E-SURGE 1.8 user`s manual

E-SURGE 1.8 user`s manual
E-SURGE 1.8 user’s manual
(MultiEvent Survival Generalized Estimation).
A program for fitting Multievent models.
CHOQUET Rémi, NOGUE Erika
Biostatistics and Population Biology group, cefe-umr 5175,
1919 Route de Mende, 34293, Montpellier, France
September 2011
This manual is to be cited as:
Choquet, R., Nogué, E. (2011). e-surge 1-8 user’s manual. cefe, UMR 5175, Montpellier, France.
(http://ftp.cefe.cnrs.fr/biom/soft-cr/).
A regular paper about e-surge is:
R. Choquet, L. Rouan, and R. Pradel. (2009). “Program e-surge: A software application for fitting
Multievent models.” In David L. Thomson, Evan G. Cooch, and Michael J. Conroy, editors, Modeling
Demographic Processes in Marked Populations, volume 3 of Environmental and Ecological Statistics,
pages 845-865. Springer.
Made with LATEX.
Contents
Contents
1
List of Tables
4
List of Figures
4
1 Introduction
1
2 Models
3
2.1
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Multievent models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.3
Umbrella models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Decomposition in Elementary Steps
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Age dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.4
Constrained models
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.5
The link function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.6
Unequal time intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.7
Maximum likelihood estimation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.8
Factorizing the likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.9
Individual covariates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.10 Independent and identically distributed (i.i.d.) random effect . . . . . . . . . . . . . . .
11
For individuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
For groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.11 Conditionality on the first occasion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
Open and closed population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
Occupancy models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.12 Non-linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3 GEPAT
15
3.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.2
A combined formulation of the Arnason-Schwarz model . . . . . . . . . . . . . . . . . .
16
3.3
The separate formulation for the Arnason-Schwarz model . . . . . . . . . . . . . . . . .
18
3.4
A version of the Arnason-Schwarz model with site fidelity parametrization . . . . . . . .
19
Defining sets of intermediate states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
Matrices and pattern matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
GEPAT in practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.5
1
2
CONTENTS
3.6
The vector of biological parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 GEMACO
5
22
23
4.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
4.2
Keywords for main effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
How is the MDL phrase time interpreted by GEMACO ?. . . . . . . . . . . . . . . . . .
24
How is the MDL phrase age interpreted by GEMACO ?. . . . . . . . . . . . . . . . . . .
25
How is the MDL phrase group interpreted by GEMACO ?. . . . . . . . . . . . . . . . .
25
How are the MDL phrases f rom and to interpreted by GEMACO ? . . . . . . . . . . .
27
4.3
Combining effects with operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
4.4
External covariates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.5
Individual covariates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
A dedicated keyword xind
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
How act operators on xind ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.6
Aggregation of parameters: lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.7
Aggregation of parameters: the aggregation operator . . . . . . . . . . . . . . . . . . . .
32
4.8
Keyword ”others” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
4.9
Shortcuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
Definition of shortcuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
Shortcuts in practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
4.10 Redundancy in matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
4.11 Random effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
4.12 A list of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
4.13 Aggregating mathematical parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
4.14 Non-linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
Data input
39
5.1
The BIOMECO format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
5.2
The MARK format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
5.3
The HEADED format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
The format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
The Header Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
A headed format example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
Automatic shortcuts/clusters creation and use . . . . . . . . . . . . . . . . . . . . . . . .
42
5.4
File of external covariates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
5.5
File of time intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
5.6
Selecting steps for unequal time interval . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
CONTENTS
5.7
File of initial values fixed values
3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 A short session
7
8
44
47
6.1
Main window of E-SURGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
6.2
Opening a session . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
6.3
Building matrix patterns using GEPAT . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
6.4
Building constraint using GEMACO . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
6.5
Changing Initial Values or Fixing Values . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
6.6
Setting optimization parameters and running the model . . . . . . . . . . . . . . . . . .
59
6.7
Output of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
Advanced tools for numerical issues
63
7.1
Initial values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
7.2
Catchpole Morgan Freeman approach for redundancy
64
. . . . . . . . . . . . . . . . . . .
Advanced tools for output
65
8.1
State dependent probabilities of CR histories . . . . . . . . . . . . . . . . . . . . . . . .
65
8.2
Non-parametric Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
8.3
The Viterbi and the Counting algorithms . . . . . . . . . . . . . . . . . . . . . . . . . .
67
8.4
Tests for environmental covariates in presence of a random effect . . . . . . . . . . . . .
71
9 Interpreting the output
71
9.1
CAS output file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
9.2
File heading (L1-L9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
9.3
Information about the model (L11-L21) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
9.4
Minimization (L23-L26 and L355-L426) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
9.5
Deviance, AIC and related topics (L35-L44 and L431- L502) . . . . . . . . . . . . . . . . .
73
9.6
Beta estimates (L355-L426) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
9.7
MLEs of parameters and standard errors (L134-L348) . . . . . . . . . . . . . . . . . . . .
74
10 A few warnings
75
10.1 Local minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
10.2 Saddle point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
10.3 Additive models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
10.4 Generalized logit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
11 Acknowledgements
76
12 Conditions of external use
76
13 Output text file
77
Bibliography
84
Index
89
List of Tables
1
Umbrella models of e-surge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2
Constraint matrix associated to time and generated by gemaco. . . . . . . . . . . . . . . .
24
3
Phrase age interpreted by gemaco. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
4
Constraint matrices associated to time and group generated by gemaco . . . . . . . . . . .
26
5
Keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
6
Constraint matrices associated to time.group and time + group generated by gemaco . . .
29
7
Models in notation of [25] vs. in gemaco . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
8
Description of the biomeco file format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
9
Description of the mark file format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
10
Description of the headed file format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
11
Description of the external covariates file . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
12
Description of the file of time intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
13
Description of a IVFV file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
14
Multiple random values from IVFV file
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
1
Definition of the UM from the GM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2
First capture in Multievent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3
Conditionality options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
4
Semi-Markov options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
5
Decomposition of the conditional event probabilities . . . . . . . . . . . . . . . . . . . . . .
16
6
Diagram describing the AS site-fidelity model . . . . . . . . . . . . . . . . . . . . . . . . . .
20
7
Shortcut creation interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
8
Set equality between mathematical parameters . . . . . . . . . . . . . . . . . . . . . . . . .
37
9
Set equality between mathematical parameters . . . . . . . . . . . . . . . . . . . . . . . . .
37
10
Menu for the choice of the survival: After exiting gemaco, the user must select a continuous
List of Figures
function thank to a new menu. Several hazard functions are available as well as a nonparametric function (i.e. full age dependant survival) and a geometrical distribution (i.e.
constant survival)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
38
LIST OF FIGURES
5
11
Menu for setting the initial values of the continuous function. . . . . . . . . . . . . . . . . .
38
12
headed format : Covariate selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
13
headed format : Cluster automatic creation . . . . . . . . . . . . . . . . . . . . . . . . . .
43
14
Menu for setting steps for UTI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
15
Set the steps for UTI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
16
General organization of e-surge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
17
e-surge main screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
18
Opening a new session . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
19
Loading data file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
20
Changing model status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
21
Window structure of the gepat interface . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
22
Pattern in gepat for the survival . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
23
Pattern in gepat for the fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
24
Pattern in gepat for the movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
25
Pattern in gepat for capture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
26
Keeping Initial State probabilities in e-surge . . . . . . . . . . . . . . . . . . . . . . . . . .
54
27
Window structure of the gemaco interface . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
28
Initial Values Fixed Values window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
29
Advanced Numerical Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
30
“Give model rank” window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
31
Output of e-surge I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
32
Output of e-surge II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
33
Save probability of each histories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
34
First four capture histories extracted from the file ’histories.tmp’.
. . . . . . . . . . . . . .
66
35
Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
36
Viterbi algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
37
Counting algorithm
70
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
Introduction
e-surge (which stands for MultiEvent Generalized Survival Estimation) is a program for fitting Multievent models [37] to capture-recapture(CR) data. Multievent models are an extension of multistate
models in which observations do not necessarily correspond to states. Several programs exist for CR
analysis (e.g., [23, 50, 9]) but e-surge is the first general program for Multievent models. e-surge
also incorporates a new and extremely flexible way of defining the transition probabilities; because of
this it is useful even when multievent considerations do not apply (for example in multistate models,
see [15], [47]).
Because the observations in Multievent models do not necessarily correspond to individual states,
they can handle state uncertainty. As a consequence, they provide a general framework for problem
such as:
• Heterogeneity of capture, survival or any parameter of interest [35, 37, 34].
• Determination of the sex when sex is not available [33, 40].
• Memory model [22, 1, 37, 43].
• Animal epidemiology model [12].
In addition e-surge can handle models conditional to the first occasion. So, it provides a natural
framework for
• Stop over duration [38].
• Closed population [52].
• Occupancy models [31].
e-surge benefits from the experience gained in developing m-surge [9], a program for multistate
CR analysis. m-surge introduced a powerful language for describing the set of multistate CR models,
reduced statistics for any classes of age, and advanced numerical algorithms. e-surge has similar
capabilities for maximum likelihood estimation of complex age and time-dependent models with linear
constraints among parameters, in a generalized linear model fashion. Its features include:
• A tool for defining general models. A general model is mainly defined by the structure of the
transition matrix and the encounter matrix. In m-surge and mark transition probabilities are
defined either directly, or in terms of survival and transition conditional on survival. e-surge
is unique in permitting more than these two steps in defining transition and encounter matrices
and initial state vector. We call this feature DES for Decomposition in Elementary Steps. The
transition and encounter matrices and the initial state vector are constructed using a pattern
generator called gepat.
2
CHAPTER 1. INTRODUCTION
• A powerful model description language. Constrained models are built using a language interpreted
by a generator of constrained matrices (called also design matrices) called gemaco. This powerful
language is similar to those used in general statistical software packages such as SAS, R, Genstat
or GLIM; for instance, the formula t+g generates a model with additive effects of time and group.
gemaco avoids tedious and error-prone matrix manipulations.
• Advanced convergence diagnostics. Convergence to the maximum likelihood estimator is a very
sensitive issue in Multievent models. In e-surge, the user gains a greater control over convergence
through a choice of non-linear solvers and of starting options including the results of previous
models, random initial values and multiple random initial values. In addition, warnings are issued
if the program stops at a saddle-point rather than a minimum.
• The detection of redundant parameters. e-surge analyzes the likelihood in the neighborhood of
the point of convergence and lists the parameters that are apparently redundant. Redundancy
can then be double-checked by drawing profile likelihood curves.
e-surge is freely downloadable from http://ftp.cefe.cnrs.fr/biom/Soft-CR/. The program is
constantly improved and new capabilities are added. Although e-surge has been extensively tested by
many people using a variety of pilot data sets, we cannot totally exclude the presence of bugs. We are
grateful to you for reporting any problems by e-mail to [email protected]
The purpose of this manual is to provide practical instructions for using e-surge 1.8 along with
some underlying theory. We assume familiarity with the basic notions of CR methodology. We recommend reading: [52] for a general review of CR models, [25] for an overview of constrained models
and generalized linear model philosophy in CR analyses, [27] for a review of multistate models, [37] for
a description of Multievent models, [17, 7] for individual and group random effects. The rest of the
manual presents:
• The notation and models covered by e-surge (chapter 2)
• The generator of general model (gepat; chapter 3)
• The language and tools for building constrained models (gemaco; chapter 4)
• Data input (chapter 5)
• An example of a session with e-surge (chapter 6)
• Some advanced tools (chapters 7 and 8)
• The outputs (chapter 9)
• A few warnings (chapter 10).
3
2
Models
2.1
Notation
Our presentation of Multievent models will use the following general notations and follow as much as
possible those of [37]:
N
the number of states.
U
the number of events.
K
the number of occasions.
A
the maximum age class.
NG
the number of groups.
LI, LT, LB
the number of steps in the decomposition for initial state, transition and event.
i = 1, . . . , N
the index of the previous (or departure) state.
j = 1, . . . , N
the index of the current (or arrival) state.
u = 1, . . . , U
the index of the current event.
k = 1, . . . , K
the occasion index.
a = 1, . . . , A
the index of current age classes.
ng = 1, . . . , N G
the index of the current group.
` = 1, . . . , LI, LT , or LB
the elementary step index.
E = {e1 , ..., eN }
the set of states, where eN = † for the death.
Ω = {v1 , ..., vU }
the set of events, where v1 = ’not seen’.
All transition matrices are written with i as row index and j as column index, following the Markov chain
convention in which transitions are from rows to columns, rather than the column-to-row convention
used in matrix population models. Encounter matrices use j (denoting the state) as row index and u
(denoting an event) as column index.
2.2
Multievent models
The time-dependent Multievent model assumes that individuals move independently among a finite set
E of states over a finite number K of sampling occasions, and that successive states obey a Markov
chain. The successive states occupied by an individual are not observed directly. Rather, at each
occasion k, one member of a finite set Ω of events is observed. The event observed at occasion k is
assumed to depend only on the unobserved underlying state of the individual at that occasion.
Unlike traditional practice in CR (but similar to [16, 37] and consistent to Markov Chain property
[29]), the dead (†) is explicitly included in E. By convention, in e-surge it appears last in the list of
states. Similarly, the event “not seen” is explicitly included in Ω, in which it appears first.
4
CHAPTER 2. MODELS
Multievent models are defined in terms of three kinds of parameters: initial state probabilities π,
transition probabilities φ, and encounter probabilities b. For group g, we have
• πik,g the probability of being in state ei when first encountered at index of time k,
• φk,a,g
the probability of being in state ej at index of time k + 1 if in state ei at index of time k,
ij
for the interval a since first capture.
• bk,a,g
the probability of event vu for an animal in state ej at index of time k, at occasion a since
ju
first capture (including first capture),
• Π = (πi ) denotes the (1 × N ) vector of initial state probabilities,
• Φ = (φij ) denotes the (N × N ) matrix of unconditional transition probabilities, i.e. the matrix
of probabilities that an individual moves from one state to another state over a time interval.
• B = (bju ) denotes the (N × U ) matrix of event probabilities.
Together, (Π, Φ, B) define the general model (GM) under which an umbrella model (UM) retained by
Goodness of Fit can be fitted (see Figure 1 for the relation between GM and UM). These matrices are
row stochastic, and are called, respectively, the full initial state vector, the full transition matrix, and
the full event matrix in e-surge. The relation of each of these matrices to the classical CAS model
and to the memory model are given in [37]. These models belong to the class of Hidden Markov Models
(HMM, see for example [29, 2]).
Figure 1: Definition of the umbrella model(UM) under the general model (GM) by setting specific
variation in parameter.
2.3
Umbrella models
An important but not always obvious notion in model selection is the umbrella model (UM). The UM is
a general model with specified variation in parameters, which is later subjected to constraints to define
2.3. UMBRELLA MODELS
5
biological hypotheses of interest. In other words, the UM is the most general model that can be fitted,
and the one within which all other models examined are nested.
The UM depends on the settings of several main options in e-surge. Because there are several
choices for these options, there are several potential UMs in e-surge and it is possible to shift from one
to another during a session. However, this should be done with care because the meaning of a model
may be lost if the underlying UM is not remembered.
There are six potential sources of variation in the parameters:
1. groups, i.e. permanent categories of individuals, such as sexes or species, or discrete unconnected
study sites,
2. age, i.e., number of occasions or intervals elapsed since first capture,
3. time,
4. state of departure,
5. state of arrival,
6. current event.
In the UM, parameters are always allowed to vary freely over time and among groups. Only the number
of states and the number of age classes can be set to different values.
Decomposition in Elementary Steps
It is sometimes useful to define the full initial state, full transition, and/or full encounter matrices as
arising from a sequence of “life processes”. The familiar decomposition of the transition matrix into
