Rachel MacKay Altman

Mixed Hidden Markov Models: An Extension of the Hidden Markov Model to the Longitudinal Data Setting

Hidden Markov models (HMMs) are a useful tool for capturing the behavior of overdispersed, autocorrelated data. These models have been applied to many different problems, including speech recognition, precipitation modeling, and gene finding and profiling. Typically, HMMs are applied to individual stochastic processes; HMMs for simultaneously modeling multiple processes—as in the longitudinal data setting—have not been widely studied. In this article I present a new class of models, mixed HMMs (MHMMs), where I use both covariates and random effects to capture differences among processes. I define the models using the framework of generalized linear mixed models and discuss their interpretation. I then provide algorithms for parameter estimation and illustrate the properties of the estimators via a simulation study. Finally, to demonstrate the practical uses of MHMMs, I provide an application to data on lesion counts in multiple sclerosis patients. I show that my model, while parsimonious, can describe the heterogeneity among such patients.

Is Cortisol Excretion Independent of Menstrual Cycle Day? A Longitudinal Evaluation of First Morning Urinary Specimens

Background Cortisol is frequently used as a marker of physiologic stress levels. Using cortisol for that purpose, however, requires a thorough understanding of its normal longitudinal variability. The current understanding of longitudinal variability of basal cortisol secretion in women is very limited. It is often assumed, for example, that basal cortisol profiles do not vary across the menstrual cycle. This is a critical assumption: if cortisol were to follow a time dependent pattern during the menstrual cycle, then ignoring this cyclic variation could lead to erroneous imputation of physiologic stress. Yet, the assumption that basal cortisol levels are stable across the menstrual cycle rests on partial and contradictory evidence. Here we conduct a thorough test of that assumption using data collected for up to a year from 25 women living in rural Guatemala. Methodology We apply a linear mixed model to describe longitudinal first morning urinary cortisol profiles, accounting for differences in both mean and standard deviation of cortisol among women. To that aim we evaluate the fit of two alternative models. The first model assumes that cortisol does not vary with menstrual cycle day. The second assumes that cortisol mean varies across the menstrual cycle. Menstrual cycles are aligned on ovulation day (day 0). Follicular days are assigned negative numbers and luteal days positive numbers. When we compared Models 1 and 2 restricting our analysis to days between −14 (follicular) and day 14 (luteal) then day of the menstrual cycle did not emerge as a predictor of urinary cortisol levels (p-value >0.05). Yet, when we extended our analyses beyond that central 28-day-period then day of the menstrual cycle become a statistically significant predictor of cortisol levels. Significance The observed trend suggests that studies including cycling women should account for day dependent variation in cortisol in cycles with long follicular and luteal phases.

Number of Children and Telomere Length in Women: A Prospective, Longitudinal Evaluation.

Life history theory (LHT) predicts a trade-off between reproductive effort and the pace of biological aging. Energy invested in reproduction is not available for tissue maintenance, thus having more offspring is expected to lead to accelerated senescence. Studies conducted in a variety of non-human species are consistent with this LHT prediction. Here we investigate the relationship between the number of surviving children born to a woman and telomere length (TL, a marker of cellular aging) over 13 years in a group of 75 Kaqchikel Mayan women. Contrary to LHT’s prediction, women who had fewer children exhibited shorter TLs than those who had more children (p = 0.045) after controlling for TL at the onset of the 13-year study period. An “ultimate” explanation for this apparently protective effect of having more children may lay with human’s cooperative-breeding strategy. In a number of socio-economic and cultural contexts, having more chilren appears to be linked to an increase in social support for mothers (e.g., allomaternal care). Higher social support, has been argued to reduce the costs of further reproduction. Lower reproductive costs may make more metabolic energy available for tissue maintenance, resulting in a slower pace of cellular aging. At a “proximate” level, mechanisms involved may include the actions of the gonadal steroid estradiol, which increases dramatically during pregnancy. Estradiol is known to protect TL from the effects of oxidative stress as well as increase telomerase activity, an enzyme that maintains TL. Future research should explore the potential role of social support as well as that of estradiol and other potential biological pathways in the trade-offs between reproductive effort and the pace of cellular aging within and among human as well as in non-human populations.

