Table Of ContentResource Selection by Animals
RESOURCE SELECTION BY
ANIMALS
Statistical design and analysis for field studies
Bryan F. J. Manly
Department of Mathematics and Statistics
University of Otago, New Zealand
Lyman L. McDonald
WEST Inc., Wyoming, USA
Dana L. Thomas
Department of Mathematical Sciences
University of Alaska, USA
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
First edition 1993
© 1993 Bryan F. J. Man1y, Lyman L. McDonald and Dana L. Thomas
Originally published by Chapman and Hali in 1993
Softcover reprint of the hardcover 1s t edition 1993
Typeset in 10/12 pt Times by Expo Holdings, Malaysia
ISBN 978-94-010-4680-0 ISBN 978-94-011-1558-2 (eBook)
DOI 10.1007/978-94-011-1558-2
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Contents
Preface ix
List of symbols xi
1. Introduction to resource selection studies 1
1.1 Motivation and definitions
1.2 The data for resource selection studies 4
1.3 Sampling designs 4
1.4 Sampling protocols 8
1.5 Indices of selection 9
1.6 Hypothesis tests and confidence intervals 11
1. 7 Discussion 13
2. Examples of the use of resource selection functions 14
2.1 Introduction 14
2.2 Examples 15
2.3 Sample designs 29
2.4 Assumptions 29
3. Statistical modelling procedures 32
3.1 Simple sample comparisons 32
3.2 Testing coefficients of individual X variables 33
3.3 Log-likelihood chi-squared tests 33
3.4 Analysis of residuals 34
3.5 Multiple tests and confidence intervals 35
4. Studies with resources defined by several categories 36
4.1 Introduction 36
4.2 Sampling designs and protocols 36
4.3 Ratios of random variables 37
4.4 Chi-squared tests 39
4.5 Design I with known proportions of available resource units 40
4.6 Tests on proportions of used units 42
VI Contents
4.7 Inferences concerning selection ratios 46
4.8 Comparison of selection ratios 48
4.9 Design I with estimated proportions of available resource units 49
4.10 Design II with sampling protocol A 54
4.11 Census of available resource units 54
4.12 Sample of available resource units 61
4.13 Design III with sampling protocol A 65
4.14 Census of available resource units 65
4.15 Sample of available resource units 66
4.16 Discussion 67
5. Estimating a resource selection probability function from a census
of resource units using logistic regression 70
5.1 Introduction 70
5.2 Estimating the logistic function 71
5.3 Discussion 82
6. Estimating resource selection probability functions from a census
of resource units at several points in time using the proportional
hazards model 89
6.1 Introduction 89
6.2 Estimating the proportional hazards function 91
6.3 Discussion 95
7. Estimating a resource selection function from samples of resource
units using proportional hazards and log-linear models 99
7.1 Introduction 99
7.2 Expected sample frequencies for resource units 100
7.3 The proportional hazards model 101
7.4 Small proportion of resource units used 102
7.5 Only samples of unused resource units available 103
7.6 Estimating parameters from sample data 103
7.7 Inferences with sample data 106
7.8 Recommendations 122
8. Estimating a resource selection function from two samples of
resource units using logistic regression and discriminant function
methods 125
8.1 Logistic regression 125
8.2 Linear discriminant function analysis 130
8.3 Quadratic discriminant function analysis 135
8.4 Discussion 135
Contents vii
9. General log-linear modelling 137
9.1 Introduction 137
9.2 Fitting models to data 138
10. Analysis of the amount of use 145
10.1 Introduction 145
10.2 Analysis of counts of the amount of use 146
10.3 Analysis of continuous measures of the amount of use 154
10.4 Mixtures of zeros and positive measures of the amount of use 155
11. The comparison of selection for different types of resource unit 158
11.1 Variances for estimates and their differences 158
References 165
Author index 173
Subject index 176
Preface
We have written this book as a guide to the design and analysis of field studies
of resource selection, concentrating primarily on statistical aspects of the
comparison of the use and availability of resources of different types. Our in
tended audience is field ecologists in general and wildlife biologists in particular
who are attempting to measure the extent to which real animal populations are
selective in their choice of food and habitat. As such, we have made no attempt
to address those aspects of theoretical ecology that are concerned with how
animals might choose their resources if they acted in an optimal manner.
The book is based on the concept of a resource selection function, where this
is a function of characteristics measured on resource units such that its value for
a unit is proportional to the probability of that unit being used. We argue that
this concept leads to a unified theory for the analysis and interpretation of data
on resource selection and can replace many ad hoc statistical methods that have
been used in the past.
The book has the following structure: Chapters 1 and 2 provide a review of
statistical methods that have been used to study resource selection and some
examples of how different types of study can be analysed in terms of resource
selection functions; Chapter 3 gives a brief introduction to a range of statistical
techniques (mainly used in Chapters 5 to 10) that are useful for modelling data;
Chapter 4 deals with the particularly important special case where the resources
being studied are in categories of habitat or food types; Chapters 5 and 6 deal
with situations where the resource consists of a finite number of identified units
and it is known which of these are used and which are not used; Chapters 7 and
8 deal with situations where samples of resource units of different types are
taken (e.g. a sample of used units and an independent sample of unused units);
Chapter 9 gives an introduction to log-linear modelling in the context of
resource selection studies; Chapter 10 gives some suggestions on how to analyse
data when the amount of use of resource units is recorded rather than simply
whether or not they are used; and, finally, Chapter 11 covers methods for
assessing the accuracy of estimates from resource selection functions, as well as
differences between these estimates, and ratios of these estimates.