survival and transition conditional on survival (implemented in m-surge) is an example of this, but in
some cases more steps may be involved.
In an approach similar to that used for periodic matrix population models ([3], Chapter 14) but
between two dates or at one occasion, the full matrices are written as products of elementary matrices.
Π =
Φ =
B =
LI
Y
`=1
LT
Y
`=1
LB
Y
Π(`)
Φ(`)
(1)
B (`)
`=1
The intermediate states involved in the sequence of life processes may not be the same as the basic set
of states in the model. Thus the elementary matrices need not be square.
6
CHAPTER 2. MODELS
For example, in [21], a new parameterization of CAS models is presented in which movement among
sites is described in terms of the probability of leaving the site of origin and the probability of settling in
the destination site conditional on leaving. This parameterization is then used to address the influence
of local perturbations on site fidelity and settlement decisions of emigrants in a subdivided population of
Black-headed Gulls, Larus ridibundus. This two-step processes can be expressed by a product of three
elementary probability matrices, the first describing survival, the second describing the probability
of emigration conditional on survival (i.e., fidelity to the site of origin), and the third describing the
probability of the destination site conditional on emigration.
Such decompositions can be used to model any multi-step processes. For example, the memory model
can be defined this way in either separate or combined formulation [10]. In e-surge, the structures of
the elementary matrices are defined by a tool called gepat (for GEnerator of PATtern, see chapter 3).
Age dependence
In the UM, survival, transitions, and encounters may depend on age (i.e., time since first capture, not
necessarily true chronological age). The user specifies an oldest relevant age class; all animals this age
or older are combined into a single age-class. While restricting the range of ages restricts the range of
models that can be fitted, it may greatly save memory and reduce computation time.
For transitions or survival, common choices for the maximum relevant age are 1 (A = 1), which
implies no age effect, and 2 (A = 2), which creates a model in which the first age class is contrasted to
older animals (this is particularly useful when animals are marked as young). Setting A = 2 can also
be used to treat transience [41].
Specifying age-dependence in encounters is slightly more complicated. In multistate (as opposed to
multievent) applications, all calculations are conditional on the first encounter and hence the probability
of that first encounter is not estimated. In multievent formulations, the first encounter may be an event
rather than a state, and thus e-surge has the option of modelling the probability of the initial event
(see Figure 2). Therefore, e-surge always considers at least two age classes for encounters, allowing
the first event probability (first class of age) to be modelled or not. Thus if one chooses a maximum
age A = 1, which implies no age effect, e-surge creates 2 classes for first and next encounter. If one
sets A = 2, e-surge creates 3 classes(with age) for events.
In e-surge, parameters are ordered in memory according to their type as follows, with the leftmost
indices varying first and the rightmost last.
π:
current state > time > group > step
φ:
previous state > next state > time > age > group > step
b:
previous state > current event > time > age > group > step
2.4. CONSTRAINED MODELS
7
Figure 2: Two individual histories and their associated probabilities
Capture is modeled by a constant first capture rate P r(1) vs a constant recapture rate P r(2) . Here P r(1)
represents the first class of ’age’ of event and P r(2) represents the second class of ’age’ of event. The
index for one age class in event is the number of occasions spent since first capture plus one.
A 6= 1
A=1
Πtime,group,(step)
Φtime,age=A,group,(step)
B time,age=A+1,group,(step)
Πtime,group,(step)
Φtime,...,group,(step)
B time,age=2,group,(step)
Table 1: Variations considered in the parameters of the umbrella models of e-surge
The type of variation is represented by upper indices for time, age, group and step. A is the number of
age classes for transition.
2.4
Constrained models
Model-building in e-surge (as in m-surge and mark) proceeds by imposing linear constraints on
the parameters of the umbrella model, in the spirit of generalized linear models [25]. The vector θ of
”biological parameters” (parameters of direct interest to the biologist e.g., θ = (π, φ, b), organized as
a vector) is expressed as a linear transformation of a vector β of ”mathematical parameters”. To keep
the biological parameters, which are probabilities, in their permissible range (0,1), a link function f is
generally applied (see 2.5):
f (θ) = Xβ
(2)
8
CHAPTER 2. MODELS
or, equivalently,
θ = f −1 (Xβ)
(3)
The matrix X is a ”matrix of constraints”. It can be a genuine design matrix in the case of a designed
experiment. In general, it expresses hypotheses about the dependence of the parameters on stage (of
departure or arrival), age (since first capture), time, group, and/or covariates. The design matrix is
built by the program gemaco (GEnerator of MAtrices of COnstraints) see Section 4), using the
model definition language described below. Often X will contain both discrete indicator 0/1 variables
(for equality constraints) and continuous covariates (e.g., effort or weather covariates). An overview of
linear constraints in CR models with a single state is given by [25], linear constraints in multistate are
considered in [5].
An important difference in the application of gemaco in e-surge, as compared to m-surge, is
that the gemaco keywords (“from”, “to”, etc.) in e-surge refer to the elementary matrices. The rows
(from) and columns (to) in these matrices do not necessarily correspond to the states in the model (e.g.,
in the encounter matrix, the columns refer to events, not states). Care is thus required in writing down
the gemaco specification.
2.5
The link function
The biological parameters θ are probabilities, and hence must lie within the interval [0, 1]. To satisfy
this constraint but still allow the optimization routines to work with mathematical parameters β that
range over (−∞, +∞), a ”link function” is applied to the parameters. The link function is a one-to-one
continuous transformation. In practice, very small or very large β values are transformed into θ = 0 or
θ = 1, so in practice some estimates may fall on the boundary of parameter space. e-surge provides
two link functions; the generalized logit and the identity link.
Transition probabilities must not only lie within the unit interval, but must sum on each row to
1. Consider two transition probabilities on the same row of Φ, φ11 and φ12 . The transformations
φ11 = logit−1 (θ11 ) and φ12 = logit−1 (θ12 ) will assure that φ11 and φ12 are both in the unit interval,
but will not guarantee that φ11 + φ12 ≤ 1. The generalized (or multinomial) logit, denoted as logitgen,
ensures that all parameters and their sum are within (0, 1). For the first N-1 transitions parameters
among the (φij )j=1,...,N , this transformation is defined as
logitgen(φij ) = log
1−
φij
PN −1
k=1
φik
, j = 1, . . . , N − 1
(4)
For N non null transition probabilities on a row, the logitgen transformation is applied to the N − 1
parameters. When N = 2, the generalized logit reduces to the logit. With more than two states, some
additive effects cannot be modelled meaningfully with the generalized logit, because parallelism is not
as clearly defined as with the logit link. In such cases, the identity link may be used in e-surge. A
2.6. UNEQUAL TIME INTERVALS
9
specific algorithm is then used to keep estimates in range, but this algorithm is slower and is not used
as the default option.
2.6
Unequal time intervals
e-surge permits calculations based on unequal time intervals for any steps of transition Φ(l) (see Equation 1) of your choice (see section 5.5 and 5.6 for practical use). Each row of the considered transition
matrix should not contain more than one parameter and its complementary. The implementation of
power of matrices should be considered to allow any kind of transition. This feature is not presently
available. If survival, the true parameter (estimated by e-surge) is then survival per unit of time
denoted s instead of survival over the whole interval denoted S. The length of the time interval ∆t is
used to back-calculate the estimate of survival over the interval: S = s∆t .
2.7
Maximum likelihood estimation
The likelihood of a model is proportional to the probability of the data given that model. The basic
unit of data in e-surge is the capture history; reduced-form data descriptions like the m-array are not
available in general for multievent models. Thus the likelihood calculation depends on the application of
the transition probabilities to individual capture histories. (See [37] for a presentation of this approach
or [16] for an ad hoc method to deal with uncertain states). Let h = (o1 , . . . , oK ) be a capture history
with first encounter at time e, event (ok )k=1,...,K has any value between 0 and U and β a vector of
parameters. Then
P (h|β) = Πe D B e,1 (·, oe )
K
Y
Φk−1,k−e D B k,k−e+1 (·, ok )
!
1N
(5)
k=e+1
where B t,a (·, ot ) is the ot th column of the encounter matrix B at time t and age a and D(x) is a matrix
with x on the diagonals and zeros elsewhere, and 1N is a N column-vector of ones.
Assuming that individuals are independent, the likelihood for the entire set of capture histories is
obtained as the product of the likelihoods for each history,
L(β) = C
Y
P (h|β)nh
h
where C is a constant and nh is the number of copies of capture history h in the data set.
The maximum likelihood estimation(MLE) algorithm is as follows
1. Select an initial value for the vector β of mathematical parameters.
2. Calculate the vector of biological parameters θ = f −1 (Xβ).
(6)
10
CHAPTER 2. MODELS
3. Calculate the elementary matrices, and (as the product of the elementary matrices) each of the
full matrices Π, Φ, and B.
4. Use the full matrices to calculate the probability P (h|β) of each capture history according to
Equation 5.
5. Calculate the relative deviance
Dev(β) = −2 (log L(β) − log C) = −2
X
nh log P (h|β)
h
6. Iterate steps 2–5 in a Quasi-Newton minimization algorithm or a Expectation-Maximization (EM)
algorithm or by-product algorithms updating the vector of mathematical parameters β to decrease
Dev(β), until convergence.
7. Obtain in turn the MLE’s and the deviance and various by-products of Maximum Likelihood
estimation.
2.8
Factorizing the likelihood
Multievent CR calculations condition on the first capture of the individual. Because of this, nonobservable states (including the ’dead’ state) cannot appear as initial states, and hence they are eliminated from the vector Π by setting the corresponding probabilities to zero. The remaining, non-zero
initial state probabilities can be hard to estimate if they are time-varying. However, when one initial
observable state corresponds to a single type of event, and when the first event probabilities are independent from later event probabilities, then Π and B ·,1 can be estimated independently of the other
parameters. In fact, we can show that there exists a unique decomposition (β 1 , β 2 ) of the parameter
vector β, such that
Dev(β) = Dev(β 1 ) + Dev(β 2 )
(7)
where
Dev(β 1 ) = −2
X
nh ∗ log(Πe D B e,1 (·, oe ) 1N ),
(8)
h
Dev(β 2 ) = −2
X
h
nh ∗ log(xeh
K
Y
Φk−1,k−e D B k,k−e+1 (·, ok )
!
1N ),
(9)
k=e+1
with β 1 the vector of mathematical parameters linked to the initial state probabilities Π and first
encounter probabilities B ·,1 ,
β 2 is the vector of mathematical parameters linked to the transition probabilities Φ and subsequent
encounter probabilities B ·,2:A+1 and
(
1 if (Πe D(B e,1 (·, oe ))1N )i > 0
(xeh )i =
0
else
i = 1, . . . , N.
(10)
2.9. INDIVIDUAL COVARIATES
11
Internal note When one state corresponds to a single type of event at recapture then we can easily
demonstrate than the m-array for time dependent model (or extended m-array for age dependent model,
[9]) is a set of sufficient statistics for the recapture part of the model.
2.9
Individual covariates
e-surge can handle individual covariates. Considering the general form of GLM f (Θ) = Xβ with
β the vector of fixed effects is computationally demanding because of the dimension of the problem
with so many potential effects. Thus, we have implemented the following restricted form of GLM by
constraining separately the two following sets of effect.
• Set of effect 1: time, age, cohort and group effects.
• Set of effect 2: individual effect.
The result is the following form of GLM implemented in e-surge.
f (Θn ) = X0 β0 + Xn β1
(11)
with Xn : individual-specific matrices of individual covariates, they are never stored in the computer
because of the memory size needed but rather they are computed each time.
2.10
Independent and identically distributed (i.i.d.) random effect
The class of mixed effects models that E-SURGE may consider can be expressed in the form of generalized linear mixed models (GLMM). Considering the general form of GLMM f (Θ) = Xβ + Zb with β the
vector of fixed effects and b the vector of random effects, we have implemented the following restricted
form of GLMM by constraining separately the set of effects 1 and 2.
f (Θn ) = X0 β0 + Xn β1 +
P
X
p=1
Zp bp +
PX
+Q
Zp,n bp,n n = 1, . . . , N I
(12)
p=P +1
where bp ∈ Rsp and bp,n ∈ R are random effects given by
(
bp ∼ N (0, σl2 × Isp ), p = 1, . . . , P,
bp,n ∼ N (0, σp2 ), p = P + 1, . . . , P + Q.
(13)
bp : random effects associated to the set of effects 1 (p = 1, . . . , P ),
sp : number of levels of the random effect l (sp = N G for a group random effects),
bp,n : individual random effects assuming that individuals are independent (p = P + 1, . . . , P + Q).
12
CHAPTER 2. MODELS
Matrices Zp,n contain either 0, 1 or values of individual covariates and are never stored. Because we
assume that individuals are independent then covariance matrices for each random effect are diagonal.
We use this property to implement efficient algorithms for individuals and groups random effects, see
[7, 17].
For individuals
E-SURGE can handle mixed models with individual random effects only (P = 0) [7, 17]:
f (Θn ) = X0 β0 + Xn β1 +
Q
X
Zp,n bp,n
(14)
p=1
with bp,n in the form of equation (13). There is no limit for the number of random effects that we can
build. However for Q > 2 the fitting step may be time consuming.
Example 1: Survival varying with individual covariates and random effect. The following model has
been used in [19] with a constant survival across time but dependent on an individual covariates (the
body weight denoted by m) and from an individual random effect (bn ).
logit(φn ) = β0 + β1 mn + bn , n = 1, . . . , N I
(15)
where bn ∼ N (0, σb2 ), (i.i.d.).
For groups
E-SURGE can also handle mixed models with group random effects only (Q = 0) [7]:
f (Θn ) = X0 β0 + Xn β1 +
P
X
Zp bp
(16)
p=1
with bp in the form of equation (13). There is no limit for the number of random effects that we can
build. However for P > 2 the fitting step may be time consuming.
Example 2: We consider a basic model where recapture rates vary with a group random effect.
logit(pg ) = β0 + bg , g = 1, . . . , N G
(17)
where bg ∼ N (0, σb2 ), (i.i.d).
2.11
Conditionality on the first occasion
Since the version 1.7, e-surge can handled models whose probabilities are written conditional on the
first occasion rather that conditional on the first encounter. So that, it provides a natural framework
2.11. CONDITIONALITY ON THE FIRST OCCASION
13
for stop over duration (SOD), closed population and occupancy models. By default, the conditionality
is on the first encounter (Conditional on 1st Capture). To handle an open or a closed population model
with conditionality on the first Occasion, choose the option Conditional on 1st Occasion. To handle a
occupancy model, choose the option Occupancy (see Figure 3).
Figure 3: Conditionality options
Important note: Neither individual covariates nor random effects are currently implemented in
e-surge for models conditional on the first occasion. But a trick can be used to get round this problem
for occupancy models. To do that, add a one to each observation of the data set, so that each first ”nonzero event” occurs at the first occasion. Select then the option Markovian states only > Conditional on
1st Capture. By this way, we can do all models with individual covariates and random effects described
respectively in sections 2.9 and 2.10.
Open and closed population
The conditional probability of history h denoted PC (h) is defined by
PC (h|β) =
P (h|β)
,
1 − P (∅|β)
(18)
with
K
Y
P (h|β) = ΠD B 1 (·, o1 )
!
Φk−1 D B k (·, ok )
1N
k=2
where P (∅|β) is the probability of an individual to remain unseen at all occasions (i.e. the probability
of a history to be empty). Since the time spend since the first capture is not defined, we set A = 1 and
the age indices vanish in the previous formulae. The likelihood is:
L(β) =
Y
PC (h|β)nh
(19)
h
As by-products, the estimated number of individuals present or passing though the site is
P
h nh
.
1 − P (∅|β̂)
In the particular case of a closed population model, it is assumed that no new individuals enter in
the area, that no individuals leave the area and that there is no mortality. So the survival parameters
14
CHAPTER 2. MODELS
has to be fixed to one in the model. For violation of closed population model assumptions and ways to
manage them, see [52].
In the case of an open population, we can for example model the SOD when the probability to leave
the area is time-dependent only [46].
Occupancy models
In occupancy models [30], we follow patches over time. As we know where patches are, the capture rate
is equal to one. Thus the observation ’0’ has no longer to be called ’not seen’. Rather the observation
’0’ gives an information about the state of patches. In that context, an empty history is the absurd
situation where a know patch is never visited. The mathematical consequence is that
P (∅|β) = 0.
As e-surge allows you to deal with imperfect detection thus at least all models described in [31] can
be fit.
Note: e-surge do not currently manage empty event (patches which are not visited at some
occasions). However if empty events are distributed randomly then an additional parameter can be
used to take into account this lack of information.
2.12
Non-linear model
e-surge is now able to fit a continuous function, for example associated to a hazard function. To that
purpose, a semi-Markov formulation of the CJS model has been developed in [11]:
• to consider continuous function (which are parametric) in relation to the age of the individual,
• to deal with left censoring (defined in the headed format, 5.3) to allow individual to start at
different ages.
As an original contribution of regular function (with well-defined second derivative), we can estimate
the onset of senescence using geometrical property [11]. To handle such a model, choose the option
Models > Markovian & semi-Markovian states > Conditional on 1st Capture (see Figure 4).
Figure 4: Semi-Markov option
15
3
Flexible generation of a general model: GEPAT
3.1
Overview
gepat (for GEnerator of PATtern of elementary matrices) makes it possible to generate the GM using
the elementary matrices in Equation 1 under which the UM is defined (see Figure 1). Denoting this
model as DES for Decomposition in Elementary Steps and denoting the number of elementary matrices
of each type LI, LT, LB, we have DES(LI, LT, LB).
This feature was chosen in the context of uncertainty (see [33, 37]) to allow
• models described by Equation 1 to be expressed as linear models,
• application of constraint on each parameter separately by a language, here gemaco.
This approach allows the specification of complex models while avoiding non-linear constraints algorithm for parameter in-range. Non-linear constraints can then be handled in a very efficient way with
unconstrained algorithms.
It is also helpful in multistate problems without uncertain states, because it allows biologists to
specify models in more details. Developing life cycle models in terms of such “lower-level” parameters has a long tradition in various branches of population biology (e.g., [3]), but it is now also used
for encounter probabilities and initial probabilities. For some recent examples, in a CR context, see
[21, 13, 48, 34, 24, 45].
By default in e-surge, LI = LT = LB = 1. Examples of such one-step process (DES(1,1,1)) are
the combined CAS model, or the combined memory model.
However, by setting LI, LT and/or LB to values greater than one and using gepat, it is possible
to define the pattern of each of the elementary matrices at each step, it becomes possible to fit many
models difficult or impossible to estimate elsewhere.
Here we present several examples to clarify the steps in the analysis.
Another strong characteristic of the implementation of DES in e-surge is that Φ(l) with 1 ≤ l ≤ LT
define a transition matrix from E(l − 1) to E(l) where E(l) can be another set of states than E.
The same feature is also available for initial state and encounter matrices, which is entirely new for
the model in CR. An example is draw in Figure 5.
We will see that it is now possible to generate new general model in the context of linear model in
CR.
16
CHAPTER 3. GEPAT
Figure 5: Decomposition of the conditional event probabilities: For sex determination in [33], two steps
are needed for encounter to generate the GM. In the diagram, S represents the set of states males or
females, O the intermediate set of events and O’ the effective set of events.
3.2
A combined formulation of the Arnason-Schwarz model
In this formulation, transition probabilities combine both survival and movement among states conditional on survival. This is a typical case of DES(1,1,1). In the CAS model the state of an observed
individual is always known without error. Thus, with 2 sites, the set of states is
E = {1, 2, †}
and the set of events is
Ω = {’not seen’, ’seen in 1’, ’seen in 2’}.
The initial state matrix at occasion k is
Πk =
πk 1 − πk
(20)
Note that in e-surge the state † is always removed from the full initial state vector because individuals
are all still alive at first release occasion.
The transition matrix at occasion k maps individuals from E to E:

