Random Effects in AD Model Builder - available

Random Effects in AD Model Builder - available
Random Effects in
AD Model Builder
ADMB-RE User Guide
Version 11.4 (2015-05-22)
Revised manual (2015-01-07)
Hans Skaug & David Fournier
admb-project.org
ADMB Foundation, Honolulu.
This is the manual for AD Model Builder with Random Effects (ADMB-RE) version 11.4.
c 2004, 2006, 2008, 2009, 2011, 2013, 2014, 2015 Hans Skaug & David Fournier
Copyright The latest edition of the manual is available at:
http://admb-project.org/documentation/manuals
Contents
Contents
1 Introduction
1.1
Summary of features . . . . . . . . . .
1.1.1 Model formulation . . . . . . . .
1.1.2 Computational basis of admb-re
1.1.3 The strengths of admb-re . . . .
1.1.4 Program interface . . . . . . . . .
1.1.5 How to obtain admb-re . . . . .
ii
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1-1
1-2
1-2
1-3
1-3
1-3
1-3
2 The Language and the Program
2.1
What is ordinary admb? . . . . . . . . . . . . .
2.1.1 Writing an admb program . . . . . . . . .
2.1.2 Compiling an admb program . . . . . . .
2.1.3 Running an admb-program . . . . . . . .
2.1.4 admb-ide: easy and efficient user interface
2.1.5 Initial values . . . . . . . . . . . . . . . . .
2.2
Why random effects? . . . . . . . . . . . . . . .
2.2.1 Statistical prerequisites . . . . . . . . . . .
2.2.2 Frequentist or Bayesian statistics? . . . . .
2.2.3 A simple example . . . . . . . . . . . . . .
2.3
A code example . . . . . . . . . . . . . . . . . .
2.3.1 Parameter estimation . . . . . . . . . . . .
2.4
The flexibility of admb-re . . . . . . . . . . . .
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2-1
2-1
2-2
2-2
2-3
2-4
2-4
2-4
2-4
2-5
2-5
2-6
2-8
2-8
3 Random Effects Modeling
3.1
The objective function . . . . . . . . . . .
3.2
The random effects distribution (prior) . .
3.2.1 Scaling of random effects . . . . . . .
3.3
Correlated random effects . . . . . . . . .
3.3.1 Large structured covariance matrices
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3-1
3-1
3-2
3-2
3-3
3-4
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ii
3.4
Non-Gaussian random effects . . . . . . . . .
3.4.1 Can a, b, and c be estimated? . . . . .
3.5
Built-in data likelihoods . . . . . . . . . . .
3.6
Phases . . . . . . . . . . . . . . . . . . . . .
3.7
Penalized likelihood and empirical Bayes . .
3.8
Building a random effects model that works
3.9
mcmc . . . . . . . . . . . . . . . . . . . . .
3.10 Importance sampling . . . . . . . . . . . . .
3.11 reml (Restricted maximum likelihood) . . .
3.12 Improving performance . . . . . . . . . . . .
3.12.1 Reducing the size of temporary files . .
3.12.2 Exploiting special model structure . . .
3.12.3 Limited memory Newton optimization
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4 Exploiting special structure (Separability)
4.1
The first example . . . . . . . . . . . . . . . . . .
4.2
Nested or clustered random effects: block diagonal
4.2.1 Gauss-Hermite quadrature . . . . . . . . . .
4.3
State-space models: banded H . . . . . . . . . . .
4.4
Crossed random effects: sparse H . . . . . . . . .
4.5
Gaussian priors and quadratic penalties . . . . . .
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3-4
3-6
3-6
3-6
3-8
3-9
3-10
3-10
3-11
3-12
3-12
3-13
3-13
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4-1
4-2
4-4
4-5
4-6
4-7
4-7
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A-1
A-1
A-1
A-2
A-5
A-6
A-7
A-7
A-8
A-9
A-10
A-11
A-11
A-11
A-12
A-12
A Example Collection
A.1 Non-separable models . . . . . . . . . . . . . . . . . . .
A.1.1 Mixed-logistic regression: a Winbugs comparison
A.1.2 Generalized additive models (gams) . . . . . . .
A.1.3 Semi-parametric estimation of mean and variance
A.1.4 Weibull regression in survival analysis . . . . . . .
A.2 Block-diagonal Hessian . . . . . . . . . . . . . . . . . .
A.2.1 Nonlinear mixed models: an nlme comparison . .
A.2.2 Pharmacokinetics: an nlme comparison . . . . .
A.2.3 Frequency weighting in admb-re . . . . . . . . .
A.2.4 Ordinal-logistic regression . . . . . . . . . . . . .
A.3 Banded Hessian (state-space) . . . . . . . . . . . . . .
A.3.1 Stochastic volatility models in finance . . . . . . .
A.3.2 A discrete valued time series: the polio data set .
A.4 Generally sparse Hessian . . . . . . . . . . . . . . . . .
A.4.1 Multilevel Rasch model . . . . . . . . . . . . . . .
B Differences between admb and admb-re?
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B-1
iii
C Command Line options
C-1
D Quick References
D-1
D.1 Compiling admb programs . . . . . . . . . . . . . . . . . . . . . . . . . . . D-1
References
Index
References-1
Index-1
iv
Preface
A comment about notation:
Important points are emphasized with a star
F like this.
Please submit all comments and complaints by email to [email protected]
1
Chapter 1
Introduction
This document is a user’s guide to random-effects modelling in AD Model Builder (admb).
Random effects are a feature of admb, in the same way as profile likelihoods are, but
are sufficiently complex to merit a separate user manual. The work on the random-effects
“module” (admb-re) started around 2003. The pre-existing part of admb (and its evolution)
is referred to as “ordinary” admb in the following.
Before you start with random effects, it is recommended that you have some experience
with ordinary admb. This manual tries to be self-contained, but it is clearly an advantage
if you have written (and successfully run) a few tpl files. Ordinary admb is described in
the admb manual [4], which is available from admb-project.org. If you are new to admb,
but have experience with C++ (or a similar programming language), you may benefit from
taking a look at the quick references in Appendix D.
admb-re is very flexible. The term “random effect” seems to indicate that it only can
handle mixed-effect regression type models, but this is very misleading. “Latent variable”
would have been a more precise term. It can be argued that admb-re is the most flexible
latent variable framework around. All the mixed-model stuff in software packages, such
as allowed by R, Stata, spss, etc., allow only very specific models to be fit. Also, it is
impossible to change the distribution of the random effects if, say, you wanted to do that.
The nlmixed macro in sas is more flexible, but cannot handle state-space models or models
with crossed random effects. Winbugs is the only exception, and its ability to handle
discrete latent variables is a bit more flexible than is admb-re’s. However, Winbugs does
all its computations using mcmc exclusively, while admb-re lets the user choose between
maximum likelihood estimation (which, in general, is much faster) and mcmc.
An important part of the admb-re documentation is the example collection, which is
described in Appendix A. As with the example collections for admb and autodif, you
will find fully worked examples, including data and code for the model. The examples have
been selected to illustrate the various aspects of admb-re, and are frequently referred to
throughout this manual.
1-1
1.1
Summary of features
Why use AD Model Builder for creating nonlinear random-effects models? The answer
consists of three words: “flexibility,” “speed,” and “accuracy.” To illustrate these points. a
number of examples comparing admb-re with two existing packages: nlme, which runs on
R and S-Plus, and Winbugs. In general, nlme is rather fast and it is good for the problems
for which it was designed, but it is quite inflexible. What is needed is a tool with at least the
computational power of nlme yet the flexibility to deal with arbitrary nonlinear randomeffects models. In Section 2.4, we consider a thread from the R user list, where a discussion
took place about extending a model to use random effects with a log-normal, rather than
normal, distribution. This appeared to be quite difficult. With admb-re, this change takes
one line of code. Winbugs, on the other hand, is very flexible, and many random-effects
models can be easily formulated in it. However, it can be very slow. Furthermore, it is
necessary to adopt a Bayesian perspective, which may be a problem for some applications.
A model which runs 25 times faster under admb than under Winbugs may be found in
Section A.1.1.
1.1.1
Model formulation
With admb, you can formulate and fit a large class of nonlinear statistical models. With
admb-re, you can include random effects in your model. Examples of such models include:
• Generalized linear mixed models (logistic and Poisson regression).
• Nonlinear mixed models (growth curve models, pharmacokinetics).
• State space models (nonlinear Kalman filters).
• Frailty models in survival analysis.
• Nonparametric smoothing.
• Semiparametric modelling.
• Frailty models in survival analysis.
• Bayesian hierarchical models.
• General nonlinear random-effects models (fisheries catch-at-age models).
You formulate the likelihood function in a template file, using a language that resembles C++.
The file is compiled into an executable program (on Linux or Windows). The whole C++
language is to your disposal, giving you great flexibility with respect to model formulation.
1-2
1.1.2
Computational basis of admb-re
• Hyper-parameters (variance components, etc.) estimated by maximum likelihood.
• Marginal likelihood evaluated by the Laplace approximation, (adaptive) importance
sampling or Gauss-Hermite integration.
• Exact derivatives calculated using Automatic Differentiation.
• Sampling from the Bayesian posterior using mcmc (Metropolis-Hastings algorithm).
• Most of the features of ordinary admb (matrix arithmetic and standard errors, etc.)
are available.
• Sparse matrix libraries, useful for Markov random fields and crossed random effects,
are available.
1.1.3
The strengths of admb-re
• Flexibility: You can fit a large variety of models within a single framework.
• Convenience: Computational details are transparent. Your only responsibility is to
formulate the log-likelihood.
• Computational efficiency: admb-re is up to 50 times faster than Winbugs.
• Robustness: With exact derivatives, you can fit highly nonlinear models.
• Convergence diagnostic: The gradient of the likelihood function provides a clear convergence diagnostic.
1.1.4
Program interface
• Model formulation: You fill in a C++-based template using your favorite text editor.
• Compilation: You turn your model into an executable program using a C++ compiler
(which you need to install separately).
• Platforms: Windows, Linux and Mac.
1.1.5
How to obtain admb-re
admb-re is a module for admb. Both can be obtained from admb-project.org.
1-3
Chapter 2
The Language and the Program
2.1
What is ordinary admb?
admb is a software package for doing parameter estimation in nonlinear models. It combines
a flexible mathematical modelling language (built on C++) with a powerful function minimizer
(based on Automatic Differentiation). The following features of admb make it very useful
for building and fitting nonlinear models to data:
• Vector-matrix arithmetic and vectorized operations for common mathematical functions.
• Reading and writing vector and matrix objects to a file.
• Fitting the model is a stepwise manner (with “phases”), where more and more parameters become active in the minimization.
• Calculating standard deviations of arbitrary functions of the model parameters by the
“delta method.”
• mcmc sampling around the posterior mode.
To use random effects in admb, it is recommended that you have some experience in writing
ordinary admb programs. In this section, we review, for the benefit of the reader without
this experience, the basic constructs of admb.
