A multistate process is a stochastic process that can take on a finite number of states K
where each state describes the current condition (Jewell 2005; Kalbfleisch and Prentice 2002). In a general form where we allow time to be continuous, we define Y(t) to be the
state of the process at time t, Y(t)∈ {1,2, . . . , K} and t ≥0. A Markov process is the
with no covariates, the transition rate from state i toj for an individual who is in state i at time t− is given by dΛij(t) = P Y(t− +dt) =j|Y(u),0≤u < t, Y(t− ) =i = P Y(t− +dt) =j|Y(t− ) = i
for all Y(u),0≤u < t, j 6=i. The Markov assumption is that the process is memoryless
in that only the current occupied state is need to specify the transition rates. Transition rates are allowed to depend on t, the amount of time since the beginning of the study.
In a general multistate model, subjects are allowed to transition from any state k to
another state k0
. Multistate growth models impose some restrictions on the multistate model. For one, growth models are unidirectional. That is, state transitions occur in a distinct order, only moving from statekto statek+1,k = 1, . . . , K−1. Also, all subjects
begin at the same initial state, k = 1. Traditional survival analysis, where subjects only
move from “at risk” to “failed” states is a simple example of a unidirectional multistate process with K = 2.
In a regular Markov model, we are able to observe the states directly so that the state transition probabilities are the only parameters of interest. In a hidden Markov model the complete state history of a multistate processY(t) is not available at every time point
t, but variables that are influenced by the state are observed. The influencing variables
can be linked to underlying latent progress variables that are indicative of the waiting time spent in a state (Dunson and Baird 2002). In a longitudinal study, the state may be observed at several time points for each subject but the exact transition times are only known to occur within an interval. In a cross-sectional design, the state information is
only available at one time point. We are concerned with the cross-sectional case where state transition times are interval censored and developmental progress covariates are measured once for each subject.
We consider multistate growth models where the moment the process begins, called the initiation time, is not known. The earliest approaches for multistate models with unknown initiation times relied on the restrictive Markov assumption (Kalbfleisch and Lawless 1985), which is not appropriate in our application. Semi-Markov models for fitting interval censored data with unknown initiation times (Satten and Sternberg 1999) have been developed that allow transition rates to depend on the amount of time spent in the state. In the fetal growth example, we consider the LMP date to be a known, pre- initiation time and conception the unknown initiation time. With this fixed time scale, it would also be possible to use the semi-Markov methods of DeGruttola and Lagakos (1989) or Sternberg and Satten (1999) by considering the pre-initiation time to be the first stage in the network. If the waiting times in different states are independent, then the semi-Markov assumption will hold. We avoid the semi-Markov assumption and allow individual transition rates between states to be influenced by a subject-specific latent variable. Our approach is most similar to Dunson and Baird (2002).
3
Computation
When proposing structural equation models, it is particularly important to consider which parameters in the model are identifiable and which must be constrained. In many cases, the marginal likelihood can be examined to determine how the data informs about each parameter. In a Bayesian analysis, fitting MCMC models can be somewhat of an art form so that alternative strategies may be needed to achieve dependable results (Gelfand and Sahu 1999). For example, including parameters that are non-identifiable, with ap- propriately vague priors, can help improve convergence. In this section, we first provide a brief introduction to Bayesian methods. We then define identifiability and Bayesian iden- tifiability and explain one way in which including non-identifiable parameters in a model can improve computational performance while introducing a more general class of con- ditionally conjugate priors. This section is concluded by describing a data augmentation algorithm useful for fitting probit and multivariate probit models.
3.1
Bayesian Methods for Data Analysis
Bayesian methods are a useful tool for the applied statistician with applications in a wide range of areas due to the development of MCMC methods and the corresponding expansion of computing power. The details of a Bayesian approach to data analysis (Carlin and Louis 2000; Gelman et al. 2004) are well beyond the scope of this dissertation. Instead we will summarize a few important concepts that serve as the foundation for
following sections.
The flexibility of the Bayesian framework allows it to deal with very complex analyt- ical problems while using relatively simple conceptual methods. Bayesian methods are based on estimating the posterior distribution of p parameters θ given the data x and
hyperparameters η, f(θ|x,η). Using Bayes’ Theorem,
f(θ|x,η)∝L(θ)π(θ|η) (3.1)
whereL(θ) =f(x,θ|η) represents the likelihood andπ(θ|η) the prior distribution onθ.
Inference is then based on the posterior distribution of parameters θi.
It is often difficult to directly calculate the complete joint posterior distribution, f(θ|x,η), when p is not small. Gibbs sampling (Gelfand and Smith 1990; Geman and
Geman 1984) is a commonly used MCMC method to draw elements ofθindividually or in small groups. The Gibbs sampler generates values from the joint posterior distribution by iteratively drawing samples from the complete conditional distributions f(θi|θj,x,η, i6=
j) for each i = 1, . . . , p0
≤ p. After sufficient iterations, draws from the individual
complete conditionals will eventually converge to being draws from the desired posterior distributionf(θ|x,η). When conjugate priors are chosen, it is often easy to sample from
the complete conditionals because they are of known forms. For complete conditionals of unknown form, sampling can be done using the Metropolis-Hastings Algorithm (Chib and Greenberg 1995; Hastings 1970) with proposal densities generated from adaptive rejection samplings (ARS; Gilks and Wild 1992) or ARMS (ARS with a metropolis step; Gilks et al. 1995. Iterations continue until the parameters are judged to converge by a variety of diagnostic techniques.