The procedures that we discuss for estimating resource selection functions
include some (such as logistic regression) that are standard enough to be avail
able in some of the popular statistical packages, and others (such as fitting a
proportional hazards model to sample data) that are by no means standard. Even
x Preface
with standard analyses there are often some complications involved in setting up
resource selection data in order to use a computer program to estimate a resource
selection function. Therefore, we have found it convenient to produce a
computer program that carries out most of the calculations described in Chapters
5 to 8. This program does not cover the methods described in Chapter 4 for
categorical data since we feel that these are simple to set up in a spreadsheet
program such as LOTUS 123. General log-linear modelling as discussed in
Chapter 9 is also omitted since we believe that there are already good computer
programs available for this purpose.
The program, which is called RSF (Resource Selection Functions), runs on
IBM PC and compatible computers with at least 640k of ram. It is written in
FORTRAN and uses the MAXLIK algorithm for maximum likelihood esti
mation of models for count data (Reed and Schull, 1968; Reed, 1969; Manly,
1985, p. 433). For more information contact WEST Inc., 1406 South Greeley
Highway, Cheyenne, Wyoming 82007, USA (telephone 307-634-1756; fax 307-
637-6981).
We acknowledge the help of our colleagues in writing this book. Particularly,
we acknowledge Dan Reed of the Alaska Department of Fish and Game who
assisted with early versions of Chapter 4, Ed Arnett, C. R. Bantock, W. F. Harris
and Tom Ryder who provided us with raw data from their studies, and Diana
Craig who read over the final manuscript. However, any errors or omissions are
entirely our responsibility.
BFJM, LLMc and DLT
Denver, January 1992
List of Symbols
Symbol Definition
I Number of categories of resource units
m+ Size of a sample of available resource units
mi Number of available units in category i in a sample of available
resource units
7Ti Proportion of the population of available units that are in category i
7Ti mJm+, sample proportion of available units in category i
u+ Size of a sample of used resource units
ui Number of units in category i in a sample of used units
oi uJu+, sample proportion of used units in category i
fi Proportion of the available category i items that are used
wi oJiri, the forage ratio (also called the selection ratio and the
preference index)
Ei (oi-iri)/(oi + ir), lvlev's electivity index
Li oi-iri, Strauss' linear selection index
Qi [oi(l-iri)]/[ iri(l-oi)], Jacobs' first selection index
Di (oi-iri)/(oi + iri-2oi7Ti), Jacobs' second selection index
ai (o J iri)/'i,( oJ iri), Chesson's selection index
Bi (uJmi)/1:(uJmi), Manly's standardized selection index with used
1
resource units replenished
B i log(l-fi)/Ilog(l-fi), Manly's standardized selection index with
2
used resource units not replenished
Wi (/'i.fi, Vanderploeg and Scavia's first selection index
E*i (Wi- l/1)/(Wi + 1/1), Vanderploeg and Scavia's second selection
index
SI MAX[(ImJm+)-(IoJo+)], Rondorff et al. selection intensity for
continuous data
w*(x) Resource selection probability function for a single period of
selection where resource units are characterized by the values x =
(x,, Xz, •.. , Xp) for variables XI tO Xp
w(x) Resource selection function, which is w*(x) multiplied by an
unknown positive constant
w*(x, t) Resource selection probability function for selection from time 0 to
timet
w(x, t) Resource selection function for selection from time 0 to time t
xii List of symbols
q/(x, t) Probability that a resource unit is not used ('survives') in the time
interval 0 to t
cf>(x, t) c/J* (x, t) multiplied by an unknown positive constant
{3j Coefficient of the variable Xi in a resource selection or survival
function
Estimate of f3i
Standard error of the estimator of f3i
Log-likelihood chi-squared statistic for measuring the goodness of fit
of a model for data
Pearson chi-squared statistic for measuring the goodness of fit of a
model for count data
Standardized residual for the ith observation in a set of data
f.Lx Population mean of the random variable X
sx Sample standard deviation of the variable X
rx Y Sample Pearson correlation between variables X and Y
var(X) Variance of the random variable X
cov(X, Y) Covariance of random variables X and Y
A+ Size of a finite population of available resource units
Ai Number of the A+ units that are in category i
Ui Number of used resource units in category i for the population
U+ Number of used resource units in all categories
w*i Proportion of the Ai available resource units that are used
W*i Estimate of w~
za Value that is exceeded with probability a by a standard normal
random variable
E(X) Expected (mean) value of a random variable X
uii Number of category i resource units used by animal j
ui+ Number of category i resource units used by all sampled animals
u+i Number of resource units used by animal j
u++ Total number of units used by all sampled animals
wii Selection ratio for the jth animal and the ith resource category
~ii Proportion of the resources available to animal j that are in category i
7Tii Sample estimate of 7Tii
S Number of selection episodes
E>ii Probability that a type i resource unit will be used in the time interval
(tj-1' tj)
P. Probability of sampling an available resource unit
P t) Probability of sampling a used resource unit at time t
u (
Prr (t) Probability of sampling an unused resource unit at time t
ui (t) Number of resource units of type i found in a sample of used units
taken at time t
iii (t) Number of resource units of type i found in a sample of unused units
taken at time t