Model fitting with admb has three stages: 1) model formulation, 2) compilation and 3)
program execution. The model fitting process is typically iterative: after having looked at
the output from stage 3, one goes back to stage 1 and modifies some aspect of the program.
2-1
2.1.1
Writing an admb program
To fit a statistical model to data, we must carry out certain fundamental tasks, such as
reading data from file, declaring the set of parameters that should be estimated, and finally
giving a mathematical description of the model. In admb, you do all of this by filling in
a template, which is an ordinary text file with the file-name extension .tpl (and hence
the template file is known as the tpl file). You therefore need a text editor—such as vi
under Linux, or Notepad under Windows—to write the tpl file. The first tpl file to which
the reader of the ordinary admb manual is exposed is simple.tpl (listed in Section 2.3
below). We shall use simple.tpl as our generic tpl file, and we shall see that introduction
of random effects only requires small changes to the program.
A tpl file is divided into a number of “sections,” each representing one of the fundamental
tasks mentioned above. See Table 2.1 for the required sections.
Name
Purpose
DATA_SECTION
Declare “global” data objects, initialization from file.
PARAMETER_SECTION
Declare independent parameters.
PROCEDURE_SECTION
Specify model and objective function in C++.
Table 2.1: Required sections.
More details are given when we later look at simple.tpl, and a quick reference card is
available in Appendix D.
2.1.2
Compiling an admb program
After having finished writing simple.tpl, we want to convert it into an executable program.
This is done in a DOS-window under Windows, and in an ordinary terminal window under
Linux. To compile simple.tpl, we would, under both platforms, give the command:
$ admb -r simple
Here, $ is the command line prompt (which may be a different symbol, or symbols, on your
computer), and -r is an option telling the program admb that your model contains random
effects. The program admb accepts another option, -s, which produces the “safe” (but slower)
version of the executable program. The -s option should be used in a debugging phase, but
it should be skipped when the final production version of the program is generated.
The compilation process really consists of two steps.
In the first step, simple.tpl is converted to a C++ program by a translator called tpl2rem
in the case of admb-re, and tpl2cpp in the case of ordinary admb (see Appendix D). An
error message from tpl2rem consists of a single line of text, with a reference to the line in
2-2
the tpl file where the error occurs. If successful, the first compilation step results in the
C++ file simple.cpp.
In the second step, simple.cpp is compiled and linked using an ordinary C++ compiler
(which is not part of admb). Error messages during this phase typically consist of long
printouts, with references to line numbers in simple.cpp. To track down syntax errors, it
may occasionally be useful to look at the content of simple.cpp. When you understand
what is wrong in simple.cpp, you should go back and correct simple.tpl and re-enter
the command admb -r simple. When all errors have been removed, the result will be an
executable file, which is called either simple.exe under Windows or simple under Linux.
The compilation process is illustrated in Section D.1.
2.1.3
Running an admb-program
The executable program is run in the same window as it was compiled. Note that data are
not usually part of the admb program (e.g., simple.tpl). Instead, data are being read from
a file with the file name extension .dat (e.g., simple.dat). This brings us to the naming
convention used by admb programs for input and output files: the executable automatically
infers file names by adding an extension to its own name. The most important files are listed
in Table 2.2. You can use command line options to modify the behavior of the program at
File name
Input
Contents
simple.dat Data for the analysis
simple.pin Initial parameter values
Output
simple.par Parameter estimates
simple.std Standard deviations
simple.cor Parameter correlations
Table 2.2: File naming convention.
runtime. The available command line options can be listed by typing:
$ simple -?
(or whatever your executable is called in place of simple). The command line options that
are specific to admb-re are listed in Appendix C, and are discussed in detail under the
various sections. An option you probably will like to use during an experimentation phase
is -est, which turns off calculation of standard deviations, and hence reduces the running
time of the program.
2-3
2.1.4
admb-ide: easy and efficient user interface
The graphical user interface to admb by Arni Magnusson simplifies the process of building
and running the model, especially for the beginner [1]. Among other things, it provides
syntax highlighting and links error messages from the C++ compiler to the .cpp file.
2.1.5
Initial values
The initial values can be provided in different ways (see the ordinary admb manual). Here,
we only describe the .pin file approach. The .pin file should contain legal values (within the
bounds) for all the parameters, including the random effects. The values must be given in
the same order as the parameters are defined in the .tpl file. The easiest way of generating
a .pin file with the right structure is to first run the program with a -maxfn 0 option (for
this, you do not need a .pin file), then copy the resulting .p01 file into .pin file, and then
edit it to provide the correct numeric values. More information about what initial values for
random effects really mean is given in Section 3.7.
2.2
Why random effects?
Many people are familiar with the method of least squares for parameter estimation. Far
fewer know about random effects modeling. The use of random effects requires that we
adopt a statistical point of view, where the sum of squares is interpreted as being part of a
likelihood function. When data are correlated, the method of least squares is sub-optimal,
or even biased. But relax—random effects come to rescue!
The classical motivation of random effects is:
• To create parsimonious and interpretable correlation structures.
• To account for additional variation or overdispersion.
We shall see, however, that random effects are useful in a much wider context, for instance,
in non-parametric smoothing.
2.2.1
Statistical prerequisites
To use random effects in admb, you must be familiar with the notion of a random variable
and, in particular, with the normal distribution. In case you are not, please consult a
standard textbook in statistics. The notation u ∼ N (µ, σ 2 ) is used throughout this manual,
and means that u has a normal (Gaussian) distribution with expectation µ and variance σ 2 .
The distribution placed on the random effects is called the “prior,” which is a term borrowed
from Bayesian statistics.
2-4
A central concept that originates from generalized linear models is that of a “linear predictor.” Let x1 , . . . , xp denote observed covariates (explanatory variables), and let β1 , . . . , βp
be the corresponding regression parameters to be estimated. Many of the examples in this
manual involve a linear predictor ηi = β1 x1,i + · · · + βp xp,i , which we also will write in vector
form as η = Xβ.
2.2.2
Frequentist or Bayesian statistics?
A pragmatic definition of a “frequentist” is a person who prefers to estimate parameters by
the method of maximum likelihood. Similarly, a “Bayesian” is a person who uses mcmc
techniques to generate samples from the posterior distribution (typically with noninformative priors on hyper-parameters), and from these samples generates some summary statistic,
such as the posterior mean. With its -mcmc runtime option, admb lets you switch freely
between the two worlds. The approaches complement each other rather than being competitors. A maximum likelihood fit (point estimate + covariance matrix) is a step-1 analysis.
For some purposes, step-1 analysis is sufficient. In other situations, one may want to see
posterior distributions for the parameters. In such situations, the established covariance
matrix (inverse Hessian of the log-likelihood) is used by admb to implement an efficient
Metropolis-Hastings algorithm (which you invoke with -mcmc).
2.2.3
A simple example
We use the simple.tpl example from the ordinary admb manual to exemplify the use of
random effects. The statistical model underlying this example is the simple linear regression
Yi = axi + b + εi ,
i = 1, . . . , n,
where Yi and xi are the data, a and b are the unknown parameters to be estimated, and
εi ∼ N (0, σ 2 ) is an error term.
Consider now the situation where we do not observe xi directly, but rather observe
Xi = xi + ei ,
where ei is a measurement error term. This situation frequently occurs in observational
studies, and is known as the “error-in-variables” problem. Assume further that ei ∼ N (0, σe2 ),
where σe2 is the measurement error variance. For reasons discussed below, we shall assume
that we know the value of σe , so we shall pretend that σe = 0.5.
Because xi is not observed, we model it as a random effect with xi ∼ N (µ, σx2 ). In
admb-re, you are allowed to make such definitions through the new parameter type called
random_effects_vector. (There is also a random_effects_matrix, which allows you to
define a matrix of random effects.)
2-5
1. Why do we call xi a “random effect,” while we do not use this term for Xi and Yi
(though they clearly are “random”)? The point is that Xi and Yi are observed directly,
while xi is not. The term “random effect” comes from regression analysis, where it
means a random regression coefficient. In a more general context. “latent random
variable” is probably a better term.
2. The unknown parameters in our model are: a, b, µ, σ, σx , and x1 , . . . , xn . We have
agreed to call x1 , . . . , xn “random effects.” The rest of the parameters are called “hyperparameters.” Note that we place no prior distribution on the hyper-parameters.
3. Random effects are integrated out of the likelihood, while hyper-parameters are estimated by maximum likelihood. This approach is often called “empirical Bayes,” and
will be considered a frequentist method by most people. There is however nothing
preventing you from making it “more Bayesian” by putting priors (penalties) on the
hyper-parameters.
4. A statistician will say, “This model is nothing but a bivariate Gaussian distribution
for (X, Y ), and we don’t need any random effects in this situation.” This is formally
true, because we could work out the covariance matrix of (X, Y ) by hand and fit
the model using ordinary admb. This program would probably run much faster, but
it would have taken us longer to write the code without declaring xi to be of type
random_effects_vector. However, more important is that random effects can be used
also in non-Gaussian (nonlinear) models where we are unable to derive an analytical
expression for the distribution of (X, Y ).
5. Why didn’t we try to estimate σe ? Well, let us count the parameters in the model:
a, b, µ, σ, σx , and σe . There are six parameters total. We know that the bivariate
Gaussian distribution has only five parameters (the means of X and Y and three free
parameters in the covariate matrix). Thus, our model is not identifiable if we also try
to estimate σe . Instead, we pretend that we have estimated σe from some external
data source. This example illustrates a general point in random effects modelling: you
must be careful to make sure that the model is identifiable!
2.3
A code example
Here is the random effects version of simple.tpl:
DATA_SECTION
init_int nobs
init_vector Y(1,nobs)
init_vector X(1,nobs)
2-6
PARAMETER_SECTION
init_number a
init_number b
init_number mu
vector pred_Y(1,nobs)
init_bounded_number sigma_Y(0.000001,10)
init_bounded_number sigma_x(0.000001,10)
random_effects_vector x(1,nobs)
objective_function_value f
PROCEDURE_SECTION
f = 0;
pred_Y=a*x+b;
// This section is pure C++
// Vectorized operations
// Prior part for random effects x
f += -nobs*log(sigma_x) - 0.5*norm2((x-mu)/sigma_x);
// Likelihood part
f += -nobs*log(sigma_Y) - 0.5*norm2((pred_Y-Y)/sigma_Y);
f += -0.5*norm2((X-x)/0.5);
f *= -1; // ADMB does minimization!
Comments
1. Everything following // is a comment.
2. In the DATA_SECTION, variables with a init_ in front of the data type are read from
file.
3. In the PARAMETER_SECTION
• Variables with a init_ in front of the data type are the hyper-parameters, i.e.,
the parameters to be estimated by maximum likelihood.
• random_effects_vector defines the random effect vector. (There is also a type
called random_effects_matrix.) There can be more than one such object, but
they must all be defined after the hyper-parameters are—otherwise, you will get
an error message from the translator tpl2rem.
• Objects that are neither hyper-parameters nor random effects are ordinary programming variables that can be used in the PROCEDURE_SECTION. For instance,
we can assign a value to the vector pred_Y.
