A semiparametric transition model with latent traits for longitudinal multistate data

SUMMARY: We propose a general multistate transition model. The model is developed for the analysis of repeated episodes of multiple states representing different health status. Transitions among multiple states are modeled jointly using multivariate latent traits with factor loadings. Different types of state transition are described by flexible transition-specific nonparametric baseline intensities. A state-specific latent trait is used to capture individual tendency of the sojourn in the state that cannot be explained by covariates and to account for correlation among repeated sojourns in the same state within an individual. Correlation among sojourns across different states within an individual is accounted for by the correlation between the different latent traits. The factor loadings for a latent trait accommodate the dependence of the transitions to different competing states from a same state. We obtain the semiparametric maximum likelihood estimates through an expectation-maximization (EM) algorithm. The method is illustrated by studying repeated transitions between independence and disability states of activities of daily living (ADL) with death as an absorbing state in a longitudinal aging study. The performance of the estimation procedure is assessed by simulation studies.     

Enhancement of Photosynthesis by Alternated Light Beams and a Kinetic Model

Using the alga Chlorella and 2 light beams of defined wavelength, we have found that short period (1 sec) alternation gives all characteristics of enhancement of net oxygen evolution observed when the beams are superimposed. With increasing period of alternation (1-20 see) decay in enhancement is more rapid at higher intensities.We have developed a kinetic model for the current hypothesis of 2 photoacts operating in series and separated by electron transport reactions. The model is applicable at low light intensity such that rate is governed only by intensities and the fractions of reaction centers open for each photoact. Potentials and pool sizes of intermediates were taken from current estimates. Wavelength dependency of the 2 photoacts was taken from enhancement spectra for superimposed light beams. Analog computer treatment of the kinetic equations gave predictions of chromatic transients and alternated enhancement which are in reasonable but not complete agreement with experimental observations.     

Covariate Adjustment of Event Histories Estimated from Markov Chains: The Additive Approach

Markov chain models are frequently used for studying event histories that include transitions between several states. An empirical transition matrix for nonhomogeneous Markov chains has previously been developed, including a detailed statistical theory based on counting processes and martingales. In this article, we show how to estimate transition probabilities dependent on covariates. This technique may, e.g., be used for making estimates of individual prognosis in epidemiological or clinical studies. The covariates are included through nonparametric additive models on the transition intensities of the Markov chain. The additive model allows for estimation of covariate-dependent transition intensities, and again a detailed theory exists based on counting processes. The martingale setting now allows for a very natural combination of the empirical transition matrix and the additive model, resulting in estimates that can be expressed as stochastic integrals, and hence their properties are easily evaluated. Two medical examples will be given. In the first example, we study how the lung cancer mortality of uranium miners depends on smoking and radon exposure. In the second example, we study how the probability of being in response depends on patient group and prophylactic treatment for leukemia patients who have had a bone marrow transplantation. A program in R and S-PLUS that can carry out the analyses described here has been developed and is freely available on the Internet.     

Sampling errors create bias in Markov models for community dynamics: the problem and a method for its solution

Repeated, spatially explicit sampling is widely used to characterize the dynamics of sessile communities in both terrestrial and aquatic systems, yet our understanding of the consequences of errors made in such sampling is limited. In particular, when Markov transition probabilities are calculated by tracking individual points over time, misidentification of the same spatial locations will result in biased estimates of transition probabilities, successional rates, and community trajectories. Nonetheless, to date, all published studies that use such data have implicitly assumed that resampling occurs without error when making estimates of transition rates. Here, we develop and test a straightforward maximum likelihood approach, based on simple field estimates of resampling errors, to arrive at corrected estimates of transition rates between species in a rocky intertidal community. We compare community Markov models based on raw and corrected transition estimates using data from Endocladia muricata-dominated plots in a California intertidal assemblage, finding that uncorrected predictions of succession consistently overestimate recovery time. We tested the precision and accuracy of the approach using simulated datasets and found good performance of our estimation method over a range of realistic sample sizes and error rates.     