φk11 φk12 1 − φk11 − φk12



k
k
k
k 
Φk = 
 φ21 φ22 1 − φ21 − φ22 
0
0
1
Rows and columns of Φ both correspond to states.
(21)
3.2. A COMBINED FORMULATION OF THE ARNASON-SCHWARZ MODEL
The event matrices for the first capture and for subsequent captures, at occasion k, are


0 1 0



B k,1 = 
0
0
1


1 0 0


1 − pk1 pk1 0


k
k 
B k,2 = 
 1 − p2 0 p2 
1
0 0
17
(22)
(23)
B maps individuals from E to Ω; the rows of B thus correspond to states and the columns correspond
to events. Because first captures are not modelled in the CAS model, B k,1 says that at its first capture
an individual in state 1 will be encountered in ’seen in 1’ with probability 1, etc. For later captures,
there is a probability pk1 of being seen in 1 at occasion k, and so on.
The matrices Π, Φ and B are row-stochastic (i.e., the sum of each row equals one). Thus one of
the entries in each row is redundant, and need not be estimated. In many cases, some of the entries
are fixed equal to 0, and also need not be estimated. The specification of which entries are redundant,
which are to be estimated, and which are fixed at 0 is done with a pattern matrix associated with each
of the initial state, transition, and encounter matrices. The entries of the pattern matrix corresponding
to redundant entries are set equal to character ’*’ (there will be one such entry in each row). The entries
of the pattern matrix corresponding to parameters to be estimated are set equal to any letter from the
alphabet. The entries corresponding to fixed zero values are set equal to character ’-’.
This is confusing, but can be made clear by example. Denote the pattern matrices by P Π, P Φ,
and P B. Then the pattern matrices for the combined CAS model are
Πk =
PΠ = π ∗
πk 1 − πk

φk11 φk12 1 − φk11 − φk12



k
k
k − φk 
Φk = 
φ
φ
1
−
φ
22
21
22 
 21
0
0
1

0 1 0

1 − pk1 pk1

k
B k,2 = 
 1 − p2
1
0
0
φ
φ

PΦ = 
 φ
φ
∗


∗ 

− − ∗




B k,1 = 
 0 0 1 
1 0 0


(24)
∗
p
−




PB = 
 ∗ − p 
∗ − −
0


pk2 

0

∗
(25)
p
(26)
−




PB = 
∗
−
p


∗ − −
(27)
Note 1 Each elementary matrix has only one pattern constant across age. So the pattern of each
encounter elementary matrix is the same for first and next encounters.
18
CHAPTER 3. GEPAT
Note 2 Factorisation of the likelihood can be done for this model (see section 2.8), permitting
probablities π to be estimated separately from probabilities φ and p.
3.3
The separate formulation for the Arnason-Schwarz model
In this formulation, survival and movement conditional on survival are separated; this is a typical case
of DES(1,2,1). The two set of states remain constant across the two steps life processes
E(0) = {’site 1’,’site 2’,†}
E(1) = {’site 1’,’site 2’,†}
and the set of events Ω is the same as in the combined model. The initial state matrix and its corresponding pattern matrix are
Πk =
πk 1 − πk
PΠ =
π ∗
(28)
There are now two elementary transition matrices, one corresponding to survival and one to transitions
(noted ψ) conditional on survival, each of which has its own pattern matrix:

sk1

Φk,(1) = 
 0
0

Φk,(2)
0
1 − sk1


sk2 1 − sk2 

0
1
k
k
ψ11
1 − ψ11

k
k
= 
ψ22
 1 − ψ22
0
0

− ∗
s




P Φ(1) = 
 − s ∗ 
− − ∗



0
ψ ∗ −




P Φ(2) = 
0 

 ∗ ψ − 
1
− − ∗
(29)
(30)
There is one elementary detection matrices, constant at the first capture and time varying to recapture,
with a pattern matrix corresponding to both

0 1 0




B k,1 = 
0
0
1


1 0 0

1 − pk1 pk1

k
B k,2 = 
 1 − p2
1
0
0
(31)
0


pk2 

0

∗
p
−




PB = 
 ∗ − p 
∗ − −
(32)
Note that as in the combined CAS model, probabilities π can be estimated separately from probabilities ψ and p.
3.4. A VERSION OF THE ARNASON-SCHWARZ MODEL WITH SITE FIDELITY
PARAMETRIZATION
3.4
19
A version of the Arnason-Schwarz model with site fidelity
parametrization
Now we consider a version of the Arnason-Schwarz model in which the probability of transition conditional on survival is further subdivided into a probability of leaving the site (the complement of site
fidelity), and a probability of moving to each other site conditional on leaving [21]. This is a DES(1,3,1)
general model.
With 3 sites, assuming that if an animal is seen its state is known without error, the set of events
(i.e., the results of observations) is
Ω = {’not seen’, ’seen at 1’, ’seen at 2’, ’seen at 3’}
Defining sets of intermediate states.
In the classical separate formulation of the Arnason-Schwarz model, the set of possible states for an
individual is the same for both elementary matrices (survival and transition conditional on survival). In
general, however, there may be a different set of states at each of the elementary steps. In constructing
the elementary matrices and their states, it may be helpful to use a directed acyclic graph (DAG) used
in many area to represent relations between items. Figure 6 shows the formulation of the Grosbois
model. In this formulation, states are denoted as numbered nodes on a row. Each step in the life
process is represented by a subsequent row, and the possible transitions are denoted by arrows. The
initial set of states is repeated at the bottom of the graph.
In the Grosbois site fidelity model, the sets of states are
E(0) = {’site 1’,’site 2’,’site 3’,†}
E(1) = {’site 1’,’site 2’,’site 3’,†}
E(2) = {’staying in 1’,’leaving 1’,’staying in 2’,’leaving 2’,
’staying in 3’,’leaving 3’,†}
(33)
Matrices and pattern matrices
The initial state matrix and its pattern matrix are
Πk = π1k π2k 1 − π1k − π2k
PΠ =
π π ∗
(34)
The first elementary transition matrix (for survival) maps from E(0) to E(1) , and hence is of dimension
4 × 4:

k,(1)
Φ

sk1
0
0
1 − sk1

 0

=
 0

0
sk2
0
0
sk3
0
0

1 − sk2 


k
1 − s3 

1

(1)
PΦ
s
− − ∗



 − s − ∗ 


=

 − − s ∗ 


− − − ∗
(35)
20
CHAPTER 3. GEPAT
Figure 6: DAG of the Grosbois model: We use a directed acyclic graph (commonly abbreviated to DAG
in the statistical literature) to describe life processes model inside one occasion for the AS site-fidelity
model, showing the transitions for survival, fidelity given survival, and destination given movement.
Transition probabilities are shown on the pathways originating in Site 1 and for the dead; the transition
probabilities on the other arrows follow the same pattern. The row-stochastic matrix Φ(1) projects from
row 1 to row 2 of the graph. The matrix Φ(2) projects from row 2 to row 3, and the matrix Φ(3) projects
from row 3 to row 4 (i.e., form row 3 back to row 1).
The second elementary matrix (for site fidelity given survival) maps from E(1) to E(2) , and hence is
4 × 7. Letting fi be the probability of remaining in site i given survival, we have

Φk,(2)
f1 1 − f1

 0

=
 0

0
0
0

0
0
0
0

0 


0 

1
0
f2 1 − f2
0
0
0
0
f3 1 − f3
0
0
0
0
0

P Φ(2)
f
∗
− − − − −



 − − f ∗ − − − 


=

 − − − − f ∗ − 


− − − − − − ∗
(36)
The third elementary matrix (for movement conditional on emigration) maps from E(2) back to E(0) ,
and so is of dimension 7 × 4:
k,(3)
Φ

1
0







=






0
k
ψ12
0
1
k
ψ21
0
0
0
k
ψ31
k
1 − ψ31
0
0



k
1 − ψ12
0 


0
0 


k
1 − ψ21
0 

1
0 


0
0 








=






0
0
0
1
P Φ(3)
∗
− − −


− 


− ∗ − − 


ψ − ∗ − 

− − ∗ − 


ψ ∗ − − 

− ψ
∗
− − −
∗
(37)
3.5. GEPAT IN PRACTICE
21
The event matrices B map from the set E(0) of states to the set Ω of events, and thus are of dimension
4 × 4.

B k,1
B



 0 0 1 0 


= 

 0 0 0 1 


1 0 0 0

k,2
0 1 0 0
1 − pk1 pk1

 1 − pk

2
= 
 1 − pk3

1
(38)

0
0
0
pk2
0
0
0
0

0 


k
p3 

0

− −


 ∗ − p −

PB = 
 ∗ − − p

∗ − − −






∗
p
(39)
Important note: the choice of states for the intermediate transitions is not always unique.
There may be more than one equivalent way to group individuals, and at the present the only advice we
can give is to determine from the structure of the model what information needs to be kept at any one
step in order to define the probability of subsequent transitions. For example, in the models described
here, the future transitions of a dead individual (these transitions are boring; the individual just remains
dead) do not depend on which state the individual died from. Thus E(1) includes only one dead state.
But the future transitions of individuals that leave a site do depend on what site the individual left
from. Thus E(2) must include separate states for individuals who left from site 1, left from site 2, and
left from site 3. In any case, the numbering of the states is completely arbitrary; changing the numbers
simply exchanges rows and columns of the elementary matrices.
3.5
GEPAT in practice
gepat is a tool for defining the pattern matrices P Π, P Φ, and P B. In the current version of gepat,
the user enters the number of steps (LI, LT, LB) for each kind of parameter. For each step of each
parameter, the user enters a pattern matrix, by rows. The matrix


t11 · · · t1J

 . .
.
. . ... 
T =
 .

tI1 · · · tIJ
would be entered in gepat using a graphical interface. See section 6.3 for details.
(40)
22
3.6
CHAPTER 3. GEPAT
The vector of biological parameter
Components of the vector of biological parameter θ are dependent upon the definition of pattern matrices. Are considered as biological parameters the set of elements which can be potentialy constrained,
i.e. the set of elements which are not labelled by a ’-’ in pattern matrices. Let considered as a example
P Φ(1) (in equation (29), the vector θ restricted to the survival matrix Φ(1) for a given occasion and a
given age is
θ = (s1 , s2 , 1 − s1 , 1 − s2 , 1)0 .
(41)
Each element of θ in (41) is labelled in the pattern matrix (29) either by a letter (s1 , s2 ) or by a ’*’
(1 − s1 , 1 − s2 , 1). Now, the vector θ, restricted to the survival but with full variation in time and age
with 2 occasions of recapture(K = 3), becomes:
1,1
1,1
1,1
2,1 2,1
2,1
2,1
θ = (s1,1
1 , s2 , 1 − s1 , 1 − s2 , 1, s1 , s2 , 1 − s1 , 1 − s2 , 1,
2,2
0
s12,2 , s22,2 , 1 − s2,2
1 , 1 − s2 , 1) .
(42)
where upper indices stand for time and age, where lower indice stands for site, and where v 0 is the
transpose of v.
23
4
Constrained models made easy: GEMACO
The definition of models in terms of constraints on each of the elementary matrices is carried out with
gemaco [5], which is also part of m-surge [9].
The most important new aspect in e-surge is that the gemaco keywords “from” and “to, current”
now refer to the rows and columns of the elementary matrices, rather than the full matrix.
4.1
Overview
With gemaco, one of the salient features of e-surge or m-surge, you will be able to generate easily
the constraint matrix X associated with the model. For the sake of simplicity, a different sub-matrix of
constraints is defined for each type of initial, transition and encounter parameter. Overall, there is LI +
LT + LB sub-matrices of constraints. e-surge assembles them to do the overall matrix of constraints.
For instance with a DES(1,2,1) general model (see section 2.3), with matrices X 1 , X 2 , X 3 , X 4 associated
to inital states, survivals, movements and events probabilities respectively, the overall matrix is, in block
matrix notation:

X
0
0
0
 1
 0 X
0
0
2

X=
 0
0 X3 0

0
0
0 X4




,


associated to the block vector

θ1



 θ 
 2 
θ=
.
 θ3 


θ4
The main step in defining a matrix of constraint for one type of parameter consists in typing a phrase
using the Model Definition Language (or mdl for short). This phrase will be interpreted by gemaco
to build X automatically. The mdl language is based on reserved keywords for various effects, such as
time (t) or group (g), and operators. This language expands the tensor notation for analysis of variance
models ([51], see [32] p.41) adapted to and advocated for cjs models by [25]. Several other steps, some
of which are optional, to build constrained models, will be examined later. We recommend that you
carefully read the presentation of the mdl and work through the examples to progressively learn how
to ”speak mdl ”. You will soon realize that gemaco along with its mdl offers very wide possibilities
that make the building of nearly any biologically meaningful model a fairly easy task.
24
CHAPTER 4. GEMACO
4.2
Keywords for main effects
In capture-recapture modeling, several classical effects, such as time, age and group, have been widely
used to explain variability in the data [25]. In the mdl of gemaco, these effects are represented
by reserved keywords, with synonyms to facilitate writing models. The effects and their associated
keywords are described in Table 5. These effects are here considered by themselves, i.e., as main effects
in an analysis of variance sense. They can also be combined as seen in the next paragraph. As a first
example of the capabilities of gemaco, let us assume we want to run a cjs-type model with survival
constant over time but varying among groups, and recapture probability varying with time only. All
one needs to do then is to define the structure for survival and recapture probabilities to be g and t,
respectively, exactly as in the tensor notation of this model (φg , pt ). The time-dependent cjs model is
written as (φt , pt ).
How is the MDL phrase time interpreted by GEMACO ?.
Let us consider with time variation in survival probability over 2 geographical sites(3 states) and K = 3
occasions (i.e. 2 intervals). The vector of survival parameters θ 2 is defined by Equation 42.
Defining the model as time or for short t creates a matrix X 2 with as many rows as components
in θ 2 , in the same order. Columns in Table 2 correspond to the time index and values(0/1) correspond
to indicator variables for time. The constraint matrix X (left part) is generated by gemaco according














X2 = 












1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
1
1
0
0
0
F
1
2
1
2
3
1
2
1
2
3
1
2
1
2
3



























|
To
1
2
3
3
3
1
2
3
3
3
1
2
3
3
3
T
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
A
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
{z
Coordinates
G
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
}
Table 2: Phrase time interpreted by gemaco.
to the component of the vector θ 2 described in Equation 42. The coordinates (F,To,T,A,G) of the
4.2. KEYWORDS FOR MAIN EFFECTS
25
components correspond respectively to (From, To, Time, Age, Group) and are displayed in the right
part.
How is the MDL phrase age interpreted by GEMACO ?.
Defining the model as a (for age) creates a matrix X 2 with A columns when the umbrella model contains
A classes of age, with A = K − 1. X 2 is given in Table 3 with A = 2 for θ 2 given by Equation 42.