2-7
• The objective function should be defined as the last variable.
4. The PROCEDURE_SECTION basically consists of standard C++ code, the primary purpose
of which is to calculate the value of the objective function.
• Variables defined in DATA_SECTION and PARAMETER_SECTION may be used.
• Standard C++ functions,
P 2 as well as special admb functions, such as norm2(x)
(which calculates
xi ), may be used.
• Often the operations are vectorized, as in the case of simple.tpl
• The objective function should be defined as the last variable.
• admb does minimization, rather than optimization. Thus, the sign of the loglikelihood function f is changed in the last line of the code.
2.3.1
Parameter estimation
We learned above that hyper-parameters are estimated by maximum likelihood, but what
if we also are interested in the value of the random effects? For this purpose, admb-re
offers an “empirical Bayes” approach, which involves fixing the hyper-parameters at their
maximum likelihood estimates, and treating the random effects as the parameters of the
model. admb-re automatically calculates “maximum posterior” estimates of the random
effects for you. Estimates of both hyper-parameters and random effects are written to
simple.par.
2.4
The flexibility of admb-re
Say that you doubt the distributional assumption xi ∼ N (µ, σx2 ) made in simple.tpl, and
that you want to check if a skewed distribution gives a better fit. You could, for instance,
take
xi = µ + σx exp(zi ),
zi ∼ N (0, 1).
Under this model, the standard deviation of xi is proportional to, but not directly equal
to, σx . It is easy to make this modification in simple.tpl. In the PARAMETER_SECTION, we
replace the declaration of x by
vector x(1,nobs)
random_effects_vector z(1,nobs)
and in the PROCEDURE_SECTION we replace the prior on x by
f = - 0.5*norm2(z);
x = mu + sigma_x*exp(z);
2-8
This example shows one of the strengths of admb-re: it is very easy to modify models.
In principle, you can implement any random effects model you can think of, but as we shall
discuss later, there are limits to the number of random effects you can declare.
2-9
Chapter 3
Random Effects Modeling
This chapter describes all admb-re features except those related to “separability,” which are
dealt with in Chapter 4. Separability, or the Markov property, as it is called in statistics, is
a property possessed by many model classes. It allows admb-re to generate more efficient
executable programs. However, most admb-re concepts and techniques are better learned
and understood without introducing separability. Throughout much of this chapter, we will
refer to the program simple.tpl from Section 2.3.
3.1
The objective function
As with ordinary admb, the user specifies an objective function in terms of data and parameters. However, in admb-re, the objective function must have the interpretation of being a
(negative) log-likelihood. One typically has a hierarchical specification of the model, where
at the top layer, data are assumed to have a certain probability distribution conditionally
on the random effects (and the hyper-parameters), and at the next level, the random effects
are assigned a prior distribution (typically, normal). Because conditional probabilities are
multiplied to yield the joint distribution of data and random effects, the objective function
becomes a sum of (negative) log-likelihood contributions, and the following rule applies:
F The order in which the different log-likelihood contributions are added to the objective
function does not matter.
An addition to this rule is that all programming variables have their values assigned before
they enter in a prior or a likelihood expression. Winbugs users must take care when porting
their programs to admb, because this is not required in Winbugs.
The reason why the negative log-likelihood is used is that for historical reasons, admb
does minimization (as opposed to maximization). In complex models, with contributions
to the log-likelihood coming from a variety of data sources and random effects priors, it
is recommended that you collect the contributions to the objective function using the -=
operator of C++, i.e.,
3-1
f -= -nobs*log(sigma_x) - 0.5*norm2((x-mu)/sigma_x);
By using -= instead of +=, you do not have to change the sign of every likelihood expression—
which would be a likely source of error. When none of the advanced features of Chapter 4 are
used, you are allowed to switch the sign of the objective function at the end of the program:
f *= -1; // ADMB does minimization!
so that in fact, f can hold the value of the log-likelihood until the last line of the program.
It is OK to ignore constant terms (0.5 log(2π), for the normal distribution) as we did in
simple.tpl. This only affects the objective function value, not any other quantity reported
in the .par and .std files (not even the gradient value).
3.2
The random effects distribution (prior)
In simple.tpl, we declared x1 , . . . , xn to be of type random_effects_vector. This statement tells admb that x1 , . . . , xn should be treated as random effects (i.e., be the targets
for the Laplace approximation), but it does not say anything about what distribution the
random effects should have. We assumed that xi ∼ N (µ, σx2 ), and (without saying it explicitly) that the xi s were statistically independent. We know that the corresponding prior
contribution to the log-likelihood is
−n log(σx ) −
1 X
(xi − µ)2
2σx2 i=1
with admb implementation
f += -nobs*log(sigma_x) - 0.5*norm2((x-mu)/sigma_x);
Both the assumption about independence and normality can be generalized—as we shortly
will do—but first we introduce a transformation technique that forms the basis for much of
what follows later.
3.2.1
Scaling of random effects
A frequent source of error when writing admb-re programs is that priors get wrongly specified. The following trick can make the code easier to read, and has the additional advantage
of being numerically stable for small values of σx . From basic probability theory, we know
that if u ∼ N (0, 1), then x = σx u + µ will have a N (µ, σx2 ) distribution. The corresponding
admb code would be
f += - 0.5*norm2(u);
x = sigma_x*u + mu;
3-2
(This, of course, requires that we change the type of x from random_effects_vector to
vector, and that u is declared as a random_effects_vector.)
The trick here was to start with a N (0, 1) distributed random effect u and to generate
random effects x with another distribution. This is a special case of a transformation.
Had we used a non-linear transformation, we would have gotten an x with a non-Gaussian
distribution. The way we obtain correlated random effects is also transformation based.
However, as we shall see in Chapter 4, transformation may “break” the separability of the
model, so there are limitations as to what transformations can do for you.
3.3
Correlated random effects
In some situations, you will need correlated random effects, and as part of your problem,
you may want to estimate the elements of the covariance matrix. A typical example is
mixed regression, where the intercept random effect (ui ) is correlated with the slope random
effect (vi ):
yij = (a + ui ) + (b + vi ) xij + εij .
(If you are not familiar with the notation, please consult an introductory book on mixed
regression, such as [9].) In this case, we can define the correlation matrix
1 ρ
C=
,
ρ 1
and we want to estimate ρ along with the variances of ui and vi . Here, it is trivial to ensure
that C is positive-definite, by requiring −1 < ρ < 1, but in higher dimensions, this issue
requires more careful consideration.
To ensure that C is positive-definite, you can parameterize the problem in terms of
the Cholesky factor L, i.e., C = LL0 , where L is a lower diagonal matrix with positive
diagonal elements. There are q(q − 1)/2) free parameters (the non-zero elements of L) to be
estimated, where q is the dimension of C. Since C is a correlation matrix, we must ensure
that its diagonal elements are unity. An example with q = 4 is
PARAMETER_SECTION
matrix L(1,4,1,4)
init_vector a(1,6)
init_bounded_vector B(1,4,0,10)
// Cholesky factor
// Free parameters in C
// Standard deviations
PROCEDURE_SECTION
int k=1;
L(1,1) = 1.0;
for(i=2;i<=4;i++)
3-3
{
L(i,i) = 1.0;
for(j=1;j<=i-1;j++)
L(i,j) = a(k++);
L(i)(1,i) /= norm(L(i)(1,i)); // Ensures that C(i,i) = 1
}
Given the Cholesky factor L, we can proceed in different directions. One option is to use the
same transformation-of-variable technique as above: Start out with a vector u of independent
N (0, 1) distributed random effects. Then, the vector
x = L*u;
has correlation matrix C = LL0 . Finally, we multiply each component of x by the appropriate
standard deviation:
y = elem_prod(x,sigma);
3.3.1
Large structured covariance matrices
In some situations, for instance, in spatial models, q will be large (q = 100, say). Then it is
better to use the approach outlined in Section 4.5.
3.4
Non-Gaussian random effects
Usually, the random effects will have a Gaussian distribution, but technically speaking, there
is nothing preventing you from replacing the normality assumption, such as
f -= -nobs*log(sigma_x) - 0.5*norm2((x-mu)/sigma_x);
with a log gamma density, say. It can, however, be expected that the Laplace approximation
will be less accurate when you move away from normal priors. Hence, you should instead use
the transformation trick that we learned earlier, but now with a non-linear transformation.
A simple example of this yielding a log-normal prior was given in Section 2.4.
Say you want x to have cumulative distribution function F (x). It is well known that
you achieve this by taking x = F −1 (Φ(u)), where Φ is the cumulative distribution function
of the N (0, 1) distribution. For a few common distributions, the composite transformation
F −1 (Φ(u)) has been coded up for you in admb-re, and all you have to do is:
1. Define a random effect u with a N (0, 1) distribution.
2. Transform u into a new random effect x using one of something_deviate functions
described below.
3-4
where something is the name of the distribution.
As an example, say we want to obtain a vector x of gamma distributed random effects
(probability density xa−1 exp(−x)/Γ(a)). We can then use the code:
PARAMETER_SECTION
init_number a
init_number lambda
vector x(1,n)
random_effects_vector u(1,n)
objective_function_value g
PROCEDURE_SECTION
g -= -0.5*norm2(u);
for (i=1;i<=n;i++)
x(i) = lambda*gamma_deviate(u(i),a);
// Shape parameter
// Scale parameter
// N(0,1) likelihood contr.
See a full example here.
Similarly, to obtain beta(a, b) distributed random effects, with density f (x) ∝ xa−1 (1 −
b−1
x) , we use:
PARAMETER_SECTION
init_number a
init_number b
PROCEDURE_SECTION
g -= -0.5*norm2(u);
for (i=1;i<=n;i++)
x(i) = beta_deviate(u(i),a,b);
// N(0,1) likelihood contr.
The function beta_deviate() has a fourth (optional) parameter that controls the accuracy
of the calculations. To learn more about this, you will have to dig into the source code.
You find the code for beta_deviate() in the file df1b2betdev.cpp. The mechanism for
specifying default parameter values are found in the source file df1b2fun.h.
A third example is provided by the “robust” normal distribution with probability density
1
1
2
2
f (x) = 0.95 √ e−0.5x + 0.05 √ e−0.5(x/c)
2π
c 2π
where c is a “robustness” parameter which by default is set to c = 3 in df1b2fun.h. Note that
this is a mixture distribution consisting of 95% N (0, 1) and 5% N (0, c2 ). The corresponding
admb-re code is
PARAMETER_SECTION
init_number sigma
number c
// Standard deviations (almost)
3-5
PROCEDURE_SECTION
g -= - 0.5*norm2(u); // N(0,1) likelihood contribution from u’s
for (i=1;i<=n;i++)
{
x(i) = sigma*robust_normal_mixture_deviate(u(i),c);
}
3.4.1
Can a, b, and c be estimated?
As indicated by the data types used above,
F a and b are among the parameters that are being estimated.
F c cannot be estimated.
It would, however, be possible to write a version of robust_normal_mixture_deviate where
also c and the mixing proportion (fixed at 0.95 here) can be estimated. For this, you need
to look into the file df1b2norlogmix.cpp. The list of distribution that can be used is likely
to be expanded in the future.
