Winbugs Thesis

Bayesian Modelling of The Indirect Effectsof Insecticides on Yellowhammer Chick

Survival

Xiaosi Wang

Dissertation submitted for the MSc in Data Analysis, Networks

and Nonlinear Dynamics, Department of Mathematics, University

of York, UK

August 2005

Contents

List of Figures iv

List of Tables v

Preface vi

Acknowledgements viii

1 Introduction of The Project 11.1 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Experiments on The Yellowhammer . . . . . . . . . . . . . . . . . . . 41.3 A Non-Bayesian Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Bayesian Method in A Relevant Area . . . . . . . . . . . . . . . . . . . . . 101.5 Motivations & Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Bayesian Approaches 142.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Bayes’ Theorem for General Quantities . . . . . . . . . . . . . . . . . . . . 162.3 Bayesian Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 Analysis of Binary Data . . . . . . . . . . . . . . . . . . . . . . . . 182.3.2 Inference with Normal Distribution . . . . . . . . . . . . . . . . . . 202.3.3 Point Estimation & Interval Estimation . . . . . . . . . . . . . . . . 21

2.4 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.2 Markov Chain Monte Carlo . . . . . . . . . . . . . . . . . . . . . . 242.4.3 Introduction of WinBUGS . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Data Analysis and Model Handling in WinBUGS 293.1 Formatting The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 The Analysis of Failure Time Data . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 Failure Time Distributions . . . . . . . . . . . . . . . . . . . . . . . 333.2.2 Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

i

Contents ii

3.2.3 Survival Analysis in WinBUGS . . . . . . . . . . . . . . . . . . . . 373.3 Models Considering Time Dependence . . . . . . . . . . . . . . . . . . . . 393.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Discussions on Failure Time Data Models 434.1 Survival Models Based on The Exponential Prior . . . . . . . . . . . . . . 43

4.1.1 Models Based on Insecticide Quantities . . . . . . . . . . . . . . . . 444.1.2 Models Based on Time . . . . . . . . . . . . . . . . . . . . . . . . . 454.1.3 Models Based on Block Structure . . . . . . . . . . . . . . . . . . . 504.1.4 Analysis of The Posterior Summaries & Convergence of MCMC Sam-

pling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Survival Models Based on The Weibull Prior . . . . . . . . . . . . . . . . . 54

4.2.1 Models Based on Non-Informative Priors . . . . . . . . . . . . . . . 554.2.2 Models Based on Informative Priors . . . . . . . . . . . . . . . . . . 64

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Discussions on Time Dependence Models 695.1 Model I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2 Model II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Further Developments 74

Conclusions 75

Appendix I 77

Appendix II 83

Bibliography 92

List of Figures

1.1 Three patterns of influences on food chains . . . . . . . . . . . . . . . . . . 21.2 A GIS Map of Nests with 200m Radius Foraging Areas . . . . . . . . . . . 6

2.1 The comparison between the MCMC and the common Markov Chain . . . 242.2 An example of using the Sample Monitoring tool . . . . . . . . . . . . . . 27

3.1 Original data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Insecticide data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1 The model fit of the daily survival rate over the different proportions ofinsecticides from exponential model . . . . . . . . . . . . . . . . . . . . . . 44

4.2 The model fit of the nestling period survival probability over different pro-portions of insecticides from the exponential model . . . . . . . . . . . . . 45

4.3 The model fit of the chick daily survival rate over time . . . . . . . . . . . 464.4 The trend of the chick daily survival rate over time . . . . . . . . . . . . . 474.5 The model fit of the nestling period survival probability over time . . . . . 484.6 The trend of the nestling period survival probability over time . . . . . . . 484.7 The model fit of the probability density of chick survival life time . . . . . 494.8 The trend of the probability density of chick survival life time . . . . . . . 504.9 The model fit of nestling daily survival rate against different blocks . . . . 514.10 The convergence of two chains run for t[10] . . . . . . . . . . . . . . . . . . 534.11 The model fit of the chick daily survival rate over different proportions of

insecticides (the Weibull model, under non-informative prior distributions) 574.12 The model fit of the nestling period survival rate over different proportions

of insecticides (the Weibull model, under non-informative prior distributions) 574.13 The model fit of the daily survival rate over time (the Weibull model, under

non-informative prior distributions) . . . . . . . . . . . . . . . . . . . . . . 584.14 The trend of the daily survival rate over time (the Weibull model, under

non-informative prior distributions) . . . . . . . . . . . . . . . . . . . . . . 594.15 The model fit of the individual chick survival rate to days (the Weibull

model, under non-informative prior distributions) . . . . . . . . . . . . . . 594.16 the trend of the individual chick survival rate to days (the Weibull model,

under non-informative prior distributions) . . . . . . . . . . . . . . . . . . 60

iii

List of Figures iv

4.17 The model fit of the probability density of the nestling death time (theWeibull model, under non-informative prior distributions) . . . . . . . . . . 61

4.18 The trend of the probability density of the nestling death time (the Weibullmodel, under non-informative prior distributions) . . . . . . . . . . . . . . 61

4.19 The multiple chains of some monitored variables which have reached con-vergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.20 The posterior densities of four monitored variables of the 17th nestling . . 634.21 The pairs of chains for bn which have clearly not reached convergence under

very vague priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.22 The model fit of nestling daily survival rate over time (the Weibull, under

standard normal distribution priors) . . . . . . . . . . . . . . . . . . . . . . 664.23 The model fit of the chick daily survival rate over different proportions of

insecticides (the Weibull model, under standard normal distribution priors) 664.24 The pairs of chains for bn which have reached reasonable convergence under

informative priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.1 Data format for model I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.2 The autocorrelation functions of the daily survival rate on the 9th day and

the 10th day respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.3 The daily survival rate of chicks from 20th May, 2002 to 3rd June, 2002(Block

One) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.4 The convergence for parameters αc and βc respectively . . . . . . . . . . . 725.5 The dynamics of the convergence for parameters αc and βc respectively . . 73

List of Tables

1.1 Bobwhite Data-Knox County, 1991 . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Conjugate Distributions Table . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 Explanations of Input Data . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1 Posterior summaries from the Exponential Model . . . . . . . . . . . . . . 524.2 A random subset of the posterior summaries and the MC errors of some

monitored variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3 A random subset of the posterior summaries and the MC errors of some

monitored variables from the Weibull Model under very vague priors . . . . 564.4 A random subset of the posterior summaries and the MC errors of some

monitored variables from the Weibull Model under informative priors . . . 644.5 A random subset of the posterior summaries and the MC errors of some

monitored variables from the Weibull Model under informative priors . . . 65

5.1 Input data for model II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

v

Preface

This paper concerns a very important and valuable ecological problem. In Britain, in recent

years, the populations of many species of birds, especially farmland birds, are declining.

The poisoning of farmland birds in the past was a direct effect of agricultural chemical

treatments. However, after the disappearance of poisonous or sub-poisonous insecticides,

one of the possible reasons for the continued decline in bird populations can be attributed

to the indirect effects of pesticides acting through the food chain.

An extensive series of farmland experiments on this issue have been completed by the

Central Science Laboratory (CSL), an executive agency of the Department for Environ-

ment Food and Rural Affairs (DEFRA), UK, which provides appropriate research and

development to safeguard the food supply and to protect the environment. Although sev-

eral interesting findings have been reported in some recent papers, only classical frequentist

techniques involving Generalized Linear Models have been applied. A more novel statis-

tical method, Bayesian inference, is expected to be used to make further developments.

Until now, few attempts of using Bayesian approaches have been made in the survival

nestling analysis. Therefore, a scientific publication will be possible if Bayesian models are

successful.

As a result, a project was promoted by CSL to model the indirect effects of insecticides

on the yellowhammer (Emberiza citrinella) chick survival conditions by using Bayesian

statistics. It was then provided to the MSc course, Data Analysis, Networks and Nonlinear

Dynamics, the University of York, 2004-2005 as a student placement for the dissertation.

vi

Preface vii

The author took the placement and was under the supervision of senior bio-statistician

Alistair Murray, the team leader of the Applied Mathematics and Statistics Department,

CSL and Dr. Peter M Lee, the University of York.

The Bayesian approaches were absolutely new to the author and a specific method

supposed to be used in the project was proved not feasible after quite a long time due to

several avoidless reasons. However, a variety of new models are completed at last with a

lot of hard work behind.

The structure of the dissertation is as following:

Firstly, the ecological experiments and the relevant statistical approaches are introduced

to give a general idea of the project.

Secondly, the main features of the Bayesian methods related to the project are pre-

sented. WinBUGS, a brilliant software for Bayesian analysis, is selected as the main tool

for building Bayesian models, whilst the language, R, is also applied in Chapter 5.

More importantly, detailed introductions of the data reformatting, the mathematical

analysis of data and the model handling in WinBUGS are provided in Chapter 3.

According to different statistical modelling perspectives, model discussions are elabo-

rated in Chapter 4 and 5 separately. These two chapters demonstrate the new achievements

of the project, and therefore are the most important parts in the paper.

Finally, the summary at the end of each chapter has listed the most important features

of that chapter.

Acknowledgements

My thanks go firstly to my supervisors, Alistair Murray, Central Science Laboratory and

Dr. Peter Lee, the University of York. Alistair explored the student placement for the

project and has provided me with valuable supervision throughout the whole process. Peter

has provided me precious expertise on Bayesian Approaches and WinBUGS programmes.

As his last student before his retirement, I hope my work is a nice present.

I am also grateful to ornithologists, Justin Hart and Dr. Tim Milsom of CSL. They

have provided me expertise in ecological issues. Justin also helped me enter the data into

different data bases which was a time-consuming work.

I should also say thanks to Dr. David Spiegelhalter and Dr. Nicky Best for their

extremely useful course on WinBUGS.

I would also like to give my thanks to Professor Mike Smith. His unselfishness helped

me overcome my family misfortune both mentally and substantially.

Finally, my thanks go to my parents and my sincere friends, Anna Armstrong, Kit Fan,

Lieven Clarisse, Mahlet Getachew, and Marina Theodoropoulou. Their support in every

aspect has encouraged me to accomplish the MSc course.

viii

Chapter 1

Introduction of The Project

1.1 An Overview

Since the middle of the 1980s, the populations of many farmland birds in the UK, including

the yellowhammer (Emberiza citrinella), have declined. The reductions of arthropods,

seeds and weeds on arable farmlands caused by intensive agricultural practices are related

to the population decline. However, the potential mechanisms are still unknown ([Boatman

N. D. et al, 2004]; [Hart, J. D. et al, 2005]).

The simultaneously increased pesticide applications are suspected to have impacted

on the bird populations. Due to the withdrawal of the organochlorine insecticides in the

1950s and 1960s, little evidence of lethal or sub-lethal poisoning of chicks or provisioning

adults which is termed as direct effects of pesticides is observed in the last three decades.

Therefore, more attention has been focused on the indirect effects of pesticides. For exam-

ple, various insects which are taken as food by nestlings of most bird species during the

breeding season are reduced by insecticides, and subsequently the populations of chicks

shrink because of the lack of food supplies. In other words, it is a mechanism that works

by operating on food chains. Although the influence has been considered in general, the

1

1.1 An Overview 2

extent to which the survival conditions of arable farmland birds are affected by this kind

of agricultural treatments is still unidentified. Three patterns of the impact have been

suspected:

• the arthropod population is depleted by insecticides

• the supply of plants which are used as hosts by arthropods is reduced by herbicides

• the weed species which provide either green matter or seeds for herbivorous andseed-eating species respectively is eliminated by herbicides

These routes are shown in Figure 1.1.

insecticides

arthropod reduction

reduction of surviving chicks

herbicides

weed reduction


herbicides

plant reduction


arthropod reduction

Figure 1.1: Three patterns of influences on food chains

1.1 An Overview 3

Although it was concluded that at least 11 species are possibly affected by increased use

of pesticides, sufficient evidence for this, and elaborate data analysis, were only available

on the grey partridge (Perdix perdix ) [Campbell, L. H. et al, 1997]. This lack of research is

due to field work required in the relevant experiments being expensive and labour-intensive.

In order to stop the trend of decline and to promote healthier farming practices, a series

of large-scale field experiments were designed and carried out by scientists from the Central

Science Laboratory between 2000 and 2003 to study the indirect effects of insecticides on

farmland birds (pattern 1, Figure 1.1) on three bird species: the yellowhammer, the skylark,

and the corn bunting, of these the yellowhammer was emphasized because of three main

reasons:

Multiple foraging areas It not only forages in the arable crops, but also forages in field

margins. Therefore, the field margins included in the study areas do not need to be

considered separately.

Invertebrates dominated nestling diet A range of arthropods comprises the main

nestling diet even though semi-ripe cereal grain is also consumed later in the breeding

season. Thus, insecticides without herbicides and fungicides can also influence the

food supplies.