X2 = 












1
1
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0



























F
1
2
1
2
3
1
2
1
2
3
1
2
1
2
3
To
1
2
3
3
3
1
2
3
3
3
1
2
3
3
3
T
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
A
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
G
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Table 3: Constraint matrix associated to age and generated by gemaco: The constraint matrix X 2 (left
part) is generated by gemaco according to the component of the vector θ 2 described in Equation 42.
The coordinates (F,To,T,A,G) of the components correspond respectively to (From, To, Time, Age,
Group) and are displayed in the right part.
Note that if A = 1 then a = a(1), else if A = 2 then a = a(1, 2) as a(2) = a(2 : K − 1) (see
section 4.6) else if A = 3 then a = a(1, 2, 3).
How is the MDL phrase group interpreted by GEMACO ?.
With 2 groups, the vector of survival parameters θ 2 becomes
θ 2 = (s1,1,1
, s1,1,1
, 1 − s1,1,1
, 1 − s1,1,1
, 1, s12,1,1 , s22,1,1 , 1 − s12,1,1 , 1 − s22,1,1 , 1,
1
2
1
2
s12,2,1 , s22,2,1 , 1 − s12,2,1 , 1 − s22,2,1 , 1,
s1,1,2
, s1,1,2
, 1 − s1,1,2
, 1 − s21,1,2 , 1, s12,1,2 , s22,1,2 , 1 − s12,1,2 , 1 − s22,1,2 , 1,
1
2
1
s12,2,2 , s22,2,2 , 1 − s12,2,2 , 1 − s22,2,2 , 1)0 .
(43)
and defining the model as t for time or g for group leads to matrices X 2 and Y with twice as many
rows, see Table 4.
26
CHAPTER 4. GEMACO





























X2 = 




























1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
1
1
0
0
0
0
0
0
0
0
1
1
0
0
0
1
1
0
0
0


1

 1



 0



 0



 0



 1



 1





 0



 0



 0



 1



 1



 0



 0




 and Y =  0

 0



 0



 0



 0



 0



 0





 0



 0



 0



 0



 0



 0



 0



 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
1
1
0
0
0
1
1
0
0
0


























































F
1
2
1
2
3
1
2
1
2
3
1
2
1
2
3
1
2
1
2
3
1
2
1
2
3
1
2
1
2
3
To
1
2
3
3
3
1
2
3
3
3
1
2
3
3
3
1
2
3
3
3
1
2
3
3
3
1
2
3
3
3
T
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
A
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
G
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
2
2
Table 4: Phrases time and group interpreted by gemaco: Constraint matrices respectively X 2 and
Y (left part) are generated by gemaco according to the component of the vector θ 2 described in
Equation 43. The coordinates (F,To,T,A,G) of the components correspond respectively to (From, To,
Time, Age, Group) and are displayed in the right part.
4.3. COMBINING EFFECTS WITH OPERATORS
27
How are the MDL phrases f rom and to interpreted by GEMACO ?
Effects f rom and to take their meaning only when there are several states (more than 2 without the
state † ). When f rom, or, for short, f is applied to survival-transition probabilities , matrices Φ of
Equation 21 will be equal to:

φ1 φ1 1 − 2 × φ1



 φ2 φ2 1 − 2 × φ2 


0 0
1
(remember! rows = previous state, columns = next state)
If the structure to is used, the survival-transition probabilities will be equal to:

φ1 φ2 1 −
P
i=1,2 φi



 φ1 φ2 1 − P

i=1,2 φi 

0 0
1
It will be seen that the two main effects (f rom, to) can in turn be combined to form model with
other effects. The keywords (d, od, ld, ud) described in Table 5 correspond to specific combinations of
categories of f rom and to.
4.3
Combining effects with operators
Two operators can be used to combine effects to generate more complex models. Let a and b be two
factors with ma and mb categories, respectively.
•
Dot product (.): ‘a.b’ is the product column by column of a by b, i.e. the set of all combinations of categories of the factors a and b, i.e. a model with interaction. The result a.b is a factor
with ma x mb categories. This dot product is the “crossing operator” of [32, pp 48–70].
•
Sum (+): ‘a+b’ joins the columns of a and b. If the intercept (constant column equal to one)
is obtained as linear combination of the variables in a and also of those in b, the first column
of b is suppressed to avoid linear redundancy. The result a+b has then ma + mb − 1 columns.
Otherwise, all the columns of a and b are kept.
For θ 2 given by Equation 43, one obtains for t.g and t + g, respectively, the matrices in Table 6.
The dot and sum operators have a well-known role in single-state models: for instance the cjs
model run independently by group will be denoted as (φg·t , pg·t ). The dot operator is very useful in
combination with the f rom and to effects when there is more than one state (s > 1): f rom.to applied
to the survival-transition matrix Equation 21 induces a variation by rows and columns, i.e. a matrix
28
CHAPTER 4. GEMACO
Table 5: Effects and keywords used in the Model Definition Language(mdl) of gemaco. Phrases in
mdl are interpreted in gemaco to build the matrices of constraints X.
Effects
Keywords and synonyms
Comments
Constant or Intercept
intercept, i
To obtain constant parameters
Categorical variation over time (”factor” with K-1a or Kb
levels)
Time
time, t
a
b
Age
age, a
Categorical variation over age (time elapsed since first
capture) (”factor” with K-1a or Kb levels). More refined
age variations are introduced later
a
b
Cohorta
a
Obtained only with A =
K − 1 classes of age
cohort, c
for transition
for initial state and for encounter
for transition
for encounter
Categorical variation between cohorts (batches of individuals released for the first time with a mark on a same
occasion) (”factor” with K-1 a or K b levels)
a
b
for transition
for initial state and for encounter
Group
group, g
Categorical variation between groups
Departure state (”from”) or
previous state (capture)
from, f,previous, p
Forces rows in elementary matrices to differ.
Arrival state (”‘to”’) or current state (encounter)
to, next, n, current
Forces columns in elementary matrices to differ
Diagonal
d
Constant diagonal terms in a s x s matrix of parameters
Off-Diagonal
od
Constant off-diagonal terms in a s x s matrix of parameters
Upper diagonals
ud
Constant terms in each upper diagonal of a s x s matrix
Lower diagonals
ld
Constant terms in each lower diagonal of a s x s matrix
First encounter
firste
Constant first encounter(i.e. a(1))
Next encounter
laste
Age independent next encounter(i.e. a(2 : A + 1))
Covariates
x
See section 4.4
Individual covariates
xind
See section 4.5
Individual
ind
See section 4.11
4.3. COMBINING EFFECTS WITH OPERATORS





























X2 = 




























1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
1
1
0
0
0


1

 1



 0



 0



 0



 0



 0





 0



 0



 0



 0



 0



 0



 0




 and Y =  0

 1



 1



 0



 0



 0



 0





 0



 0



 0



 0



 0



 0



 0



 0
0
29
0
0
0
0
0
1
1
0
0
0
1
1
0
0
0
0
0
0
0
0
1
1
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
1
1
0
0
0
1
1
0
0
0


























































F
1
2
1
2
3
1
2
1
2
3
1
2
1
2
3
1
2
1
2
3
1
2
1
2
3
1
2
1
2
3
To
1
2
3
3
3
1
2
3
3
3
1
2
3
3
3
1
2
3
3
3
1
2
3
3
3
1
2
3
3
3
T
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
A
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
G
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
2
2
Table 6: Phrases time.group and time + group interpreted by gemaco. Constraint matrices respectively X 2 and Y are generated by gemaco according to the component of the vector θ 2 described in
Equation 43. The coordinates (F, To, T, A, G) of the components correspond respectively to (From,
To, Time, Age, Group) and are displayed in the right part.
with all elements different:

β1 β3 1 − β1 − β3



 β2 β4 1 − β2 − β4 


0 0
1
Several effects can be combined using these operators since in a.b and a + b, a and b can themselves be
model formulae.
The dot operator has priority over the + operator. This order can be changed using brackets as for
instance in [a + t].g.
30
CHAPTER 4. GEMACO
4.4
External covariates
x1
!
. The X 2
x2
matrix (associated to θ 2 given byEquation42) corresponding to a linear effect of this time-dependent
1 x1


 1 x1 




 0 0 


 0 0 




 0 0 




 1 x2 




 1 x2 



covariate within model t is X = 
 0 0 , generated by the phrase i + t ∗ x.


 0 0 




 0 0 


 1 x 
2 



 1 x2 




 0 0 


 0 0 


0 0
Thus, the matrix product of a factor by an external covariate is a way of replacing this factor by
Let us assume that a time-dependent covariate x is available as a column vector x =
the linear effect of the covariate. Contrary to the dot (“.”) and sum (“+”) operators, the “*” operator
which is indeed to the traditional matrix product is neither commutative nor associative. The default
priority order of operations is (+ < . <*), and, as above, can be changed using brackets (e.g., [a.t] ∗ x).
Several covariates related to different effects can be used simultaneously provided they are prepared
in a same file with a specific format (see section 5.4). They are then used as x(1), x(2),. . . (e.g.,
i + t ∗ x(1) + t ∗ x(2)).
4.5
Individual covariates
A dedicated keyword xind
Syntax : xind(list) : consider a list of individual covariates given in the input capture-recapture file
(see chapter 5).
Example: We would like to have the survival depending on two individual covariates x(1) and x(2) .
Index n is for individual n.
(2)
S = i + xind(1, 2) builds a model where, logit(φ(n) ) = β0 + β1 × x(1)
n + β 2 × xn
4.6. AGGREGATION OF PARAMETERS: LISTS
31
How act operators on xind ?
Matrix product ∗ allows one slope associated to a set of covariates.
Syntax: effect ∗ xind(list(levels of effect))
Example: We want to build a model where only one slope is associated to a set of individual
covariates varying with occasion k (1 < k < K − 1).
S = i + t ∗ xind(1 K − 1) builds a model where, logit(φk,n ) = β0 + β1 × xkn
Dot product . allows one effect to act on a set of covariates.
Syntax: effect.xind(list)
Example: We want to build a model where a different slope is associated to a covariate at each
occasion.
S = i + t.xind builds a model where, logit(φk,n ) = β0 + βk × xn
Operator > allows to restrict the effect of a covariate to a set of occasions, groups,etc...
Syntax: effect > xind(list).
Example: We want to build a model where the covariate applies only to occasions 1, 3 and
5.
S = i + t(1, 3, 5) > xind builds a model where,
logit(φk6=(1,3,5),n ) = β0 and logit(φk=(1,3,5),n ) = β0 + β1 × xn
4.6
Aggregation of parameters: lists
e-surge offers several possibilities of “grouping parameters” in the broad sense. First, one often needs
to build effects that are less complex than full dependence on time, age or any other factor. Such effects
on models are obtained by lumping categories. For instance in an analysis of European Dipper data over
6 years, floods decreased survival in years 2 and 3 [25]. The resulting X 2 matrix is obtained by lumping
years 2 and 3 on the one hand, and year 1, 4, 5, and 6 on the other hand. This is done in e-surge
using lists of categories, each list corresponding to a set of categories lumped together. In the Dipper
example, the model formula to reduce the variation over time to two levels is t(1 4 5 6, 2 3). Similarly,
over 7 occasions, to distinguish the first year after capture from the other ones, as two age-classes, one
32
CHAPTER 4. GEMACO
will use a(1, 2 3 4 5 6 7) or a(1, 2 : 7). The overall syntax keyword(list1, list2, . . . , listm) where lists
(defined below) are separated by commas generates a factor with m levels. Each list can be either:
• a list of integers “j k l. . . ”
Example: a(1) is the first age class. a(1 2) aggregates the first two age classes.
• a discrete interval “i:j” as a shortcut for the set of integers {i,i+1,. . . ,j}.
Example: a(3 : 7) is equivalent to a(3 4 5 6 7).
• a series of indices i : k : j as a shortcut for the set of integers {i, i+k, i+2k,. . . }, in which two
successive integers are separated by the step k.
Example: a(3 : 2 : 7) and a(3 : 2 : 8) are both equivalent to a(3 5 7).
• A composite list using “;” and square brackets, according to the syntax list = [list1; list2; . . .]
Example: a([2 : 3; 6 : 7; 10]) is equivalent to a(2 3 6 7 10).
The syntax keyword(list) constrains all parameters in the list to be equal, and leaves the parameters corresponding to the other categories of the effect in keyword unconstrained. In terms of the
corresponding X matrix, it sums the columns in the list to produce a single column and leaves the
other unchanged. If only two age-classes (A = 2) are used for the umbrella model then the model can
be re-written a(1, 2) instead of a(1, 2 : 7). The + operator can also be used in this context: the formula
t(1 4 5 6, 2 3) in the dipper example above is equivalent to the formula t(1 4 5 6) + t(2 3). In this type of
combination, when one wants to keep a series of consecutive categories (or equivalently, of factor levels)
distinct, one can also use the sign “ ” to replace the list with commas as separators. The overall syntax
is keyword(. . . , i j, . . .) which constrains all levels between i and j to be different.
Example: t(1 : 3, 4 6)=t(1 2 3, 4, 5, 6) forces levels 1, 2, 3 to be equal and keeps levels 4, 5, 6 different.
4.7
Aggregation of parameters: the aggregation operator
Lists make it possible only to aggregate parameters within a same main effect. The aggregation operator “&” makes it possible to aggregate parameters corresponding to categories of different effects, i.e.
which cannot be handled within a same list.
•
Aggregation (&): The syntax a&b sums each column of the matrix corresponding to a (which
can be effect1(list1)) with each column of the matrix corresponding to b (which can be effect2(list2)).
If the numbers of columns are not equal then the last columns of the effect with the largest number
of columns are kept unchanged. It is particularly useful and most commonly used with a single
column in each of the terms aggregated, to form a single new column as in the following example.
4.8. KEYWORD ”OT HERS”
33
Example: f (1).to(1)&to(2) applied to the combined survival transition for two states builds the
following constraint {φ11 = φ12 = φ22 }.
The default priority order of operations is (+ < &<. <*).
4.8
Keyword ”others”
Assume we are modeling data with two groups and three occasions of recapture, with two mathematical
parameters defined by t(1, 2).g(1) and that we want to constrain all the other biological parameters to
be equal to a third mathematical parameter. This third parameter may be defined by t(3).g(1)&g(2).
The overall model definition will thus be t(1, 2).g(1) + t(3).g(1)&g(2). Using the keyword others makes
this simpler. The model can be simply defined as t(1, 2).g(1) + others.
Important note: This keyword must always be used at the end of the sentence as “model+others”.
This keyword is particularly useful for multievent models when many parameters have to be fixed
to a same value. In this case, one first defines the mathematical parameters of interest and then simply
add the keyword others to account for all remaining parameters.
4.9
Shortcuts
Definition of shortcuts
In order to keep model definitions as simple and readable as possible, e-surge makes it possible to use
shortcuts. The user associates a shortcut name to an expression written with the mdl via a graphical
interface (see Figure 7).
A shortcut name begins by a letter followed by any letters or figures (Ex : sex for g(1,2)). A
shortcut can be combined to another shortcut (Ex : sex.t for g(1, 2).t). Then, gemaco substitutes
every occurrence of the shortcut name by the equivalent expression.
The syntax for addressing shortcut levels or of any part of a sentence is (see section 4.7):
shortcut(list1 , list2 , ...)
Shortcuts in practice
Let us consider for instance data consisting of individuals marked as juveniles and as adults. Juveniles
are stored in group one and adults in group two. Individuals are considered as juveniles only during
their first year and thereafter become adults.
We can create two shortcuts; Juv for [a(1).g(1)] and Ad for [a(2 : 5).g(1)&g(2)]. For a model in which
survival is different for juveniles and adults and is constant over time, one simply writes Ad + Juv.
34
CHAPTER 4. GEMACO
Figure 7: Shortcut creation interface
gemaco automatically replaces the sentence Ad + Juv by [a(1).g(1)] + [a(2 : 5).g(1)&g(2)]. We can
test, as usual, for effects affecting separately juveniles and adults. For example, the sentence Ad.t + Juv
considers a time-dependent survival for adults only.
The shortcut ”realage” be defined as [a(1).g(1) + a(2 : 5).g(1)&g(2)] with 2 levels. We can build models
Ad + Juv and Ad.t + Juv respectively with the two phrases realage and realage(1).t + realage(2).
4.10
Redundancy in matrices
Each matrix Π, Φ, B is row-stochastic, i.e., the sum of each row is equal to one. Thus, for a matrix of
size R x S, fewer than R x (S-1) parameters out of the R x S parameters have to be estimated. One
redundant parameter has to be chosen for each row. This is open to user’s choice, based on a pattern
matrix T made of character. The T matrix of size R x S is made of ’*’, ’-’ and alphabetical letter,
with rows corresponding to previous states and columns to next state (or current event), as usual. For
each non zero elements of the row-stochastic matrix (either Π, Φ, B), trs is equal to any alphabetical
letter except for one element per row that is set to ’*’ to define the position of the redundant param-
4.11. RANDOM EFFECTS
35
eter for this row. Again, there must be a single star (’*’) in each row. For each element structurally
egal to 0 (either Π, Φ, B), trs is equal to character ’-’. For instance, if we want to use the parameters {ψ11 , ψ12 , ψ21 , ψ22 , ψ32 , ψ33 } of Equation 30 adapted to 3 states instead of using
 the parameters

∗ ψ ψ −


 ψ ∗ ψ − 


{ψ12 , ψ13 , ψ21 , ψ23 , ψ31 , ψ32 }, we have to change the transition pattern matrix from 