3.5
Built-in data likelihoods
In the simple simple.tpl, the mathematical expressions for all log-likelihood contributions
where written out in full detail. You may have hoped that for the most common probability
distributions, there were functions written so that you would not have to remember or look
up their log-likelihood expressions. If your density is among those given in Table 3.1, you
are lucky. More functions are likely to be implemented over time, and user contributions are
welcomed!
We stress that these functions should be only be used for data likelihoods, and in fact,
they will not compile if you try to let X be a random effect. So, for instance, if you have
observations xi that are Poisson distributed with expectation µi , you would write
for (i=1;i<=n;i++)
f -= log_density_poisson(x(i),mu(i));
Note that functions do not accept vector arguments.
3.6
Phases
A very useful feature of admb is that it allows the model to be fit in different phases. In
the first phase, you estimate only a subset of the parameters, with the remaining parameters
3-6
Density
Expression
µx
Poisson
Neg. binomial
Γ(x+1)
e−µ
µ = E(X), τ =
V ar(X)
E(X)
Parameters
Name
µ>0
log_density_poisson
µ, τ > 0
log_negbinomial_density
Table 3.1: Distributions that currently can be used as high-level data distributions (for
data X) in admb-re. The expression for the negative binomial distribution is omitted, due
to its somewhat complicated form. Instead, the parameterization, via the overdispersion
coefficient, is given. The interested reader can look at the actual implementation in the
source file df1b2negb.cpp
.
being fixed at their initial values. In the second phase, more parameters are turned on, and
so it goes. The phase in which a parameter becomes active is specified in the declaration of
the parameter. By default, a parameter has phase 1. A simple example would be:
PARAMETER_SECTION
init_number a(1)
random_effects_vector b(1,10,2)
where a becomes active in phase 1, while b is a vector of length 10 that becomes active in
phase 2. With random effects, we have the following rule-of-thumb (which may not always
apply):
Phase 1 Activate all parameters in the data likelihood, except those related to random
effects.
Phase 2 Activate random effects and their standard deviations.
Phase 3 Activate correlation parameters (of random effects).
In complicated models, it may be useful to break Phase 1 into several sub-phases.
During program development, it is often useful to be able to completely switch off a
parameter. A parameter is inactivated when given phase −1, as in
PARAMETER_SECTION
init_number c(-1)
The parameter is still part of the program, and its value will still be read from the pin file,
but it does not take part in the optimization (in any phase).
For further details about phases, please consult the section “Carrying out the minimization in a number of phases” in the admb manual [4].
3-7
3.7
Penalized likelihood and empirical Bayes
The main question we answer in this section is how are random effects estimated, i.e., how
are the values that enter the .par and .std files calculated? Along the way, we will learn a
little about how admb-re works internally.
By now, you should be familiar with the statistical interpretation of random effects.
Nevertheless, how are they treated internally in admb-re? Since random effects are not
observed data, then they have parameter status—but we distinguish them from hyperparameters. In the marginal likelihood function used internally by admb-re to estimate
hyper-parameters, the random effects are “integrated out.” The purpose of the integration
is to generate the marginal probability distribution for the observed quantities, which are X
and Y in simple.tpl. In that example, we could have found an analytical expression for
the marginal distribution of (X, Y ), because only normal distributions were involved. For
other distributions, such as the binomial, no simple expression for the marginal distribution
exists, and hence we must rely on admb to do the integration. In fact, the core of what
admb-re does for you is automatically calculate the marginal likelihood during its effort to
estimate the hyper-parameters.
The integration technique used by admb-re is the so-called Laplace approximation [11].
Somewhat simplified, the algorithm involves iterating between the following two steps:
1. The “penalized likelihood” step: maximizing the likelihood with respect to the random
effects, while holding the value of the hyper-parameters fixed. In simple.tpl, this
means doing the maximization w.r.t. x only.
2. Updating the value of the hyper-parameters, using the estimates of the random effects
obtained in item 1.
The reason for calling the objective function in step 1, a penalized likelihood, is that the
prior on the random effects acts as a penalty function.
We can now return to the role of the initial values specified for the random effects in
the .pin file. Unless you use the command line option -noinit these values are not used
(but dummy values must nevertheless be provided in the .pin file). Each time step 1 above
is performed the maximization of the penalized likelihood is always initiated at x = 0. If
the command line option -noinit is used the value from the .pin file is used the first time
step 1 above is performed. The remaining times the value from the previous optimum is
used as the starting value.
Empirical Bayes is commonly used to refer to Bayesian estimates of the random effects,
with the hyper-parameters fixed at their maximum likelihood estimates. admb-re uses
maximum aposteriori Bayesian estimates, as evaluated in step 1 above. Posterior expectation
is a more commonly used as Bayesian estimator, but it requires additional calculations, and
is currently not implemented in admb-re. For more details, see [11].
3-8
The classical criticism of empirical Bayes is that the uncertainty about the hyper-parameters
is ignored, and hence that the total uncertainty about the random effects is underestimated.
admb-re does, however, take this into account and uses the following formula:
−1
0
2
∂u
∂u
∂ log p(u |, data; θ)
cov(θ)
(3.1)
+
cov(u) = −
0
∂u∂u
∂θ
∂θ
where u is the vector of random effect, θ is the vector of hyper-parameters, and ∂u/∂θ is
the sensitivity of the penalized likelihood estimator on the value of θ. The first term on the
r.h.s. is the ordinary Fisher information based variance of u, while the second term accounts
for the uncertainty in θ.
3.8
Building a random effects model that works
In all nonlinear parameter estimation problems, there are two possible explanations when
your program does not produce meaningful results:
1. The underlying mathematical model is not well-defined, e.g., it may be over-parameterized.
2. You have implemented the model incorrectly, e.g., you have forgotten a minus sign
somewhere.
In an early phase of the code development, it may not be clear which of these is causing
the problem. With random effects, the two-step iteration scheme described above makes it
even more difficult to find the error. We therefore advise you always to check the program
on simulated data before you apply it to your real data set. This section gives you a recipe
for how to do this.
The first thing you should do after having finished the tpl file is to check that the
penalized likelihood step is working correctly. In admb, it is very easy to switch from a
random-effects version of the program to a penalized-likelihood version. In simple.tpl,
we would simply redefine the random effects vector x to be of type init_vector. The
parameters would then be a, b, µ, σ, σx , and x1 ,. . . , xn . It is not recommended, or even
possible, to estimate all of these simultaneously, so you should fix σx (by giving it a phase −1)
at some reasonable value. The actual value at which you fix σx is not critically important,
and you could even try a range of σx values. In larger models, there will be more than one
parameter that needs to be fixed. We recommend the following scheme:
1. Write a simulation program (in R, S-Plus, Matlab, or some other program) that generates data from the random effects model (using some reasonable values for the parameters) and writes to simple.dat.
2. Fit the penalized likelihood program with σx (or the equivalent parameters) fixed at
the value used to simulate data.
3-9
3. Compare the estimated parameters with the parameter values used to simulate data.
In particular, you should plot the estimated x1 , . . . , xn against the simulated random
effects. The plotted points should center around a straight line. If they do (to some degree of approximation), you most likely have got a correct formulation of the penalized
likelihood.
If your program passes this test, you are ready to test the random-effects version of the
program. You redefine x to be of type random_effects_vector, free up σx , and apply your
program again to the same simulated data set. If the program produces meaningful estimates
of the hyper-parameters, you most likely have implemented your model correctly, and you
are ready to move on to your real data!
With random effects, it often happens that the maximum likelihood estimate of a variance
component is zero (σx = 0). Parameters bouncing against the boundaries usually makes
one feel uncomfortable, but with random effects, the interpretation of σx = 0 is clear and
unproblematic. All it really means is that data do not support a random effect, and the
natural consequence is to remove (or inactivate) x1 , . . . , xn , together with the corresponding
prior (and hence σx ), from the model.
3.9
mcmc
There are two different mcmc methods built into admb-re: -mcmc and -mcmc2. Both
are based on the Metropolis-Hastings algorithm. The former generates a Markov chain on
the hyper-parameters only, while the latter generates a chain on the joint vector of hyperparameters and random effects. (Some sort of rejection sampling could be used with -mcmc
to generate values also for the random effects, but this is currently not implemented). The
advantages of -mcmc are:
• Because there typically is a small number of hyper-parameters, but a large number of
random effects, it is much easier to judge convergence of the chain generated by -mcmc
than that generated by -mcmc2.
• The -mcmc chain mixes faster than the -mcmc2 chain.
The disadvantage of the -mcmc option is that it is slow, because it relies on evaluation of
the marginal likelihood by the Laplace approximation. It is recommended that you run
both -mcmc and -mcmc2 (separately), to verify that they yield the same posterior for the
hyper-parameters.
3.10
Importance sampling
The Laplace approximation may be inaccurate in some situations. The accuracy may be
improved by adding an importance sampling step. This is done in admb-re by using the
3-10
command line argument -is N seed, where N is the sample size in the importance sampling
and seed (optional) is used to initialize the random number generator. Increasing N will give
better accuracy, at the cost of a longer run time. As a rule-of-thumb, you should start with
N = 100, and increase N stepwise by a factor of 2 until the parameter estimates stabilize.
By running the model with different seeds, you can check the Monte Carlo error in your
estimates, and possibly average across the different runs to decrease the Monte Carlo error.
Replacing the -is N seed option with an -isb N seed one gives you a “balanced” sample,
which in general, should reduce the Monte Carlo error.
For large values of N, the option -is N seed will require a lot of memory, and you will
see that huge temporary files are produced during the execution of the program. The option
-isf 5 will split the calculations relating to importance sampling into 5 (you can replace
the 5 with any number you like) batches. In combination with the techniques discussed in
Section 3.12.1, this should reduce the storage requirements. An example of a command line
is:
lessafre -isb 1000 9811 -isf 20 -cbs 50000000 -gbs 50000000
The -is option can also be used as a diagnostic tool for checking the accuracy of the
Laplace approximation. If you add the -isdiag (print importance sampling), the importance
sampling weights will be printed at the end of the optimization process. If these weights do
not vary much, the Laplace approximation is probably doing well. On the other hand, if a
single weight dominates the others by several orders of magnitude, you are in trouble, and
it is likely that even -is N with a large value of N is not going to help you out. In such
situations, reformulating the model, with the aim of making the log-likelihood closer to a
quadratic function in the random effects, is the way to go. See also the following section.
3.11
reml (Restricted maximum likelihood)
It is well known that maximum likelihood estimators of variance parameters can be downwards biased. The biases arise from estimation of one or more mean-related parameters.
The simplest example of a reml estimator is the ordinary sample variance
n
1 X
(xi − x̄)2 ,
s =
n − 1 i=1
2
where the divisor (n−1), rather than the n that occurs for the maximum likelihood estimator,
accounts for the fact that we have estimated a single mean parameter.