A longitudinal model for magnetic resonance imaging lesion count data in multiple sclerosis patients

Magnetic resonance imaging (MRI) data are routinely collected at multiple time points during phase 2 clinical trials in multiple sclerosis. However, these data are typically summarized into a single response for each patient before analysis. Models based on these summary statistics do not allow the exploration of the trade-off between numbers of patients and numbers of scans per patient or the development of optimal schedules for MRI scanning. To address these limitations, in this paper, we develop a longitudinal model to describe one MRI outcome: the number of lesions observed on an individual MRI scan. We motivate our choice of a mixed hidden Markov model based both on novel graphical diagnostic methods applied to five real data sets and on conceptual considerations. Using this model, we compare the performance of a number of different tests of treatment effect. These include standard parametric and nonparametric tests, as well as tests based on the new model. We conduct an extensive simulation study using data generated from the longitudinal model to investigate the parameters that affect test performance and to assess size and power. We determine that the parameters of the hidden Markov chain do not substantially affect the performance of the tests. Furthermore, we describe conditions under which likelihood ratio tests based on the longitudinal model appreciably outperform the standard tests based on summary statistics. These results establish that the new model is a valuable practical tool for designing and analyzing multiple sclerosis clinical trials.

The ex-pat effect: presence of recent Western immigrants is associated with changes in age at first birth and birth rate in a Maya population from rural Guatemala

Abstract Background: Economic transitions expose indigenous populations to a variety of ecological and cultural challenges, especially regarding diet and stress. These kinds of challenges are predicted by evolutionary ecological theory to have fitness consequences (differential reproduction) and, indeed, are often associated with changes in fertility dynamics. It is currently unclear whether international immigration might impact the nature of such an economic transition or its consequences for fertility. Aim: To examine measures of fertility, diet and stress in two economically transitioning Maya villages in Guatemala that have been differentially exposed to immigration by Westerners. Subjects and methods: This study compared Maya women’s ages at first birth and birth rates between villages and investigated whether these fertility indicators changed through time. It also explored whether the villages differed in relation to diet and/or a proxy of stress. Results: It was found that, in the village directly impacted by immigration, first births occurred earlier, but birth rate was slower. In both villages, over the sampled time window, age at first birth increased, while birth rate decreased. The villages do not differ significantly in dietary indicators, but the immigration-affected village scored higher on the stress proxy. Conclusion: Immigration can affect fertility in host communities. This relationship between immigration and fertility dynamics may be partly attributable to stress, but this possibility should be evaluated prospectively in future research.

MRI-based clinical trials in relapsing–remitting MS: new sample size calculations based on a longitudinal model

Background: Sample sizes for magnetic resonance imaging (MRI)-based clinical trials in multiple sclerosis (MS) generally assume that lesion counts are reasonably described by the negative binomial (NB) model. Objective: This study aimed to assess the appropriateness of the NB model for lesion count data and to provide sample sizes for placebo-controlled, MRI-based clinical trials in relapsing–remitting MS using a more realistic model. Methods: The fit of the NB model in each arm of five MS clinical trials was assessed using Pearson’s chi-squared statistic. Required sample sizes associated with various tests of treatment effect were estimated by simulating data from a new, longitudinal model for repeated lesion count data on individual patients. Results: Evidence (p < 0.05) against the NB model was found in at least one arm of four of the five trials. If a trial is designed using this model but the resulting clinical data do not follow its assumptions then this trial can be seriously under-powered for assessing differences in mean lesion counts. Conclusion: Sample sizes based on the longitudinal model are more realistic and often smaller than those previously reported using the NB model.

