Fetal growth is important both clinically and in epidemiologic studies relating growth to later pregnancy and developmental outcomes. To date, large cohort studies have re- lied on the last menstrual period (LMP) and onset of bleeding to determine the time of miscarriage. This approach is centrally flawed because it ignores the developmental state of the fetus prior to the loss. Development may stop days to weeks prior to the onset of bleeding so that, for example, a loss classified as a miscarriage in the 10th week of gestation may have been the loss of an appropriately-grown 10 week fetus, or a fetus with development arrested at 7 weeks. Coupling cross-sectional ultrasound information with the ability to estimate growth provides an opportunity to estimate probable devel- opmental state prior to miscarriage. This can help researchers more accurately identify which insults must have occurred prior to the pregnancy loss and rule out exposures that occur after development arrest but before the onset of bleeding.
Another complication in determining the miscarriage date is uncertainty in pregnancy dating. After the occurrence of a positive pregnancy test, time of conception is tradi- tionally dated as two weeks after the LMP. Such dating is notoriously imprecise due to variation in follicular phase length, the time between menses and ovulation, and, in rarer cases, missed menses (Kramer et al. 1988; Savitz et al. 2002). However, the distribution of the follicular phase length has been well characterized (Wilcox et al. 2001) and can be used to obtain more realistic estimates of the conception time. With these difficulties in mind, our goal in the early pregnancy example analysis is two-fold: (1) to determine whether there exists variation in fetal growth before 20 weeks gestation, and (2) if varia-
tion in growth exists, to determine if growth is associated with the probability of future pregnancy loss. In particular, if (2) holds the current practice of using ultrasound dating in early pregnancy loss studies may need to be re-evaluated.
This vexing problem can be posed statistically using a latent multistate growth process with unknown initiation time. In a multistate model the current condition of a subject is summarized by assignment to one of a finite number of K states (Jewell 2005). A
growth model assumes that all subjects begin at the same initial state and advance unidirectionally through subsequent states. That is, all subjects progress in order from state k to k+ 1, k = 1, . . . , K−1 without skipping any states or regressing to previous
states. This article focuses on the problem of estimating such models using cross-sectional data in which transition times are not observed. For each subject, we have information at one time point on the current state as well as measurements of developmental progress that are surrogates of the amount of time spent in a state. Also, the moment the growth process begins, referred to as the initiation time, is not known, though some time point before initiation is known.
Multistate growth models with interval censored transition times appear in several applications. Often the initiation time is also unknown but not considered in the analysis. One example involves studying tumor growth in mice. Typically, a mouse is exposed to a carcinogenic compound and a tumor may develop some time following the exposure. At a later date, the mouse is sacrificed to determine how far the tumor has progressed (Albert and Shih 2003; Dewanji and Kalbfleisch 1986; Ryan and Orav 1988). In these studies, the investigators analyze time from exposure rather than time from tumor initiation. We focus on measuring growth as a function of time since initiation and demonstrate that,
in our application, failing to account for the unknown initiation time can have a large impact on the analysis. Other examples of multistate models include AIDS progression (DeGruttola and Lagakos 1989), breast cancer (Duffy et al. 1995), and the development of uterine fibroids (Dunson and Baird 2002). Our research is motivated by the Right from the Start (RFTS) study of embryonic development.
RFTS is a prospective cohort study that identified women who were planning to conceive or in early pregnancy (Promislow et al. 2004). Investigators enrolled women when they had a positive pregnancy test, and then promptly provided an early first trimester ultrasound. At the time of the ultrasound, each fetus was assigned to one of three states based on the presence or absence of important developmental features: (1) only gestational sac present, (2) fetal pole present without regular cardiac activity, and (3) fetal pole with normal cardiac activity. Lack of cardiac activity is a natural state of development that every fetus experiences for hours to days while the heart forms, becomes detectable by ultrasound, and before a heart rate is established. It is not necessarily indicative of a problem pregnancy. The ultrasound also provides two measurements of developmental progress, the mean gestational sac diameter and fetal pole length. Both of these covariates have been studied extensively and found to be strongly associated with time since LMP (Filly and Hadlock 2000; Hadlock et al. 1992).
We develop a latent variable method for incorporating development progress variables using structural equations (Joreskog 1970) in a multistate growth model. Our approach differs from previous analyses that used developmental progress covariates directly. For example, Albert and Shih (2003) jointly model tumor onset and growth after onset using a single tumor volume measurement. When multiple measures of developmental progress
are available, previous direct approaches have attempted to categorize disease severity into a set of levels (Craig et al. 1999). The number, volume and location of a tumor could be used to assign subjects into severity levels ranging from high to low, although defining the cut points for the levels may be unclear. Instead, we follow a latent variable approach similar to Dunson and Baird (2002) that naturally combines multiple measurements of developmental progress into the underlying latent growth concept. We allow the latent variable to be continuous, which is more flexible than the direct approach in that it does not pre-specify arbitrary severity boundaries in the classification procedure. We are also able to consider associations between the latent variable and future outcomes. For example, in our early pregnancy analysis we identify an important relationship between latent growth and the risk of pregnancy loss by 20 weeks.
Bayesian methods are well-suited to the embryonic growth application because abun- dant prior knowledge is available. We propose statistical models in a specific form that allows prior information about the time from LMP to conception and time to state transitions to be readily incorporated. Available methodology using Markov models (Kalbfleish and Lawless 1985) would not be appropriate in our application because it assumes that the transition rates are independent of the amount of time spent in state. Semi-Markov models have also been developed for fitting interval censored data with unknown initiation times. While Satten and Sternberg (1999) allow the transition rate to depend on the time spent in the current state, they do not incorporate surrogates of developmental progress that are functions of time since initiation in addition to state membership. Bayesian methods that consider surrogates of developmental progress for interval censored data have been proposed (Dunson and Baird 2002), but do not allow
the developmental progress covariates to be functions of an unknown initiation time. In addition, Dunson and Baird (2002) treat the multistate growth model as the primary outcome of interest, while we additionally consider joint modeling of growth with a future pregnancy outcome.
Section 4.3 describes the underlying data structure, the general stochastic and latent variable models, and the specific models we used in the RFTS example analysis. In section 4.4 we provide a Bayesian approach to fitting such a model with the necessary conditional distributions in Appendix A. Section 4.5 contains the application of our methods to a study of early pregnancy and section 4.6 discusses the results.