There are many ways of deriving the reml correction, but in the current context, the most
natural explanation is that we integrate the likelihood function (note: not the log-likelihood)
with respect to the mean parameters—β, say. This is achieved in admb-re by defining β
as being of type random_effects_vector, but without specifying a distribution/prior for
the parameters. It should be noted that the only thing that the random_effects_vector
3-11
statement tells admb-re is that the likelihood function should be integrated with respect to
β. In linear-Gaussian models, the Laplace approximation is exact, and hence this approach
yields exact reml estimates. In nonlinear models, the notion of reml is more difficult, but
reml-like corrections are still being used. For linear-Gaussian models, the reml likelihood
is available in closed form. Also, many linear models can be fitted with standard software
packages. It is typically much simpler to formulate a hierarchical model with explicit latent
variables. As mentioned, the Laplace approximation is exact for Gaussian models, so it does
not matter what way you do it.
An example of such a model is found here. To make the executable program run efficiently,
the command line options -nr 1 -sparse should be used for linear models. Also, note that
reml estimates can be obtained, as explained in Section 3.11.
3.12
Improving performance
In this section, we discuss certain mechanisms you can use to make an admb-re program
run faster and more smoothly.
3.12.1
Reducing the size of temporary files
When admb needs more temporary storage than is available in the allocated memory buffers,
it starts producing temporary files. Since writing to disk is much slower than accessing memory, it is important to reduce the size of temporary files as much as possible. There are several
parameters (such as arrmblsize) built into admb that regulate how large of memory buffers
an admb program allocates at startup. With random effects, the memory requirements increase dramatically, and admb-re deals with this by producing (when needed) six temporary
files. (See Table 3.2.): The table also shows the command line arguments you can use to
File name
Command line option
f1b2list1
-l1 N
f1b2list12
-l2 N
f1b2list13
nf1b2list1
-l3 N
-nl1 N
nf1b2list12
-nl2 N
nf1b2list13
-nl3 N
Table 3.2: Temporary file command line options.
manually set the size (determined by N) of the different memory buffers.
3-12
When you see any of these files starting to grow, you should kill your application and
restart it with the appropriate command line options. In addition to the options shown above,
there is -ndb N, which splits the computations into N chunks. This effectively reduces the
memory requirements by a factor of N —at the cost of a somewhat longer run time. N must
be a divisor of the total number of random effects in the model, so that it is possible to split
the job into N equally large parts. The -ndb option can be used in combination with the -l
and -nl options listed above. The following rule-of-thumb for setting N in -ndb N can be
used: if there are a total of m random effects in the model, one should choose N such that
m/N ≈ 50. For most of the models in the example collection (Chapter 3),this choice of N
prevents any temporary files being created.
Consider this model as an example. It contains only about 60 random effects, but does
rather heavy computations with these. As a consequence, large temporary files are generated.
The following command line
$ ./union -l1 10000000 -l2 100000000 -l3 10000000 -nl1 10000000
takes away the temporary files, but requires 80Mb of memory. The command line
$ ./union -est -ndb 5 -l1 10000000
also runs without temporary files, requires only 20Mb of memory, but runs three times
slower.
Finally, a warning about the use of these command line options. If you allocate too
much memory, your application will crash, and you will (should) get a meaningful error
message. You should monitor the memory use of your application (using “Task Manager”
under Windows, and the command top under Linux) to ensure that you do not exceed the
available memory on your computer.
3.12.2
Exploiting special model structure
If your model has special structure, such as grouped or nested random effects, state-space
structure, crossed random effects or a general Markov strucure you will benefit greatly from
using the techniques described in Section 4 below. In this case the memory options in
Table 3.2 are less relevant (although sometimes useful), and instead the memory use can be
controlled with the classical AD Model Builder command line options -cbs, -gbs etc.
3.12.3
Limited memory Newton optimization
The penalized likelihood step (Section 3.7), which forms a crucial part of the algorithm used
by admb to estimate hyper-parameters, is by default conducted using a quasi-Newton optimization algorithm. If the number of random effects is large—as it typically is for separable
models—it may be more efficient to use a “limited memory quasi-Newton” optimization algorithm. This is done using the command line argument -ilmn N, where N is the number of
steps to keep. Typically, N = 5 is a good choice.
3-13
Chapter 4
Exploiting special structure
(Separability)
A model is said to be “separable” if the likelihood can be written as a product of terms, each
involving only a small number of random effects. Not all models are separable, and for small
toy examples (less than 50 random effects, say), we do not need to care about separability.
You need to care about separability both to reduce memory requirements and computation
time. Examples of separable models are
• Grouped or nested random effects
• State-space models
• Crossed random effects
• Latent Markov random fields
The presence of separability allows admb to calculate the “Hessian” matrix very efficiently.The
Hessian H is defined as the (negative) Fisher information matrix (inverse covariance matrix)
of the posterior distribution of the random effects, and is a key component of the Laplace
approximation.
How do we inform admb-re that the model is separable? We define SEPARABLE_FUNCTIONs
in the PROCEDURE_SECTION to specify the individual terms in the product that defines the
likelihood function. Typically, a SEPARABLE_FUNCTION is invoked many times, with a small
subset of the random effects each time.
For separable models the Hessian is a sparse matrix which means that it contains mostly
zeros. Sparsity can be exploited by admb when manipulating the matrix H, such as calculating its determinant. The actual sparsity pattern depends on the model type:
• Grouped or nested random effects: H is block diagonal.
• State-space models: H is a banded matrix with a narrow band.
4-1
• Crossed random effects: unstructured sparsity pattern.
• Latent Markov random fields: often banded, but with a wide band.
For block diagonal and banded H, admb-re automatically will detect the structure from
the SEPARABLE_FUNCTION specification, and will print out a message such as:
Block diagonal Hessian (Block size = 3)
at the beginning of the phase when the random effects are becoming active parameters. For
general sparsity pattern the command line option -shess can be used to invoke the sparse
matrix libraries for manipulation of the matrix H.
4.1
The first example
A simple example is the one-way variance component model
yij = µ + σu ui + εij ,
i = 1, . . . , q,
j = 1, . . . , ni
where ui ∼ N (0, 1) is a random effect and εij ∼ N (0, σ 2 ) is an error term. The straightforward (non-separable) implementation of this model (shown only in part) is
PARAMETER_SECTION
random_effects_vector u(1,q)
PROCEDURE_SECTION
for(i=1;i<=q;i++)
{
g -= -0.5*square(u(i));
for(j=1;j<=n(i);j++)
g -= -log(sigma) - 0.5*square((y(i,j)-mu-sigma_u*u(i))/sigma);
}
The efficient (separable) implementation of this model is
PROCEDURE_SECTION
for(i=1;i<=q;i++)
g_cluster(i,u(i),mu,sigma,sigma_u);
SEPARABLE_FUNCTION void g_cluster(int i, const dvariable& u,...)
g -= -0.5*square(u);
for(int j=1;j<=n(i);j++)
g -= -log(sigma) - 0.5*square((y(i,j)-mu-sigma_u*u)/sigma);
4-2
where (due to lack of space in this document) we’ve replaced the rest of the argument list
with ellipsis (...).
It is the function call g_cluster(i,u(i),mu,sigma,sigma_u) that enables admb-re to
identify that the posterior distribution factors (over i):
(n
)
q
i
Y
Y
p(u | y) ∝
p ui | yij
i=1
j=1
and hence that the Hessian is block diagonal (with block size 1). Knowing that the Hessian is
block diagonal enables admb-re to do a series of univariate Laplace approximations, rather
than a single Laplace approximation in full dimension q. It should then be possible to fit
models where q is in the order of thousands, but this clearly depends on the complexity of
the function g_cluster.
The following rules apply:
F The argument list in the definition of the SEPARABLE_FUNCTION should not be broken
into several lines of text in the tpl file. This is often tempting, as the line typically
gets long, but it results in an error message from tpl2rem.
F Objects defined in the PARAMETER_SECTION must be passed as arguments to g_cluster.
There is one exception: the objective function g is a global object, and does not need to
be an argument. Temporary/programming variables should be defined locally within
the SEPARABLE_FUNCTION.
F Objects defined in the DATA_SECTION should not be passed as arguments to g_cluster,
as they are also global objects.
The data types that currently can be passed as arguments to a SEPARABLE_FUNCTION are:
int
const dvariable&
const dvar_vector&
const dvar_matrix&
with an example being:
SEPARABLE_FUNCTION void f(int i, const dvariable& a, const dvar_vector& beta)
The qualifier const is required for the latter two data types, and signals to the C++ compiler
that the value of the variable is not going to be changed by the function. You may also come
across the type const prevariable&, which has the same meaning as const dvariable&.
There are other rules that have to be obeyed:
F No calculations of the log-likelihood, except calling SEPARABLE_FUNCTION, are allowed
in PROCEDURE_SECTION. Hence, the only allowed use of parameters defined in PARAMETER_SECTION
is to pass them as arguments to SEPARABLE_FUNCTIONs. However, evaluation of sdreport
numbers during the sd_phase, as well as MCMC calculations, are allowed.
4-3
This rule implies that all the action has to take place inside the SEPARABLE_FUNCTIONs.
To minimize the number of parameters that have be passed as arguments, the following
programming practice is recommended when using SEPARABLE_FUNCTIONs:
F The PARAMETER_SECTION should contain definitions only of the independent parameters
(those variables whose type has a init_ prefix) and random effects, i.e., no temporary
programming variables.
All temporary variables should be defined locally in the SEPARABLE_FUNCTION, as shown
here:
SEPARABLE_FUNCTION void prior(const dvariable& log_s, const dvariable& u)
dvariable sigma_u = exp(log_s);
g -= -log_s - 0.5*square(u(i)/sigma_u);
See a full example here. The orange model has block size 1.
4.2
Nested or clustered random effects:
block diagonal H
In the above model, there was no hierarchical structure among the latent random variables
(the us). A more complicated example is provided by the following model:
yijk = σv vi + σu uij + εijk ,
i = 1, . . . , q,
j = 1, . . . , m,
k = 1, . . . , nij ,
where the random effects vi and uij are independent N (0, 1) distributed, and εijk ∼ N (0, σ 2 )
is still the error term. One often says that the us are nested within the vs.
Another perspective is that the data can be split into independent clusters. For i1 6= i2 ,
yi1 jk and yi2 jk are statistically independent, so that the likelihood factors are at the outer
nesting level (i).
To exploit this, we use the SEPARABLE_FUNCTION as follows:
PARAMETER_SECTION
random_effects_vector v(1,q)
random_effects_matrix u(1,q,1,m)
PROCEDURE_SECTION
for(i=1;i<=q;i++)
g_cluster(v(i),u(i),sigma,sigma_u,sigma_v,i);
Each element of v and each row (u(i)) of the matrix u are passed only once to the separable
function g_cluster. This is the criterion admb uses to detect the block diagonal Hessian
structure. Note that v(i) is passed as a single value while u(i) is passed as a vector to the
SEPARABLE_FUNCTION as follows:
4-4
SEPARABLE_FUNCTION void g_cluster(const dvariable& v,const dvar_vector& u,...)
g -= -0.5*square(v);
g -= -0.5*norm2(u);
for(int j=1;j<=m;j++)
for(int k=1;k<=n(i,j);k++)
g -= -log(sigma) - 0.5*square((y(i,j,k)
-sigma_v*v - sigma_u*u(j))/sigma);
F For a model to be detected as “Block diagonal Hessian,” each latent variable should be
passed exactly once as an argument to a SEPARABLE_FUNCTION.