Practical considerations when analyzing discrete survival times using the grouped relative risk model

The grouped relative risk model (GRRM) is a popular semi-parametric model for analyzing discrete survival time data. The maximum likelihood estimators (MLEs) of the regression coefficients in this model are often asymptotically efficient relative to those based on a more restrictive, parametric model. However, in settings with a small number of sampling units, the usual properties of the MLEs are not assured. In this paper, we discuss computational issues that can arise when fitting a GRRM to small samples, and describe conditions under which the MLEs can be ill-behaved. We find that, overall, estimators based on a penalized score function behave substantially better than the MLEs in this setting and, in particular, can be far more efficient. We also provide methods of assessing the fit of a GRRM to small samples.

Child mortality, hypothalamic-pituitary-adrenal axis activity and cellular aging in mothers

Psychological challenges, including traumatic events, have been hypothesized to increase the age-related pace of biological aging. Here we test the hypothesis that psychological challenges can affect the pace of telomere attrition, a marker of cellular aging, using data from an ongoing longitudinal-cohort study of Kaqchikel Mayan women living in a population with a high frequency of child mortality, a traumatic life event. Specifically, we evaluate the associations between child mortality, maternal telomere length and the mothers’ hypothalamic-pituitary-adrenal axis (HPAA), or stress axis, activity. Child mortality data were collected in 2000 and 2013. HPAA activity was assessed by quantifying cortisol levels in first morning urinary specimens collected every other day for seven weeks in 2013. Telomere length (TL) was quantified using qPCR in 55 women from buccal specimens collected in 2013. Results: Shorter TL with increasing age was only observed in women who experienced child mortality (p = 0.015). Women with higher average basal cortisol (p = 0.007) and greater within-individual variation (standard deviation) in basal cortisol (p = 0.053) presented shorter TL. Non-parametric bootstrapping to estimate mediation effects suggests that HPAA activity mediates the effect of child mortality on TL. Our results are, thus, consistent with the hypothesis that traumatic events can influence cellular aging and that HPAA activity may play a mediatory role. Future large-scale longitudinal studies are necessary to confirm our results and further explore the role of the HPAA in cellular aging, as well as to advance our understanding of the underlying mechanisms involved.

Comment on “Nasal bone length: prenasal thickness ratio: a strong 2D ultrasound marker for Down syndrome”

I have read with interest the article by Szabó et al., which discusses the potential of fetal nasal bone length (NBL) and prenasal thickness (PT) as markers for Down syndrome. The authors claim that the NBL : PT ratio is a better marker than the PT : NBL ratio because the false positive rate (FPR) is apparently lower when predictions are based on the former. I would like to clarify that, statistically, the two ratios in question provide exactly the same information with regard to Down syndrome. Specifically, because the PT :NBL ratio can be computed by taking the inverse of the NBL : PT ratio and vice versa, one ratio cannot be more informative than the other. The differences between the two ratios (in terms of FPRs when predicting Down syndrome) are thus not due to the ratios themselves but rather to the specific method used for classifying the pregnancies. In particular, the authors apparently selected a ‘normal’ range for the markers in the euploid fetuses. Fetuses with marker values falling outside that range were then predicted to have Down syndrome. The normal range appears to have been chosen somewhat arbitrarily; however, its definition is critical as it determines the performance of the test. In fact, inspection of Figure 3B suggests that if the upper boundary of the range were slightly higher in the PT : NBL case, the sensitivity would remain the same while the FPR would actually be lower than in the NBL : PT case. In other words, the conclusions regarding the performance of the two tests would be the opposite of those reported by the authors. I would also like to emphasize the importance of using an independent data set for assessing the predictive ability of a given method. In this case, the authors used the same data set both for formulating their predictive methods and for estimating the sensitivities and FPRs of these methods. As a result, the apparent performances of their methods are likely better than the actual performances. If a separate data set is not available, methods such as cross-validation can be used to obtain a more realistic assessment of performance.