Sufficient abundance for sampling Although it is a species with marked decline, it is

still wide-spread on farmlands. This situation enables us to collect enough samples

for study

Treatments in these experiments included supplementary seed supplies in winter to

enhance the sufficiency of winter food and increases in the times of normal insecticide

applications to depress the invertebrate food resources. Three sites, each of which were

composed of four blocks with 1km radius were involved in the experiments. They were

located in Hampshire, Lincolnshire, and North Yorkshire, on arable or mixed farms. The

1.2 The Experiments on The Yellowhammer 4

data were collected from different aspects, including insecticide use, invertebrate availabil-

ity, foraging, chick condition, growth rate and chick survival.

The data of the insecticide inputs and chick survival conditions of the yellowhammer

which have been grouped and subsequently used in the previous analysis are provided to

the student placement for deeper and further Bayesian analysis, a method which can be

seen as a novel interpretation into the study of bird reproductive performances under para-

meter controls (insecticide applications). The earlier analysis was conducted by traditional

statistical methods and therefore the data were not in the an appropriate form at the be-

ginning of the placement. Reformatting the data (see Section 3.1, Chapter 3) was needed.

Although the experiments carried out were on different sites within successive years, they

are designed as replicated studies. Therefore, once a statistical model based on one data

set from a specific site in a certain year is applicable, it can be quickly translated into

studies on other data sets.

1.2 The Experiments on The Yellowhammer

The yellowhammer is a typical farmland bird species which is suffering from population

decline. Fundamental statistical studies have provided some evidences on the relationships

between the abundance of arthropods which was heavily reduced by insecticides and the

physical mass of chicks. As a development of the current achievements, modelling the

impact of summer insecticide inputs on the yellowhammer chick survival conditions from

hatching to fledging from different perspectives are the major tasks for the project, in-

cluding studies of the daily survival rate of chicks, the nestling period survival rate of an

individual chick, the direct relations of the daily survival rate and insecticide quantities

and so forth.

The data used in the paper were from four blocks of farmlands at Castle Howard


site, North Yorkshire, 2002. The breeding season, a critical phase in the life circle of

the yellowhammer, is between the late April and early August. Hence, this period is

chosen for the data collection. During the time under monitoring, all active nests located

in the study area were found and each nest was visited every 2-4 days but not daily to

eliminate disturbance. Chick conditions were recorded from different aspects, including

general condition (alive or dead) at each visit, physical mass at certain ages and so on.

Particularly, a chick was always weighted to determine its age when it was found for the

first time.

The data of insecticides were recorded from farming practices. All insecticides applied

here belonged to the pyrethroid family which have no short or chronic poisonous effects on

chicks. Most fields on block 1 and 4 were sprayed only once at a normal quantity whereas

those on block 2 and 3 received twice normal insecticide treatments. Conventionally, blocks

treated twice at different time were referred to as ‘extra’ while those treated only once

were referred to as ‘normal’. There was no evident difference between ‘extra’ and ‘normal’

blocks in general [Boatman N. D. et al, 2004] mainly because the ‘extra’ only meant an

second treatment with a normal quantity and the first spray was also at a normal level

of insecticides but on a much earlier date. During the interval between two sprays, the

population of most arthropods had recovered rapidly. As a result, the block structure was

ignored and only the data of pesticide quantities related to individual nests were used.

In general, 90% of the foraging flights were found to take place in a 200m-radius circle

area around a single yellowhammer nest. Hence nests were located on a Geographic In-

formation Systems (GIS ) map and the proportions of fields in such a foraging area with

insecticides on were calculated. Such a method is shown in Figure 1.2. Note that different

colors only denote different crops. Thus, it has nothing to do with the proportions of

insecticide treated area.


Figure 1.2: A GIS Map of Nests with 200m Radius Foraging Areas

The data of insecticides are available for most nests. However, a few nests failed shortly

after they had been found various unusual reasons unrelated to the pesticide treatments,

so no insecticide data are recorded for such cases. For example, all chicks in nest No. 254

died in an accident as the nest was not stably built and fell off the tree when there was

heavy wind.

The previous data analysis mainly reported some evidences on the relationships be-

tween insecticide use and chick food invertebrates, between chick condition and chick food

invertebrates, between chick growth rate and chick survival. However, some crucial analy-

ses such as the daily survival rate of yellowhammer chicks from hatching to fledging, and

the relationship between the insecticide use and chick survival were not studied. Statistical

models to solve such problems are built in this project.

1.3 A Non-Bayesian Method 7

1.3 A Non-Bayesian Method

In bird studies, a widely-used statistical method is called Mayfield Method, which is named

after Harold Mayfield, a professor at Cornell University, whose two papers ([Mayfield,

1961]; [Mayfield, 1975]) presented an easy-handled statistical method for estimating daily

survival rate (dsr) of bird nests. Further modifications and developments were made by

other statisticians ([Johnson, 1979]; [Hensler and Nichols, 1981]). As a result, the initial

method was formalized into a standard mathematical method. It has been concluded that

the Mayfield formula was actually a Maximum Likelihood Estimator (MLE) of the dsr. A

brief introduction is presented here in order to carry out the comparison between ‘Bayesian’

and ‘Mayfield’ (See Chapter 4 and 5).

To start the discussion of ‘Mayfield’, some assumptions are necessary:

• a nest survives from one day to the next with the same probability throughout the

nest life

• the complete period to success (the nesting stage) are of the same length for all nests

• all nests being observed have the same probability to survive from one day to the

next

• the exact date of a nest’s success or failure is known

• the nests under monitoring comprise a random sample of the population of nests

under consideration

• all nests are independent

The formula derivations introduced here are mainly based on [Aebischer, 1999] and [Hensler

and Nichols, 1981]. Suppose that a nest is found at a particular nesting stage, i.e. egg-

laying, incubation or brood-rearing. Let ωk denote the probability that it is found on


day k (encounter probability) within the total nesting days K. It is evident that all the

probabilities form a vector ~ω = (ω1, ω2 . . . ωK). The nest is observed for n days, and its

outcome o is recorded as failure (0) or success (1). The quantity of interest is the probability

that an active nest on one day survives to the next (daily survival rate s). Let f denote

the conditional probability of the binary outcome, i.e. either failure or success. It can be

expressed in a function of s (the likelihood function) given the conditions described above:

f(s|o, n, K, ~ω) = (ωK−n+1sn)o

[

sn−1(1 − s)K−n+1∑

k=1

ωk

]1−o

If M nests are found, the mth nest is monitored for nm days with outcome om. The joint

probability of the outcomes given that all the nests are independent is:

M∏

m=1

f(s|om, nm, Km, ~ωm)

So the log-likelihood function is:

l(s|~o, ~n, ~K, ~ω) = ln

[

M∏

m=1

f(s|om, nm, Km, ~ωm)

]

=M∑

m=1

(nm + om − 1) ln s +M∑

m=1

(1 − ym) ln(1 − s)+

M∑

m=1

ln

ωom

Km−nm+1,m

(

Km−nm+1∑

k=1

ωkm

)1−ym

= (N + O − M) ln p + (M − O) ln(1 − p)+

M∑

m=1

ln g(om, nm, Km, ωm)


where

N =

M∑

m=1

nm;

O =M∑

m=1

om;

g(om, nm, Km, ωm) = ωom

Km−nm+1,m(

Km−nm+1∑

k=1

ωkm)1−ym

Here N denotes the total number of nest-days under monitoring, and O represents the

successful number of nests.

To solve the equation

dl(s|~o, ~n, ~K, ~ω)

dp= 0

we get the maximum likelihood estimator s of s:

s =N + O − M

N

Furthermore, the asymptotic variance can be estimated by

V ar(s) =−1

E[

d2l(s|~o,~n, ~K,~ω)ds2

] , when O < M

Hence,

V ar(s) =s(1 − s)

N, (~s 6= 1)

For O = M , 1 is a boundary estimate of s and subsequently the approximate confidence

interval of the variance is obtained by that of the minimum p-value whose 95% confidence

interval includes 1. For further extensions on multi-way comparisons and generalized linear

regression models, please refer to [Aebischer, 1999].

1.4 Bayesian Method in A Relevant Area 10

Although it is hinted in [Mayfield, 1975] that this method can be generalized to calculate

the survival rate of eggs or nestlings, existing differences between chicks and nests can cause

some problems if it is translated directly. For instance, the ages of nests are usually left-

truncated, or right-truncated, or even double-interval truncated because they are usually

not found until the incubation starts, implying the building period of the nest is unknown.

It is even worse that if the nest is still active when the observation stops because this

implies that the finishing age of the nest is missing, too. It is a different situation for

chicks: the ages of the chicks are derived from their physical mass after being weighed

when they are first found.

It has also been noticed ([He, 2003]) that large biased estimation of daily survival rate

might be given by Mayfield methodology. This is mainly because some of the assumptions

are hardly to be satisfied in practice. Therefore, uncertainties in the fates of nests are hardly

modeled in Mayfield formula which to some extents reflects the generic disadvantages of

the frequentist perspective. That is why Bayesian methods are strongly needed.

1.4 Bayesian Method in A Relevant Area

The idea for applying Bayesian methods to the analysis of the yellowhammer survival rate

was prompted by a high-profile paper [He, 2003], from Dr. Zhechong He, University of

Missouri-Columbia, US. Bayesian inferences were used to model the age-specific survival

rate of the bird nests with double-interval censoring of the active nest life time. There have

been further developments of Dr. He’s method which is finished in a new paper [Cao and

He, 2005].

In that paper, the data of 36 active northern bobwhite nests were used to demonstrate

the Bayesian method. The survival time of each nest was assumed equally likely, 25 days.

Nest conditions were grouped into different catalogues at each visit:

1.4 Bayesian Method in A Relevant Area 11

(1) destroyed or abandoned nests were considered failed

(2) nests with at least one egg hatched or one chick fledged inside were assumed successful

(3) other nests were called active nests

The uncertainty involved here included data truncation. Usually, nests were not visited

daily but less frequently, i.e. 2-4 days interval to avoid disturbance which was the same as

what happened in the CSL experiments. Moreover, nests were usually not discovered until

the breeding performances began, so the encounter age of individual nest was unknown.

On the contrary, ornithologists at CSL weighed the individual chick to determine its age in

order to reduce the uncertainty. This means more information is available for our project.

Table 1.1 shows the data format from the paper [He, 2003]:

Table 1.1: Bobwhite Data-Knox County, 1991

Encounter age Observed daysOutcome Minimum Maximum Minimum Maximum

0 1 23 2 21 1 1 24 240 1 22 3 50 1 22 3 51 1 1 24 24

From Table 1.1, we can see that if a nest failed, the encounter age was supposed to be

within a range restricted by the observation days. On the other hand, if it was successful,

the encounter age was derived from the observation days under the assumption that the life

length of each nest is same. This is why that the addition of the minimum of the observed

days and the maximum of the encounter age is an invariant number, 25. Obviously, in the

real world, the number of the total nesting days is not fixed but may vary within a small

range. For example, both the incubation and the nestling periods of the yellowhammer

last 10-15 days respectively. Therefore, the fledging age of the yellowhammer chicks has

1.5 Motivations & Difficulties 12

to be assumed equal if we would like to use the same model. Having been discussed with

the ecologists at CSL, the hypothesis is considered inappropriate for our project.

It is worth mentioning that the results given by the model of [He, 2003] were demon-

strated to be better than those from classical Mayfield using some simulated known values.

Therefore, the advantages of Bayesian methods have been established.

1.5 Motivations & Difficulties

Although the profile of the dissertation only emphasized the modelling of the relationship

between the insecticide applications and nestling survival conditions, scientists in CSL are

also very interested in the Bayesian results of the survival probability over time because

those results from the ‘Mayfield’ are under an assumption that the daily hazard rate is

a constant which in some sense implies an unnatural restriction. Therefore, modelling

different survival properties against time axis by applying Bayesian ideas has motivated

the author to seek the solutions.

Several difficulties arising from the missing data and the relevant the Bayesian methods

have been conquered. Due to the irregular visits of nests, a lot of information was miss-

ing. As mentioned above, in daily survival analysis, the exact time of the deaths usually

measured in days are compulsory. However, the information would be unavailable if some

days were skipped between the days when it was still alive and when the chick was found

dead.

The exact same Bayesian models from the paper [He, 2003] were suggested to be trans-

lated into the project at the beginning of the student placement. However, after repeated

comparisons of He’s model and the problems we need to solve, several fundamental dif-

ferences were found out, preventing us to follow He’s footsteps. Moreover, because very

complex computations, several new theorems and a distinct data form (Table 1.1) were

1.6 Summary 13

used in [He, 2003], nearly one month was spent in investigating the feasibility of applying

those methods to the CSL data. The author, Dr. Zhechong He and her student, Jing Cao,

who has made some new developments, were contacted, and after considerable delay in

their responding, feedback from them also suggested that models using different methods

should be built. The integrals that were required to be solved to implement their method

are complex and non-standard and would have taken too long to re-implement. Dr. He

was unable to make the software available at this time although a commitment to do so

was made in [He, 2003]; it is to be hoped this will eventually be done. I am grateful for

their feedback and advice.