To ensure that you have not broken this rule, you should look for an message like this at run
time:
Block diagonal Hessian (Block size = 3)
The “block size” is the number of random effects in each call to the SEPARABLE_FUNCTION,
which in this case is one v(i) and a vector u(i) of length two. It is possible that the groups
or clusters (as indexed by i, in this case) are of different size. Then, the “Block diagonal
Hessian” that is printed is an average.
The program could have improperly been structured as follows:
PARAMETER_SECTION
random_effects_vector v(1,q)
random_effects_matrix u(1,q,1,m)
PROCEDURE_SECTION
for(i=1;i<=q;i++)
for(j=1;j<=m;j++)
g_cluster(v(i),u(i,j),sigma,sigma_u,sigma_u,i);
but this would not be detected by admb-re as a clustered model (because v(i) is passed
multiple times), and hence admb-re will not be able to take advantage of block diagonal
Hessiand, as indicated by the absence of runtime message
Block diagonal Hessian (Block size = 3)
4.2.1
Gauss-Hermite quadrature
In the situation where the model is separable of type “Block diagonal Hessian,” (see Section 4), Gauss-Hermite quadrature is available as an option to improve upon the Laplace
approximation. It is invoked with the command line option -gh N, where N is the number
4-5
of quadrature points determining the accuracy of the integral approximation. For a block
size of 1, the default choice should be N = 10 or greater, but for larger block sizes the
computational and memory requirements very quickly limits the range of N . If N is chosen
too large admb-re will crash witout giving a meaningful error message. To avoid admb-re
creating large temporary files the command line options -cbs and -gbs can be used.
The -gh N option should be preferred over importance sampling (-is).
4.3
State-space models: banded H
A simple state space model is
yi = ui + i ,
ui = ρui−1 + ei ,
where ei ∼ N (0, σ 2 ) is an innovation term. The log-likelihood contribution coming from the
state vector (u1 , . . . , un ) is
n
X
(ui − ρui−1 )2
1
exp −
,
log √
2σ 2
2πσ
i=2
where (u1 , . . . , un ) is the state vector. To make admb-re exploit this special structure,
we write a SEPARABLE_FUNCTION named g_conditional, which implements the individual
terms in the above sum. This function would then be invoked as follows
for(i=2;i<=n;i++)
g_conditional(u(i),u(i-1),rho,sigma);
See a full example here.
Above, we have looked at a model with a univariate state vector. For multivariate state
vectors, as in
yi = ui + vi + i ,
ui = ρ1 ui−1 + ei ,
vi = ρ2 vi−1 + di ,
we would merge the u and v vectors into a single vector (u1 , v1 , u2 , v2 , . . . , un , vn ), and define
random_effects_vector u(1,m)
where m = 2n. The call to the SEPARABLE_FUNCTION would now look like
for(i=2;i<=n;i++)
g_conditional(u(2*(i-2)+1),u(2*(i-2)+2),u(2*(i-2)+3),u(2*(i-2)+4),...);
where the ellipsis (...) denotes the arguments ρ1 , ρ2 , σe , and σd .
4-6
4.4
Crossed random effects: sparse H
The simplest instance of a crossed random effects model is
yk = σu ui(k) + σv vj(k) + εk ,
i = 1, . . . n,
where u1 , . . . , un and v1 , . . . , vM are random effects, and where i(k) ∈ {1, N } and j(k) ∈
{1, M } are index maps. The ys sharing either a u or a v will be dependent, and in general,
no complete factoring of the likelihood will be possible. However, it is still important to
exploit the fact that the us and vs only enter the likelihood through pairs (ui(k) , vj(k) ). Here
is the code for the crossed model:
for (k=1;k<=n;k++)
log_lik(k,u(i(k)),v(j(k)),mu,s,s_u,s_v);
SEPARABLE_FUNCTION void log_lik(int k, const dvariable& u,...)
g -= -log(s) - 0.5*square((y(k)-(mu + s_u*u + s_v*v))/s);
If only a small proportion of all the possible combinations of ui and vj actually occurs in the
data, then the posterior covariance matrix of (u1 , . . . , un , v1 , . . . , vM ) will be sparse. When
an executable program produced by admb-re is invoked with the -shess command line
option, sparse matrix calculations are used.
This is useful not only for crossed models. Here are a few other applications:
• For the nested random effects model, as explained in Section 4.2.
• For reml estimation. Recall that reml estimates are obtained by making a fixed
effect random, but with no prior distribution. For the nested models in Section 4.2,
and the models with state-space structure of Section 4.3, when using reml, admb-re
will detect the cluster or time series structure of the likelihood. (This has to do with the
implementation of admb-re, not the model itself.) However, the posterior covariance
will still be sparse, and the use of -shess is advantageous.
4.5
Gaussian priors and quadratic penalties
In most models, the prior for the random effect will be Gaussian. In some situations, such
as in spatial statistics, all the individual components of the random effects vector will be
jointly correlated. admb contains a special feature (the normal_prior keyword) for dealing
efficiently with such models. The construct used to declaring a correlated Gaussian prior is
random_effects_vector u(1,n)
normal_prior S(u);
4-7
The first of these lines is an ordinary declaration of a random effects vector. The second
line tells admb that u has a multivariate Gaussian distribution with zero expectation and
covariance matrix S, i.e., the probability density of u is
h(u) = (2π)− dim(S)/2 det(S)−1/2 exp − 12 u0 S −1 u .
Here, S is allowed to depend on the hyper-parameters of the model. The part of the code
where S gets assigned its value must be placed in a SEPARABLE_FUNCTION.
F The log-prior log (h (u)) is automatically subtracted from the objective function. Therefore, the objective function must hold the negative log-likelihood when using the
normal_prior.
See a full example here.
4-8
Appendix A
Example Collection
This section contains various examples of how to use admb-re. Some of these has been
referred to earlier in the manual. The examples are grouped according to their “Hessian
type” (see Section 4). At the end of each example, you will find a Files section containing
links to webpages, where both program code and data can be downloaded.
A.1
Non-separable models
This section contains models that do not use any of the separability stuff. Sections A.1.2
and A.1.3 illustrate how to use splines as non-parametric components. This is currenlty a
very popular technique, and fits very nicely into the random effects framework [10]. All the
models except the first are, in fact, separable, but for illustrative purposes (the code becomes
easier to read), this has been ignored.
A.1.1
Mixed-logistic regression: a Winbugs comparison
Mixed regression models will usually have a block diagonal Hessian due to grouping/clustering of the data. The present model was deliberately chosen not to be separable, in order to
pose a computational challenge to both admb-re and Winbugs.
Model description
Let y = (y1 , . . . , yn ) be a vector of dichotomous observations (yi ∈ {0, 1}), and let u =
(u1 , . . . , uq ) be a vector of independent random effects, each with Gaussian distribution
(expectation 0 and variance σ 2 ). Define the success probability πi = Pr(yi = 1). The
following relationship between πi and explanatory variables (contained in matrices X and
Z) is assumed:
πi
= Xi β + Zi u,
log
1 − πi
A-1
where Xi and Zi are the ith rows of the known covariates matrices X (n × p) and Z (n × q),
respectively, and β is a p-vector of regression parameters. Thus, the vector of fixed-effects is
θ = (β, log σ).
Results
The goal here is to compare computation times with Winbugs on a simulated data set.
For this purpose, we use n = 200, p = 5, q = 30, and values of the the hyper-parameters,
as shown in the Table A.1. The matrices X and Z were generated randomly with each
element uniformly distributed on [−2, 2 ]. As start values for both AD Model Builder and
bugs, we used βinit,j = −1 and σinit = 4.5. In bugs, we used a uniform [−10, 10 ] prior on
βj and a standard (in the bugs literature) noninformative gamma prior on τ = σ −2 . In
AD Model Builder, the parameter bounds βj ∈ [−10, 10 ] and log σ ∈ [−5, 3 ] were used in
the optimization process. On the simulated data set, AD Model Builder used 27 seconds to
β1
β2
β3
β4
β5
σ
True values
0.0000
0.0000
0.0000
0.0000
0.0000
0.1000
admb-re
0.0300 −0.0700
0.0800
0.0800 −0.1100
0.1700
Std. dev.
0.1500
0.1500
0.1500
0.1400
0.1600
0.0500
Winbugs
0.0390 −0.0787
0.0773
0.0840 −0.1041
0.1862
Table A.1: True values
converge to the optimum of likelihood surface. On the same data set, we first ran Winbugs
for 5, 000 iterations. The recommended convergence diagnostic in Winbugs is the GelmanRubin plot (see the help files available from the menus in Winbugs), which require that
two Markov chains are run in parallel. From the Gelman-Rubin plot, it was clear that
convergence appeared after approximately 2, 000 iterations. The time taken by Winbugs to
generate the first 2, 000 was approximately 700 seconds.
See the files here.
A.1.2
Generalized additive models (gams)
Model description
A very useful generalization of the ordinary multiple regression
yi = µ + β1 x1,i + · · · + βp xp,i + εi ,
is the class of additive model
yi = µ + f1 (x1,i ) + · · · + fp (xp,i ) + εi .
(A.1)
A-2
Here, the fj are “nonparametric” components that can be modelled by penalized splines.
When this generalization is carried over to generalized linear models, and we arrive at the
class of gams [6]. From a computational perspective, penalized splines are equivalent to
random effects, and thus GAMs fall naturally into the domain of admb-re.
For each component fj in equation (A.1), we construct a design matrix X such that
fj (xi,j ) = X(i) u, where X(i) is the ith row of X and u is a coefficient vector. We use the
R function splineDesign (from the splines library) to construct a design matrix X. To
avoid overfitting, we add a first-order difference penalty [3]
X
− λ2
(uk − uk−1 )2 ,
(A.2)
k=2
to the ordinary glm log-likelihood, where λ is a smoothing parameter to be estimated. By
viewing u as a random effects vector with the above Gaussian prior, and by taking λ as a
hyper-parameter, it becomes clear that GAM’s are naturally handled in admb-re.
Implementation details
• A computationally more efficient implementation is obtained by moving λ from the
penalty term to the design matrix, i.e., fj (xi,j ) = λ−1 X(i) u.
• P
Since equation (A.2) does not penalize the mean of u, we impose the restriction that
k=1 uk = 0 (see union.tpl for details). Without this restriction, the model would
be over-parameterized, since we already have an overall mean µ in equation (A.1).
• To speed up computations, the parameter µ (and other regression parameters) should
be given “phase 1” in admb, while the λs and the us should be given “phase 2.”
