Modeling physical growth using mixed effects models

This article demonstrates the use of mixed effects models for characterizing individual and sample average growth curves based on serial anthropometric data. These models are advancement over conventional general linear regression because they effectively handle the hierarchical nature of serial growth data. Using body weight data on 70 infants in the Born in Bradford study, we demonstrate how a mixed effects model provides a better fit than a conventional regression model. Further, we demonstrate how mixed effects models can be used to explore the influence of environmental factors on the sample average growth curve. Analyzing data from 183 infant boys (aged 3–15 months) from rural South India, we show how maternal education shapes infant growth patterns as early as within the first 6 months of life. The presented analyses highlight the utility of mixed effects models for analyzing serial growth data because they allow researchers to simultaneously predict individual curves, estimate sample average curves, and investigate the effects of environmental exposure variables. Am J Phys Anthropol, 2013. © 2012 Wiley Periodicals, Inc.     

Analysis of current status data with missing covariates

Statistical inference based on right-censored data for the proportional hazards (PH) model with missing covariates has received considerable attention, but interval-censored or current status data with missing covariates has not yet been investigated. Our study is partly motivated by the analysis of fracture data from the 2005 National Health Interview Survey Original Database in Taiwan, where the occurrence of fractures was interval censored and the covariate osteoporosis was not reported for all residents. We assume that the data are realized from a PH model. A semiparametric maximum likelihood estimate implemented by a hybrid algorithm is proposed to analyze current status data with missing covariates. A comparison of the performance of our method with full-cohort analysis, complete-case analysis, and surrogate analysis is made via simulation with moderate sample sizes. The fracture data are then analyzed.     

不同额外随机效应对估计协方差函数的影响

将6个世代的686头SD-Ⅱ系猪的生长记录资料用于研究,分别将窝效应和个体效应作为额外随机效应时对估计加性遗传和永久环境协方差函数的影响.配合将年龄的勒让德多项式作为自变量的随机回归模型,用平均信息约束最大似然(AIREML)法估计.结果表明,窝效应只能反映部分永久环境效应,估计协方差函数时需配合全阶多项式;而将个体效应作为额外随机效应,降阶配合即可.降阶配合涉及的参数较少,同时可以缓和协方差估值间的差异.对协方差函数的意义和应用做了讨论。 Abstract:Growth records of 686 pigs of SD- Il Swine Line over 6 generations were used to study the influence of litter effect and animal effect,which was used as different additional random effect,on estimating additive genetic and permanent environmental covariance functions.A random regression model with Legendre polynomials of age as independent variables was used to estimate the covariance functions by restricted maximum likelihood employing the average information algorithm (AIREML).The results showed that litter effect only reflected part of the permanent environmental effects and full order fit was necessary to estimate the covariance functions.However,a reduced order fit was feasible using animal effect as additional random effect.Moreover,it involved less parameters and smoothed out differences in estimates of the covariances.Significance and application of covariance functions are discussed.     

Fitting a regression model for genotype-by-environment data on heading dates in grasses by methods for nonlinear mixed models

The analysis of agricultural crop variety trials is usually complicated by the presence of genotype-by-environment interaction. A number of methods and models have been proposed to tackle this problem. One of the most common methods is the regression approach due to Yates and Cochran (1938, Journal of Agricultural Science 28, 556-580), in which performances of genotypes in the environments are regressed onto environmental means. The underlying regression model contains a multiplicative term with two unknown parameters (one for genotypes and one for environments). In the present paper, the model is modified by exchanging the role of genotypes and environments. Various diagnostic plots show that this modified model is adequate for a data set on heading dates in the grass species Dactylis glomerata. If environments are considered as a random factor while genotypes are taken as fixed, the model falls into the class of nonlinear mixed models. Recently, a number of procedures have been suggested for this class of models, which are based on first-order Taylor series expansion. Alternatively, the model can be estimated by maximum likelihood. This paper discusses the application of these methods for estimating parameters of the model.     