1.6 Summary

The information provided in this chapter will be used throughout the paper.

Firstly, an overview of the ecological problem is provided. The lack of evidence on the

indirect effects of pesticides on other bird species except grey partridge (Perdix perdix ),

has motivated ecologists to search new evidence on other bird species, such as the yel-

lowhammer, on which the data analysis in the project is based.

Secondly, important concepts and detailed descriptions of the experiments on the yel-

lowhammer are presented. Some relevant statistical methods are briefly introduced. Al-

though they are not analytically used in this project, they have played a very important

role in helping the author understand the project and bio-statistical modelling.

Finally, the motivations and difficulties of the project are mentioned.

Chapter 2

Bayesian Approaches

The core issues of the Bayesian analysis related to the project are introduced in this chap-

ter. In probability theory, both continuous and discrete conditions have to be discussed.

However, regarding the fact that in practice, daily collected survival data are treated as a

continuous series in bird studies, Bayesian ideas will be illustrated within the continuous

literature. The discussions presented in this chapter are mainly based on two books: [Lee,

2004] and [Spiegelhalter, D. J. et al, 2004].

2.1 Background

The traditional probability theories are based on searching the long-run properties of ran-

dom events. This is referred to as the frequency interpretation of probability. Consequently,

the standard statistical methods are developed from the frequentist perspective. For ex-

ample, “although there is still no universal agreements on the definition of probability,

most people might agree the following statement: Probability represents the times of the

occurring of a random event θ in the repeated trials after a ‘long time’. ”—[Spiegelhalter,

D. J. et al, 2004] We may use a mathematical notation P (θ) to represent Probability. This

14

2.1 Background 15

definition describes the frequency with which specific events take place.

However, in the last two decades, more and more statisticians have started interpreting

probability theory from another perspective. “As [Lindley, 2000] addressed, we do not

need to assume that the rules of probability are objective, but we can derive it from our

individual feelings of the uncertainty. For example, a gambling game in Macao is carried

out like this: two dice are covered instantaneously outside of the gamblers’ sight after being

tossed up to the sky. Then the players are asked to guess the different combinations of

the dice. The fact that the event has happened means the participants bet their fortunes

only depending on their subjective feelings, i.e.personal probabilities.”—[Spiegelhalter, D.

J. et al, 2004] A subjective interpretation of probability is given in such situation. It is

vital that your probability for a event represents your relationship to that event which has

nothing to do with the objective property of the event itself.

Bayesian statistical methods are based on the subjective view of probability introduced

above. This kind of thinking in some sense reflects the dynamics and the natural process

of knowledge learning and accumulation of human beings. Bayesian inference has obtained

more and more applications since it is a very comprehensive and robust method. In the

past, the properties of estimators usually only include unbiasedness, minimum variance,

efficiency, consistency, and sufficiency. In recent years, a special attention has been given

to a statistical property called robustness. An estimator is said to be robust if the results

given by a statistical model maintain stable after new samples are pooled in. Traditional

methods are known to be weak in robustness.

Bayesian approach begins its life with a simple theorem, which was named Bayes’

Theorem after Thomas Bayes, a nonconformist minister from the small English town of

Tunbridge Wells. It first appeared in one of his posthumous publications in 1763. We

have learnt the simple equation form of the theorem in the first course of mathematical

statistics, however, in practice, a form for general quantities is preferred.

2.2 Bayes’ Theorem for General Quantities 16

2.2 Bayes’ Theorem for General Quantities

Firstly, suppose that we are interested in a series of unknown parameters

Θ = (θ1, θ2, . . .), Θ ∈ RP

Meanwhile, we might have some personal beliefs or guesses of the probability density

function pdf of Θ before we see the data:

P (Θ)

which also represents the dimension of Θ. It is given a statistical term, the prior distribu-

tion. Prior means a statistician’s hypothesis of the unidentified quantities before the data

were available. Then a sequence of data was observed

B = (B1, B2, . . .).

A conditional pdf can be used to express how likely to have current values of ~B in terms

of the current values of ~θ:

P (B|Θ).

This is usually referred to as the likelihood function of Θ. The next step is to update the

prior by the likelihood. Recall that there is a rule concerning the conditional distribution

in probability theory:

P (Θ|B)P (B) = P (Θ,B) = P (B|Θ)P (Θ).

2.3 Bayesian Inference 17

Therefore,

P (Θ|B) =P (B|Θ)P (Θ)

P (B)(2.1)

which is the simplest form of Bayes’ theorem. Note that P (B) is just a denominator to

make sure that∫

P (θ|B)dθ = 1, and its value is usually not concerned unless alternative

models are compared. Thus, a proportional form which only regards the terms including

Θ is commonly used in practice:

P (Θ|B) ∝ P (B|Θ)P (Θ) (2.2)

It is a significant improvement that the parameters of interest have been successfully mod-

ified in the light of the data. The pdf on the left hand side of the proportion is called the

posterior distribution reflecting how the data change our initial state of knowledge. When

we obtain new data, the current posterior distribution becomes the prior for the new sam-

ple. Our knowledge of the parameters increase in such process and can always be modified

by the movements of the parameters because they are not constants but variables.

In theory, statistical inferences can be derived from the entire posterior distribution.

However, all information is encapsulated in the posterior distribution with a very compact

form, and a mathematical formula is too simple to convey the information in a useful form.

Therefore, a range of helpful summaries should be supplemented. In the next section, a

detailed discussion is carried out to demonstrate how to make inferences.

2.3 Bayesian Inference

In the study of statistics, summarizing the operating mechanisms behind the observed data

is a main task needed to be solved. The process of making such statements about the given

physical systems is called inference.


The fact that Bayesian inference allows us to combine the substantive information

we have with our personal beliefs about the unknown parameters could lead to some

conclusions that we have expected naively to get from statistics. For instance, about

the traditional 95% confidence interval, “all we can say is that if we carry out similar

procedures time after time then the unknown parameters will lie in the confidence intervals

we construct 95% of the time. It appears that the books we look at are not answering

the questions that naturally occur to a reader, and that instead they answer some rather

recondite questions which no-one is likely to want to ask.”—[Lee, 2004] While on the

contrary, in Bayesian analysis, “the 95% Highest Posterior Density Interval (HPDI ) really

does mean an interval in which the statistician is justified in thinking that there is a 95%

probability of finding the unknown parameter.”—[Lee, 2004] This interpretation is more

natural and easy to be handled.

2.3.1 Analysis of Binary Data

Although the likelihood of binary data is from a discrete sample space, the parameter θ

of interest is usually given a continuous prior distribution. The simplest assumption is to

suppose the possible values of θ following a uniform distribution, and hence P (θ) = 1 for

0 6 θ 6 1. Using the proportional form of Bayes’ Theorem (2.2),

P (θ|B) ∝ θk(1 − θ)n−k × 1

= θ(k+1)−1(1 − θ)(n−r+1)−1

= Beta(k + 1, n − r + 1)

where k is the number of events occurred, n is the total number of trials. The posterior is

taking the form of Beta distribution. “It immediately suggests that we can summarize the

posterior distribution in terms of mean and variance of Beta distribution, and make prob-


ability statements based on what we know about it (for example, many common statistical

packages will calculate tail area probabilities for the Beta distribution).”—[Spiegelhalter,

D. J. et al, 2004]

Instead of a uniform prior, Beta(a, b) can be used as a prior and the following analysis

can be obtained

Prior ∝ θa−1(1 − θ)b−1

Likelihood ∝ θk(1 − θ)n−k

Posterior ∝ θa−1(1 − θ)b−1θk(1 − θ)n−k

= Beta(a + k, b + n − k)

The prior is said to be conjugate to the likelihood if the prior and posterior are from the

same family of distributions. This has the advantage that prior parameters can usually be

interpreted as a prior sample. Some common conjugate families are shown in Table 2.1:

Table 2.1: Conjugate Distributions Table

Prior Likelihood PosteriorNormal Normal NormalBeta Binomial Beta

Gamma Poisson GammaDirichlet multinomial Dirichlet

However, the conjugate priors do not exist for all likelihood, and always be restrictive.

Hence, the uniform or normal distribution with large variance are usually used as priors,

especially when little information of the shape of the likelihood distribution is available.


2.3.2 Inference with Normal Distribution

In many circumstances, it is appropriate to suppose that the likelihood follows a normal

distribution which is also the case in the project. From the conjugate family, the prior

distribution has the form

P (θ) = N

(

θ | µ,σ2

n

)

(2.3)

where µ is the prior mean, σ is the likelihood standard deviation, and n is the sample size

of the prior. As n → 0, the variance becomes very large and the distribution becomes very

‘flat’. Thus, the limit form is essentially identical to a uniform distribution over (−∞,∞)

which is known as a non-informative distribution.

Assume a normal prior

θ ∼ N

(

µ,σ2

n

)

(2.4)

and the likelihood

B ∼ N

(

µ,σ2

l

)

(2.5)

where l is the sample size of real data set. Applying Bayes’ theorem

P (θ|B) ∝ P (θ)P (B|θ) (2.6)

∝ e−l(B−θ)2

2σ2 × e−n(µ−θ)2

2σ2 (2.7)

The terms without θ can be ignored in the proportional form of Bayes’ theorem. After

rearranging the terms according to θ, the posterior distribution can be obtained

θ|B ∼ N

(

θ |nµ + lB

n + l,

σ2

n + l

)

(2.8)

The expression (2.7) is important, “It says that the posterior mean nµ+lB

n+lis weighted

average of the prior mean µ and the likelihood B, weighted by their precisions, and therefore


is always a compromise between the two. The posterior variance (1/precision) is based on

an implicit size equivalent to the sum of the prior ‘sample size’ n and the sample size of

the data l: thus, when combining sources of evidence from the prior and the likelihood,

we add precisions and hence always decrease the uncertainty.”—[Spiegelhalter, D. J. et al,

2004]

If n tends to 0, the prior approaches a uniform distribution and the posterior distribution

is totally dominated by the data, i.e. the posterior will have the same shape with the

likelihood.

2.3.3 Point Estimation & Interval Estimation

In many cases, posterior summaries, such as point estimates and interval estimates are

generally needed. In the project, summary statistics are used at every second. Therefore,

some specific terms in Bayesian inference must be introduced.

Point Estimates The mean, median and mode derived from the posterior distribution

are given the same interpretations as those from the classical frequentist theory. It

is usually preferable to report all the three measures of location of distributions. In

the project, median is suggested to plot the exact model fit in order to avoid the

influence from the tails of skewed distributions whereas the mean value is preferred

to give the plot of the trend.

Interval estimates Usually the interval estimates are termed credible intervals or poste-

rior intervals in Bayesian inference to differ from the traditional confidence interval.

Three types of intervals exist in Bayesian literature, in which the Highest Posterior

Density Interval (HPDI or HPD) is the most commonly used.

Suppose a continuous parameter θ ∈ (−∞, +∞) and a posterior conditional on

generic data B.

2.4 Computational Methods 22

One-sided “A one sided upper 95% interval would be expressed as (θl, +∞), where

P (θ < θl|B) = 0.05.”

Two-sided (equal tail area) “A two-sided 95% interval with equal probability in

each tail area would comprise (θl, θu), where P (θ < θl|B) = 0.025, and P (θ >

θu|B) = 0.975.”

Highest Posterior Density “If the posterior distribution is skewed, then a two-

sided interval with equal tail areas will generally contain some parameter values

that have lower posterior probability than values out side the interval. An HPDI

does not have this property - it is adjusted so that the probability ordinates at

each end of the interval are identical, and hence it is also the narrowest possible

interval containing the required probability. ”

2.4 Computational Methods

Although the theory of Bayesian inference is developed elaborately, the fact that the re-

quired complex and high dimension integrations can hardly be solved analytically has

restricted the applications of the method in the past two hundred years. Until the last two

decades, a computer-based method, Markov Chain Monte Carlo (MCMC), has provided

an alternative way to solve the problem. Considering the facts that MCMC is default and

packed in WinBUGS and building proper statistical models is much more important and

meaningful according to this project, only a short introduction is given to this numerical

method.