 ψ ψ ∗ − 


− − − ∗


ψ ψ ∗ −


 ψ ψ ∗ − 


which is the default, to 

 ∗ ψ ψ − 


− − − ∗
4.11
Random effects
For random effects, we extend the mdl for fixed effects [5]. Therefore, we introduce a new built-in keyword factor denoted ind for individual random effects, and implement random effects for groups with
the keyword random, which translate fixed effects into random effects. These additions fit naturally
into E-SURGE’s model specification syntax. However contrary to traditional effect like time, age, group,
direct addressing of levels of ind (one level corresponding to one individual) is not currently allowed.
We extend also the operator + to concatenate fixed effect and random effect to generate mixed models
of the form (14,16). Examples include:
The phrase i + weight + ind models equation (15).
The phrase i + random(group) models equation (17).
More generally, two general forms of phrase are currently allowed
phrase1 + phrase2.ind for equation (14) and
phrase1 + random(phrase2) for equation (16),
where phrase1 and phrase2 are any general phrases for fixed effects.
Note: The phrase random(ind) is equivalent to the phrase ind.
36
CHAPTER 4. GEMACO
4.12
A list of models
The mdl in gemaco has been considerably expanded over the notation originally proposed by [25]. It
is flexible and powerful even for cjs models. A comparison between the two notations for a few models
frequently used is provided in Table 7.
Table 7: Correspondence between models in the notation of [25] and in the mdl in gemaco [5].
Model in the notation of [25]
For survival in gemaco
For event in gemaco
t∗g
t+g
a2 + t
a2 ∗ t
a2 ∗ g
a2 + g
t+m
t∗m
g∗m
g + ma
t.g
t+g
a(1, 2 : A) + t
a(1, 2 : A).t
a(1, 2 : A).g
a(1, 2 : A) + g
−
−
−
f irste + nexte.t.g
f irste + nexte.[t + g]
−
−
−
−
f irste + nexte.[a(2, 3 : A + 1) + t]
f irste + a(2, 3 : A + 1).t
f irste + a(2, 3 : A + 1).g
a
For models with trap effect(denoted m), the data must be
decomposed according to [36, 8]
4.13
firste+nexte.[a(2,3:A+1)+g]a
−
a
same
a(1)+a(2:A+1).[a(2,3:A+1)+g]
as
Aggregating mathematical parameters
gemaco allows you aggregate levels inside or between a effect inside a step defined in gepat (for example
the survival). But it is impossible with gemaco to set equality between parameters of different type. In
e-surge, we can do it by aggregating mathematical parameters. To do this, select the option Setting >
Set equality between parameters of various types after defining the model in gemaco but before ivfv
(see Figure 8).
e-surge ask you lists of integer at which constraint will apply (see Figure 9).
4.14
Non-linear model
When the option Models > Markovian & semi-Markovian states > Conditional on 1st Capture is
selected, steps in gemaco and in ivfv interfaces are two-fold. In gemaco, first defined the linear model
a usually (but using the intercept for the survival) then exit. A new menu appears, see Figure 10. Several
continuous functions associated to different hazard functions are available [11], select one of them. In
ivfv, fix the only one parameter link to the survival to 1, fix also the relevant capture rate (like the
4.14. NON-LINEAR MODEL
37
Figure 8: Set equality between mathematical parameters
Figure 9: Set equality between param. of various types: User’s should enter lists of integer at which
constraint will apply like ’{1 4:6} {2:3}’. Here mathematical parameters 1, 4, 5 and 6 are set equals as
well as mathematical parameters 2 and 3. Lists must be separated by a space (i.e. } ’space’ {).
38
CHAPTER 4. GEMACO
Figure 10: Menu for the choice of the survival: After exiting gemaco, the user must select a continuous
function thank to a new menu. Several hazard functions are available as well as a non-parametric
function (i.e. full age dependant survival) and a geometrical distribution (i.e. constant survival)
Figure 11: Menu for setting the initial values of the continuous function.
first one corresponding to f irste to 1) then exit. A new menu appears, see Figure 11, for setting the
initial values of the continuous function.
39
5
Data input
CR data typically consist of recapture history data (e1 . . . ek . . . eK ) with associated number of animals
(eff1 eff2 . . . effN G ). A negative value for effng means that the animals were removed at the occasion of
last capture. e-surge recognizes three file formats for CR data input, the biomeco, the mark where
numbers (instead of letters) are used as labels for states and the headed format. This implies that 9
states at most can be handled using the mark format. Any number of states can be considered with
the biomeco or the headed format.
5.1
The BIOMECO format
This format stems for the statistical ecology software biomeco [28]. It makes it possible to label rows
and columns via external files. This may be advantageous for proper retrieval of CR data, using, e.g.,
individual band numbers as labels for rows when there is one row per individual. The filename ( dummy
sign $ ) can be used if you do not want to create specific label files. The biomeco format, described in
Table 8 (see also [39]) can also be used in input and output in u-care.
NH K+NG
Filename1
Filename2
e1,1 a e12 . . . e1,K eff1,1 b . . . eff1,N G
..
.
eN H,1 eN H,2 . . . eN H,K effN H,1 . . . effN H,N G
a
b
integers enh,k must be separated by spaces(not tabular)
real numbers effnh,ng must be separated by spaces(not tabular)
Filename1 and Filename2 : names of files with row and column labels,respectively,
K : number of capture occasions,
NG : number of groups,
NH : number of capture histories,
K+NG : total number of columns in the file.
Table 8: Description of the biomeco format as applied to CR data
enh,k is either 0, if the individual nh is not seen at occasion k, or u, if individual nh is seen in the
event u, (u = 1, . . . , U ) at occasion k. The set of values (enh,1 enh,2 . . . enh,K ) is the capture history
enh,. and the associated vector effnh,ng is the number of animals with history enh,. (nh = 1, . . . , N H) in
group ng (ng = 1, . . . , N G). Negative value for eff means that animals are removed immediately after
last capture.
40
CHAPTER 5.
DATA INPUT
Remarks:
• e-surge does not allow external filenames. Only the dummy sign $ is accepted in place of
filename1 and filename2. We plan to use such names for labels of rows and columns respectively
in further versions.
5.2
The MARK format
Alternatively, the input data file can be in mark format [50] described in Table 9.
e1,1 a e1,2 . . . e1,K eff1,1 b . . . eff1,N G ;
..
.
eN H,1 eN H,2 . . . eN H,K effN H,1 . . . effN H,N G ;
a
b
integers enh,k are written without data separator
real numbers effnh,ng must be separated by spaces(not tabulator)
Table 9: Description of the mark format
e-surge asks then the number of columns containing covariates. By default, this value is zero (no
external variable) as such covariates are not presently handled by e-surge. Note that the enh,k have to
be contiguous, i.e. not separated by a blank (in contrast to the biomeco format). e-surge uses only
digits and not letters for states and the maximum number of states with the mark format in e-surge
is presently 9.
5.3
The HEADED format
The format
This format is a more general format in the sense that it includes the two previous formats with an
explicit label for each column. Using meaningful names as labels may be advantageous for proper
retrieval of CR data. Numbers are used as labels for events thus the number of events is not limited.
e-surge uses only digits and not letters for states.
Note that the enh,k have to be separated by a blank or a tabular if the number of states is higher than
9.
Comments are accepted and should be written between ’/*’ and ’*/’.
5.3. THE HEADED FORMAT
41
Header line with formatted column names
e1,1 a e1,2 . . . e1,K eff1,1 b . . . eff1,N G c1,1 c . . . c1,N C i1,1 d i1,2
..
.
eN H,1 eN H,2 . . . eN H,K effN H,1 . . . effN H,N G cN H,1 . . . cN H,S iN H,1 iN H,2
a
integers or reals enh,k must be separated by spaces or tabulars if number of states is higher than 9
real numbers effnh,ng must be separated by spaces or tabular
c
numbers ou letter cnh,nc must be separated by spaces or tabular
d
real numbers inh,t must be separated by spaces or tabular
b
K : number of capture occasions,
NG : number of groups,
NH : number of capture histories,
NC : number of covariables,
2 : number of informative variables (censoring variables),
The Header line contains the K+NG+S+2 formatted column labels.
Table 10: Description of the headed format
The Header Line
The header line contains a label for each column to permit e-surge to read several kind of data. The
label syntax is :
[format indicator] key-word [label]
• The key-words
Several kind of data are allowed and classified according to key-words.
Variable Type
Key-Word
Definition
History ei,j
H:
Recapture History data
Sample Size effi,k
S:
Associated number of animals
Left Censoring ii,1
LC:
Right Censoring ii,2
Covariable ci,h
RC:
COV:
When the animal was censored before the first capture :



 real age at the first capture if left censoring
=
-1 if left censoring but unknown age


 0 if no left censoring.
(
-1 if the animal is removed at the last capture
=
0 if no right censoring.
Explanatory Variables (predictors), ID.
• The format-indicator
The format-indicator allows e-surge to identify the variable format. By default, when the formatindicator is not mentionned, variables are considered as numeric. To read character variables, it is
42
CHAPTER 5.
DATA INPUT
necessary to specify the format-indicator $ as in SAS.
S:SampleSize
$COV:Sex
COV:weight
55
Male
1.2
12
Female
2.3
41
Female
0.9
• The label
A label is mandatory for all covariables and optional for the others. It is a word beginning by a letter
followed by any letters or figures. The covariable labels will be used to build the model in e-surge.
Important note: The left censoring is only implemented for the Models > Markovian & semiMarkovian states > Conditional on 1st Capture option. In that case, for programming convenience,
it is necessary to add as much columns of 0 (for occasions) as the value of the maximal age of left
censoring.
A headed format example
H:C1
H:
H:C3
H:C4
S:
COV:weight
COV:height
LC:dead
RC:
$COV:sex
$COV:Couple
1
0
12
0
1
24.2
31
0
0
male
A
0
0
4
1
1
18.2
28
5
-1
female
A
0
0
0
1
1
33.7
52
0
0
female
B
0
0
2
12
1
10.1
33
0
0
male
B
0
0
1
4
1
10.7
18
4
0
male
C
Automatic shortcuts/clusters creation and use
When a qualitative covariate is find in the file and selected by the user (see Figure 12), a shortcut (see
Figure 7) or a cluster (see Figure 13) is automatically build according to the number of modalities:
• Shortcuts
If the number of modalities (levels) is lower or equal to 50, a shortcut is created which can be
used as a fixed or a random effect.
• Clusters
If the number of modalities (levels) is strictly larger than 50, a cluster is created by grouping
individuals by modality. It can be used only as a ”cluster” random effect (currently not available
but very soon available).
5.3. THE HEADED FORMAT
43
Figure 12: Covariate selection interface
Figure 13: Cluster interface
44
5.4
CHAPTER 5.
DATA INPUT
File of external covariates
The format of the file is described in Table 11.
n is the number of external covariates xi stored as successive rows of ki values.
n
k1 k2 . . . kn
x11 x12 . . . x1k1
..
.
xn1 xn2 . . . xnkn
Table 11: Description of the external covariates file
5.5
File of time intervals
The file of time intervals is a row of K − 1 real numbers as in Table 12.
t1 a t2 . . . tK−1
a
real numbers ti must be separated by spaces(not tabulator)
K − 1 : number of time interval,
ti : time interval 1 ≤ i ≤ K − 1
Table 12: Description of the file of time intervals
5.6
Selecting steps for unequal time interval
e-surge allows the user to select steps at which unequal time intervals apply. One or more steps can
be considered. By default, unequal time intervals apply to step one. This can be change by selecting
Unequal time intervals: # of steps in the menu Setting(see Figure 14). A dialog box appears (see
Figure 15) asking for the steps.
5.7
File of initial values fixed values
The structure input file for Initial Values and Fixed Values in e-surge is described in Table 13.
5.7. FILE OF INITIAL VALUES FIXED VALUES
45
Figure 14: Select steps from the main menu
Figure 15: UTI applied to transition: User’s should enter a list of integer at which unequal time interval
will apply like ’1 3’. By default, unequal time intervals apply to step 1.
46
CHAPTER 5.
DATA INPUT
nI nT nB
########## INIT ##########
typ1 β1
..
.
typnI βnI
########## TRANSITION ##########
typnI+1 βnI+1
..
.
typnI+nT βnI+nT
########## ENCOUNTER ##########
typnI+nT +1 βnI+nT +1
..
.
typnI+nT +nB βnI+nT +nB
Table 13: Description of the file of Initial and Fixed values
βi is a real and typi is an integer