The Wage-union data
The data, which are available from Statlib, contain information for each of 534 workers about
whether they are members (yi = 1) of a workers union or are not (yi = 0). We study the
probability of membership as a function of six covariates. Expressed in the notation used by
the R (S-Plus) function gam, the model is:
union ~ race + sex + south + s(wage) + s(age) + s(ed), family=binomial
Here, s() denotes a spline functions with 20 knots each. For wage, a cubic spline is
used, while for age and ed, quadratic splines are used. The total number of random effects
that arise from the three corresponding u vectors is 64. Figure A.1 shows the estimated
nonparametric components of the model. The time taken to fit the model was 165 seconds.
A-3
0.4
0.3
0.0
0.1
0.2
P(union)
0.3
0.2
0.0
0.1
P(union)
0.4
0.5
df = 9
0.5
df = 8
0
10
20
30
40
wages
20
30
40
50
60
age
0.3
0.2
0.0
0.1
P(union)
0.4
0.5
df = 9
5
10
15
education
Figure A.1: Probability of membership as a function of covariates. In each plot, the remaining covariates are fixed at their sample means. The effective degrees of freedom (df) are also
given [6].
A-4
Extensions
• The linear predictor may be a mix of ordinary regression terms (fj (x) = βj x) and
nonparametric terms. admb-re offers a unified approach to fitting such models, in
which the smoothing parameters λj and the regression parameters βj are estimated
simultaneously.
• It is straightforward in admb-re to add “ordinary” random effects to the model, for
instance, to accommodate for correlation within groups of observations, as in [8].
See the files here.
A.1.3
Semi-parametric estimation of mean and variance
Model description
An assumption underlying the ordinary regression
yi = a + bxi + ε0i
is that all observations have the same variance, i.e., Var(ε0i ) = σ 2 . This assumption does
not always hold, e.g., for the data shown in the upper panel of Figure A.2. This example is
taken from [10].
It is clear that the variance increases to the right (for large values of x). It is also clear
that the mean of y is not a linear function of x. We thus fit the model
yi = f (xi ) + σ(xi )εi ,
where εi ∼ N (0, 1), and f (x) and σ(x) are modelled nonparametrically. We take f to be a
penalized spline. To ensure that σ(x) > 0, we model log [σ(x)], rather than σ(x), as a spline
function. For f , we use a cubic spline (20 knots) with a second-order difference penalty
−λ2
20
X
(uj − 2uj−1 + uj−2 )2 ,
k=3
while we take log [σ(x)] to be a linear spline (20 knots) with the first-order difference penalty
(see equation A.2).
Implementation details
Details on how to implement spline components are given in Example A.1.2.
• Parameters associated with f should be given “phase 1” in admb, while those associated
with σ should be given “phase 2.” The reason is that in order to estimate the variation,
one first needs to have fitted the mean part.
A-5
• In order to estimate the variation function, one first needs to have fitted the mean
part. Parameters associated with f should thus be given “phase 1” in admb, while
those associated with σ should be given “phase 2.”
−0.4
−0.8
y
0.0
LIDAR data
400
450
500
550
600
650
700
600
650
700
x
0.00 0.05 0.10 0.15
sigma
sigma(x)
400
450
500
550
x
Figure A.2: lidar data (upper panel) used by [10], with fitted mean. Fitted standard
deviation is shown in the lower panel.
See the files here.
A.1.4
Weibull regression in survival analysis
Model description
A typical setting in survival analysis is that we observe the time point t at which the death
of a patient occurs. Patients may leave the study (for some reason) before they die. In this
case, the survival time is said to be “censored,” and t refers to the time point when the patient
left the study. The indicator variable δ is used to indicate whether t refers to the death of the
patient (δ = 1) or to a censoring event (δ = 0). The key quantity in modelling the probability
distribution of t is the hazard function h(t), which measures the
R t instantaneous death rate at
time t. We also define the cumulative hazard function Λ(t) = 0 h(s) ds, implicitly assuming
A-6
that the study started at time t = 0. The log-likelihood contribution from our patient is
δ log(h(t)) − H(t). A commonly used model for h(t) is Cox’s proportional hazard model, in
which the hazard rate for the ith patient is assumed to be on the form
hi t = h0 (t) exp(ηi ),
i = 1, . . . n.
Here, h0 (t) is the “baseline” hazard function (common to all patients) and ηi = Xi β, where
Xi is a covariate vector specific to the ith patient and β is a vector of regression parameters.
In this example, we shall assume that the baseline hazard belongs to the Weibull family:
h0 (t) = rtr−1 for r > 0.
In the collection of examples following the distribution of Winbugs, this model is used
to analyse a data set on times to kidney infection for a set of n = 38 patients (see Kidney:
Weibull regression with random effects in the Examples list at The Bugs Project). The
data set contains two observations per patient (the time to first and second recurrence of
infection). In addition, there are three covariates: age (continuous), sex (dichotomous), and
type of disease (categorical, four levels). There is also an individual-specific random effect
ui ∼ N (0, σ 2 ). Thus, the linear predictor becomes
ηi = β0 + βsex · sexi + βage · agei + βD xi + ui ,
where βD = (β1 , β2 , β3 ) and xi is a dummy vector coding for the disease type. Parameter
estimates are shown in Table A.2. See the files here.
β0
βage
admb-re
−4.3440
Std. dev.
BUGS
Std. dev.
β1
β2
β3
βsex
r
σ
0.0030
0.1208 0.6058 −1.1423 −1.8767
1.1624
0.5617
0.8720
0.0137
0.5008 0.5011
0.4754
0.1626
0.2970
−4.6000
0.8962
0.0030
0.0148
0.1329 0.6444 −1.1680 −1.9380
0.5393 0.5301
0.8335
0.4854
1.2150
0.1623
0.6374
0.3570
0.7729
Table A.2: Parameter estimates for Weibull regression with random effects.
A.2
Block-diagonal Hessian
This section contains models with grouped or nested random effects.
A.2.1
Nonlinear mixed models: an nlme comparison
Model description
The orange tree growth data was used by [9, Ch. 8.2] to illustrate how a logistic growth
curve model with random effects can be fit with the S-Plus function nlme. The data contain
A-7
measurements made at seven occasions for each of five orange trees. See Table A.3.
tij
yij
Time when the j th measurement was made on tree i.
Trunk circumference of tree i when measured at time tij .
Table A.3: Orange tree data.
The following logistic model is used:
yij =
φ1 + ui
+ εij ,
1 + exp [− (tij − φ2 ) /φ3 ]
where (φ1 , φ2 , φ3 ) are hyper-parameters, ui ∼ N (0, σu2 ) is a random effect, and εij ∼ N (0, σ 2 )
is the residual noise term.
Results
Parameter estimates are shown in Table A.4. The difference between the estimates obφ1
φ2
φ3
admb-re
192.1
727.9
348.1
Std. dev.
15.7
35.2
191.0
722.6
nlme
σ
7.843 31.65
27.1 1.013
344.2
σu
10.26
7.846 31.48
Table A.4: Parameter estimates.
tained with admb-re and nlme is small. The difference is caused by the fact that the
two approaches use different approximations to the likelihood function. (admb-re uses the
Laplace approximation, and for nlme, the reader is referred to [9, Ch. 7].)
The computation time for admb was 0.58 seconds, while the computation time for nlme
(running under S-Plus 6.1) was 1.6 seconds.
See the files here.
A.2.2
Pharmacokinetics: an nlme comparison
Model description
The “one-compartment open model” is commonly used in pharmacokinetics. It can be described as follows. A patient receives a dose D of some substance at time td . The concentration ct at a later time point t is governed by the equation
ct = D
exp − Cl
(t − td )
V
V
A-8
where V and Cl are parameters (the so-called “Volume of concentration” and the “Clearance”). Doses given at different time points contribute additively to ct . This model has been
fitted to a data set using the S-Plus routine nlme, see [9, Ch. 6.4], with the linear predictor
log (V ) = β1 + β2 W + uV ,
log (Cl) = β3 + β4 W + uCl ,
2
where W is a continuous covariate, while uV ∼ N (0, σV2 ) and uCl ∼ N (0, σCl
) are random
effects. The model specification is completed by the requirement that the observed concentration y in the patient is related to the true concentration by y = ct + ε, where ε ∼ N (0, σ 2 )
is a measurement error term.
Results
Estimates of hyper-parameters are shown in Table A.5. The differences between the estimates
β1
β2
β3
β4
σ
σV
σCl
admb-re
Std. dev.
−5.99 0.622 −0.471 0.532 2.72 0.171 0.227
0.13 0.076
0.067 0.040 0.23 0.024 0.054
nlme
−5.96
0.620 −0.485
0.532
2.73
0.173
0.216
Table A.5: Hyper-parameter estimates: pharmacokinetics.
obtained with admb-re and nlme are caused by the fact that the two methods use different
approximations of the likelihood function. admb-re uses the Laplace approximation, while
the method used by nlme is described in [9, Ch. 7].
The time taken to fit the model by admb-re was 17 seconds, while the computation
time for nlme (under S-Plus 6.1) was 7 seconds.
See the files here.
A.2.3
Frequency weighting in admb-re
Model description
Let Xi be binomially distributed with parameters N = 2 and pi , and further assume that
pi =
exp(µ + ui )
,
1 + exp(µ + ui )
(A.3)
A-9
where µ is a parameter and ui ∼ N (0, σ 2 ) is a random effect. Assuming independence, the
log-likelihood function for the parameter θ = (µ, σ) can be written as
l(θ) =
n
X
log p(xi ; θ) .
(A.4)
i=1
In admb-re, p(xi ; θ) is approximated using the Laplace approximation. However, since xi
only can take the values 0, 1, and 2, we can rewrite the log-likelihood as
l(θ) =
2
X
nj log p(j; θ) ,
(A.5)
j=0
where nj is the number xi being equal to j. Still, the Laplace approximation must be used
to approximate p(j; θ), but now only for j = 0, 1, 2, as opposed to j = 1, . . . n, as above. For
large n, this can give large savings.
To implement the log-likelihood (A.5) in admb-re, you must organize your code into a
SEPARABLE_FUNCTION (see the section “Nested models” in the admb-re manual). Then you
should do the following:
• Formulate the objective function in the weighted form (A.5).
• Include the statement
!! set_multinomial_weights(w);
in the PARAMETER_SECTION. The variable w is a vector (with indexes starting at 1)
containing the weights, so in our case, w = (n0 , n1 , n2 ).
See the files here.
A.2.4
Ordinal-logistic regression
Model description
In this model, the response variable y takes on values from the ordered set {y (s) , s = 1, . . . , S−
1}, where y (1) < y (2) < · · · < y (S) . For s = 1, . . . , S − 1, define Ps = P (y ≤ y (s) ) and
κs = log[Ps /(1 − Ps )]. To allow κs to depend on covariates specific to the ith observation
(i = 1, . . . , n), we introduce a disturbance ηi of κs :
P (yi ≤ y (s) ) =
exp(κs − ηi )
,
1 + exp(κs − ηi )
s = 1, . . . , S − 1.
with
ηi = Xi β + uji ,
where Xi and β play the sample role, as in earlier examples. The uj (j = 1, . . . , q) are
independent N (0, σ 2 ) variables, and ji is the latent variable class of individual i.