2.4.1 The Problem

All the mathematical derivations presented here are mainly based on the tutorial paper

[Brooks, 1998]. Recall that P (Θ) is the prior distribution and P (B|Θ) is the likelihood


function, the posterior is usually written in a proportional form

P (Θ|B) ∝ P (B|Θ)P (Θ)

So the normalizing factor is given by

∫

P (B|θ)P (θ)dθ (2.9)

Suppose now Θ = (θ1, θ2), the marginal posterior distribution might be of interests:

P (θ1|B) =

∫

P (θ1, θ2|B)dθ2 (2.10)

More commonly, any summary of the posterior distribution, for example moments, quan-

tiles, HPDI, etc are legitimate for Bayesian inference. All these features can be related to

the posterior expectation of functions of θ. For instance, the posterior expectation of a

function g(θ) is

E[g(θ)|B] =

∫

g(θ)P (θ|B)dθ (2.11)

No matter the constant of proportionality (2.9), the marginal density (2.10), or the pos-

terior expectation (2.11), the ability to solve the integrations which are often complex

and high dimensional. It is extremely hard to obtain analytical solutions. Alternatively,

MCMC is able to simulate the posterior density. Hence, the difficulty in computation

is substantially conquered. This breakthrough in numerical methods has accelerated the

boom of Bayesian applications.


2.4.2 Markov Chain Monte Carlo

The idea of MCMC sampling is simple. There is an elaborate description in [Brooks,

1998], “Suppose we have some distribution P (θ), θ ∈ S ⊆ RP , which is known as only

up to some multiplicative constant. We commonly refer to this as the target distribution.

If P is sufficiently complex that we cannot sample from it directly, an indirect method

for obtaining samples from P is to construct an aperiodic and irreducible Markov Chain

with state space S, and whose stationary (or invariant) distribution is P (θ), as discussed

in Smith and Roberts (1993), for example. Then, if we run the chain for sufficiently long,

simulated values from the chain can be treated as a dependent sample from the target

distribution and used as a basis for summarizing important features of P . ”

The relationship between the MCMC approach and the usual Markov Chain theory is

shown in Figure 2.1. The stationary distribution of the Markov Chain is known, and the

transition distribution is the target.

Transition Distribution

Stationary Distribution

Common Markov Chain Theory Markov Chain Monte Carlo

Figure 2.1: The comparison between the MCMC and the common Markov Chain

In order to make the introduction more convenient and to keep consistency with

[Brooks, 1998], the Markov chain transition distribution is denoted by K since it is more

often referred to as Markov chain transition kernel. The conditional distribution of the


current state (θ(current)) given the previous state (θ(previous)) is K(θ(previous), θ(current)) con-

sequently.

“The main theorem underpinning the MCMC method is that any chain which is ir-

reducible and aperiodic will have a unique stationary distribution, and that the t-step

transition kernel will ‘converge’ to that stationary distribution as t → ∞. Thus, to gen-

erate a chain with stationary distribution P , we need only to find transition kernels K

that satisfy these conditions and for which PK = P , i.e. K is such that, given an ob-

servation θ(previous) ∼ P (θ(previous)), if θ(current) ∼ K(θ(previous), θ(current)), then θ(previous) ∼

P (θ(previous)), also. ”—[Brooks, 1998]

In practice, different kernels would be used to update the distinct components of the

whole chain instead of a single form of transition distribution. The most popular up-

date algorithm is the Gibbs sampling, which is a special case of the Metropolis-Hastings

updating. It is possible to combine them to construct a Markov chain which performs

appropriately. The combination in WinBUGS is called the Metropolis-within-Gibbs algo-

rithm. However, their details are beyond the scope of the paper. Detailed discussions of

these update schemes can be found in [Gilks and Wild, 1992], [Gelman and Rubin, 1992],

[Gilks, W. R. et al, 1996] and [Gamerman, 1997].

2.4.3 Introduction of WinBUGS

WinBUGS is a free software for Bayesian analysis, especially for complex Bayesian models.

It is built up by David Spiegelhalter, Andrew Thomas, Nicky Best, and Dave Lunn, all

of whom are working in the area of public health. The initial purpose for developing

WinBUGS is to use Bayesian approaches to solve problems in clinic trials and health-

care evaluation. Its brilliance have been acknowledged by more and more statisticians

([Congdon, 2001];[Congdon, 2003];[Lee, 2004]). As a result, there are more attempts to use

WinBUGS in other research areas, such as economics, biology, ecology and so on, which is


just the case of this paper. The author was sent by CSL to take a short course in WinBUGS

in Medical Research Council Biostatistics Unit, Cambridge, run by Dr. Spiegelhalter, and

Dr. Best. The precious expertise from Dr. Spiegelhalter has definitely accelerated the

process of modelling.

The main advantages of WinBUGS are that model specification in WinBUGS is fairly

straightforward if you are skilled with Bayesian analysis and different updating algorithms

for using MCMC method are automatically chosen to sample from the posterior distrib-

ution. The disadvantages are that “it cannot be integrated with a traditional statistical

package for data manipulation, exploratory analysis and so forth since it is designed as a

‘stand-alone’ program”—[Spiegelhalter, D. J. et al, 2004] and the speed for the convergence

of MCMC sampling can be extremely slow if the data set is huge.

Some basic notations which having contributed to the models in the paper are shown

below, “

• <- represents logical dependence, e.g.

m <- a+b*x

• ∼ represents stochastic dependence, e.g.

r ~ dunif(a, b)

• Can use arrays and loops, e.g.

for (i in 1:n){

r[i] ~ dbin(p[i], n[i])

p[i] ~ dunif(0, 100)

}

• Some functions can appear on the left-hand-side of an expression, e.g.

logit(p[i])<- a+b*x[i]

log(m[i])<- c+d*y[i]

2.5 Summary 27

• The normal is parameterized in terms of its mean and precision = 1/variance = 1/sd2

• Functions cannot be used as arguments in distributions

”—[Spiegelhalter and Best, 2005].

Some tools in WinBUGS are very helpful. For example, Figure 2.2 shows a posterior

density of an element of a random variable by using the Sample Monitoring tool.

Posterior distribution of anelement of a random variable

Figure 2.2: An example of using the Sample Monitoring tool

More applications will be combined with model discussions. Finally, it should be men-

tioned that it was concluded [Celeux, 2005] that the DIC tool [Spiegelhalter, D. J. et al,

2002] in WinBUGS, which is famous for comparing the models is not naturally defined for

the missing data models. Therefore, it cannot be applied in the project.

2.5 Summary

The comparison between the novel Bayesian perspective and the classical frequentist theory

is given as the background, but not emphasized.

2.5 Summary 28

The proportional form of Bayes’ theorem is the backbone of the whole theory, and

therefore is introduced in detail. Comparatively, the Bayesian methods for inference are

the most important part in this chapter. The difficulties in computation were solved by

MCMC simulation, and hence a brief introduction is presented. Finally, a short description

is given to WinBUGS, a brilliant software which is used throughout the project.

Chapter 3

Data Analysis and Model Handling

in WinBUGS

3.1 Formatting The Data

The quality and suitability of the statistical models in some sense are determined by the

way that the data are handled. Moreover, data structures for Bayesian inferences are

usually different from those for traditional statistical models. Therefore, it is important

to format the original data into appropriate forms for distinct models to be built for the

project.

Originally, all the data about chicks were recorded in the ornithologist’s observation

diaries, i.e. field work diary. Usually, the data of chick conditions in an specific nest would

be collected twice a week if the first time for recording was early in that week. However,

if a nest was visited on Thursday or Friday, the next visit would be delayed to the next

Monday. Consequently, more information was lost in such a condition. The original data

form was shown in Figure 3.1:

29

3.1 Formatting The Data 30

Figure 3.1: Original data

The information about the percentages of the 200m-radius foraging area around an

individual nest sprayed with insecticides were arranged into two groups in accordance with

the timing: the proportions of the 200m-radius foraging area sprayed with insecticides

6 20 days before hatching and the proportions of the 200m-radius foraging area sprayed

with insecticides > 20 days before hatching. Such data are shown in Figure 3.2.

Figure 3.2: Insecticide data

3.2 The Analysis of Failure Time Data 31

In [Boatman N. D. et al, 2004], it was mentioned that there was no significant relation-

ship between the brood reduction and the proportions of the 200m-radius foraging area

sprayed with insecticides > 20 days before hatching, so those data (Column “more20”,

Figure 3.2) are discarded in the current analysis. In the light of the expertise and expe-

rience from bio-statisticians and ornithologists, the exact values of the proportions of the

200m-radius foraging area sprayed with insecticides ≤ 20 days before hatching are grouped

into <= 25% and > 25% to simplify the calculation.

3.2 The Analysis of Failure Time Data

Usually, the survival analysis is grouped into the analysis of failure time data. Generally

speaking, the analysis of data when the response of interest is the duration until a single

event occurs, is called the analysis of failure time data. The events, such as the deaths of

chicks, the reappearance of an specific illness etc., are defined by temporary or permanent

changes of states. According to our project, the deaths of chicks are obviously the latter

condition. On the other hand, such events are also generically called failures. More impor-

tantly, “such data can be studied in terms of the distribution of waiting times or in terms

of the rate of change between states in given time intervals.”—[Congdon, 2001]

Therefore, the questions we would like to answer can be described under the literature

of the analysis of failure time data:

• the impact of the covariates (the proportions of the foraging area sprayed with in-

secticides, blocks) on the survival rate or the length of survival

• the characteristics of the survival rate (the rate of no transition between states), for

example, how it changed with time spent in the current state

The whole theory can be found in the book [Kalbfleisch and Prentice, 2002].


Let T denote a random survival time of an individual chick from hatching, i.e. age

of chick, with probability density f(t). Then the probability of an individual chick which

died before time t from hatching is known as the failure function F (t),and expressed as

F (t) = P (T < t), 0 < t < ∞ (3.1)

Two most useful functions are given as definitions:

Definition 3.2.1 The survival function S(t) is the probability of an individual chick which

survives beyond time t.

S(t) = P (T > t) = 1 − F (t), 0 < t < ∞ (3.2)

Definition 3.2.2 The hazard rate is the chance of the death of an individual chick in the

interval (t, t + ∆t) given the survival until t. Hence

h(t) =f(t)

S(t)(3.3)

If it is very short, the function represents an instantaneous hazard rate. If it represents

one day, the hazard rate is a daily hazard rate (dhr) and also known as the daily mortality

rate (dmr). Therefore, the daily survival rate (dsr) is

dsr = 1 − dhr (3.4)

All the notations will be used in the paper to avoid dull English. It also follows that the

cumulative hazard

H(t) =

∫ t

0

h(ν)dν (3.5)


Hence, the survival function (S ) can be written as

S(t) = e−H(t) (3.6)

The probability density can be written as

f(t) = h(t)e[−H(t)] (3.7)

It must be mentioned that in the project, the survival function (S ) is known as the

nestling period survival probability or the single chick survival rate to days.

3.2.1 Failure Time Distributions

From the Bayesian perspective, a prior distribution of the daily hazard rate before the

action that the data are pooled in the models is strongly needed. A simplest hypothesis

is to suppose the survival lifetime follows the exponential distribution. Consequently, the

prior assumption of the hazard rate is h(t) = µ, µ is a constant. It will be dominated and

modified by the likelihood of the data even though it is assumed invariant in the prior.

Therefore, the survival function and the density function of T become, respectively,

S(t) = e−µt (3.8)

f(t) = µe−µt (3.9)

from 3.6 and 3.7. Recall that a constant daily hazard rate is also used by traditional

Mayfield. However, it is treated as a random variable in Bayesian literature which can

be modified by the data. The results which will be discussed later have shown significant

advantages of the Bayesian methods.

Having been obtained from the exponential models, the posterior distribution of the


daily hazard rate has been found a slight increase trend. Therefore, Weibull distribution

is applied to set up a more informative prior distribution for the survival time T .

Two-parameter Weibull distribution W (µ, γ)(µ, γ > 0) allows for a power dependence

of the hazard function on time, where µ and γ are scale and shape parameters respectively

[Congdon, 2003]. Weibull hazard is expressed as

h(t) = µγtγ−1 (3.10)

The survival function

S(t) = e−µtγ (3.11)

The probability density function

f(t) = µγtγ−1e−µtγ (3.12)

The hazard is monotonically increasing for γ > 1, decreasing for γ < 1. In the project, an

empirical value γ = 1.4 is used.

3.2.2 Regression Models

In order to model and determine the relationship between insecticide applications and yel-

lowhammer chick survival rate, regression models are necessary. The explanatory variables

include the proportions of the 200m-radius foraging area sprayed with insecticides 6 20

days before hatching and blocks which are non-informative.

A vector of covariates x with two dimensions (treatments and blocks) is available on

each chick. In general, it is possible for x to include both qualitative and quantitative

variables. Qualitative variables are adopted in the project following the expertise in ecology

in CSL. The first variable is the proportions of the 200m-radius foraging area sprayed with


insecticides before hatching which have been grouped to 6 25% foraging area and > 25%

foraging area whilst the second variable is the number of blocks (1, 2, 3, 4). It was also

reported in the previous paper [Boatman N. D. et al, 2004] that two blocks received normal

insecticide applications whereas the other two blocks received extra insecticide applications.

It looks like that such information can be used to set up two groups to form an independent

covariate. However, the actual procedure included a first spray in late May with normal

farmland practice quantity and a second spray in late June with normal quantity as well.