Sources of Variability in Net Primary Production Predictions at a Regional Scale: A Comparison Using PnET-II and TEM 4.0 in Northeastern US Forests

Because model predictions at continental and global scales are necessarily based on broad characterizations of vegetation, soils, and climate, estimates of carbon stocks and fluxes made by global terrestrial biosphere models may not be accurate for every region. At the regional scale, we suggest that attention can be focused more clearly on understanding the relative strengths of predicted net primary productivity (NPP) limitation by energy, water, and nutrients. We evaluate the sources of variability among model predictions of NPP with a regional-scale comparison between estimates made by PnET-II (a forest ecosystem process model previously applied to the northeastern region) and TEM 4.0 (a terrestrial biosphere model typically applied to the globe) for the northeastern US. When the same climate, vegetation, and soil data sets were used to drive both models, regional average NPP predictions made by PnET-II and TEM were remarkably similar, and at the biome level, model predictions agreed fairly well with NPP estimates developed from field measurements. However, TEM 4.0 predictions were more sensitive to regional variations in temperature as a result of feedbacks between temperature and belowground N availability. In PnET-II, the direct link between transpiration and photosynthesis caused substantial water stress in hardwood and pine forest types with increases in solar radiation; predicted water stress was relieved substantially when soil water holding capacity (WHC) was increased. Increasing soil WHC had little effect on TEM 4.0 predictions because soil water storage was already sufficient to meet plant demand with baseline WHC values, and because predicted N availability under baseline conditions in this region was not limited by water. Because NPP predictions were closely keyed to forest cover type, the relative coverage of low- versus high-productivity forests at both fine and coarse resolutions was an important determinant of regional NPP predictions. Therefore, changes in grid cell size and differences in the methods used to aggregate from fine to coarse resolution were important to NPP predictions insofar as they changed the relative proportions of forest cover. We suggest that because the small patches of high-elevation spruce-fir forest in this region are substantially less productive than forests in the remainder of the region, more accurate NPP predictions will result if models applied to this region use land cover input data sets that retain as much fine-resolution forest type variability as possible. The differences among model responses to variations in climate and soil WHC data sets suggest that the models will respond quite differently to scenarios of future climate. A better understanding of the dynamic interactions between water stress, N availability, and forest productivity in this region will enable models to make more accurate predictions of future carbon stocks and fluxes. Received 19 June 1998; accepted 25 June 1999.     

Compensatory evolution in RNA secondary structures increases substitution rate variation among sites

There is growing evidence that interactions between biological molecules (e.g., RNA-RNA, protein-protein, RNA-protein) place limits on the rate and trajectory of molecular evolution. Here, by extending Kimura's model of compensatory evolution at interacting sites, we show that the ratio of transition to transversion substitutions (kappa) at interacting sites should be equal to the square of the ratio at independent sites. Because transition mutations generally occur at a higher rate than transversions, the model predicts that kappa should be higher at interacting sites than at independent sites. We tested this prediction in 10 RNA secondary structures by comparing phylogenetically derived estimates of kappa in paired sites within stems (kappa(p)) and unpaired sites within loops (kappa(u)). Eight of the 10 structures showed an excellent match to the quantitative predictions of the model, and 9 of the 10 structures matched the qualitative prediction kappa(p) > kappa(u). Only the Rev response element from the human immunovirus (HIV) genome showed the reverse pattern, with kappa(p) < kappa(u). Although a variety of evolutionary forces could produce quantitative deviations from the model predictions, the reversal in magnitude of kappa(p) and kappa(u) could be achieved only by violating the model assumption that the underlying transition (or transversion) mutation rates were identical in paired and unpaired regions of the molecule. We explore the ability of the APOBEC3 enzymes, host defense mechanisms against retroviruses, which induce transition mutations preferentially in single-stranded regions of the HIV genome, to explain this exception to the rule. Taken as a whole, our findings suggest that kappa may have utility as a simple diagnostic to evaluate proposed secondary structures.     

Effect of optimization criterion on spinal force estimates during asymmetric lifting

Optimization-based muscle force prediction models of the lumbar region are used in research and ergonomic practice, usually as a subroutine of a job analysis software package. Various optimization criteria have been put forward for use in rationally selecting a set of muscle forces to satisfy moment equilibrium, including the sum of cubed muscle contraction intensities and a double linear programming procedure for minimizing the spinal compression force and maximum muscle contraction intensity. A laboratory study was conducted to determine whether these two model formulations produce significantly different estimates of spinal forces for a dynamic asymmetric lift. Although statistically significant differences were found between the predictions of the two models, the difference in peak spinal compression force was only 1.1%.     

A generalized mover-stayer model for panel data

A generalized mover-stayer model is described for conditionally Markov processes under panel observation. Marginally the model represents a mixture of nested continuous-time Markov processes in which sub-models are defined by constraining some transition intensities to zero between two or more states of a full model. A Fisher scoring algorithm is described which facilitates maximum likelihood estimation based only on the first derivatives of the transition probability matrices. The model is fit to data from a smoking prevention study and is shown to provide a significant improvement in fit over a time-homogeneous Markov model. Extensions are developed which facilitate examination of covariate effects on both the transition intensities and the mover-stayer probabilities.     