See the files here.
A-10
A.3
Banded Hessian (state-space)
Here are some examples of state-space models.
A.3.1
Stochastic volatility models in finance
Model description
Stochastic volatility models are used in mathematical finance to describe the evolution of
asset returns, which typically exhibit changing variances over time. As an illustration, we use
a time series of daily pound/dollar exchange rates {zt } from the period 01/10/81 to 28/6/85,
previously analyzed by [5]. The series of interest are the daily mean-corrected returns {yt },
given by the transformation
yt = log zt − log zt−1 − n−1
n
X
(log zt − log zt−1 ).
i=1
The stochastic volatility model allows the variance of yt to vary smoothly with time. This
is achieved by assuming that yt ∼ N (µ, σt2 ), where σt2 = exp(µx + xt ). The smoothly varying
component xt follows the autoregression
xt = βxt−1 + εt ,
εt ∼ N (0, σ 2 ).
The vector of hyper-parameters is for this model is thus (β, σ, µ, µx ).
See the files here.
A.3.2
A discrete valued time series: the polio data set
Model description
A time series of monthly numbers of poliomyelitis cases during the period 1970–1983 in the
U.S. was analyzed by [12]. We make a comparison to the performance of the Monte Carlo
Newton-Raphson method, as reported in [7]. We adopt their model formulation.
Let yi denote the number of polio cases in the ith period (i = 1, . . . , 168). It is assumed
that the distribution of yi is governed by a latent stationary AR(1) process {ui } satisfying
ui = ρui−1 + εi ,
where the εi ∼ N (0, σ 2 ). To account for trend and seasonality, the following covariate vector
is introduced:
!
2π
2π
2π
2π
i
, cos
i , sin
i , cos
i , sin
i
.
xi = 1,
1000
12
12
6
6
A-11
Conditionally on the latent process {ui }, the counts yi are independently Poisson distributed with intensity
λi = exp(xi 0 β + ui ).
Results
Estimates of hyper-parameters are shown in Table A.6.
β1
admb-re
Std. dev.
β2
β3
β4
β5
β6
ρ
σ
0.242 −3.81 0.162 −0.482 0.413 −0.0109 0.627 0.538
0.270
2.76 0.150
0.160 0.130
0.1300 0.190 0.150
Kuk & Cheng [7] 0.244 −3.82
0.162 −0.478
0.413 −0.0109
0.665
0.519
Table A.6: Hyper-parameter estimates: polio data set.
We note that the standard deviation is large for several regression parameters. The
admb-re estimates (which are based on the Laplace approximation) are very similar to the
exact maximum likelihood estimates, as obtained with the method of [7].
See the files here.
A.4
A.4.1
Generally sparse Hessian
Multilevel Rasch model
The multilevel Rasch model can be implemented using random effects in admb. As an
example, we use data on the responses of 2042 soldiers to a total of 19 items (questions),
taken from [2]. This illustrates the use of crossed random effects in admb. Furthermore,
it is shown how the model easily can be generalized in admb. These more general models
cannot be fitted with standard glmm software, such as “lmer” in R.
See the files here.
A-12
Appendix B
Differences between admb and admb-re?
• Profile likelihoods are now implemented also in random effects models, but with the
limitation that the likeprof_number can only depend on parameters, not random
effects.
• Certain functions, especially for matrix operations, have not been implemented.
• The assignment operator for dvariable behaves differently. The code
dvariable y = 1;
dvariable x = y;
will make x and y point to the same memory location (shallow copy) in admb-re.
Hence, changing the value of x automatically changes y. Under admb, on the other
hand, x and y will refer to different memory locations (deep copy). If you want to
perform a deep copy in admb-re you should write:
dvariable y = 1;
dvariable x;
x = y;
For vector and matrix objects admb and admb-re behave identically in that a shallow
copy is used.
B-1
Appendix C
Command Line options
A list of command line options accepted by admb programs can be obtained using the
command line option -?, for instance,
$ simple -?
Those options that are specific to admb-re are printed after line the “Random effects options
if applicable.” See Table C.1. The options in the last section (the sections are separated by
horizontal bars) are not printed, but can still be used (see earlier).
C-1
Option
Explanation
-nr N
maximum number of Newton-Raphson steps
-imaxfn N
maximum number of evals in quasi-Newton inner optimization
-is N
set importance sampling size to N for random effects
-isf N
set importance sampling size funnel blocks to N for random effects
-isdiag
print importance sampling diagnostics
-hybrid
do hybrid Monte Carlo version of mcmc
-hbf
set the hybrid bounded flag for bounded parameters
-hyeps
mean step size for hybrid Monte Carlo
-hynstep
number of steps for hybrid Monte Carlo
-noinit
do not initialize random effects before inner optimzation
-ndi N
set maximum number of separable calls
-ndb N
set number of blocks for derivatives for random effects (reduces temporary
file sizes)
-ddnr
use high-precision Newton-Raphson for inner optimization for banded
Hessian case only, even if implemented
-nrdbg
verbose reporting for debugging Newton-Raphson
-mm N
do minimax optimization
-shess
use sparse Hessian structure inner optimzation
-l1 N
set size of buffer f1b2list1 to N
-l2 N
set size of buffer f1b2list12 to N
-l3 N
set size of buffer f1b2list13 to N
-nl1 N
set size of buffer nf1b2list1 to N
-nl2 N
set size of buffer nf1b2list12 to N
-nl3 N
set size of buffer nf1b2list13 to N
Table C.1: Command line options.
C-2
Appendix D
Quick References
D.1
Compiling admb programs
To compile model.tpl in a DOS/Linux terminal window, type
admb [-r] [-s] model
where the options
-r is used to invoke the random effect module
-s yields the “safe” version of the executable file
There are two stages of compilation:
• Translate TPL to C++: tpl2cpp or tpl2rem
• Build executable from C++ code (using GCC, Visual C++, etc.)
D-1
modif
y
TPL
source code
Translator
inspe
& mo ct
dify
Error message from tpl2xxx
tpl2cpp
tpl2rem
Error in line 26
C++
source code
inspe
ct
Error message from
C++ compiler
C++ compiler
Excutable file
D-2
The ADMB primer
ADMB project
test.tpl
File hierarchy
Compilation
admb [-re] test
Executable file
test.exe
Input files
Data file
test.dat
Output files
Initial values
test.pin
Estimates
test.par
Standard deviations
test.std
Corelation matrix
test.cor
Report file
test.rep
TPL file
Two blanks indentation
of each line.
init_int n
init here means
”read from dat-file”.
init here means
independent variable, i.e.
parameter to be estimated.
Starting value read from
pin-file.
Declaration of data objects, and programming
variables derived from these.
DATA_SECTION
Valid index range: 2,…,n
init_number a
init_vector y(2,n)
init_matrix z(1,n,2,10)
int M
vector x(1,n)
Row range: 1,…,n
Col. range: 2,…, 10
Declaration of independent parameters, random
effects and variables derived from these.
PARAMETER_SECTION
Bounds: 0 < beta < a
init_number alpha
init_bounded_number beta(0,a,2)
Valid indices: 3,…,8
Bounds: 0 < mu(i) < a
Activated in phase 3
init_bounded_vector mu(3,8,0,a,3)
Parameter beta
activated in
phase 2,
random_effects_vector u(1,n,2)
Assigns value to the
log-likelihood function,
in terms of data,
parameters and random
effects.
Pure C++ code.
Contribution from N(0,1)
variables placed on random
effects
Temporary variable
defined in
PARAMETER SECTION
C++ incremental operator
objective_function_value g
Vector of random
effects. Valid index
range: 1,…,n.
Activated in phase 2.
PROCEDURE_SECTION
g = 0.5*norm2(u);
Objective function / negative loglikelihood
ADMB function that calculates the
for(int i=2;i<=n;i++)
squared norm of the vector u.
{
tmp = beta*(y(i)-u(i)-alpha);
Likelihood contribution
g += log(beta)
from y(i), corresponding
to assumption that y(i) is
+ 0.5*square(tmp/beta);
normally distributed with
mean u(i)-alpha and
standard deviation beta.
D-3
References
[1] ADMB Foundation. ADMB-IDE: Easy and efficient user interface. ADMB Foundation
Newsletter, 1(3):1–2, July 2009. 2-4
[2] H. Doran, D. Bates, P. Bliese, and M. Dowling. Estimating the multilevel Rasch model:
With the lme4 package. Journal of Statistical Software, 20(2):1–17, 2007. A-12
[3] P. Eilers and B. Marx. Flexible smoothing with B-splines and penalties. Statistical
Science, 11:89–121, 1996. A-3
[4] D. Fournier. An Introduction to AD Model Builder. ADMB Foundation, Honolulu,
2011. 1-1, 3-7
[5] A.C. Harvey, E. Ruiz, and N. Shephard. Multivariate stochastic variance models. Review
of Economic Studies, 61:247–264, 1994. A-11
[6] T.J. Hastie and R.J. Tibshirani. Generalized Additive Models, volume 43 of Monographs
on Statistics and Applied Probability. Chapman & Hall, London, 1990. A-3, A-4
[7] A. Y. C. Kuk and Y. W. Cheng. Pointwise and functional approximations in Monte
Carlo maximum likelihood estimation. Statistics and Computing, 9:91–99, 1999. A-11,
A-12
[8] X. Lin and D. Zhang. Inference in generalized additive mixed models by using smoothing
splines. Journal of the Royal Statistical Society of Britain, 61:381–400, 1999. A-5
[9] José C. Pinheiro and Douglas M. Bates. Mixed-Effects Models in S and S-PLUS. Statistics and Computing. Springer, 2000. 3-3, A-7, A-8, A-9
[10] David Ruppert, M.P. Wand, and R.J. Carroll. Semiparametric Regression. Cambridge
University Press, 2003. A-1, A-5, A-6
[11] H. Skaug and D. Fournier. Automatic approximation of the marginal likelihood in nonGaussian hierarchical models. Computational Statistics & Data Analysis, 56:699–709,
2006. 3-8
References-1
[12] S. L. Zeger. A regression-model for time-series of counts. Biometrika, 75:621–629, 1988.
A-11
References-2
Index
command line options
admb-re-specific, C-1
crossed effects, 4-7
gam, A-3
Gauss-Hermite quadrature, 4-5
hyper-parameter, 2-6
importance sampling, 3-10
limited memory quasi-Newton, 3-13
linear predictor, 2-5
mcmc, 3-10
nonparametric estimation
splines, A-2
variance function, A-5
penalized likelihood, 3-9
phases, 3-6
prior distributions
Gaussian priors, 4-7
random effects, 2-4
correlated, 3-3
Laplace approximation, 3-8
random effects matrix, 2-5
random effects vector, 2-5
reml, 3-11, 4-7
splines
difference penalty, A-3
state-space models, 4-6
temporary files
f1b2list1, 3-12
reducing the size, 3-12
tpl file
compiling, 2-2, 2-3
writing, 2-2
Index-1
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