The invertebrate groups that were important in the diet of yellowhammer chicks might

have recovered [Boatman N. D. et al, 2004]. Hence, the summaries (normal, extra) of

blocks are non-informative, i.e. it is improper if they are mixed with the information of

the proportions to act as an independent regression variable.

The next step is to set up appropriate regression functions. Generally speaking, the

hazard rate usually depends on both time t and explanatory variables x. Therefore, re-

gression models are obtained by allowing the hazard rate to be a function of covariates.

Let b = (b1, . . . , bk) denote the regression parameters, and G is the specified function. For

the exponential distribution, the function can be written as

h(t, x) = µG(bx)

Where µ is the constant. The most natural form of G in survival analysis [Kalbfleisch and

Prentice, 2002] is

G(y) = ey

since it can guarantee that G(bx) > 0 for all possible x. Therefore the model of hazard

rate becomes

h(t, x) = µe(bx) (3.13)


In practice, a slight modification is made: the log survival time is used to set up the

regression model instead of hazard rate to simplify the calculation. This is demonstrated

in the models for the project. In terms of the log survival time

L = ln t (3.14)

The regression equation can be written as

L = b0 + bx (3.15)

where the inception b0 = − ln µ.

Similarly, the GLMs are built for the Weibull distribution. The conditional hazard is

h(t, x) = µγtγ−1e(bx) (3.16)

Alternatively, the linear regression on the log survival time is given as

L = b0 − bx (3.17)

where b0 = − ln µ, and b = µ−1b. Note that there is a difference between the sign in front

the regression coefficients which should be given cautions when analyzing the posterior

summaries.

Finally, it should be mentioned that Linear and Logistic regressions will not work for

right-truncation [Vittinghoff, E. et al, 2005]. For example, logistic regressions are untenable

because equal length of observation time is required. However, right-censoring of the data

caused different lengths of monitoring time.


3.2.3 Survival Analysis in WinBUGS

In the model specification part, a for-loop in which all the functions are specified is used

to go through all the individuals in the sample. For each stochastic node, exponential

distribution is routinely implemented as

t[i] ∼ dexp(µ[i])

while the Weibull is

t[i] ∼ dweib(r, µ[i])

where i is the loop variable and r is the Weibull shape parameter. Then the daily mortality

rate, the daily survival rate, the nestling period survival probability and the probability

density of chick survival time are built up respectively. Finally, the log of mu[i] is written

as a linear link of the covariates.

The coefficients of the covariates were subsequently given very vague, medium non-

informative and informative priors. The reasons and discussions will be given in the next

chapter. The main difficulty in the project is that right-censoring caused missing data. In

WinBUGS, missing data are treated as unknown parameters. Hence, the inferences are

based on the joint posterior distribution of parameters and missing data given the observed

data and prior distribution.

In WinBUGS, “Censoring is denoted using the notation I(lower, upper) e.g.

y ∼ ddist(θ)I(lower, upper)

would denote a quantity y from distribution dist with parameters θ, which had been

observed to lie between lower and upper.”—[User Manual of WinBUGS]

In the project, x is the survival time t of yellowhammer chicks while the lower and


upper are denoted by tmin and tmax respectively. Furthermore, starting values must be

given to the MCMC sampler of the stochastic node t. Two initial files are denoted as

tinitial1 and tinitial2. Three representative types of input data are shown in Table 3.1 to

illustrate how the missing data problem is solved in real models.

Table 3.1: Explanations of Input Data

t tmin tmax tinitial1 tinitial2NA 13 300 14 200NA 13 300 14 200NA 13 300 14 200NA 13 300 14 200NA 12 15 12 15NA 8 11 8 11NA 5 8 5 8NA 1 3 1 37 0 300 NA NA7 0 300 NA NA

The data of four chicks in nest No.247 are shown in the first four rows. All of them

fledged when they were age 13 (i.e. on the 13th day including the hatch day). It is common

for chicks in a specific nest to fledge or to be predated on the same day. Therefore, the death

of each chick should occur between 13 days and an unknown later date since ornithologists

stopped the monitoring once they had observed the outcome of the nest (i.e. the right

truncation happened). In WinBUGS, tmax can be given an arbitrary number larger than

tmin because fledged individuals only contribute to the denominator of the daily hazard

rate from hatching to fledging and the period beyond the nestling duration will be discarded

automatically if the study is restricted to the chick period from hatching to fledging. In

the models, a reasonable value , the average life length of yellowhammer (about 300 days),

is given to tmax. Theoretically, the model can also be used to estimate the chick survival

rate within 300-day average life length even though the estimates are not accurate.

3.3 Models Considering Time Dependence 39

From the 5th to the 8th row, information about four chicks in the nest No.250 is

provided. They all died from starvation, but none of the failure dates was known exactly.

Therefore uncertainty was involved. Since the death of an individual chick happened

between the last observation when it was still alive and the first observation when it was

found to have failed, the corresponding chick ages on these two days are filled into columns

tmin and tmax respectively.

In the two conditions above, the exact age of failure was missing, so the values in the

column of the exact t must be denoted as NA. Furthermore, the most popular method to

check the convergence of the MCMC simulation is to run two chains simultaneously with

widely differing starting values. For missing data models, the widely differing initial values

have to be restricted between the lower and upper bounds Table (3.1).

Occasionally, the dates on which the chicks died were known exactly. It was the case

for the chicks in nest No.304. The relevant data are filled in the last two rows of Table

3.1. In such a condition, the missing data structure will be ignored in WinBUGS. Hence,

the exact chick survival days are filled in the column of t. Two arbitrary numbers have to

be filled in the columns of tmin and tmax to make sure the exact t is in the range. For

the two nodes, they are not stochastic nodes any more, so no MCMC is applied and initial

values are denoted as NA.

3.3 Models Considering Time Dependence

Sometimes, it is of interest that if the chicks monitored in May may have different daily

survival rate with those found in June. Therefore, two models (I and II) considering the

time dependence are built up. For these models, some assumptions are needed to deal

with missing data problems. Therefore, these models are not as appropriate as the failure

data models introduced in the last section. However, as models from a totally different

3.3 Models Considering Time Dependence 40

perspective of survival modelling, these models deserve discussion.

Two assumptions are needed for model I:

• The total number of chicks on day t is known

• The number of chicks survived from day t-1 is known

The idea is simple: the number of survival chicks on day t was part of the total number of

chicks on day t-1. Hence, a natural model would be to assume that

Ns, t ∼ Binomial(dsr, Nt−1)

where Ns, t denotes the number of nestlings on day t surviving from day t-1, Nt−1 denotes

the total number of chicks on day t-1, and dsr is the daily survival rate as usual.

Logistic regression is applied to adopt the covariate t:

logit(dsr) = α0 + α1 ∗ t (3.18)

The regression parameters α0 and α1 are given very vague priors

αn ∼ Normal(0.0, 1.0E − 6)

The data used here are from block one, Castle Howard, 2002. Since not many yel-

lowhammer nestlings are found until after the middle of May, the period from the 18th

May to the 3rd June, lasting for 17 days is studied. Another advantage for using informa-

tion from that period is that most fields on block one received the insecticide treatment

only once on 20th May. Therefore, it should be reflected on the daily survival rate if there

were effects from the insecticide treatment.

In this model, missing data have to be entered under some assumptions. One assump-

3.4 Summary 41

tion from [Johnson, 1979] is suggested by [He, C. Z. et al, 2001]: if days were skipped

between the last visit when it was still found alive and the first observation of the death,

the 40% point is assigned as the day of failure. However, a modification is made after

discussing with ecologists that the previous day before the first observation of the failure is

supposed to be the day of death. Another problem is that the sample size is comparatively

small because the total number of chicks found on one block within a specific duration is

limited. Therefore, the results are relatively large biased. In spite of this, the model still

can provide some useful information. The discussion of the results will be carried out in

the next chapter.

Model II, with a much more complex structure, is built to find out not only the daily

hazard rate but also the probability that a nestling was still found alive on date ta was

subsequently found died on a later day td. It is naturally assumed that the values of the

number of chicks found alive on ta was subsequently found died on td for each specific ta fol-

low a multinomial distribution with proportions Pta, td denoting the probability mentioned

above. The regression equation is the same as equation 3.3.

Daily collected data are also required for this model. Because of the time limit of the

placement, real data are not available for the model. Therefore, only a simulated data

structure taking a form of 5 × 6 matrix is applied. Model discussions will be presented in

the next chapter. For more models considering time dependence in the animal conservation

study, good references include [Brooks, S. P. et al, 2000], [Besbeas, 2002], [Brooks, S.P. et

al, 2004],and [King, R. et al, 2005].

3.4 Summary

This chapter introduces the kernel methods of data analysis involved in the project.

First we discuss how to arrange the data to make them more convenient for Bayesian

3.4 Summary 42

analysis. Then the core issues of model buildings from different aspects are elaborately

described.

Failure time models have successfully solved the missing data problem mainly because

uncertainty modelling is available from Bayesian perspective. Therefore, the main problem

with the data involved in the project is primarily solved. These models are elaborately

built and are the most important models for the posterior inferences carried out in the

next chapter. Some very good estimate are given by these models. It is worth noting that

no approximations on the missing data are needed in these models. Therefore, they have

highlighted Bayesian approaches compared to traditional Mayfield method.

Moreover, new models from a different logical aspect are built. They consider time

dependence. No specific functions in WinBUGS are needed for them. However, they are

not comprehensively developed because of the time limit of the student placement.

Chapter 4

Discussions on Failure Time Data

Models

All the random variables are monitored in WinBUGS, including the posterior summaries

of the variables and the regression coefficients. The discussions begin with the exponential

model. The model comparison between Exponential and Weibull will be carried out in

the Weibull model part. The data from the experiment site, Castle Howard, 2002 (four

blocks) are pooled in. In the sample, the longest nestling period of a individual chick is

15 days. Therefore, the study is restricted to a 15 day range. The default burn-in number

and follow up iterations are 5000 and 20000 respectively.

4.1 Survival Models Based on The Exponential Prior

It takes 132s to run the program on a 1.24 GHz personal computer. Moreover, some longer

sampling chains are also used to test the robustness of the model.

43

4.1 Survival Models Based on The Exponential Prior 44

4.1.1 Models Based on Insecticide Quantities

In this project, the main goal is to find out whether there is indirect effects of insecticides

on the survival of the yellowhammer chicks. Therefore, this is modeled at the first place

by combining the data of the proportions of the foraging area (200m-radius) sprayed with

insecticides 6 20 days before hatching and the data of survival chicks. A negative rela-

tionship between them has been found out. Figure 4.1 plots the posterior means (dots) of

the daily survival rate against two treatment groups.

1.0 1.5 2.0

0.8

0.85

0.9

0.95

daily

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Insecticide applications1: <= 25% 200m-radius foraging area was sprayed2: > 25% 200m-radius foraging area was sprayed

posterior means of the monitored variablelinear fitting

Figure 4.1: The model fit of the daily survival rate over the different proportions of insec-ticides from exponential model

It has been shown that chicks that were in the nest of which more than 25% of the

foraging circle area was sprayed with insecticides had a much lower daily survival rate than

those in foraging area of which less than 25% of the foraging area was treated. The mean

of the dsr the former is a number less than 0.925 whereas that of the latter is about 0.95.

Furthermore, the analysis on the nestling period survival rate (S ) also shows a negative

relationship which is illustrated in Figure 4.2.




1.0 1.5 2.0

0.0

0.25

0.5

0.75

1.0

sing

le s

urvi

val r

ate

to d

ays

Figure 4.2: The model fit of the nestling period survival probability over different propor-tions of insecticides from the exponential model

4.1.2 Models Based on Time

The direct relationship between insecticides and nestling survival is found out in the last

section. In fact, detailed information, i.e. how the insecticides impact on the chick survival,

has to be obtained by modelling variables of interest over time.

The model fit of the chick daily survival rate (dsr) against time t is shown in Figure

4.3. The posterior medians and the 95% Highest Posterior Density Intervals are plotted.

As we can see, the daily survival rate was between about less than 0.80 and 1.00. The

model fit curve oscillates heavily before chick age 10. For such a phenomenon to occur

before chick age 6, especially the steep drop which took place around age 2, one most likely

ecological explanation is that yellowhammer chicks suffered from the starvation caused by

the reduction of arthropod abundance after the insecticide applications [Boatman N. D. et

al, 2004].


