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BITE: Bayesian Intensity Estimator
Reference Manual
⁄Tommi Harkanen˜ ˜
March 24, 2004
cThe BITE software and this documentation are ° Copyright 2000 by
Tommi Harkanen, Rolf Nevanlinna Institute. All rights reserved. Repro-˜ ˜
ductions for personal use are allowed. See also flle copyright.txt. The
software and the documentation are provided without warranty of any kind.
⁄Rolf Nevanlinna Institute, P.O.B. 4, FIN-00014 University of Helsinki, Finland, E-mail:
[e-mail], tel. [phone], FAX [phone].
1Contents
1 Introduction 5
2 Terminology 6
2.1 Filtering and censoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Step functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Covariates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Installation and description of commands 12
3.1 Installation and running BITE . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Basic elements of the BITE syntax . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Setting simulation limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Loading data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.5 Setting prior variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.6 Initializing model functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.7 Saving Markov chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.8 Intensity model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.9 Monitoring updating of model functions . . . . . . . . . . . . . . . . . . . . 19
3.10 Calculating posterior expectation . . . . . . . . . . . . . . . . . . . . . . . . 19
3.11 Graphical output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.12 Error messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Prior distributions 23
4.1 Scalar parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Covariates and responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Model functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3.1 Gamma prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
24.3.2 Increasing gamma prior . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3.3 Time independent prior . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Markov chain Monte Carlo 27
6 Examples 27
6.1 Heart transplant survival data . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.1.1 Model and estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.1.2 Data flles for BITE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.1.3 Command flle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.2 Survival analysis with a frailty model . . . . . . . . . . . . . . . . . . . . . . 32
6.3 Dental caries study I: standard frailty model . . . . . . . . . . . . . . . . . 36
7 The output of BITE 41
8 Discussion 43
A The grammar of BITE 45
3List of Figures
1 An example of an Markovian model. . . . . . . . . . . . . . . . . . . . . . . 6
2 An example of an intensity function ‚(t), and a possible realization from a
corresponding Poisson process. . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 An example of a step function . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 A template for a hierarchical intensity model accepted by BITE. . . . . . . . 11
5 Example of point process data. . . . . . . . . . . . . . . . . . . . . . . . . . 16
6 The estimates of the baseline hazards ‚ and f of the heart transplant data. 290
7 Baseline hazard estimate of the kidney example. . . . . . . . . . . . . . . . 33
List of Tables
1 Poisson-likelihood contributions and the notations of the parameters in BITE. 12
2 Basic tokens of BITE grammar. . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Internal operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Available dependencies of (prior) distributions. . . . . . . . . . . . . . . . . 24
5 The probability distributions recognized by BITE. . . . . . . . . . . . . . . . 24
6 Estimates of the regression coe–cients in the heart transplant data. . . . . 29
7 Data flles of the heart transplant example. . . . . . . . . . . . . . . . . . . . 30
8 The estimates of the Bayesian model with a nonparametric baseline hazard. 34
9 The data flles of the kidney example. . . . . . . . . . . . . . . . . . . . . . . 36
10 Data flles of caries I example. . . . . . . . . . . . . . . . . . . . . . . . . . . 37
11 Some software packages for Bayesian inference. . . . . . . . . . . . . . . . . 43
12 The tokens of BITE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
41 Introduction
Markov chain Monte Carlo (MCMC) methods have had an profound efiect on statistical
methodology based on Bayesian inference (see, for example, Gelman et al. (1995)): Analyticds provide solutions only to simple models, thus the development of e–cient comput-
ers allowed statisticians to apply numerical methods to solve more complicated (and more
realistic) problems, and other n methods are convenient only in low-dimensional
problems.
Some software packages like BUGS (see Gilks et al. (1994)) have been developed for
Bayesian modelling, but they are generally unable to cope with nonparametric intensity
models. The variety of difierent (Bayesian) models is huge, thus building for all of them a
general software package which is both easy to use, exible and e–cient may be close to
impossible in the near future.
This package has been developed for estimating a subset of nonparametric Bayesian
intensity models which provide a exible framework for modelling various life history phe-
nomena with continuous distributions, such as survival analysis and event-history data. For
example, analyses on heart transplant data, see section 6.1, dental caries data, see Harka-˜ ˜
nen et al. (2000a) and H˜arkanen˜ et al. (2000b), infections in kidneys (see section 6.2), and
treatment efiects on leukemia, see Harkanen (2000). For review of the theoretical back-˜ ˜
ground of intensity models see for example, Arjas (1989) or Andersen et al. (1993) which
contains more examples about intensity models, and classical estimation procedures.
Section 2 contains an introduction to the notations. The response is a point process in
one-dimensional space (time) which may be independently and non-informatively censored
or flltered. The probabilistic model for the point process responses is a Poisson process
(which will be presented in section 2.2) parametrized by a non-negative intensity function
which is here approximated by a piecewise constant function (see subsection 2.3). The class
of piecewise constant functions provides a good framework for intensity models, because
addition or multiplication of those functions results again a piecewise constant function. A
time independent intensity (yielding a homogenous Poisson process) is a special case of a
piecewise constant function.
Some elements of the point process can be random, and also time independent covariates
and startpoints can be given prior distributions with some limitations. This topic, as well
as the prior distributions for the intensity functions, will be introduced in section 4.
Variants of Metropolis-Hastings algorithm (which is described in Gilks et al. (1996)
among other MCMC methods) are used for generating a sample from a Markov chain which
has the posterior distribution as the limiting (invariant) distribution. Posterior expectations
can then be approximated by appropriate averages of the sample. This approach (as well
as most numerical methods) has drawbacks, which will be discussed more in section 5.
Section 3 describes the syntax of the commands which can be used for describing (in-
tensity) models, and some examples about using BITE are presented in section 6. Output
of BITE is presented in section 7. Discussion in section 8 lists some software for Bayesian
inference.
5
6
2 Terminology
The basic unit for models is an individual i = 0; 1;:::;N¡ 1, where N is the number
of individuals. The individuals can experience events (T;X), where T is the event time
and X the event type. The event types are called marks: the response is a marked point
process (MPP) with a flnite mark space X2 E :=fE ; j2Jg, where the index setJj
contains M elements. The response data is considered to be an N£M matrix whose
elements contain the marked point process observations of individual i and of markij
j: the elements are more complicated than just real numbers of an ordinary matrix, and
they are described in more detail below. In order to keep the notation more readable, the
observations corresponding to individual i and mark j are denoted by D := .ij
For example, in a simple survival analysis the mark space has only one element (death)
with just one event time for each individual, and after death the individual is no longer in
the risk set. In a repeating events case, for example when observing epileptic seiz