Dealing with Uncertainty in Spatially Explicit Population Models

It has been argued that spatially explicit population models (SEPMs) cannot provide reliable guidance for conservation biology because of the difficulty of obtaining direct estimates for their demographic and dispersal parameters and because of error propagation. We argue that appropriate model calibration procedures can access additional sources of information, compensating the lack of direct parameter estimates. Our objective is to show how model calibration using population-level data can facilitate the construction of SEPMs that produce reliable predictions for conservation even when direct parameter estimates are inadequate. We constructed a spatially explicit and individual-based population model for the dynamics of brown bears (Ursus arctos) after a reintroduction program in Austria. To calibrate the model we developed a procedure that compared the simulated population dynamics with distinct features of the known population dynamics (=patterns). This procedure detected model parameterizations that did not reproduce the known dynamics. Global sensitivity analysis of the uncalibrated model revealed high uncertainty in most model predictions due to large parameter uncertainties (coefficients of variation CV 0.8). However, the calibrated model yielded predictions with considerably reduced uncertainty (CV 0.2). A pattern or a combination of various patterns that embed information on the entire model dynamics can reduce the uncertainty in model predictions, and the application of different patterns with high information content yields the same model predictions. In contrast, a pattern that does not embed information on the entire population dynamics (e.g., bear observations taken from sub-areas of the study area) does not reduce uncertainty in model predictions. Because population-level data for defining (multiple) patterns are often available, our approach could be applied widely.     

A Method for Estimating the Predictive Power in a Model of a Biological System with Low Sensitivity to Parameters

The ambiguity of parameter estimates for the model of a biological system may be due to low sensitivity of the model to perturbations of input data (parameters), which mathematically reflects biological mechanisms of robustness. We developed a novel method for estimating the predictive power of a model with the ambiguity of parameter estimates. The predictions are understood as a correct reproduction of the system behavior by the model when changing input data and parameters. The method is based on the relative sensitivity analysis of the fitted model to stiff parameters of the predicted model. The application principles of our approach are demonstrated using a model for the formation of the mRNA expression pattern of the hb gene in the Drosophila embryo and its ability to predict the hb pattern in the Kr null mutant. The nonlinear nature of the system is simulated by a saturating sigmoid function, which is the cause of low sensitivity. Our method allows us to estimate the predictive power of the model and uncover the causes of poor predictions, as well as choose the relevant level of the model detail in terms of predictions.     

Thermal sensation and thermophysiological responses to metabolic step-changes

This study investigated the effect on thermal perception and thermophysiological variables of controlled metabolic excursions of various intensities and durations. Twenty-four subjects were alternately seated on a chair or exercised by walking on a treadmill at a temperature predicted to be neutral at sedentary activity. In a second experimental series, subjects alternated between rest and exercise as well as between exercise at different intensities at two temperature levels. Measurements comprised skin and oesophageal temperatures, heart rate and subjective responses. Thermal sensation started to rise or decline immediately (within 1 min) after a change of activity, which means that even moderate activity changes of short duration affect thermal perceptions of humans. After approximately 15–20 min under constant activity, subjective thermal responses approximated the steady-state response. The sensitivity of thermal sensation to changes in core temperature was higher for activity down-steps than for up-steps. A model was proposed that estimates transient thermal sensation after metabolic step-changes. Based on predictions by the model, weighting factors were suggested to estimate a representative average metabolic rate with varying activity levels, e.g. for the prediction of thermal sensation by steady-state comfort models. The activity during the most recent 5 min should be weighted 65%, during the prior 10–5 min 25% and during the prior 20–10 min 10%.     

Phenomenological vs. biophysical models of thermal stress in aquatic eggs

Predicting species responses to climate change is a central challenge in ecology. These predictions are often based on lab‐derived phenomenological relationships between temperature and fitness metrics. We tested one of these relationships using the embryonic stage of a Chinook salmon population. We parameterised the model with laboratory data, applied it to predict survival in the field, and found that it significantly underestimated field‐derived estimates of thermal mortality. We used a biophysical model based on mass transfer theory to show that the discrepancy was due to the differences in water flow velocities between the lab and the field. This mechanistic approach provides testable predictions for how the thermal tolerance of embryos depends on egg size and flow velocity of the surrounding water. We found support for these predictions across more than 180 fish species, suggesting that flow and temperature mediated oxygen limitation is a general mechanism underlying the thermal tolerance of embryos.     