The steep drop that happened around chick age 9 is possibly because the risk of preda-

tion increased when the chicks became more vocal and the adult feeding visits to the nest

became more frequent after chick age 8. After about chick age 9, the curve becomes much

smoother. It declines slightly at first, then reaches a local minimum at about chick age 12,

increases again until a local maximum after age 13 and finally goes down. In the sample,

most chicks fledged at about age 13. This can explain why chicks suffered from a relatively

higher dsr at that time. In general, the model fit plot implies that the likelihood of data

has been dominating the posterior distribution. Obviously, the noise from the sample has

influenced the shape of the curve. Therefore, it is better to refer to the plot of the trend.

survival time from hatching to fledging (15 days)

daily

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t

95% posterior intervalposterior median of each element of themonitored variable

Figure 4.3: The model fit of the chick daily survival rate over time

The trend of the dsr is shown in Figure 4.4. A slightly declined trend of the daily

survival rate is observed, implying that the posterior result has changed the prior opinion

because a constant daily mortality rate is summarized in the prior distribution. Therefore,

a more informative prior, the Weibull distribution should lead to some more realistic results.



daily

sur

viva

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ethe posterior means of monitored variableand 95% posterior intervaltrend

t

Figure 4.4: The trend of the chick daily survival rate over time

Another important summary is the nestling period survival probability (S ), which is

also known as the single chick survival rate to days. It is the probability that chick survives

to a certain future date given that it has survived in the past days. The model fit and the

trend are shown in Figure 4.5 and Figure 4.6 respectively.


survival time from hatching to fledging (15 days) t

The nestling period survival probability


Figure 4.5: The model fit of the nestling period survival probability over time



the posterior means of monitored variableand 95% posterior intervaltrend

Figure 4.6: The trend of the nestling period survival probability over time

The model fit shows a vibrated curve before chick age 10. It has kept the consistency

with the plot of the daily survival rate. Therefore, similar ecological interpretations should


be applicable.

Meanwhile, the trend provides a smooth decline with two drops around nestling age

5 and age 12. These changes suggest slightly increased hazard rates around the two ages

which is also consistent with the earlier results.

In bird brood performance studies, the probability density function (pdf ) of the nestling

life time was not emphasized. However, one advantage of this summary is that it can

provide the information more visually, such as which stage of the chick age the single chick

will suffer a higher hazard rate. In other words, the corresponding ecological explanations

will be more reasonable if this summary can be provided. The model fit and the trend of

the pdf are shown in Figure 4.7 and 4.8.

95% posterior intervalposterior median of eachelement of the monitoredvariable


probability density function of survival time t

Figure 4.7: The model fit of the probability density of chick survival life time




t


Figure 4.8: The trend of the probability density of chick survival life time

It can be seen that around age 2 and age 5, chicks were exposed to relative high hazard

risks, confirming the earlier results of the dsr and the S. Moreover, it also shows that the

death of a single chick most likely happened around chick age 2, which is a new finding. It

has been known that the physical mass of the new born chicks in a specific nest is usually

not balanced. Therefore, once the food resources are not abundant, the weak ones will

probably die from starvation. Therefore, the new discovery has provided strong evidence

that the rapid reduction of arthropod groups, caused by insecticide applications, which

comprised the main diet of the yellowhammer nestlings did influence the chick survival

conditions, especially that of the new born chicks.

4.1.3 Models Based on Block Structure

Although it was shown that no significant difference between blocks was detected, detailed

estimates were not available. It might be of interest if more information can be obtained at

the block level. The model fit of daily survival rate against blocks are shown in Figure 4.9.


In the experiment, block 1 and 4 received normal treatments twice which were referred to

as ‘extra’ whereas block 2 and 3 received normal treatment only once which were referred

to as ‘normal’.

Four experiment blocks

daily

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block


Figure 4.9: The model fit of nestling daily survival rate against different blocks

In general, there was no significant difference between the first three blocks on the dsr.

However, the daily survival rate of chicks on the fourth block is much lower than those on

the other three blocks. After checking the original data, a large number of chicks found

on that block died from starvation. Unfortunately, the ecological interpretations are not

available at present.

4.1.4 Analysis of The Posterior Summaries & Convergence of

MCMC Sampling

Besides the discussion from the biometrics perspective, the mathematical analysis of the

posterior summaries obtained by using MCMC simulation is also important. Table 4.1.4


provides various statistics of the posterior summaries of the regression coefficients.

Table 4.1: Posterior summaries from the Exponential Model

node mean sd 2.5% median 97.5%b0 -3.946 7.612 -14.26 -6.229 12.82

b1[1] 1.013 5.276 -9.556 1.949 8.918b1[2] 1.756 5.278 -8.799 2.695 9.663b2[1] -0.6453 3.625 -8.066 -0.2109 5.795b2[2] -0.9388 3.645 -8.462 -0.5143 5.576b2[3] -0.8334 3.634 -8.293 -0.4005 5.622b2[4] 0.6511 3.626 -6.743 1.079 7.095

We note that the posterior means for the slope parameter b1 are positive, implying

that there is a positive relationship (Equation 3.15) between the daily hazard rate and the

insecticide treatments which is consistent with the model fit plots. Moreover, the posterior

mean of b1[2] is larger than that of b1[1], confirming the earlier conclusion again that the

daily hazard risk was higher if a greater proportion of the foraging area was sprayed. In

general, no conclusions of the relationship between the hazard rate and the index of blocks

can be derived since both positive and negative means are involved in b2. However, the

posterior mean of b2[4] suggests that chicks found on the block No.4 suffered a much higher

mortality rate than those on the other three blocks. confirming the model fit in Figure 4.9.

Checking the convergence of the chains for the monitored variables of interest is also

important. In practice, running two long chains with widely differing initial conditions

is the most popular approach. The convergence is said to have been achieved if the two

chains appear to be overlapping one another. Figure 4.10 shows a reasonable convergence

of the two chains for the tenth element of the stochastic node t. From the plot, we can

conclude that the model is appropriate as the convergence is reached rapidly.


initial values for MCMC

Figure 4.10: The convergence of two chains run for t[10]

Once the convergence is satisfactory, the burn-in iterations will be discarded. A fur-

ther number of iterations should be run to obtain samples that can be used for posterior

inferences. There is no fixed value for the number of the following iterations. However,

the Monte Carlo error (MC error) is usually used to assess the accuracy of the required

posterior distributions. It is defined as “the difference between the mean of the sampled

values (which we are using as our estimate of the posterior mean for each parameter) and

the true posterior mean”—[User Manual of WinBUGS]. In practice, it is said to be an

sufficient number of iterations if the MC error for each variable of interest is less than 5%

of the sample standard deviation. Table 4.2 reports a random subset of the posterior sum-

maries and the MC errors of some monitored variables. For a formal method to diagnose

the convergence, please refer to [Brooks and Gelman, 1998].

4.2 Survival Models Based on The Weibull Prior 54

Table 4.2: A random subset of the posterior summaries and the MC errors of some moni-tored variables

node mean sd MC error 2.5% median 97.5%S[116] 0.4233 0.08134 5.503E-4 0.2674 0.4229 0.5839S[117] 0.4237 0.08143 5.477E-4 0.268 0.4233 0.5849S[118] 0.6099 0.09249 5.338E-4 0.4212 0.6146 0.7708S[119] 0.1625 0.103 4.897E-4 0.007498 0.1541 0.3759S[120] 0.163 0.1034 5.39E-4 0.008026 0.1541 0.3771

pdf[116] 0.04263 0.004261 2.155E-5 0.03574 0.04237 0.05075pdf[117] 0.04267 0.004249 2.274E-5 0.03577 0.04243 0.05079pdf[118] 0.06238 0.01033 6.166E-5 0.04595 0.06104 0.085pdf[119] 0.01619 0.009402 4.469E-5 7.888E-4 0.01612 0.03194pdf[120] 0.01622 0.0094 4.876E-5 8.559E-4 0.01614 0.03194t[116] 8.428 0.8638 0.004302 7.065 8.391 9.913t[117] 8.419 0.8622 0.004417 7.066 8.381 9.911t[118] 4.862 1.153 0.005515 3.079 4.789 6.871t[119] 21.04 10.47 0.05434 11.25 17.82 49.27t[120] 20.98 10.41 0.05898 11.24 17.77 48.54

In Table 4.2, all the MC errors after 20000 times following-up are much smaller than

the 5% of the standard deviations. Therefore, the posterior inferences are appropriate. It is

also worth noting that the last two elements of t are estimated with large precision. This is

because, even though the survival conditions of the yellowhammer adults after fledging are

not of interest, the lack of such information and the average survival length of adult birds

which is used as the upper bound of the missing data structure have caused comparatively

large posterior standard deviations.

4.2 Survival Models Based on The Weibull Prior

Having observed a slightly decreasing trend of the daily survival rate, a more expert prior

on the individual survival time, a Weibull distribution is applied. Model comparisons

between the exponential model and the Weibull model are carried out. The discussion on


the effects of different priors will also be reported. An empirical hyperparameter γ = 1.4

is selected to run the Weibull models. The first 5000 iterations are discarded and the

posterior distributions are then inferred after a 20000-iteration run.

4.2.1 Models Based on Non-Informative Priors

In this section, a very ‘vague’ prior for the regression coefficients is used to avoid disturbing

the posterior distribution:

bn ∼ Normal(0.0, 1.0E − 4)

In fact, extensive sensitivity studies in which each of these prior values are increased by

several orders of magnitude have given essentially identical results, implying that the exact

choice of prior had little influence on the posteriors obtained. Therefore, only one of them

will be reported in Section 4.2.2. It takes 169s to run the program on a 1.24 GHz personal

computer.

In order to compare the results with those given by the exponential model, Table 4.3

reports the same subset of the posterior summaries and the MC errors of some monitored

variables as Table 4.2.


Table 4.3: A random subset of the posterior summaries and the MC errors of some moni-tored variables from the Weibull Model under very vague priors


pdf[116] 0.05937 0.005737 2.995E-5 0.04976 0.05914 0.07042pdf[117] 0.05941 0.005766 2.909E-5 0.04984 0.05913 0.07048pdf[118] 0.07543 0.009935 6.467E-5 0.05692 0.07479 0.09644pdf[119] 0.02444 0.01282 6.512E-5 0.001627 0.02519 0.04473pdf[120] 0.02441 0.01278 5.589E-5 0.001657 0.02524 0.04461t[116] 8.43 0.8606 0.004425 7.067 8.398 9.912t[117] 8.427 0.8648 0.004195 7.064 8.391 9.918t[118] 4.946 1.146 0.005735 3.099 4.911 6.888t[119] 16.56 5.244 0.02886 11.15 15.04 30.41t[120] 16.55 5.192 0.02444 11.16 15.07 30.19

Compared to Table 4.2, the 95% HPDI s have comparatively shrunk and the posterior

standard deviations have decreased in general. These are particularly true for the estimates

of t in the last two rows. Moreover, the mean of each element of a monitored variable is

much closer to each other, implying that some exceptional data values hardly shake the

model. Therefore, the posterior summaries from Weibull model are more realistic and

robust. On the other hand, the results from the exponential model are relatively more

biased, implying that the model lacks robustness.

Models Based on Insecticide Quantities

First of all, it is vital to pool the data of the insecticides into the Weibull model. Fig-

ure 4.11 shows the relationship between the proportions of the foraging area sprayed with

insecticides and the chick daily survival rate. Meanwhile, Figure 4.12 illustrates the re-

lationship between the proportions of the foraging area sprayed with insecticides and the


nestling period survival rate. Compared to Figure 4.1 and Figure 4.2, the curve slopes down

more evidently, supporting our assumption that the insecticide applications influenced the

survival conditions of the yellowhammer chicks.

daily

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Figure 4.11: The model fit of the chick daily survival rate over different proportions ofinsecticides (the Weibull model, under non-informative prior distributions)



sing

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to d

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Figure 4.12: The model fit of the nestling period survival rate over different proportionsof insecticides (the Weibull model, under non-informative prior distributions)


Models Based on Time

The models based on time are necessary. Figure 4.13 and Figure 4.14 provide the model

fit and the trend of the daily survival rate over time t respectively. In the model fit plot,

the drops are not so steep as those shown in Figure 4.3, which confirms the analysis of

Table 4.3 that the results are less influenced by some special nestlings with comparatively

extreme life lengths. Furthermore, the 95% Highest Posterior Density Intervals are much

narrower. Meanwhile, the trend shows a more evident decline than that in Figure 4.4. The

plot of the trend also shows some stages in chick age, around which relatively low daily

survival rates occurred. Such results have kept consistency with those in Figure 4.4, but

much clearer. Therefore, the same ecological explanations should work and be given more

credence.


daily

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t


Figure 4.13: The model fit of the daily survival rate over time (the Weibull model, undernon-informative prior distributions)


daily

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ethe posterior means of the monitored variabletrend


Figure 4.14: The trend of the daily survival rate over time (the Weibull model, undernon-informative prior distributions)

Similarly, the plots of model fit and the trend of the single chick survival rate to days

(Figure 4.15 and Figure 4.16) have also shown better estimates than Figure 4.5 and Figure

4.6.




