Accelerated maximum likelihood parameter estimation for stochastic biochemical systems

ABSTRACT: BACKGROUND: A prerequisite for the mechanistic simulation of a biochemical system is detailed knowledge of its kinetic parameters. Despite recent experimental advances, the estimation of unknown parameter values from observed data is still a bottleneck for obtaining accurate simulation results. Many methods exist for parameter estimation in deterministic biochemical systems; methods for discrete stochastic systems are less well developed. Given the probabilistic nature of stochastic biochemical models, a natural approach is to choose parameter values that maximize the probability of the observed data with respect to the unknown parameters, a.k.a. the maximum likelihood parameter estimates (MLEs). MLE computation for all but the simplest models requires the simulation of many system trajectories that are consistent with experimental data. For models with unknown parameters, this presents a computational challenge, as the generation of consistent trajectories can be an extremely rare occurrence. RESULTS: We have developed Monte Carlo Expectation-Maximization with Modified Cross-Entropy Method (MCEM2): an accelerated method for calculating MLEs that combines advances in rare event simulation with a computationally efficient version of the Monte Carlo expectation-maximization (MCEM) algorithm. Our method requires no prior knowledge regarding parameter values, and it automatically provides a multivariate parameter uncertainty estimate. We applied the method to five stochastic systems of increasing complexity, progressing from an analytically tractable pure-birth model to a computationally demanding model of yeast-polarization. Our results demonstrate that MCEM2 substantially accelerates MLE computation on all tested models when compared to a stand-alone version of MCEM. Additionally, we show how our method identifies parameter values for certain classes of models more accurately than two recently proposed computationally efficient methods. CONCLUSIONS: This work provides a novel, accelerated version of a likelihood-based parameter estimation method that can be readily applied to stochastic biochemical systems. In addition, our results suggest opportunities for added efficiency improvements that will further enhance our ability to mechanistically simulate biological processes.     

Approximate maximum likelihood estimation for logistic regression with covariate measurement error

In nutritional epidemiology, dietary intake assessed with a food frequency questionnaire is prone to measurement error. Ignoring the measurement error in covariates causes estimates to be biased and leads to a loss of power. In this paper, we consider an additive error model according to the characteristics of the European Prospective Investigation into Cancer and Nutrition (EPIC)‐InterAct Study data, and derive an approximate maximum likelihood estimation (AMLE) for covariates with measurement error under logistic regression. This method can be regarded as an adjusted version of regression calibration and can provide an approximate consistent estimator. Asymptotic normality of this estimator is established under regularity conditions, and simulation studies are conducted to empirically examine the finite sample performance of the proposed method. We apply AMLE to deal with measurement errors in some interested nutrients of the EPIC‐InterAct Study under a sensitivity analysis framework.     

One-step targeted maximum likelihood estimation for time-to-event outcomes

Researchers in observational survival analysis are interested in not only estimating survival curve nonparametrically but also having statistical inference for the parameter. We consider right-censored failure time data where we observe n independent and identically distributed observations of a vector random variable consisting of baseline covariates, a binary treatment at baseline, a survival time subject to right censoring, and the censoring indicator. We assume the baseline covariates are allowed to affect the treatment and censoring so that an estimator that ignores covariate information would be inconsistent. The goal is to use these data to estimate the counterfactual average survival curve of the population if all subjects are assigned the same treatment at baseline. Existing observational survival analysis methods do not result in monotone survival curve estimators, which is undesirable and may lose efficiency by not constraining the shape of the estimator using the prior knowledge of the estimand. In this paper, we present a one-step Targeted Maximum Likelihood Estimator (TMLE) for estimating the counterfactual average survival curve. We show that this new TMLE can be executed via recursion in small local updates. We demonstrate the finite sample performance of this one-step TMLE in simulations and an application to a monoclonal gammopathy data.     

Maximum likelihood estimation for stochastic first order reactions

Small sample properties of the maximum likelihood estimator for the rate constant of a stochastic first order reaction are investigated. The approximate bias and variance of the maximum likelihood estimator are derived and tabulated. If observations of the system are made at timesiτ,i=1, 2, ...,N; τ>0, the observational spacing τ which minimizes the approximate variance of the maximum likelihood estimator is found. The non-applicability of large sample theory to confidence interval derivation is demonstrated by examination of the relative likelihood. Bartlett’s method is employed to derive approximate confidence limits, and is illustrated by using simulated kinetic runs.     

Stochastic search strategy for estimation of maximum likelihood phylogenetic trees

The maximum likelihood (ML) method of phylogenetic tree construction is not as widely used as other tree construction methods (e.g., parsimony, neighbor-joining) because of the prohibitive amount of time required to find the ML tree when the number of sequences under consideration is large. To overcome this difficulty, we propose a stochastic search strategy for estimation of the ML tree that is based on a simulated annealing algorithm. The algorithm works by moving through tree space by way of a "local rearrangement" strategy so that topologies that improve the likelihood are always accepted, whereas those that decrease the likelihood are accepted with a probability that is related to the proportionate decrease in likelihood. Besides greatly reducing the time required to estimate the ML tree, the stochastic search strategy is less likely to become trapped in local optima than are existing algorithms for ML tree estimation. We demonstrate the success of the modified simulated annealing algorithm by comparing it with two existing algorithms (Swofford's PAUP* and Felsenstein's DNAMLK) for several theoretical and real data examples.     

Bayesian and maximum likelihood estimation of genetic maps

There has recently been increased interest in the use of Markov Chain Monte Carlo (MCMC)-based Bayesian methods for estimating genetic maps. The advantage of these methods is that they can deal accurately with missing data and genotyping errors. Here we present an extension of the previous methods that makes the Bayesian method applicable to large data sets. We present an extensive simulation study examining the statistical properties of the method and comparing it with the likelihood method implemented in Mapmaker. We show that the Maximum A Posteriori (MAP) estimator of the genetic distances, corresponding to the maximum likelihood estimator, performs better than estimators based on the posterior expectation. We also show that while the performance is similar between Mapmaker and the MCMC-based method in the absence of genotyping errors, the MCMC-based method has a distinct advantage in the presence of genotyping errors. A similar advantage of the Bayesian method was not observed for missing data. We also re-analyse a recently published set of data from the eggplant and show that the use of the MCMC-based method leads to smaller estimates of genetic distances.     

The EM algorithm for maximum likelihood estimation in the mover-stayer model

The discrete-time mover-stayer model (Blumen, Kogan, and McCarthy, 1955, The Industrial Mobility of Labor as a Probability Process, Ithaca, New York: Cornell University Press) is a useful model for studying changes over time in heterogeneous populations. Using the EM algorithm, we present an alternative method for obtaining maximum likelihood estimates of the parameters of the mover-stayer model, and consider an extension of the basic model to the problem of incomplete follow-up in panel studies. The models and the methods are illustrated with data from a community-based survey of changes in mental health status over a 1-year period.     

A computer program for stepwise logistic regression using maximum likelihood estimation

A computer program has been written which performs a stepwise selection of variables for logistic regression using maximum likelihood estimation. The selection procedure is based on likelihood ratio tests for the coefficients. These tests are used in a forward selection and a backward elimination at each step. The use of the program is illustrated by several examples.