A Bayesian approach to modeling associations between pulsatile hormones

Summary .  Many hormones are secreted in pulses. The pulsatile relationship between hormones regulates many biological processes. To understand endocrine system regulation, time series of hormone concentrations are collected. The goal is to characterize pulsatile patterns and associations between hormones. Currently each hormone on each subject is fitted univariately. This leads to estimates of the number of pulses and estimates of the amount of hormone secreted; however, when the signal-to-noise ratio is small, pulse detection and parameter estimation remains difficult with existing approaches. In this article, we present a bivariate deconvolution model of pulsatile hormone data focusing on incorporating pulsatile associations. Through simulation, we exhibit that using the underlying pulsatile association between two hormones improves the estimation of the number of pulses and the other parameters defining each hormone. We develop the one-to-one, driver–response case and show how birth–death Markov chain Monte Carlo can be used for estimation. We exhibit these features through a simulation study and apply the method to luteinizing and follicle stimulating hormones.     

Can quantum-bumps in photoreceptors be reconstructed from noise-data?

The method of reconstructing quantum bumps in photoreceptor cells from noise data by making use of shot noise theory is critically reviewed. The application of this method produces results irrespective of whether the conditions for reconstructing bumps by the method are satisfied or not and even irrespective of whether at high stimulus intensities quantum bumps exist or not. We argue that at high intensities the concept of quantum bumps indeed becomes physically meaningless and degenerates to a purely mathematical concept. In order to investigate the meaning of the results of the reconstruction method, we submit it to a test model for which bumps and single channel opening events can be evaluated analytically. By comparing the analytical results of the test model with that of the reconstruction method applied to the test model we find: (1) even at low intensities, the reconstructed bump values deviate from the analytical results by up to an order of magnitude due to the variability of the bumps, (2) at high intensities, the reconstruction method produces single chennel opening events rather than anything like a quantum bump. We also find, however, that there is no continuous transition from a bump at low intensities to a single channel event at high intensities.     

Pandemic velocity: Forecasting COVID-19 in the US with a machine learning & Bayesian time series compartmental model

Predictions of COVID-19 case growth and mortality are critical to the decisions of political leaders, businesses, and individuals grappling with the pandemic. This predictive task is challenging due to the novelty of the virus, limited data, and dynamic political and societal responses. We embed a Bayesian time series model and a random forest algorithm within an epidemiological compartmental model for empirically grounded COVID-19 predictions. The Bayesian case model fits a location-specific curve to the velocity (first derivative) of the log transformed cumulative case count, borrowing strength across geographic locations and incorporating prior information to obtain a posterior distribution for case trajectories. The compartmental model uses this distribution and predicts deaths using a random forest algorithm trained on COVID-19 data and population-level characteristics, yielding daily projections and interval estimates for cases and deaths in U.S. states. We evaluated the model by training it on progressively longer periods of the pandemic and computing its predictive accuracy over 21-day forecasts. The substantial variation in predicted trajectories and associated uncertainty between states is illustrated by comparing three unique locations: New York, Colorado, and West Virginia. The sophistication and accuracy of this COVID-19 model offer reliable predictions and uncertainty estimates for the current trajectory of the pandemic in the U.S. and provide a platform for future predictions as shifting political and societal responses alter its course.     

Seascape genetics: a coupled oceanographic-genetic model predicts population structure of Caribbean corals

Population genetics is a powerful tool for measuring important larval connections between marine populations [1-4]. Similarly, oceanographic models based on environmental data can simulate particle movements in ocean currents and make quantitative estimates of larval connections between populations possible [5-9]. However, these two powerful approaches have remained disconnected because no general models currently provide a means of directly comparing dispersal predictions with empirical genetic data (except, see [10]). In addition, previous genetic models have considered relatively simple dispersal scenarios that are often unrealistic for marine larvae [11-15], and recent landscape genetic models have yet to be applied in a marine context [16-20]. We have developed a genetic model that uses connectivity estimates from oceanographic models to predict genetic patterns resulting from larval dispersal in a Caribbean coral. We then compare the predictions to empirical data for threatened staghorn corals. Our coupled oceanographic-genetic model predicts many of the patterns observed in this and other empirical datasets; such patterns include the isolation of the Bahamas and an east-west divergence near Puerto Rico [3, 21-23]. This new approach provides both a valuable tool for predicting genetic structure in marine populations and a means of explicitly testing these predictions with empirical data.     