 0 if βi
typi =
1 if βi


 2 if β
i
giving the type of βi :
is an initial value
is a fixed value in the logitgen link scale(if applied)
is a fixed value in the identity scale
typi = 1 is usefull to fix mathematical parameters whereas typi = 2 is usefull to fix biological parameters
to 1 or 0.
47
6
A short session
In this chapter, we show how to use e-surge based on a version of the Conditional Arnason-Schwarz
model with site fidelity parametrization (see section 3.4). The data are those used in [22] and come
from a 6-year study of a seabird between 1984 and 1999, for the study of movements of the Canada
Geese between three sites. The following steps that are necessary to obtain the parameter estimates
are listed below:
• Open a session (i.e. a frame that will contain the specifics of the data and the analysis results for
future retrieval),
• Load the capture histories data if the session is new,
• Build a general model containing an umbrella model (based in particular on appropriate Goodnessof-fit test in u-care) using the gepat interface,
• Specify and build further constraints using the gemaco interface,
• Fix parameters and/or change initial values if needed using the ivfv interface,
• Run the model,
• Examine and interpret the results.
We will go through these steps in the following paragraphs. The general organization of e-surge is
summarized in Figure 16.
Figure 16: General organization of e-surge.
48
6.1
CHAPTER 6. A SHORT SESSION
Main window of E-SURGE
After invoking e-surge (e.g., from Windows Explorer by double-clicking on e-surge.exe), we are
presented with the main window of e-surge (see Figure 17). The window is divided in four areas
(namely Data Status, Advanced Numerical Options, Compute a Model and Output) and the toolbar
has six menus (namely Start, Data, Models, Setting, Run & See).
Figure 17: Main windows of e-surge with a toolbar and 4 distinct areas: The aspect of the window is
that before the data set has been loaded (a short description of the data is given in the ”Data status”
area) but before any model has been defined (see area ”Fit a model”). In this area, four buttons give
access to the gepat interface, the gemaco interface, the ivfv (Initial Values and Fixed Values) interface
and the run (Deviance Minimization routine, or solver). At the stage considered here, no button is
activated.
6.2
Opening a session
In the Start menu, you can select either a new or an old session (or exit from the program). If you
want to begin a new session, you first select the option open a new session. A window will appear
6.2. OPENING A SESSION
49
asking where you want to store the results during the working session. For example, use the name
firststep.mod as in Figure 18.
Figure 18: Open a new session “firststep”
(This analysis was run under a French version of Windows)
All essential pieces of information about data and computations will be stored in this session file and
it will be saved automatically for future use when you exit e-surge. Next, you must load the capture
histories data (prepared as either a biomeco, a mark or a headed format file).
Figure 19: Read the data from the Biomeco file geese.rh (This analysis was run under a French version
of Windows)
Select the option ”Open a Biomeco file” in the Data menu and select your file. In our example the
50
CHAPTER 6. A SHORT SESSION
file is geese.rh as it is shown in Figure 19.
Once you have loaded your data file, the area ”Data status” is automatically updated with a description of the data. When an old session is opened, the data set is loaded automatically and e-surge gives
a short description of the data in the ”Data status” area of the main window (see Figure 17). You can,
and must if necessary, change the number of groups, the number of states and events and the number
of age classes by pressing the Modify button (change the current values, see Figure 20 ).
Figure 20: Change the number of age classes from 5 to 1.
The choice of an appropriate number of age classes leads to a faster algorithm and is thus recommended if appropriate. Select 1 age class in Figure 20 for time dependent model. All result files will be
saved in the directory of the current session file.
6.3
Building the pattern matrices for the CAS model with site
fidelity parametrization using GEPAT
The next step is to specify in e-surge, the pattern of the matrices P Π, P Φ(1) , P Φ(2) , P Φ(3) , P B defined
respectively in equations (34,35,36,37,39). When the button gepat in the Compute a model area of
the main window (Figure 17) is pressed (or alternatively gepat in the Setting menu is selected), the
gepat window opens (Figure 21). This window has a toolbar with three menus (Parameters, InputOutput for patterns, Pre-defined Patterns), four areas (Sentence for pattern, Matrix pattern, Options,
Automatic Patterns) and three edit boxes related to the number of steps(’1’), the current step(’1’) and
its label(’IS’).
The matrix pattern P Π is by default already defined as in Equation 34 (Please note that often this
pattern has to be modified) For the four others matrix patterns, proceed step by step as follows:
6.3. BUILDING MATRIX PATTERNS USING GEPAT
51
Figure 21: Window structure of the gepat interface : The default patterns for transitions and encounters
matrices are those of the combined formulation of the cas general model. The default matrix pattern
for initial state will be kept unchanged.
• First, select Transition in the menu Parameters, change the number of steps to 3 and optionaly
the label of the current step 1 to S (or any convenient label for survival). Change the matrix
pattern to Equation 35, the result is given in Figure 22. To obtain this diagonal matrix, you can
click on the button Diagonal Matrix.
• Select the current step 2 by clicking on the adjacent right arrow. Change the label to F(or any
convenient label for fidelity). Define the size of the matrix in the options area, enter 4 for the
number of rows and 7 for the number of columns and click on the button update now. Replace
the empty matrix pattern by Equation 36, the result should be the one visualize in Figure 23.
• Select the current step 3 by clicking again one time on the right arrow. Change the label to M(or
any convenient label for settlement). Define the size of the matrix in the options area, enter 7
for the number of rows and 4 for the number of columns and click on the button update now.
Replace the empty pattern matrix by Equation 37, the result should be the same than the one
display in Figure 24.
• Last, select Event in the menu Parameters. Change the pattern matrix to Equation 39, the result
is given in Figure 25.
52
CHAPTER 6. A SHORT SESSION
Figure 22: Matrix pattern in gepat for survival corresponding to Equation 35.
Figure 23: Matrix pattern in gepat for fidelity corresponding to Equation 36.
6.3. BUILDING MATRIX PATTERNS USING GEPAT
Figure 24: Matrix pattern in gepat to Equation 37.
Figure 25: Matrix pattern in gepat for encounter corresponding to Equation 39.
53
54
CHAPTER 6. A SHORT SESSION
The general model under which the CAS model with site fidelity parametrization is now fully
specified and you can leave the GEPAT interface by clicking on the button EXIT in the lower part
of the window. However before exiting, you can save matrix patterns for an upcoming use. To that
purpose, select Save file with Patterns in the Input-Output for patterns menu.
Presently, two link functions are available in e-surge, the generalized logit and identity links. You
can choose between these two links in the Setting menu. For now, select the generalized logit link (the
default link). This action completes the specification of the general model for the current session.
Before running any model, unequal time intervals (see section 2.6) can be used by selecting in the
Setting menu the Unequal time intervals sub-menu. e-surge asked for a file of unequal time intervals
(see section 5.5). By default, unequal time intervals are applied to the first step of transition.
In our example, initial state probabilities will be estimated. The first step to compute these estimates
is to select Initial, Transition & Encounter in the menu Models | If any factorisation (see Figure 26).
See section 2.8 for details about the corresponding full likelihood. To skip Initial states probabilities
from transitions probabilities and encounter probabilities, select Transition & Encounter in the menu
Models | If any factorisation. In this case, only the partial likelihood (conditionial to the first capture)
is considered.
Figure 26: Menu to estimate or to skip initial state probabilities
6.4
Building the CAS model with site fidelity parametrization using
GEMACO
The next step is to specify more precisely the particular model to fit. This model is always nested within
the umbrella model and appropriate restrictions are implemented through constraints on parameters.
For people used to mark or surge, building constraints means creating design matrices. One great
feature of e-surge is that, constraints are specified by means of a Model Definition Language interpreted by gemaco. When the button gemaco in the compute a model area of the main window (see
Figure 17) is pressed (or alternatively gemaco in the Setting menu is selected), the gemaco window
opens (see Figure 27). This window has a toolbar with four menus (Input-Output for constraint matrix, Parameters, Parameters and Gemaco) and four areas (Model definition, Shortcuts for sentences,
Transitions pattern and Constraint matrix) in Figure 27.
6.4. BUILDING CONSTRAINT USING GEMACO
55
Figure 27: Window structure of the gemaco interface : As shown here, the initial state part of the
cas model has just been built. The notation current.t means that the initial states vary by current site
(current) and time step (t). The corresponding design matrix, automatically created by gemaco, has
popped up in the Constraint matrix area.
Constraints on each type of parameters (initial states, transitions (survival, fidelity, settlement)
and encounter) are defined in turn independently of each other. Select each parameter type from the
Parameters menu or by clicking repeatedly on the top button with the name of the currently active type
(e.g. initial state in Figure 27). For the cas model with site fidelity parametrization that we intend to
fit to the goose data set, proceed step by step as follows:
• First, select initial state, enter the string ”current.t” in the Model definition area, and validate
it by clicking out of the Model definition area. The button Call Gemaco is now activated. Click
on it or select call Gemaco in the Gemaco menu. The design matrix appears in the Constraint
matrix area. The initial state part of the model is now defined.
• Select transitions in the menu Parameters, select current step 1 corresponding to survival, enter
the string ”f.t” in the Model definition area (”f = f rom” and ”t = time”), and validate it by
clicking out of the Model definition area. The button Call Gemaco is now activated. Click on it
56
CHAPTER 6. A SHORT SESSION
or select call Gemaco in the Gemaco menu. The design matrix appears in the Constraint matrix
area. The survival part of the model is now defined.
• Select current step 2 corresponding to fidelity, enter the string ”f.t” in the Model definition area,
validate it and call gemaco.
• Select current step 3 corresponding to settlement, enter the string ”f.to.t” in the Model definition
area, validate it and call gemaco.
• Select Event in the menu Parameters, enter the string ”f irste + nexte.current.t” in the Model
definition area (”a = age”), validate it and call Gemaco. The first encounter (denoted as f irste vs
nexte for re-encounter) corresponding to the probability of capture ( include B k,1 ) will be fixed
to one later on in ivfv.
The cas model with site fidelity parametrization is now fully specified and you can leave the interface
by clicking on the button EXIT in the lower part of the window. However if you want to linger some
more within the gemaco interface, here are some indications of what you can do.
For each kind of parameters, two steps are compulsory and four steps are optional.
• (Optional) Select an external file if any external effects are involved (see section 4.4). To standardize these external covariates, select file with external variables to standardize. A file with
standardized variables is created and saved as the original file with a ”std ” suffix.
• (Optional) Define shortcuts in shortcuts for sentences before gemaco (see section 4.9 for its use).
Click on the button Add shortcut to define a new shortcut. You can also select a shortcut and
erase it by clicking on delete shortcut or you can also load and save shortcuts from the menu
Pre-defined Shortcuts. At the end of the session, shortcuts are saved with the session. With a
new session, if no shortcuts is already defined, a default file is loaded to which you can of course
add your own shortcuts.
• (Optional) Change the pattern for transitions in the Transitions pattern area (see section 4.10
for its use). The position of the zero in each row represents the parameter that is not constraint
directly. To change it, first select the row concerned and then move to the desired position the
bottom cursor.
• Enter a string in the Model definition area and validate it by clicking out of the Model definition
area (see section 4.12 for examples).
• Call gemaco by clicking on the button call Gemaco. If the model is correctly specified, the
constraint matrix appears in the top left corner of the window in the Constraint matrix area.
6.5. CHANGING INITIAL VALUES OR FIXING VALUES
57
• (Optional) Create user-defined models. Save the matrix displayed in the Constraint matrix area,
(toolbar Input-Output for constraint matrix), then you can change it using an external editor
(such as WordPad or TextPad or vim) so that the new design matrix represents the model you
would like to fit. Finally you can save this new matrix. You can then load it in m-surge (using
toolbar Input-Output for constraint matrix).
• (Optional) Build all the model together from the menu Gemaco | Call Gemaco (all phrases).
6.5
Changing Initial Values or Fixing Values of parameters (IVFV)
The ivfv interface serves to help in reaching convergence of a model by setting initial values to replace
default values or to set one or more parameters at a pre-determined value (“fixed values”). To do that :
- click on the button ivfv (“initial values, fixed values”) in the Compute a model area of the main
window (see Figure 17). A new window will appear similar to that of Figure 28. By default, each
parameter has been assigned an initial value equal to 0.5.
- Select the encounter part to fix the first encounter parameter to 1. In our example parameter 1
appears in Figure 28. Next to Beta #1 on the left is the chain “1 2 1 1 1 1”. The 1 2 sequence
means that the beta parameter corresponds to the capture probability of events 2 (seen as a 1)
for animals in state 1, the fourth indice 1 indicates the first class of age (the first capture).
- On the same line, you should replace the value 0.5 with 1. This is the new starting value for
parameter #1.
- Check the box nearby to fix this parameter. Now, parameter #1 is excluded from the optimization
process. Its value is frozen at the initial value you have just entered.
- Click on the button Exit to exit the ivfv interface.
It may also useful to fix some parameters to pre-determined values. In the cas model, the recapture
probability at the last occasion is not identifiable separately from the survival probability over the last
interval. Fixing the three last recapture probabilities to 1 does not change the results and may indeed
facilitate convergence.
There is an option Initial Value in the main window (see Figure 17) to change the way initial values
or fixing values are set. You may want to play with it and see the change in the ivfv interface. Fixing
and selecting initial values can be done either on the ]0, 1[ axis (i.e.; on the biological parameter scale
without link like θ = β), or on the real axis (i.e.; on the mathematical parameter scale f (θ) = β). In
the latter case, it is the transformed value (i.e. the logit or the generalized logit) of the parameter that
is set. This can be specified via the toolbar menu item Value space. Finally, the set of initial values
can be saved in a file and reloaded later (file option of the toolbar).
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CHAPTER 6. A SHORT SESSION
Figure 28: IVFV interface: The encounter parameter 1 is fixed to one in the [0,1] scale. This is because
the box “fixing or setting initial values of Beta ?” is activated. In the current figure the capture parameter
2 is not fixed. Iterations will start for this parameter with an initial value equal to 0.5.
6.6. SETTING OPTIMIZATION PARAMETERS AND RUNNING THE MODEL
6.6
59
Setting optimization parameters and running the model
Before running the model, it is possible to change the numerical options that govern the optimization
algorithm in the Advanced Numerical Options area of the main window (see Figure 17). Because our
example is a relatively simple model with good data, the maximum number of iterations may be reduced
to 200 (see Figure 29). Also we can set the tolerance to parameters change to 0.0000001. This is one
of the two stopping criteria of the algorithm: the lower the tolerances, the more precise the result is. If
the maximum number of iterations is small as here, we recommend you set the Convergence option to
Continue (after n cycle)?. If no stopping criterion is satisfied after n cycles of 200 iterations, e-surge
will ask you whether to go on with n cycles of 200 additional iterations or stop (here, n = 3). To run
the model, click on run which is the red button in the Compute a Model area of the main window (see
Figure 17). This button becomes active once you exit ivfv.
Figure 29: Advanced Numerical Options: The tolerance on change in parameters has been set to 10−7 ,
the maximum number of iterations is 200. If no convergence is achieved after 3 cycles of 200 iterations,
e-surge will ask whether to continue for 3 cycles of 200 more iterations (the continue after n cycle?
option has been set with n = 3) or not. The analytical gradient will be used to compute the rank
(default, rather than with the Hessian) and a detailed output of the iterations will be displayed in a
dos window. The Hessian is needed to get standard errors estimates.
Now the model is being fit. The rank of the model conditional on the data is estimated by computing the rank of a matrix composed of the gradient of each history probability collapsed together and
estimated near the mles. Getting the correct value for the rank of the model is critical for model selection and we implemented a very precise algorithm using a numerical version of the Catchpole Morgam
Freeman approach [43, 6](see also section 7.2). e-surge suggests a model rank by default that you can
60
CHAPTER 6. A SHORT SESSION
modify. Here, the rank is estimated correctly at 69 (see Figure 30).
After convergence, i.e. minimization of the deviance, the Hessian matrix can be calculated optionally
by ticking the compute Hessian option in area 2 and running again. The Hessian is the matrix of second
order derivatives of the log-likelihood and is used to approximate the variance-covariance matrix of the
estimated parameters. It is computed by a finite difference scheme using either
• the deviance, by default when the finite difference gradient is used for optimization.
• the analytical gradient, when the analytical gradient is used for optimization. The option very
efficient in m-surge is not presently recommended in e-surge because of the additional cost
needed to compute the analytical gradient for Multievent model.
The variance-covariance matrix is approximated by the generalized inverse of the Hessian matrix. The
rank of the model conditional on the data can also be estimated from the numerical rank of the Hessian
matrix [49], but this new estimate is generally less precise than the default one.
Figure 30: “Give model rank” window: The estimated rank is 69 and a model name has to be chosen
