Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Multi-state models are used in this dissertation to model panel data, also known as longitudinal
or cross-sectional time-series data. These are data sets which include units that are observed
across two or more points in time. These models have been used extensively in medical studies
where the disease states of patients are recorded over time.
A theoretical overview of the current multi-state Markov models when applied to panel data
is presented and based on this theory, a simulation procedure is developed to generate panel
data sets for given Markov models. Through the use of this procedure a simulation study
is undertaken to investigate the properties of the standard likelihood approach when fitting
Markov models and then to assess its shortcomings. One of the main shortcomings highlighted
by the simulation study, is the unstable estimates obtained by the standard likelihood models,
especially when fitted to small data sets.
A Bayesian approach is introduced to develop multi-state models that can overcome these
unstable estimates by incorporating prior knowledge into the modelling process. Two Bayesian
techniques are developed and presented, and their properties are assessed through the use of
extensive simulation studies.
Firstly, Bayesian multi-state models are developed by specifying prior distributions for the
transition rates, constructing a likelihood using standard Markov theory and then obtaining
the posterior distributions of the transition rates. A selected few priors are used in these
models. Secondly, Bayesian multi-state imputation techniques are presented that make use
of suitable prior information to impute missing observations in the panel data sets. Once
imputed, standard likelihood-based Markov models are fitted to the imputed data sets to
estimate the transition rates. Two different Bayesian imputation techniques are presented.
The first approach makes use of the Dirichlet distribution and imputes the unknown states at
all time points with missing observations. The second approach uses a Dirichlet process to
estimate the time at which a transition occurred between two known observations and then a
state is imputed at that estimated transition time.
The simulation studies show that these Bayesian methods resulted in more stable results, even
when small samples are available. / AFRIKAANSE OPSOMMING: Meerstadium-modelle word in hierdie verhandeling gebruik om paneeldata, ook bekend as
longitudinale of deursnee tydreeksdata, te modelleer. Hierdie is datastelle wat eenhede insluit
wat oor twee of meer punte in tyd waargeneem word. Hierdie tipe modelle word dikwels in
mediese studies gebruik indien verskillende stadiums van ’n siekte oor tyd waargeneem word.
’n Teoretiese oorsig van die huidige meerstadium Markov-modelle toegepas op paneeldata word
gegee. Gebaseer op hierdie teorie word ’n simulasieprosedure ontwikkel om paneeldatastelle
te simuleer vir gegewe Markov-modelle. Hierdie prosedure word dan gebruik in ’n simulasiestudie
om die eienskappe van die standaard aanneemlikheidsbenadering tot die pas vanMarkov
modelle te ondersoek en dan enige tekortkominge hieruit te beoordeel. Een van die hoof
tekortkominge wat uitgewys word deur die simulasiestudie, is die onstabiele beramings wat
verkry word indien dit gepas word op veral klein datastelle.
’n Bayes-benadering tot die modellering van meerstadiumpaneeldata word ontwikkel omhierdie
onstabiliteit te oorkom deur a priori-inligting in die modelleringsproses te inkorporeer. Twee
Bayes-tegnieke word ontwikkel en aangebied, en hulle eienskappe word ondersoek deur ’n
omvattende simulasiestudie.
Eerstens word Bayes-meerstadium-modelle ontwikkel deur a priori-verdelings vir die oorgangskoerse
te spesifiseer en dan die aanneemlikheidsfunksie te konstrueer deur van standaard
Markov-teorie gebruik te maak en die a posteriori-verdelings van die oorgangskoerse te bepaal.
’n Gekose aantal a priori-verdelings word gebruik in hierdie modelle. Tweedens word Bayesmeerstadium
invul tegnieke voorgestel wat gebruik maak van a priori-inligting om ontbrekende
waardes in die paneeldatastelle in te vul of te imputeer. Nadat die waardes ge-imputeer is,
word standaard Markov-modelle gepas op die ge-imputeerde datastel om die oorgangskoerse te
beraam. Twee verskillende Bayes-meerstadium imputasie tegnieke word bespreek. Die eerste
tegniek maak gebruik van ’n Dirichletverdeling om die ontbrekende stadium te imputeer by alle
tydspunte met ’n ontbrekende waarneming. Die tweede benadering gebruik ’n Dirichlet-proses
om die oorgangstyd tussen twee waarnemings te beraam en dan die ontbrekende stadium te
imputeer op daardie beraamde oorgangstyd.
Die simulasiestudies toon dat die Bayes-metodes resultate oplewer wat meer stabiel is, selfs
wanneer klein datastelle beskikbaar is.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/71910 |
| Date | 12 1900 |
| Creators | Muller, Christoffel Joseph Brand |
| Contributors | Mostert, Paul J., Stellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | English |
| Type | Thesis |
| Format | 253 p. : ill. |
| Rights | Stellenbosch University |
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