The importance of assessing parameter sensitivity when using biophysical models: a case study using plethodontid salamanders

Landscapes are continually changing due to numerous assaults, including habitat alteration, anthropogenic disturbances, and climate change. Understanding how species will respond to these changes is of critical importance for conservation and management. Mechanistic models, such as biophysical models (BPMs), are an increasingly popular tool to predict how local population dynamics or species’ distributions may be altered in response to environmental and climate changes. By mechanistically modeling relationships between environmental conditions, physiology and behavior, it is possible to make accurate predictions about how species may respond. However, BPMs are often difficult to implement due to lack of appropriate, species-specific data that is biologically realistic or relevant. In this study, we present a BPM for the salamander Plethodon jordani and assess how adding more biological realism has potential to alter model predictions about annual energy budgets. Additionally, we conducted local and global sensitivity analyses to evaluate the importance of accurately specifying model parameter values and functional relationships. We found that the addition of biological realism resulted in greater model complexity as well as substantially different estimates of energy balance. Correct parameterization of biophysical models is also critical, as small changes in parameter values can result in disproportionately large changes in downstream model estimates. Our model highlights the overall importance of using ecologically relevant and specific data for input parameters, as well as careful assessment of parameter sensitivity. We encourage researchers to be aware of the data they are using to parameterize BPMs, and urge the collection of system-specific data that is relevant in spatial and temporal scale. We also recommend greater and more transparent use of sensitivity analyses to provide a better understanding of the model, as well as greater confidence in model predictions.     

Finding confidence limits on population growth rates: Bootstrap and analytic methods

When predicting population dynamics, the value of the prediction is not enough and should be accompanied by a confidence interval that integrates the whole chain of errors, from observations to predictions via the estimates of the parameters of the model. Matrix models are often used to predict the dynamics of age- or size-structured populations. Their parameters are vital rates. This study aims (1) at assessing the impact of the variability of observations on vital rates, and then on model’s predictions, and (2) at comparing three methods for computing confidence intervals for values predicted from the models. The first method is the bootstrap. The second method is analytic and approximates the standard error of predictions by their asymptotic variance as the sample size tends to infinity. The third method combines use of the bootstrap to estimate the standard errors of vital rates with the analytical method to then estimate the errors of predictions from the model. Computations are done for an Usher matrix models that predicts the asymptotic (as time goes to infinity) stock recovery rate for three timber species in French Guiana. Little difference is found between the hybrid and the analytic method. Their estimates of bias and standard error converge towards the bootstrap estimates when the error on vital rates becomes small enough, which corresponds in the present case to a number of observations greater than 5000 trees.     

A simple model predicting individual weight change in humans

Excessive weight in adults is a national concern with over 2/3 of the US population deemed overweight. Because being overweight has been correlated to numerous diseases such as heart disease and type 2 diabetes, there is a need to understand mechanisms and predict outcomes of weight change and weight maintenance. A simple mathematical model that accurately predicts individual weight change offers opportunities to understand how individuals lose and gain weight and can be used to foster patient adherence to diets in clinical settings. For this purpose, we developed a one-dimensional differential equation model of weight change based on the energy balance equation paired to an algebraic relationship between fat-free mass and fat mass derived from a large nationally representative sample of recently released data collected by the Centers for Disease Control. We validate the model's ability to predict individual participants’ weight change by comparing model estimates of final weight data from two recent underfeeding studies and one overfeeding study. Mean absolute error and standard deviation between model predictions and observed measurements of final weights are less than 1.8±1.3 kg for the underfeeding studies and 2.5±1.6 kg for the overfeeding study. Comparison of the model predictions to other one-dimensional models of weight change shows improvement in mean absolute error, standard deviation of mean absolute error, and group mean predictions. The maximum absolute individual error decreased by approximately 60% substantiating reliability in individual weight-change predictions. The model provides a viable method for estimating individual weight change as a result of changes in intake and determining individual dietary adherence during weight-change studies.     

Inbreeding depression and the evolution of dispersal rates: a multilocus model

Inbreeding depression is one of the possible reasons organisms disperse. In this article, we present a two-locus model for the evolution of dispersal in the presence of inbreeding depression. The first locus codes for a modifier of the migration rate, while the second locus is a selected locus generating inbreeding depression. We express the change in frequency of the migration modifier as a function of allele frequencies and genetic associations and then use a quasi-equilibrium assumption to express genetic associations as functions of allele frequencies. Our model disentangles two effects of inbreeding depression: it gives an advantage to migrant individuals because their offspring are on average less homozygous, but it also decreases the degree of population structure, thus decreasing the strength of kin selection for dispersal. We then extend our model to include an infinite number of selected loci. When the cost of dispersal is not too high, the model predictions are confirmed by multilocus simulation results and show that inbreeding depression can have a substantial effect on the dispersal rate. For high costs of dispersal, we observe discrepancies between the model and the simulations, probably caused by associations among selected loci, which are neglected in the analysis.