Figure 4.15: The model fit of the individual chick survival rate to days (the Weibull model,under non-informative prior distributions)





Figure 4.16: the trend of the individual chick survival rate to days (the Weibull model,under non-informative prior distributions)

Finally, the model fit and the trend of the probability density of chick survival life

time are provided in Figure 4.17 and Figure 4.18 respectively. It is worth noting that The

95% HPDI s of the pdf of the survival time after chick age 13 (Figure 4.17) have suddenly

increased, suggesting that the monotone declining Weibull prior on the hazard rate is

not very suitable after that time point. Alternatively, the exponential model gives much

narrower HPDI s (Figure 4.7), implying that the daily survival rate after that chick age

was almost invariant. On the other hand, compared to the results about dsr and S, more

information has been provided by the pdf of chick survival time, confirming the earlier

conclusion that the study on the probability density of chick survival time is necessary.


95% posterior intervalposterior median of eachelement of the monitoredvariable



t

Figure 4.17: The model fit of the probability density of the nestling death time (the Weibullmodel, under non-informative prior distributions)



t


Figure 4.18: The trend of the probability density of the nestling death time (the Weibullmodel, under non-informative prior distributions)


Checking The Convergence of The MCMC Simulation

It is still important to check the MCMC simulations for the stochastic nodes of interest,

such as the daily survival rate, the nestling period survival rate and pdf of single chick

failure time. Table 4.3 reports very small MC errors. On the other hand, once the conver-

gence has been reached, samples should look like a random scatter about a stable mean

value. Such evidence is reported in Figure 4.19. Moreover, some smoothed kernel density

estimates are also available. Four posterior densities about the seventeenth nestling are

shown in Figure 4.20. Two chains are run simultaneously, therefore the total sample size

is 40000.

Var

iabl

e va

lue

Var

iabl

e va

lue

Var

iabl

e va

lue

Figure 4.19: The multiple chains of some monitored variables which have reached conver-gence


Figure 4.20: The posterior densities of four monitored variables of the 17th nestling

However, having been checked, the chains for the regression coefficients bn are still far

away from convergence. Such phenomena are shown in Figure 4.21.

b0 chains 1:2

iteration

5001 10000 15000 20000 25000

-300.0

-280.0

-260.0

-240.0

-220.0

b1[1] chains 1:2

iteration

5001 10000 15000 20000 25000

100.0

120.0

140.0

160.0

180.0

b2[1] chains 1:2

iteration

5001 10000 15000 20000 25000

80.0

100.0

120.0

140.0

160.0

Par

amet

er v

alue

Par

amet

er v

alue

Par

amet

er v

alue

Figure 4.21: The pairs of chains for bn which have clearly not reached convergence undervery vague priors


Although they do not influence the inferences which are based on the monitored vari-

ables with converged posterior sample densities, they do imply that the prior distributions

are too vague. Therefore, more informative priors should be applied and the robustness of

the Weibull model should be tested. As a result, the prior parameter values are increased

by several orders of magnitude. As mentioned above, all of them have given essentially

equal results, and therefore only one of them will be discussed in the next section as an

example.

4.2.2 Models Based on Informative Priors

Given the prior distribution,

bn ∼ Normal(0.0, 1.0)

which is in fact the standard normal distribution, the posterior summaries of the regression

coefficients and the monitored variables are reported in Table 4.4 and Table 4.5 respectively.

It takes 177s to run the model on a 1.24 GHz personal computer.

Table 4.4: A random subset of the posterior summaries and the MC errors of some moni-tored variables from the Weibull Model under informative priors

node mean sd MC error 2.5% median 97.5%b0 -2.201 0.6409 0.02564 -3.488 -2.189 -0.9946

b1[1] -1.499 0.5977 0.0226 -2.666 -1.5 -0.3334b1[2] -0.6668 0.6061 0.02214 -1.852 -0.6708 0.5043b2[1] -0.858 0.5166 0.01539 -1.898 -0.85 0.1381b2[2] -1.022 0.6016 0.01349 -2.234 -1.009 0.132b2[3] -0.9346 0.5544 0.01438 -2.044 -0.9276 0.1263b2[4] 0.5433 0.5003 0.01464 -0.4686 0.5471 1.516


Table 4.5: A random subset of the posterior summaries and the MC errors of some moni-tored variables from the Weibull Model under informative priors


pdf[116] 0.05925 0.005629 3.014E-5 0.04997 0.0589 0.07024pdf[117] 0.05923 0.005626 3.102E-5 0.04994 0.0589 0.07016pdf[118] 0.07452 0.009777 6.182E-5 0.0562 0.07404 0.09523pdf[119] 0.02444 0.01282 6.409E-5 0.001629 0.02529 0.04474pdf[120] 0.0245 0.0129 6.75E-5 0.0016 0.02541 0.0448t[116] 8.425 0.8605 0.004308 7.068 8.395 9.909t[117] 8.429 0.8597 0.004353 7.066 8.392 9.911t[118] 4.958 1.149 0.005981 3.099 4.932 6.887t[119] 16.7 5.338 0.02796 11.16 15.17 30.81t[120] 16.7 5.364 0.02825 11.15 15.16 30.69

After comparison, there is little differences in essence between the data in Table 4.5

and Table 4.3, confirming that the model is very robust since the exact choice of prior

has little shaking on the posteriors. Moreover, Figure 4.22 and Figure 4.23 also provide

identical results with the corresponding ones under non-informative priors.



daily

sur

viva

l rat

e

t


Figure 4.22: The model fit of nestling daily survival rate over time (the Weibull, understandard normal distribution priors)

daily

sur

viva

l rat

e



Figure 4.23: The model fit of the chick daily survival rate over different proportions ofinsecticides (the Weibull model, under standard normal distribution priors)

Finally, it can be seen that the MC errors are less than 5% of the standard deviations in

Table 4.4, suggesting the reasonable convergence of chains has been reached, which is also

shown in Figure 4.24. Recall that the samplings on the regression coefficients are still far

4.3 Summary 67

away (Figure 4.21)from the stable means under non-informative priors after even longer

running time. Therefore, the informative priors are preferred in practice.

Par

amet

er v

alue

Par

amet

er v

alue

Par

amet

er v

alue

Figure 4.24: The pairs of chains for bn which have reached reasonable convergence underinformative priors

4.3 Summary

One of the distinguishing features of this chapter is that detailed discussions on the pos-

terior distributions are carried out from many different aspects. Sufficient data are used

to ensure that the models can give reasonable estimates. Different prior distributions (the

4.3 Summary 68

exponential distribution, the Weibull distribution) are used to build the Bayesian models.

Under the Weibull prior of the chick survival time, different prior distributions on the re-

gression coefficients (informative and non-informative) are also used to test the robustness

of posterior inferences. Therefore, all the models can be seen as hierarchical Bayesian GLM

models.

The most important finding is that there is a negative relationship between the insec-

ticide applications and the yellowhammer chick daily survival rate. The main research

goal of the project has been achieved. Furthermore, it is also found that the insecticide

applications negatively influence the yellowhammer nestling period survival rate in general.

Models for the chick daily survival rate, the nestling period survival rate and the prob-

ability density function of the chick survival based on time are built. Many informative

results are obtained and available ecological explanations are presented. Important achieve-

ments include: the yellowhammer chicks in the sample suffered from comparatively high

hazard risk at about chick age 2 and chick age 5; the dsr of the sample was not constant

but decreased. This suggested a more reasonable Weibull prior with a monotone decline

trend should provide more realistic estimates then the exponential prior.

Results from the Weibull prior show more clear features on all the monitored variables,

implying it is more suitable than the simple exponential prior. Different priors on regression

coefficients show no impacts on the results, suggesting that the model is very robust.

Analysis of the posterior summaries and MC errors in tables are also provided. The

convergence of the MCMC simulation is checked to make sure the inference is carried out

under a stable posterior distribution.

Finally, modelling on block level shows no significant difference except block 4. However,

the reason of such a phenomenon is still unknown.

Chapter 5

Discussions on Time Dependence

Models

In general, models discussed in this chapter represent totally different logical thinking

perspective from the last chapter. However, similar results with those presented in the

last chapter are obtained, implying that the Bayesian approaches are very flexible and

powerful. Meanwhile, some new tools in WinBUGs are used in this chapter. The language,

R, is applied as well.

5.1 Model I

For model I, Table 5.1 shows the input data.

69

5.1 Model I 70

Figure 5.1: Data format for model I

Figure 5.2 shows the autocorrelation function of two elements of the variable out to

lag-50. The plots illustrate that the drops of the autocorrelation coefficients happen in a

very short time, implying the reasonable convergence of simulating chains has been reached

rapidly.

Figure 5.2: The autocorrelation functions of the daily survival rate on the 9th day and the10th day respectively

Figure 5.3 provides a plot (in R) demonstrating the decrease of the daily survival rate

from 20th May, 2002 to 3rd June, 2002. A very smooth declining curve is obtained,

confirming the results from the failure time data models that the dsr of the yellowhammer

chick found in the experiment site was not a constant.

5.2 Model II 71


daily

sur

viva

l rat

e

95% posterior interval

posterior median of eachelement of the monitoredvariable

t

Figure 5.3: The daily survival rate of chicks from 20th May, 2002 to 3rd June, 2002(BlockOne)

5.2 Model II

For Model II, only some simulated data are used to investigate the feasibility of the model

which are shown in Table 5.1. Suppose that the chick survival conditions were recorded

for six days, from day 1 to day 6. Denote each row as t1 (2 6 t1 6 6) and each column

as t2 (1 6 t2 6 6). Because the model is built to calculate the probability of survival to

days with an arbitrary beginning, each element of the matrix except those in the right

most column is used to denote the total number of chicks still found alive on t1 − 1 and

subsequently found dead on t2. The right most column is filled with the number of survival

chicks from t1 − 1 when the observation stopped. Consequently, the sum of each element

on row t1 is the total number of chicks found on t1 − 1.

5.2 Model II 72

Table 5.1: Input data for model II

13 2. 1. 3. 4. 600. 14 3. 5. 3. 500. 0. 15 3. 5. 400. 0. 0. 19 5. 300. 0. 0. 0. 5 20

Although model II is much more complex than model I, the convergence of chains

is reached rapidly (Figure 5.4). This is because very small data set is applied. Widely

differing initial values for chains are used to test the convergence.

Figure 5.4: The convergence for parameters αc and βc respectively

Alternatively, the convergence can be monitored dynamically by using the trace tool.

Figure 5.5 provides two examples.

5.3 Summary 73

Figure 5.5: The dynamics of the convergence for parameters αc and βc respectively

Because the original data are not arranged into a reasonable form for Model II, survival

analysis is not available. However, interesting findings should be available if this model

can be further developed.

5.3 Summary

In general, models in this chapter represent very novel ideas. The time dependence should

be considered in this project because the effects of insecticides decline when time passes,

i.e. it is a time dependent variable. A very exciting finding is that a declining trend of the

dsr between 20th May and 3rd June 2002, given by Model I, confirms the models in the

last chapter which arise from a totally different logical perspective. It should give more

accurate results if more data from more blocks are used. However, that was not carried

out because of the time limit of the student placement.

Model II can be used to model the nestling period survival rate with an arbitrary start

point and end point. Compared to other models in which the start point is restricted to

the hatch day, the results given by this model are more flexible and more abundant if real

data can be used.

Both the models may overestimate the underlying variables because the approximations

are used when the right censoring occurs on both death and fledging.

Further Developments

First of all, for the failure time models, non-grouped insecticide data will be applied to

give more detailed relationships between the insecticides and chick survival conditions. All

data from all experiments will be pooled in. In the current models, chicks are not differed

by their encounter ages, therefore age-specific survival rate will be studied. Similarly, the

chicks are supposed to be independent. However, the correlations between the death of

a chick in a specific nest might influence the survival conditions of the living ones. For

example, the death of one chick might give the surviving chicks a higher probability of

survival rate, based on an increased capacity of available food resources, aided with the

reduction of food competition.

Secondly, for the models with time dependence, more data from different time interval,

say July, is needed for model I and real data should be entered for model II.

Until now, WinBUGS still suffer from the convergence problems caused by huge data

sets. A computer in CSL always crashed when the author ran the WinBUGS programmes.

Similar experience also reported by other WinBUGS users. On the other hand, all data

from all sites will surely comprise an extremely huge data set because chick numbers are

much more than nest numbers. Therefore, a lower-level language instead of WinBUGS,

will be used to overcome such problems.

74

Conclusions

To demonstrate a link between the insecticides and the population decline of a specific bird

species, evidence is needed from different aspects [Boatman N. D. et al, 2004].

New and very important evidence for the indirect effects of insecticides on the yel-

lowhammer chick survival conditions has been found in the project. Therefore, more infor-

mation from different catalogues is now available on the link between the indirect effects of

insecticides and the population decline of the yellowhammer. If further work can be done,

a conclusive demonstration of that link should be very possible. Consequently, it will help

people to carry out healthier farming practice to protect the ecological system.