for the CAS model with site fidelity parametrization.
6.7
Output of results
The previously fitted models of the session, their deviances and aics are permanently displayed in the
Output area (see Figure 31). If more than one model has been run, these models are sorted from top
to bottom by increasing qaic values.
When the optimization of a new model stops, the program examines whether it has not stopped
at a saddle point in which case a warning is issued (see section 10.2). Then, it looks for redundant
parameters. This is done by analyzing the singular values of a derivative matrix ([see 43, 6] for details
and section 7.2) at 4 points in the neighborhood of the mle and at the mle itself. The estimated model
rank at each of the five points is shown in the dos window (see Figure 31) while one list of potentially
redundant parameters corresponding are listed for each point in the output file (see Figure 32). The
6.7. OUTPUT OF RESULTS
61
Figure 31: Output of e-surge I : Informations in the dos window: While a model is being fitted,
information about the iterations scrolls down the left dos window. Once convergence has been reached,
the numerical ranks of the model at 4 points in the neighborhood of the mles and at the mles itself
(5th value) are displayed. The right bottom window (Output area of the main window) permanently
displays the previously fitted models ordered by aic, with the following pieces of information: model
name, rank, deviance and aic. Once validated, the new model will take its place in this window. A
warning occurs saying that estimates are on a saddle point. To avoid this problem, run again the model
after fixing the last three capture rates to 1 and decreasing tolerances to 10− 8.
user is then prompted to validate the estimated rank and to name the model. This done, the new model
takes its place in the list of previous models maintained in the Output area of the main window (see
Figure 31).
After each model fit, the results are also saved automatically in two files named namemodel.out and
namemodel.xls in the working directory. The mouse cursor flick during the save of the Excel file. The
model name by default is model? where “?” is replaced by a number. The text file can be displayed by
selecting the model in the Output area of the main window (Figure 17), and then clicking the button
View file of results. The corresponding file with generic name *.out is opened by the editor. The Excel
file can be opened by Excel and it can be automatically laid out by clicking on Update output Excel
62
CHAPTER 6. A SHORT SESSION
Figure 32: Output of e-surge II: Result of the potentially redundant parameters : The cas model
without the last captures fixed to 1 is known to be redundant, with a fall of rank equal to the number
of states. However, with sparse data, this may be worse. Here the formal result is verified. The above
temporary window displays for each of the 5 points near the mle (see text and caption of Figure 31)
the number of singular values of the derivative matrix (see [4] below the indicated threshold (here
9.2387e − 006), the number of additional singular values below a less selective threshold and the indices
of the potentially redundant mathematical parameters (here there are none).
file shape in the Run and see menu. We recommend you associate files with a suffix .out to the editor
of your choice. Similarly, the Hessian and the estimated variance co-variance matrices are saved in the
Excel file and in a temporary file named Hessian.tmp and may be retrieved using an editor. Any model
can be retrieved by selecting a model in the Output window part and by clicking the Retrieve model
button. The model selection can also be exported to Excel using the Export to Excel button.
63
7
Advanced tools for numerical issues
Critical issues particularly for multistate models are the risk of numerical convergence to a local rather
than the global minimum and problems of parameter redundancy. Several advanced tools have been
made available in e-surge to address these issues. The diagnostic tool for parameter redundancy is
available for any Markovian models, but only for fixed effect model without an individual covariate.
7.1
Initial values
The default “constant initial values” may lead to a local minimum of the deviance. To reach the absolute
minimum, the initial value should ideally be chosen near the unknown mles. In the absence of clues
about good starting values, a possibility is to change the initial values once or repeatedly at random;
another to start from the results of a previous model. The two approaches are available in e-surge.
• Use random initial values (option Random). Repeated random initial values are particularly useful
(option Multiple Random): over several successive runs, you will most of time get convergence at
least once at the global minimum of the deviance, even if there are local minima.
• Use mles of the previous model (option From last model) as initial values. As an example, the
mles of the jmv model are more easily attained by starting from the mles of the corresponding
cas model. e-surge automatically adapts parameter values to fit the structure of the current
model.
• Use random initial values (option Multiple Random from IVFV files) defined in IVFV files(see
section 5.7 for the definition of IVFV file). To use this option, you must first define a file as
Table 14 containing the number of initial values and the name of each IVFV file.
n
name f ile1 .f ix
..
.
name f ilen .f ix
Table 14: Multiple random values from IVFV file: The first line of the file containts n the number of
starting values and the next lines n names of ivfv files.
Important note: Like in multistate models, in multievent models, no method totally guarantees
convergence to the global minimum of the deviance. Based on our experience, we recommend to use
the option EM(20)+quasi-Newton available for the option Markovian states only > Conditional on 1st
Capture.
64
CHAPTER 7.
7.2
ADVANCED TOOLS FOR NUMERICAL ISSUES
A numerical approach for redundancy
Another crucial point is parameter redundancy. In their version adapted to multistate models [18], advanced users may use the formal methods of [4] for studying parameter redundancy. The key advantage
of this method is that estimable functions of the redundant parameters are explicitly identified. This
enables the user to fix the values of some redundant parameters to render the model full rank and above
all to interpret the values of the estimable parameters confidently. When this method cannot be applied
(e.g., for complex models), redundancy can be examined by looking at estimated standard errors (see
[20]). In e-surge, a parameter not at a boundary but with a very large or null standard error is in
general redundant. However, this approach is often unreliable.
The numerical version of the cmf approach is more reliable and has been implemented in e-surge.
It was used as a tool in [43] to demonstrate that some memory model are full rank. This approach [6]
considers the properties of the numerical derivative matrix rather than those of the formal derivative
matrix. The local rank, conditional on data, is estimated as the number of non-zero singular values and
the redundant mathematical parameters are identified. The only limitation is that estimable functions
of the redundant parameters are not explicitly identified. To improve precision, the calculations are
performed at 5 points by e-surge: the first 4 are neighbors of the mles shifted out of boundaries and
the last one is the mles itself. The estimated numerical rank provided by e-surge the maximum of the
5 ranks obtained. The list of potentially redundant parameters is established as the union of indices of
potentially redundant parameters at each point where the rank is maximal. This set of indices indicates
which parameters should be considered carefully for interpretation. Recall that direct interpretation of
parameters involved in redundancy is not relevant.
The rank of a model can drop locally, although on a set of interior values of measure 0 (the probability
to draw such an interior point is zero and four times in a row almost never) ([43]). Such an increase
also happens when a parameter is estimated at a boundary (see the legend of Figure 32). This local
redundancy often occurs at the mles in which case the set of indices increases at point 5 (the mles). In
case of doubt about the identifiability of a mathematical parameter, we recommend drawing the profile
deviance. The cas model with site fidelity parametrization with 3 states and no fixed parameters has
its parameters over the last interval redundant: the last three survivals (parameters 25, 26 and 27), the
last 3 fidelity (parameters 40, 41 and 42 ) and the last 3 transitions (parameters 55, 56 and 57) and the
last 3 captures (parameters 70, 71 and 72). The last list of indices (build at the mles) contains two
more parameters, two transitions (parameters 45 and 48). This could be an instance of a local drop of
rank. However, checking the estimated values shows that this parameter is estimated at the boundary
(its estimated value is zero because there is no such transition in the data set). This is not a case of
65
redundancy.
8
Advanced tools for output
Several advanced tools have been made available in e-surge to give additional output. Only the nonparametric bootstrap is general. The other tools are currently only available for the option Markovian
states only > Conditional on 1st Capture.
8.1
State dependent probabilities of CR histories
One may wish to consider the probability of each histories or the probability of an individual conditional
to its history to be in state D at first encounter e and in state A at the last occasion K . This approach
is related to the Bayes theorem and was used in [35] to obtain a post allocation of individual animals
to classes of heterogeneity. To obtain these quantities in e-surge, select View history state dependent
probability in the menu Run & See, see Figure 33.
Figure 33: To save probability of each histories, select View history state dependent probability in the
menu Run & See
After the parameters have been estimated, e-surge creates a file named ’histories.tmp’ which contains
for each history h
P (h)
and
P (D, A, h)
P (D, A/h) = Pa=1,N
d=1,N −1 P (d, a, h)
Each conditional probabilities P(D,A/h) and the associated marginals are stored in an array like:
d>a
1
..
.
1
P (d = 1, a = 1/h)
...
N
P (d = 1, a = N/h)
P (d/h)
P (d = 1/h)
..
.
N −1
P (d = N − 1, a = 1/h)
P (a = 1/h)
...
P (d = N − 1, a = N/h)
P (a = N/h)
P (d = N − 1/h)
1
Figure 34 illustrate the output associated to the model used in chapter 6.
66
CHAPTER 8.
ADVANCED TOOLS FOR OUTPUT
Figure 34: First four capture histories extracted from the file ’histories.tmp’.
8.2. NON-PARAMETRIC BOOTSTRAP
8.2
67
Non-parametric Bootstrap
e-surge allows you to do non-parametric bootstrap. Select the option RUN & See > Run Options >
Bootstrap (see Figure 35) and set the number of iterates (by default 1). A file named ’bootstrap.txt’ is
saved containing first the list of deviances and followed by a list of vectors of mathematical parameters.
To prevent for local minima, it is highly recommended to used the Multiple Random option (section 10.1) to fit the model at each iterate of the bootstrap .
Figure 35: Selecting the non-parametric bootstrap option
8.3
The Viterbi and the Counting algorithms
The Viterbi and the counting algorithms were developed by [44] in the context of capture-recapture for
estimating the lifetime reproductive success (LRS). The Viterbi algorithm reconstitutes the life of the
individual: The most probable underlying state sequence or more generally a set of state sequences such
that the cumulative probability reached a fixed threshold. From that sequences, it is easy to calculate
the LRS or any quantitative value of interest.
To do this, select the option Run > Compute reconstituted histories (viterbi). A menu appears (See
Figure 37), the sentence ’1:173’ selects all the histories 1 to 173, the number ’4’ asks for the 4 most
probable sequences for each history. All the details (by histories) are saved in the file ’viterbi.txt’.
Originally, the counting algorithm estimates the occurrences of the hidden states in the life of the
individual. This algorithm is faster than the Viterbi algorithm but as implemented by [44] gives less
informations. So, we generalized this algorithm to evaluate the number of transition between states.
To do this, select the option Run > Count transition numbers. A menu appears (See Figure 37),
the sentence ’1:173’ selects all the histories 1 to 173, the sentence ’2:4’ selects states of interest for the
LRS (here, state 2=’1 fawn’, state 3=’2 fawns’, state 4=’3 fawns’). A resume is given by e-surge.
Numberofindividuals =
212
# of occasions in a state: mean, se and ci
===========================================
68
CHAPTER 8.
mean =
0.9386
1.2949
1.5392
0.1184
0.1279
1.0615
1.2871
1.5282
1.7914
se =
0.0790
ci(:,:,1) =
0.7829
ci(:,:,2) =
1.0943
===========================================
# of transition between states: mean, se and ci
===========================================
ans =
0.1943
0.2215
0.2311
0.2071
0.4064
0.2559
0.2155
0.3165
0.7392
0.0335
0.0279
0.0287
0.0328
0.0683
0.0319
0.0301
0.0347
0.0842
ans =
ADVANCED TOOLS FOR OUTPUT
8.3. THE VITERBI AND THE COUNTING ALGORITHMS
69
ans =
0.1284
0.1664
0.1746
0.1425
0.2717
0.1931
0.1562
0.2480
0.5732
0.2603
0.2765
0.2877
0.2717
0.5411
0.3188
0.2748
0.3850
0.9051
ans =
===========================================
full details saved in file counting.txt
From this resume, an estimate of the LRS is given by the formula 0.9386 × 1 + 1.2949 × 2 + 1.5392 × 3 =
8.1460. All the details (by histories) are saved in the file ’counting.txt’.
Important notes:
• These two algorithms are availables only for fixed effect (not individual effect).
• More elaborate sentences can be used to select histories of interest. For example, the sentence
’find(his(:,1)>=2)’ selects all histories of the first cohort.
Figure 36: Menu for the Viterbi algorithm
70
CHAPTER 8.
ADVANCED TOOLS FOR OUTPUT
Figure 37: Menu for the Counting algorithm
8.4. TESTS FOR ENVIRONMENTAL COVARIATES IN PRESENCE OF A RANDOM EFFECT71
8.4
Tests for environmental covariates in presence of a random
effect
To test for environmental covariates in presence of a time random effect whitout fitting the random
effect, there is two options[26].
• using a permutation test,
• using a ANODEV F-test ( or its t-test version when a one side test is suitable )
In both cases, build the model with environmental covariates following the usual three steps gemaco,
gepat and ivfv.
To do the permutation test, select the option, RUN > Permutation test.
To do the standard test, select the option RUN > ANODEV test. In addition to the deviances of
constant and time-dependent models, e-surge asks for the Excel file containing the time-dependent
model with estimated variances. With this last option, e-surge compute an estimates of the residual
variance.
Important remark: This option is available only for the first step of transition which is in general
the step for survival.
9
Interpreting the output
In this chapter, we will go through the output file obtained at the end of chapter 6. Some of these
informations and others are also available in the corresponding Excel file.
9.1
CAS output file
The content of the cas output file is given in chapter 13 in detail. The lines are numbered from 1 to
499 (abbreviated L1 to L499 hereafter).
9.2
File heading
(L1-L9)
Lines 1 to 4 are where information about the version of e-surge used (L1), the name of the current
output file (L3), and the data file (L4) are given. Lines 5-9 give some basic and essential information
about the data, such as the number of occasions (L5), the number of states (L6), the number of events
(L7), the number of groups (L8) and the number of age classes (L9).
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CHAPTER 9. INTERPRETING THE OUTPUT
9.3
Information about the model
(L11-L21)
Lines 10 to 68 provide information about the fitted model. The name of the model is given on L11. Lines
25 to 68 give, for each step, the matrix pattern defined in gepat and the Model Definition Language
phrase used in gemaco to build the constrained matrix. The initial values of each beta value are given
in line 18 on the real axis (i.e., after logit or generalized logit transformation).
The indices of the beta values that were fixed are given in line 20 and the corresponding fixed values
on the scale ]0,1[ appear in line 21. Only results for free beta parameters are later given from L350 to
L423.
9.4
Minimization
(L23-L26 and L355-L426)
Lines 69 to 73 give the advanced numerical options used by the unconstrained nonlinear minimization
algorithm. e-surge uses a Quasi-Newton algorithm [14], an Expectation-Maximization algorithm (EM)
[42] or a hybrid algorithm (20 iterations of EM followed by a Quasi-Newton algorithm) when the
generalized logit is used as a link function. One of these non-linear solvers can be chosen by the user
in e-surge. If available for the model under consideration, the hybrid algorithm is recommended. A
constrained nonlinear algorithm is used with the identity link. On line 69 it is noted which link function
has been used (logitgen in our case). Line 70 mentions that the gradient of the deviance has been
calculated numerically (a centered finite difference scheme applied to the deviance is used to compute
the gradient). Otherwise, the gradient is computed analytically. In lines 71 and 72, tolerances tolf
and tolx used as criteria of convergence are given. Stopping criteria recommended by [14][p347] are
used.
The minimization process gives a local minimum which is not necessarily global [27]. Running the
same model with other different initial values is currently the main method to check if another, lower
minimum of the deviance can be achieved. Research is currently underway to help with this difficult
problem. See also sections 7.1.
e-surge computes the eigenvalues of the Hessian matrix (L428 to L499), based on the singular
value decomposition (svd). These eigenvalues indicate the redundancy and reliability of the parameter
estimates. If the eigenvalues are:
1) All strictly positive: e-surge has found a local minimum of the deviance function and provides
estimates and confidence intervals for all the estimators.
2) Some negative or positive but near zero: some parameters of the model cannot be identified. In
this case, e-surge decides how many eigenvalues can be considered as equal to zero according to a
threshold.
3) Strictly negative and far from zero: e-surge did not reach a minimum and the parameter estimates
are unreliable. A warning appears if any eigenvalue is lower than −10−6 ∗ λ1 (where λ1 is the
9.5. DEVIANCE, AIC AND RELATED TOPICS (L35-L44 AND L431- L502)
73
largest eigenvalue). In this case, the advice is to set smaller tolerances and re-run the same model.
Sometimes, it is not enough and the model has some difficulties to achieve convergence, i.e. estimates
are close to the mles but numerical difficulties slow down the convergence. Fixing some parameters
is another possibility and is very efficient: for example fixing the last capture probabilities to one
for the CAS model or fixing to zero capture probabilities at occasion when there is no capture.
9.5
Deviance, AIC and related topics
(L35-L44 and L431- L502)
First, the time needed to obtain the parameter estimates and the time needed to calculate the Hessian
are respectively given in L90 and L91, together with the number of iterations(L93). Line 75 and 76 give
the deviance and the Akaike information criterion amended for overdispersion (Qaic):
QAIC = dev/ĉ + 2 ∗ rank
L84 gives ĉ (c-hat, provided by the user according to the results of gof tests; default is 1. and L79 gives
an estimate of the rank of the model conditional on the data. By default in e-surge, the rank is the
maximum of numerical rank of derivatives matrix [43]) of the summary statistics calculated at several
