrete distribution
Ti ¼1 with probability p2 with probability 1 � pð Þ
�
Note that this formulation may be easily generalisedto additional components of the mixture. For the sakeof computational efficiency we make a reparameterisa-tion assuming that 2 ¼ 1 þ �; � > 0:
In this example 1, �, �2 and p, are given independent
noninformative priors, including a uniform prior for pon 0; 1ð Þ. Then, we apply a Markov chain Monte Carlomethod, with the Gibbs sampler, using BUGS. Weobtain samples (of size 10 000) from the posterior dis-tributions of the parameters of interest. Then, we con-sider the mean and median of the samples as a summaryof the distributions. A measure of precision of thesevalues, the standard deviation and 0.95 posterior inter-vals are shown in Table 6.
We obtain that 60% of the seeds would come frommicrohabitat 1. We can also see that the seeds frommicrohabitat 1 have a slightly shorter mean time ofgermination than the seeds from microhabitat 2(Table 6). The difference in the mean time of germina-tion might involve differences in seed physiology origi-nated by contrasting microhabitats.
Table 4
Different estimations for the logistic regression model. Maximum
likelihood and BUGS (Gibbs sampler results using BUGS software)
Variable
Maximum likelihood�� SE
BUGS �� SE
constant (�0)
�0.548�0.167 �0.542�0.178seed (�1)
0.097�0.278 0.028�0.340extract (�2)
1.337�0.237 1.368�0.253interaction (�12)
�0.811�0.385 �0.792�0.426scale (�)
0.236�0.110 0.292�0.152Table 5
Simulated times of germination of seeds collected in a population with two potential microhabitats
529.0
530.0 532.0 533.1 533.4 533.6 533.7 534.1 534.8535.3
535.4 535.9 536.1 536.3 536.4 536.6 537.0 537.4537.5
538.3 538.5 538.6 539.4 539.6 540.4 540.8 542.0542.8
543.0 543.5 543.8 543.9 545.3 546.2 548.8 548.7548.9
549.0 549.4 549.9 550.6 551.2 551.4 551.5 551.6552.8
552.9 553.2386 J.M. Marın et al. / Biological Conservation 113 (2003) 379–387
7. Concluding remarks
The final emphasis in conservation biology should bedecision-making issues concerning interventions. Theintrinsically uncertain nature of conservation biologymakes Bayesian analysis a natural approach within thisfield whenever statistical problems are considered.
To some extent we have conveyed the relevance ofBayesian methods through various examples and mod-els, covering areas of wide interest including space issuesand population structure forecasting. We have outlinedsome typical problems and possible solutions under theBayesian paradigm but now much further work has tobe done. We hope that conservation scientists willemphasise Bayesian ideas in their future studies.
Acknowledgements
Research supported by grants from CICYT, CAMand URJC. We are grateful to discussions with AdrianEscudero, Jose M. Iriondo, the participants at theThreatened Plant Conservation Biology Seminar held atUniversidad Politecnica de Madrid and the referees.
References
Besag, J., Green, P., 1993. Spatial Statistics and Bayesian computation
(with discussion). Journal Royal Statistics Society B 55, 25–37.
Caswell, H., 2001. Matrix Population Models. Sinauer, Sunderland.
Crowder, M.J., 1978. Beta–binomial Anova for proportions. Applied
Statistics 27, 34–37.
Dennis, B., 1996. Should ecologists become Bayesian. Ecological
Applications 6, 1095–1103.
Diebolt, J., Robert, C., 1994. Estimation of finite mixture distributions
through Bayesian sampling. Journal Royal Statistics Society B 56,
163–175.
Ellison, A., 1996. An introduction to Bayesian inference for ecological
research and environmental decision making. Ecological Applica-
tions 6, 1036–1046.
French, S., Rıos Insua, D., 2000. Statistical Decision Theory. Arnold,
London.
Gregoire, T.G., Billinger, D.R., Diggle, P.J., Russek-Cohen, E., War-
ren, W.G., Wolfinger, R.D. (Eds.), 1997. Modelling Longitudinal
and Spatially Correlated Data: Methods, Applications, and Future
Directions. Lecture Notes in Statistics 122. Springer-Verlag, New
York.
Marın, J.M., Muller, P., Rıos Insua, D., 2002a. Spatial clustering with
multi-level mixture models. Technical Report, U. Rey Juan Carlos.
Marın, J.M., Pla, L.M., Rıos Insua, D., 2002b. Inference for some
stochastic processes related with sow farm management. Technical
Report, U. Rey Juan Carlos.
O’Hagan, A., 1994. Bayesian Inference. Arnold, London.
Osenberg, C., Sarnelle, O., Goldberg, D., 1999. Meta-analysis in
Ecology. Ecology 80, 1103–1104.
Reckhow, K., 1990. Bayesian inference in non-replicated ecological
studies. Ecology 71, 2053–2059.
Rubio, A., Escudero, A., 2000. Small-scale spatial soil–plant rela-
tionship in semi-arid gypsum environment. Plant and Soil 220,
139–150.
Schemske, D., Husband, B., Ruckelshaus, M., Goodwillie, C., Parker,
J., Bishop, J., 1994. Evaluating approaches to the conservation of
rare and endangered plants. Ecology 75, 2053–2059.
Spiegelhalter, D.J., Thomas,A. and Best, N.G., 1999. WinBUGS Ver-
sion 1.2 User Manual MRC Biostatistics Unit. http://www.mrc-
bsu.cam.ac.uk/bugs/.
Wade, P., 2000. Bayesian methods in conservation biology. Con-
servation Biology 14, 1308–1316.
West, M., Harrison, J., 1997. Bayesian Forecasting and Dynamic
Models. Springer, New York.
Table 6
Sample estimators for the parameters obtained applying a Markov chain Montecarlo method with the Gibbs sampler
Parameter
Mean S.D. 2.5% Median 97.5%p
0.5982 0.08526 0.426 0.6016 0.75541
536.7 0.9265 535.0 536.7 538.62
548.9 1.256 546.3 548.9 551.2�
3.768 0.6137 2.935 3.668 5.25J.M. Marın et al. / Biological Conservation 113 (2003) 379–387 387