Moreover, this project represents a new attempt of applying Bayesian approaches to

model chick survival conditions. No assumptions are needed for the missing data in the

Bayesian failure time models. Therefore, it is much more advanced than the traditional

Mayfield method which has been dominating this area since 1960s. Other attempts of

using Bayesian methods include summarizing the data from different aspects, modelling

more complex chick survival probabilities.

A high-profile publication is very likely in the light of the new evidence and the novel

Bayesian application.

Personally, the author has learnt a lot of the Bayesian approaches and has been im-

pressed by the power and the flexibility of such methods in the bio-statistical modelling.

Meanwhile, the author has built a good working relationship with scientists in CSL and

75

5.3 Summary 76

therefore has motivated to carry out further developments on the project in CSL.

Appendix I

The Exponential Model

model{

for(i in 1:N){

t[i] ~ dexp(mu[i]) I(tmin[i], tmax[i])

S[i] <- exp(-mu[i]*t[i])

h[i] <- mu[i]

dsr[i] <- 1-h[i]

pdf[i] <- mu[i]*S[i]

log(mu[i]) <- b0 + b1[treat[i]] + b2[block[i]]

}

b0 ~ dnorm(0.0, 0.001)

for(k in 1:4){

b1[k] ~ dnorm(0.0, 0.01)

b2[k] ~ dnorm(0.0, 0.01)

}

}

77

Appendix I 78

The Weibull Model

model{

for(i in 1:N){

t[i] ~ dweib(r, mu[i]) I(tmin[i], tmax[i])

h[i] <- mu[i]*r*pow(t[i], r-1)

dsr[i] <- 1-h[i]

S[i] <- exp(-mu[i]*pow(t[i], r))

pdf[i] <- h[i]*S[i]

log(mu[i]) <- b0 + b1[treat[i]] + b2[block[i]]

}

b0 ~ dnorm(0.0, 1.0E-4)

for(k in 1:2){

b1[k] ~ dnorm(0.0, 1.0E-4)

}

for(k in 1:4){

b2[k] ~ dnorm(0.0, 1.0E-4)

}

}

Model I

Main Model in WinBUGS

model{

for( t in 1 : T-1 ) {

Appendix I 79

Ns[t+1] ~ dbin(dsr[t],N[t])

logit(dsr[t])<- alpha0+alpha1*(t)

}

alpha0 ~ dnorm(0.0,1.0E-6)

alpha1 ~ dnorm(0.0,1.0E-6)

}

data

list(T=17, N= c(3, 10, 10, 13, 12, 11, 11, 11, 10, 11, 11, 13,

12, 8, 6, 6, 6), Ns=c(0, 2, 10, 10, 12, 11, 11, 11, 10, 9, 11, 11,

12, 8, 6, 6, 6) )

initials(1)

list(alpha0 = 0, alpha1 = 0 )

initials(2)

list(alpha0 = 500, alpha1 =500 )

R Commands for The Plot

setwd("I:/") data <- read.table("I:/Model 1 Input data.txt")

dsr <-data[,2]

plot(1:16,dsr,type="l")

plot(1:16,dsr,ylim=c(0,1),type="l")

sd <- data[,3]

hi <- dsr+1.96*sd

Appendix I 80

lo <- dsr-1.96*sd

par(new=T)

plot(1:16,hi,ylim=c(0,1),type="o")

par(new=T)

plot(1:16,lo,ylim=c(0,1),type="o")

Model II

alive[t, t]: the no. of chicks found alive on day t

dsr[t]: chick survival rate on from day t-1 to day

model{

alphac~ dnorm(0, 0.01)

betac~ dnorm(0, 0.01)

for(i in 1:T){

logit(dsr[i])<- alphac+betac*(i)

}

for(t in 1:T){

m[t, 1:(T+1)] ~ dmulti(p[t, ], alive[t])

}

for(t in 1:T){

alive[t]<-sum(m[t, ])

}

Appendix I 81

#T:total observation days -1= 5

#chicks found alive on day 0~(T=4)

are found dead on t2: 1<=t1<=(T-1), t1<t2<=T

#t1: the second day to the fifth day starting

#t2: same with t1

for(t1 in 1:T){

for(t2 in t1:T){

for(i in t1:t2){

lsur[t1, t2, i]<-log(dsr[i])

}

p[t1, t2]<- exp(sum(lsur[t1, t2, t1:t2]))*(1-dsr[t2])

}

for(t2 in 1:(t1-1)){

p[t1, t2]<-0

}

p[t1, T+1] <- 1-sum(p[t1, 1:T])

}

}

data

list(T=5,

m=structure(

.Data = c(

13, 2., 1., 3., 4., 60,

Appendix I 82

0., 14, 3., 5., 3., 50,

0., 0., 15, 3., 5., 40,

0., 0., 0., 19, 5., 30,

0., 0., 0., 0., 5, 20)

,.Dim = c(5,6)

)

)

initials

list(alphac=2, betac=-0.1)

Appendix II

Initial Values for The Failure Time Models

Inits(1)

list(

t=c(14,14,14,14,12,8,5,1,14,14,14,14,14,14,

14,14,0,14,14,0,14,3,0,14,14,14,14,14,14,14,

14,6,6,0,0,3,3,3,3,5,5,5,14,14,14,0,14,14,5,

5,2,14,14,14,14,14,14,14,14,16,16,16,16,14,

8,2,2,14,14,14,14,14,14,14,14,14,14,2,2,2,2,

2,14,14,14,14,14,3,14,14,14,3,3,0,14,14,

4,4,0,0,NA,6,11,3,3,3,14,14,3,3,NA,NA,3,

3,7,7,7,3,11,11,3,3,9,9,6)

)

Inits(2)

list(

t=c(200,200,200,200,15,11,8,3,200,200,200,

200,200,200,200,200,1,200,200,3,200,6,3,200,

200,200,200,200,200,200,200,7,7,3,3,6,6,6,6,

8,8,8,200,200,200,2,200,200,9,9,5,200,200,200,

83

Appendix II 84

200,200,200,200,200,200,200,200,200,200,9,6,6,

200,200,200,200,200,200,200,200,200,200,6,6,5,

5,5,200,200,200,200,200,6,200,200,200,5,5,3,

200,200,7,7,4,4,NA,8,14,7,7,7,200,200,7,7,NA,

NA,6,6,10,10,10,7,200,200,7,7,13,13,9)

)

Input Data for The Failure Time Models

Rectangular Structure

Data

list(N=125)

t[] tmin[] tmax[] treat[] block[]

NA 13 300 2 1

NA 13 300 2 1

NA 13 300 2 1

NA 13 300 2 1

NA 12 15 2 1

NA 8 11 2 1

NA 5 8 2 1

NA 1 3 2 1

NA 11 300 2 1

NA 11 300 2 1

NA 13 300 1 1

NA 13 300 1 1

Appendix II 85

NA 13 300 1 1

NA 12 300 1 1

NA 12 300 1 1

NA 12 300 1 1

NA 0 1 1 1

NA 13 300 2 1

NA 13 300 2 1

NA 0 3 2 1

NA 12 300 2 1

NA 3 6 2 1

NA 0 3 2 1

NA 12 300 2 1

NA 12 300 2 1

NA 12 300 1 1

NA 12 300 1 1

NA 12 300 1 1

NA 12 300 1 1

NA 11 300 2 1

NA 11 300 2 1

NA 6 7 1 1

NA 6 7 1 1

NA 0 3 1 1

NA 0 3 1 1

NA 3 6 2 1

NA 3 6 2 1

NA 3 6 2 1

Appendix II 86

NA 3 6 2 1

NA 5 8 2 1

NA 5 8 2 1

NA 5 8 2 1

NA 13 300 2 1

NA 13 300 2 1

NA 13 300 2 1

NA 0 2 2 1

NA 9 300 2 1

NA 9 300 2 1

NA 5 9 1 1

NA 5 9 1 1

NA 2 5 1 1

NA 11 300 1 1

NA 11 300 1 1

NA 11 300 1 1

NA 13 300 1 2

NA 13 300 1 2

NA 13 300 1 2

NA 13 300 1 2

NA 11 300 1 2

NA 15 300 1 2

NA 15 300 1 2

NA 15 300 1 2

NA 15 300 1 2

NA 9 300 1 2

Appendix II 87

NA 8 9 1 2

NA 2 6 1 2

NA 2 6 1 2

NA 10 300 1 3

NA 10 300 1 3

NA 10 300 1 3

NA 9 300 1 3

NA 9 300 1 3

NA 11 300 1 3

NA 11 300 1 3

NA 11 300 1 3

NA 11 300 1 3

NA 13 300 2 3

NA 2 6 2 3

NA 2 6 2 3

NA 2 5 2 3

NA 2 5 2 3

NA 2 5 2 3

NA 10 300 1 3

NA 10 300 1 3

NA 10 300 1 3

NA 13 300 1 3

NA 13 300 1 3

NA 3 6 1 3

NA 13 300 1 3

NA 13 300 1 3

Appendix II 88

NA 13 300 1 3

NA 3 5 1 4

NA 3 5 1 4

NA 0 3 1 4

NA 8 300 1 4

NA 8 300 1 4

NA 4 7 2 4

NA 4 7 2 4

NA 0 4 2 4

NA 0 4 2 4

9 0 300 2 4

NA 6 8 2 4

NA 11 14 1 4

NA 3 7 1 4

NA 3 7 1 4

NA 3 7 1 4

NA 14 300 1 4

NA 14 300 1 4

NA 3 7 1 4

NA 3 7 1 4

7 0 300 1 4

7 0 300 1 4

NA 3 6 1 4

NA 3 6 1 4

NA 7 10 1 4

NA 7 10 1 4

Appendix II 89

NA 7 10 1 4

NA 3 7 1 4

NA 11 300 1 4

NA 11 300 1 4

NA 3 7 1 4

NA 3 7 1 4

NA 9 13 1 4

NA 9 13 1 4

NA 6 9 1 4

NA 3 6 1 4

END

S-Plus Structure

list( N=126, t=c(NA,NA,NA,NA,NA,NA,

NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,




NA,15,15,15,15,NA,NA,NA,NA,NA,NA, NA,

NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,

NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,

NA,NA,NA,NA,NA,NA,9,NA,NA,NA,NA,NA,

NA,NA,NA,NA,7,7,NA,NA,NA,NA,NA,NA,

NA,NA, NA,NA,NA,NA,NA,NA),

Appendix II 90

tmin=c(13,13,13,13,12,8,5,1,11,11,13,

13,13,12,12,12,0,13,13,0,12,3,0,12,12,

12,12,12,12,11,11,6,6,0,0,3,3,3,3,5,5,

5,13,13,13,0,9,9,5,5,2,11,11,11,13,13,

13,13,11,0,0,0,0,9,8,2,2,10,10,10,9,9,

11,11,11,11,13,2,2,2,2,2,10,10,10,13,

13,3,13,13,13,3,3,0,8,8,4,4,0,0,0,6,11,

3,3,3,14,14,3,3,0,0,3,3,7,7,7,3,11,11,3,

3,9,9,6,3),

tmax=c(300,300,300,300,15,11,8,3,300,

300,300,300,300,300,300,300,1,300,300,

3,300,6,3,300,300,300,300,300,300,300,

300,7,7,3,3,6,6,6,6,8,8,8,300,300,300,

2,300,300,9,9,5,300,300,300,300,300,300,

300,300,300,300,300,300,300,9,6,6,300,

300,300,300,300,300,300,300,300,300,6,

6,5,5,5,300,300,300,300,300,6,300,

300,300,5,5,3,300,300,7,7,4,4,300,

8,14,7,7,7,300,300,7,7,300,

300,6,6,10,10,10,7,300,300,

7,7,13,13,9,6),

treat =c(4,4,4,4,4,4,4,4,4,4,3,3,3,

3,3,3,3,4,4,4,4,4,4,4,4,3,3,3,3,4,

Appendix II 91

4,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,

4,3,3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,

1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,

2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,

2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,

1,1,1,1,1,1,1,1,1,1,1),

block =c(1,1,1,1,1,1,1,1,1,1,1,1,1,

1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,

2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,

3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,

4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,

4, 4,4,4,4,4,4,4,4,4,4) )

Input Data for Model I Plot

node mean sd

dsr[1] 0.9599 0.0281

dsr[2] 0.9575 0.02669

dsr[3] 0.9548 0.02534

dsr[4] 0.9516 0.02404

dsr[5] 0.948 0.02283

dsr[6] 0.9439 0.02175

dsr[7] 0.9391 0.02091

dsr[8] 0.9337 0.02053

Appendix II 92

dsr[9] 0.9275 0.02088

dsr[10] 0.9205 0.02235

dsr[11] 0.9125 0.0253

dsr[12] 0.9033 0.02996

dsr[13] 0.893 0.03649

dsr[14] 0.8813 0.04491

dsr[15] 0.8683 0.05525

dsr[16] 0.8538 0.06742

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