neighbors (µ) of the mles (section 7.2). The algorithm to compute the rank is summarized below:
1. Choose a point µ near the mles.
2. Compute Dµ the derivatives matrix at µ.
3. Normalize Dµ by Gµ .
4. Compute U , V orthogonal matrices and E diagonal matrix such that U t Dµ Gµ V = E.
5. Estimate rank(µ) =
P
i (ei
≥ me1 ) where m is the number of columns of Dµ .
The rank can be also estimated less precisely by the numerical rank of the computed Hessian plus the
estimated number of boundary parameters, based on a threshold to decide which eigenvalues listed
in L428 to L499 can be considered as equal to zero, (as mentioned in the previous paragraph). This
threshold criterion λi ≥ ∗10−7 ∗ λ1 , where n is the size of the Hessian matrix (L78) and λ1 its largest
eigenvalue. Thus instead of the current value estimated by the numerical CMF method (section 7.2),
L79 may give the number of eigenvalues which satisfy this criterion plus the estimated number of
boundary parameters (L80). Following [49], another less severe threshold is applied in e-surge as
λi ≥ n ∗ 10−4 ∗ λ1 .
The difference between the results of the application of the two thresholds is given in L82. These
results can be used together with theoretical calculations for advanced investigations of redundancy
issue. In this example, we see from L428-L499 that the rank is at least 70, as all eigenvalues but two
(-0.00000801 0.00172081) are clearly larger than 0. However, two estimates are considered as being
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CHAPTER 9. INTERPRETING THE OUTPUT
on a boundary. [49] recommend considering such parameters as non-redundant. Furthermore, the
convergence was not achieved in this case, two eigenvalues are lower than zero. Hence the number of
non-redundant parameters is taken to be 72, which is a bad estimate for the rank.
9.6
Beta estimates
(L355-L426)
The Maximum Likelihood Estimates (mles) of the “mathematical parameters” (β̂ ) are given along with
their 95% confidence intervals and their standard errors. Fixed betas do not appear. The standard errors
(SEs) are derived from the matrix of variance-covariance computed as the inverse of the second order
derivative matrix of the likelihood or equivalently as the first order derivative matrix of the analytical
√
√
gradient of the likelihood. The 95 % confidence interval β − 1.96 SE ĉ, β + 1.96 SE ĉ relies on the
asymptotic Gaussian distribution of mles.
9.7
MLEs of parameters and standard errors
(L134-L348)
In lines 229 to 343, the mles of the ”biological parameters” θ̂ = X β̂, their confidence intervals and
standard errors are listed. To easily identify the parameters, their row number in the X matrix is given.
The concerned states, occasions, ages, groups and steps are also given in this order according to the
letters ”F To T A G S” (line 227) which refers to: From, To, Time, Age, Group, Step in order. Irrelevant
values are set to 0.
Example: ”S( 2, 2)( 4, 1)( 1 1)” means probability of survival(step 1), from state 2, at occasion 4 (i.e.,
between occasions 4 and 5), for age 1 and group 1 (there is only one age classes and one group).
Considering the number of biological parameter, users are helped by
• an Excel file copy,
• a summary of lines 229 to 343 given from lines 151 to 221.
The covariance matrix of X β̂ is obtained from that of β̂, Σ(β̂) as XΣ(β̂)X t = Σ(θ̂). Standard errors of
x = logit−1 (X β̂) are computed by the delta method. 95 % ci are obtained by back-transforming the
endpoints of 95% CI of X β̂. As a consequence, the confidence interval of the parameter xi is
j
√ k
√ logit−1 θ̂ − 1.96V (θ̂) ĉ , logit−1 θ̂ + 1.96V (θ̂) ĉ
i
i
For parameters obtained using the generalized-logit link function, the delta method is first applied
to θ = logit(logitgen−1 (X β̂)). Then, we proceed as above to obtain standard errors and confidence
75
intervals for x = logit−1 (θ).
10
A few warnings
The estimates and other results provided by e-surge are obtained via a complex numerical analysis
procedure. Users must be aware of various complications that may arise; to find improvements and
solutions is an active area of research.
10.1
Local minima
The optimization procedure used by e-surge gives a minimum of the deviance function and not necessarily the global minimum. Repeating minimization with different initial values is currently one of
the only practical solutions to this problem (see section 7.1). Another method is to use initial values
computed from the mles of a simpler model, e.g., to use mles of a time-constant model as initial values
for the optimization of a time-dependent model. This approach is available in e-surge by first running
the simpler model, and then, selecting the start from last model option in the Advanced Numerical
Options before running a finer model. This allows the new model to start from the solution of the
previous one. Our preliminary investigations with this approach yielded promising results when the
simpler model was well chosen.
10.2
Saddle point
Estimates at a saddle point are always the results of a bad convergence and/or a difficult problem. To
avoid it, several solutions may be advocated:
1. If the convergence is not attained, try to help the convergence by fixing appropriate parameters.
Some capture rates which are known to be zero may be fixed to zero. Some parameters involved
in the redundancy like the last capture rates for the cjs model may be fixed to one.
2. If the convergence is attained, reduce tolerances.
10.3
Additive models
Because of the redundancy inherent in categorical variables, the sentence t+g is reduced to t+g(2 N G).
The first column of g is automatically deleted as the sum of t and the sum of g are both equal to the
intercept. For the sentence g + t, the first column of t is deleted. However, the two formulations are
equivalent to the model t + g because the resulting X matrices will generate the same linear subspaces,
i.e., in turn, lead to the same final parameter estimates and minimal deviance value .
76
CHAPTER 12. CONDITIONS OF EXTERNAL USE
10.4
Generalized logit
Additive effects with the generalized logit do not generate the usual parallel responses. Pending further
investigations, we recommend not using additive effects with the generalized logit. However, for the
combined survival-transition formulation with one state and the separate survival-transition formulation
with two states, the logit and generalized logit coincide.
11
Acknowledgements
We warmly thank Lauriane Rouan, Christine Hunter, Stéphanie Jenouvrier, Cédric Juillet, JeanDominique Lebreton, Olivier Gimenez and members of the Biostatistics and Population Biology team
for their useful comments (and for some to be patient as being first users of e-surge) and to Hal Caswell
for his participation to the first part of the manual. This research was supported by a grant from the
’Jeunes Chercheuses et Jeunes Chercheurs’ program of the French ANR (ANR–08–JCJC–0028–01).
12
Conditions of external use
Program e-surge is property of those who wrote the software. Conditions for its use are the usual
ones:
• The software will be used for an academic or research purpose only. In particular, it will not be
used for commercial applications.
• Due acknowledgement will be made for the use of e-surge program in research reports or publications, mentioning the program as well as the publication related to the program (our preferred
option for obvious reasons) or the manual.
• The user recognizes to be aware that the software is a research product and is provided without
any expressed or implied warranty. There is no warranty of any kind concerning the fitness of the
software for any particular purpose.
77
13
Output text file
1 E-SURGE V 1.7.1,23-Sep-2010
2
3 OUTPUT FILE : C:\Erika\workshop\TUTORIALS\Ex3 - geese\CAS.out
4 Data File :C:\Erika\workshop\TUTORIALS\Ex3 - geese\Geese.rh
5 Number of occasions : 6
6 Number of states : 4
7 Number of events : 4
8 Number of groups : 1
9 Number of age classes : 1
10 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
11 Model Name : Model1
12 Model formula (or file) :
13 For Initial State:IS - Step 1 - (12): current.t
14 For Transition:M - Step 1 - (15): f.t
15 For Transition:F - Step 2 - (15): f.t
16 For Transition:T - Step 3 - (15): f.to.t
17 For Event:E - Step 1 - (16): firste+nexte.current.t
18 Init(values) -0.000000,-0.000000,-0.000000,-0.000000 ... 0.000000,
19 Init(indices) 1, 2, 3, 4, 5, 6, ... 72, 73,
20 Fix(values) 1.000000,
21 Fix(indices) 58,
22 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
23 Full Model Details
24 -------------------25 # of step for initial state :1
26 Phrase for step 1 : current.t
27 Number of shortcuts : 0
28 Pattern matrix :
29 p p *
30 Name file for covariates : defaultfile
31
32 # of step for transition : 3
33 Phrase for step 1 : f.t
34 Number of shortcuts : 0
35 Pattern matrix :
78
CHAPTER 13. OUTPUT TEXT FILE
36 y - - *
37 - y - *
38 - - y *
39 - - - *
40 Phrase for step 2 : f.t
41 Number of shortcuts : 0
42 Pattern matrix :
43 z * - - - - 44 - - z * - - 45 - - - - z * 46 - - - - - - *
47 Phrase for step 3 : f.to.t
48 Number of shortcuts : 0
49 Pattern matrix :
50 * - - 51 - d * 52 -* - 53 d - * 54 - - * 55 d * - 56 - - - *
57 Name file for covariates : defaultfile
58
59 # of step for encounter : 1
60 Phrase for step 1 : firste+nexte.current.t
61 Number of shortcuts : 0
62 Pattern matrix :
63 * b - 64 * - b 65 * - - b
66 * - - 67 Name file for covariates : defaultfile
68 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
69 Link function : logitgen
70 Explicit gradient : off
71 TOLF : 0.0000001
72 TOLX : 0.0000001
79
73 Order for the Gauss-Hermite interpolation : 15.000000
74 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
75 dev(PI)+dev(PHI,B) = 116414.612
76 QAIC = 116552.612
77
78 Number of mathematical parameters : 72.000000
79 Estimated model rank applicable to the data : 69.000000
80 Estimated number of boundary parameters : 2.000000
81
82 WARNING : there might be 0 further non identifiable parameters
83
84 c hat used for the QAIC : 1.000000
85 Model type = 0(Markovian), >0(Unavailable) : 0.000000
86 Conditionality = 0(1st capture), 1(1st occasion), 2(Closed) : 0.000000
87 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
88 Method for the Gauss-Hermite interpolation : unknown
89 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
90 Time for optimisation : 1.215743e+001 seconds
91 Time for Hessian : 2.745516e+001 seconds
92
93 Number of iterations for optimisation : 84
94 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
95 Informations about parameter identifiability :
96 --------------------------------------------------------97 69 singular values bigger than :9.2387e-006
98 May be0 more parameters are non-estimables
99 --------------------------------------------------------100 9 quantities solutions of3 partial derivatives equations,
101 made of redundant parameters (indices below) are estimables
102 25 26 27 40 41 42 55 56 57 70 71 72
103 25 26 27 40 41 42 55 56 57 70 71 72
104 25 26 27 40 41 42 55 56 57 70 71 72
......
......
137 --------------------------------------------------------138 11 quantities solutions of3 partial derivatives equations,
139 made of redundant parameters (indices below) are estimables
80
CHAPTER 13. OUTPUT TEXT FILE
140 25 26 27 40 41 42 45 48 55 56 57 70 71 72
141 25 26 27 40 41 42 45 48 55 56 57 70 71 72
142 25 26 27 40 41 42 45 48 55 56 57 70 71 72
143
144 Maximum Likelihood Estimates
145 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
146 Reduced set of parameters
147 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
148 Index Estimates | Lower & Upper 95 percent CI | S.E.
149 F To T A G S
150 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
151 Par# 1# IS( 1, 1)( 1, 1)( 1 1) | 0.224670873 0.211123372 0.238824520 0.007066957
152 Par# 2# IS( 1, 2)( 1, 1)( 1 1) | 0.597023446 0.580654624 0.613178976 0.008299832
......
......
160 Par# 16# IS( 1, 1)( 6, 1)( 1 1) | 0.202304724 0.175603208 0.231924148 0.014365353
161 Par# 17# IS( 1, 2)( 6, 1)( 1 1) | 0.450704260 0.416098003 0.485794108 0.017808202
162 Par# 19# M( 1, 1)( 1, 1)( 1 1) | 0.631613156 0.579582626 0.680751334 0.025889372
163 Par# 20# M( 2, 2)( 1, 1)( 1 1) | 0.742891360 0.703481678 0.778710814 0.019209356
......
......
175 Par# 48# M( 2, 2)( 5, 1)( 1 1) | 0.401722974 0.375928707 0.428073091 0.013313691
176 Par# 49# M( 3, 3)( 5, 1)( 1 1) | 0.623398935 0.569533095 0.674378087 0.026837606
177 Par# 54# F( 1, 1)( 1, 1)( 1 2) | 0.778638230 0.720419147 0.827634547 0.027367239
178 Par# 56# F( 2, 3)( 1, 1)( 1 2) | 0.901196378 0.876701846 0.921261837 0.011313145
......
......
190 Par# 84# F( 2, 3)( 5, 1)( 1 2) | 0.819115054 0.792772322 0.842773232 0.012748695
191 Par# 86# F( 3, 5)( 5, 1)( 1 2) | 0.735238092 0.691715302 0.774618492 0.021175829
192 Par# 90# T( 4, 1)( 1, 1)( 1 3) | 0.887027523 0.772978810 0.947661013 0.042718142
193 Par# 91# T( 6, 1)( 1, 1)( 1 3) | 0.263997269 0.178922130 0.371234867 0.049405648
......
......
204 Par# 131# T( 6, 1)( 5, 1)( 1 3) | 0.122292640 0.083089655 0.176433064 0.023559032
205 Par# 132# T( 2, 2)( 5, 1)( 1 3) | 0.974046985 0.909740540 0.992895327 0.016956592
206 Par# 143# E( 1, 2)( 1, 1)( 1 1) | 1.000000000 1.000000000 1.000000000 0.000000000
207 Par# 185# E( 1, 2)( 2, 2)( 1 1) | 0.618845204 0.548023405 0.684949372 0.035136109
81
......
......
220 Par# 214# E( 2, 3)( 6, 2)( 1 1) | 0.707113893 0.670609870 0.741134050 0.018012035
221 Par# 215# E( 3, 4)( 6, 2)( 1 1) | 0.385297408 0.339917748 0.432763412 0.023749312
222
223 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
224 Parameters
225 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
226 Index Estimates | Lower & Upper 95 percent CI | S.E.
227 F To T A G S
228 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
229 Par# 1# IS( 1, 1)( 1, 1)( 1, 1) | 0.224670873 0.211123372 0.238824520 0.007066957
230 Par# 2# IS( 1, 2)( 1, 1)( 1, 1) | 0.597023446 0.580654624 0.613178976 0.008299832
......
......
145 Par# 17# IS( 1, 2)( 6, 1)( 1, 1) | 0.450704260 0.416098003 0.485794108 0.017808202
146 Par# 18# IS( 1, 3)( 6, 1)( 1, 1) | 0.346991015 0.314411399 0.381070122 0.017026884
147 Par# 19# M( 1, 1)( 1, 1)( 1, 1) | 0.631613156 0.579582626 0.680751334 0.025889372
148 Par# 20# M( 2, 2)( 1, 1)( 1, 1) | 0.742891360 0.703481678 0.778710814 0.019209356
......
......
179 Par# 52# M( 3, 4)( 5, 1)( 1, 1) | 0.376601065 0.325621913 0.430466905 0.026837606
180 Par# 53# M( 4, 4)( 5, 1)( 1, 1) | 1.000000000 1.000000000 1.000000000 0.000000000
181 Par# 54# F( 1, 1)( 1, 1)( 1, 2) | 0.778638230 0.720419147 0.827634547 0.027367239
182 Par# 55# F( 1, 2)( 1, 1)( 1, 2) | 0.221361770 0.172365453 0.279580853 0.027367239
......
......
213 Par# 87# F( 3, 6)( 5, 1)( 1, 2) | 0.264761908 0.225381508 0.308284698 0.021175829
214 Par# 88# F( 4, 7)( 5, 1)( 1, 2) | 1.000000000 1.000000000 1.000000000 0.000000000
215 Par# 89# T( 1, 1)( 1, 1)( 1, 3) | 1.000000000 1.000000000 1.000000000 0.000000000
216 Par# 90# T( 4, 1)( 1, 1)( 1, 3) | 0.887027523 0.772978810 0.947661013 0.042718142
......
......
263 Par# 137# T( 5, 3)( 5, 1)( 1, 3) | 1.000000000 1.000000000 1.000000000 0.000000000
264 Par# 138# T( 7, 4)( 5, 1)( 1, 3) | 1.000000000 1.000000000 1.000000000 0.000000000
265 Par# 139# E( 1, 1)( 1, 1)( 1, 1) | 0.000000000 0.000000000 0.000000000 0.000000000
266 Par# 140# E( 2, 1)( 1, 1)( 1, 1) | 0.000000000 0.000000000 0.000000000 0.000000000
82
CHAPTER 13. OUTPUT TEXT FILE
......
......
342 Par# 214# E( 2, 3)( 6, 2)( 1, 1) | 0.707113893 0.670609870 0.741134050 0.018012035
343 Par# 215# E( 3, 4)( 6, 2)( 1, 1) | 0.385297408 0.339917748 0.432763412 0.023749312
344
345 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
346 Beta (Mathematical parameters)
347 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
348 Index Beta | Lower & Upper 95 percent CI | S.E.
349 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
350 Beta# 1# | +0.231137158 +0.125789960 +0.336484355 +0.053748570
351 Beta# 2# | +1.208456999 +1.118874734 +1.298039263 +0.045705237
......
......
360 Beta# 11# | -0.539523790 -0.735544318 -0.343503262 +0.100010473
361 Beta# 12# | +0.261512495 +0.103114152 +0.419910838 +0.080815481
362 #####################################################################
363 Beta# 13# | +0.539143521 +0.321060268 +0.757226773 +0.111266966
364 Beta# 14# | +1.061051094 +0.863932820 +1.258169368 +0.100570548
......
......
406 Beta# 56# | -1.970896378 -2.401089488 -1.540703268 +0.219486281
407 Beta# 57# | +3.625171738 +2.310471023 +4.939872454 +0.670765671
408 #####################################################################
409 Beta# 58# | +0.484649587 +0.192687597 +0.776611578 +0.148960199
410 Beta# 59# | -0.229668409 -0.367699756 -0.091637062 +0.070424157
......
......
422 Beta# 71# | +0.881407926 +0.710944699 +1.051871152 +0.086971034
423 Beta# 72# | -0.467123038 -0.663660781 -0.270585295 +0.100274359
424 #####################################################################
425 --------------------------426 Hessian eigenvalues
427 --------------------------428 -1.94745573
429 -1.70465435
......
83
......
498 2259.51584662
499 2296.98055316
84
BIBLIOGRAPHY
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Index
:, 31
principle, 30
;, 31
Tests, 71
[], 31
D
, 31
Data, 39
A
BIOMECO files, 39
Age model, 6
HEADED files, 40
AIC, 73
MARK files, 40
Arnason-Schwarz model
Decomposition in Elementary Steps, see DES,
combined, 16
15
separate, 18
DES, 5, 15
with site fidelity, 19
diagram, 19
Design matrix, see Constraint matrix
B
Biological parameters, 7
E
label, 50
E-SURGE
output, 74
organisation, 47
Bootstrap, 67
session, 48
Effects, see Keywords
C
combining effects, 27
Censoring
levels, 27
left, 14
lists, 28
Closed models, 13
main effects, 28
Closed population, 13
Encounter
Cluster, 42
decomposition, 15
Conditionality, 12
First, 6
Confidence intervals, 74
keywords firste & nexte, 28
by delta method, 74
Next, 6
Constrained models, 7
Expectation-Maximization (EM), 72
Constraint matrix, 7, 23
External covariates
Continuous function, 14
input file for, 44
continuous function, 36
Counting algorithm, 67
G
Covariates
GEMACO, 23
Covariate selection, 42
in practice, 54
file format, 44
General model, 4
individual, 11, 30
GEPAT, 15
89
90
INDEX
Generalized linear mixed model, 11
to,current, 23, 27, 28
GEPAT, 15, see Pattern matrix
x, 28
xind, 28, 30
biological parameters, 22
practice, 21
Global minimum, 63
L
Lifetime reproductive success, 67
Likelihood, 9
H
factorizing, 10
Hessian
MLE, 9
eigenvalues, 72
Link
I
function, 7, 8, 54
Initial state
generalized logit, 76
skipping, 54
identity, 8
Initial values
logit, 8
changing/fixing, 57
multinomial logit, 8
constant, 63
Lists, 31
from IVFV files, 63
Local minima, 75
from previous model, 63
initial values, 75
random, 63
scale, 57
M
Mathematical parameters, 7
Intermediate states, 19
aggregating, 36
IVFV
output, 74
input file for, 44
Model
additive, 75
K
information, 72
Keywords, 23, 24
retrieve, 61
age, 25, 28
cohort, 28
d,od,ud,ld, 28
N
Non-linear model, 14
firste,nexte, 28
non-linear model, 36
from, 23, 27, 28
group, 25, 28
O
ind, 28
Occupancy models, 14
intercept, 28
Operators
lists, 28
aggregation (&), 32
others, 33
covariates (*), 30
random, 11
dot product (.), 27
time, 24, 28
priority of, 32
INDEX
91
sum (+), 27
numerical CMF approach, 64
with xind, 30
of parameters, 64
Residual variance, 71
Output
area, 60
bootstrap, 67
S
Saddle point, 61, 75
Excel file, 61
how to manage ?, 72
model selection, 61
Senescence
redundancy, 62
age of, 14
saved files, 61
Shortcuts
tos window, 61
addressing levels, 33
variance-covariance matrix, 61
definition, 33
limitation, 33
P
Standard error, 74
Pattern matrix, 16
State dependent probabilities, 65
in practice, 50
input, 16, 21
U
redundancy(’*’), 34
Umbrella model, 4, 7
zero value(’-’), 34
Unequal time intervals
Permutation test, 71
application, 9
for real age, 14
Q
in practice, 44, 54
Quasi-Newton, 72
input file for, 44
R
limitation, 9
Hazard function, 14
Gompertz, 14
Mixture, 14
Siler, 14
Weibull, 14
hazard function, 36
Gompertz, 36
Mixture, 36
Siler, 36
Weibull, 36
Random Effects, 11
Redundancy
in transition matrix, 34
in X, 27
V
Viterbi algorithm, 67
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