Estimating polymorphic growth curve sets with nonchronological data

1. When we collect the growth curves of many individuals, orderly variation in the curves is often observed rather than a completely random mixture of various curves. Small individuals may exhibit similar growth curves, but the curves differ from those of large individuals, whereby the curves gradually vary from small to large individuals. It has been recognized that after standardization with the asymptotes, if all the growth curves are the same (anamorphic growth curve set), the growth curve sets can be estimated using nonchronological data; otherwise, that is, if the growth curves are not identical after standardization with the asymptotes (polymorphic growth curve set), this estimation is not feasible. However, because a given set of growth curves determines the variation in the observed data, it may be possible to estimate polymorphic growth curve sets using nonchronological data.
2. In this study, we developed an estimation method by deriving the likelihood function for polymorphic growth curve sets. The method involves simple maximum likelihood estimation. The weighted nonlinear regression and least‐squares method after the log‐transform of the anamorphic growth curve sets were included as special cases.
3. The growth curve sets of the height of cypress (Chamaecyparis obtusa) and larch (Larix kaempferi) trees were estimated. With the model selection process using the AIC and likelihood ratio test, the growth curve set for cypress was found to be polymorphic, whereas that for larch was found to be anamorphic. Improved fitting using the polymorphic model for cypress is due to resolving underdispersion (less dispersion in real data than model prediction).
4. The likelihood function for model estimation depends not only on the distribution type of asymptotes, but the definition of the growth curve set as well. Consideration of these factors may be necessary, even if environmental explanatory variables and random effects are introduced.

     

Segmented linear regression models applied to the analysis of data from a cross-sectional growth experiment

Several different methods of analysis are applied to data consisting of weight measurements, taken at specified post-treatment times, of harvested thyroids from rats given one of four treatments. Previous studies of this type of data indicated that the growth is initially rapid, and that a second phase of less rapid growth is followed by a final phase in which little additional growth occurs. The data are further characterized by increasing variance through time. The primary purpose of the analysis is to study the effect of the treatments at the end of the study period. One-way analysis of variance tests among groups are performed on each day, but the results are not particularly helpful. However, results from two-way analyses of variance (over subsets of days and groups) are consistent with the three phase model and accordingly indicate significant group differences during each. Finally, maximum likelihood methods are used to fit a three part segmented linear regression model.     

Semiparametric transformation models for interval‐censored data in the presence of a cure fraction

Mixed case interval‐censored data arise when the event of interest is known only to occur within an interval induced by a sequence of random examination times. Such data are commonly encountered in disease research with longitudinal follow‐up. Furthermore, the medical treatment has progressed over the last decade with an increasing proportion of patients being cured for many types of diseases. Thus, interest has grown in cure models for survival data which hypothesize a certain proportion of subjects in the population are not expected to experience the events of interest. In this article, we consider a two‐component mixture cure model for regression analysis of mixed case interval‐censored data. The first component is a logistic regression model that describes the cure rate, and the second component is a semiparametric transformation model that describes the distribution of event time for the uncured subjects. We propose semiparametric maximum likelihood estimation for the considered model. We develop an EM type algorithm for obtaining the semiparametric maximum likelihood estimators (SPMLE) of regression parameters and establish their consistency, efficiency, and asymptotic normality. Extensive simulation studies indicate that the SPMLE performs satisfactorily in a wide variety of settings. The proposed method is illustrated by the analysis of the hypobaric decompression sickness data from National Aeronautics and Space Administration.     

Statistical analysis of amounts and ratios in pheromone composition: a mixed model approach

Analysis of variation in pheromone amounts and ratios between individuals is usually performed separately for amounts and ratios of the different components. Non-parametric tests are regularly applied. This way of analysis is statistically correct, yet, limited for several reasons. The use of a parametric linear mixed model approach to analyze both amounts and ratios of different components at the same time is proposed. This method appears to be very flexible and may facilitate the analysis of pheromone data.     

Sample size recalculation in multicenter randomized controlled clinical trials based on noncomparative data

Many late-phase clinical trials recruit subjects at multiple study sites. This introduces a hierarchical structure into the data that can result in a power-loss compared to a more homogeneous single-center trial. Building on a recently proposed approach to sample size determination, we suggest a sample size recalculation procedure for multicenter trials with continuous endpoints. The procedure estimates nuisance parameters at interim from noncomparative data and recalculates the sample size required based on these estimates. In contrast to other sample size calculation methods for multicenter trials, our approach assumes a mixed effects model and does not rely on balanced data within centers. It is therefore advantageous, especially for sample size recalculation at interim. We illustrate the proposed methodology by a study evaluating a diabetes management system. Monte Carlo simulations are carried out to evaluate operation characteristics of the sample size recalculation procedure using comparative as well as noncomparative data, assessing their dependence on parameters such as between-center heterogeneity, residual variance of observations, treatment effect size and number of centers. We compare two different estimators for between-center heterogeneity, an unadjusted and a bias-adjusted estimator, both based on quadratic forms. The type 1 error probability as well as statistical power are close to their nominal levels for all parameter combinations considered in our simulation study for the proposed unadjusted estimator, whereas the adjusted estimator exhibits some type 1 error rate inflation. Overall, the sample size recalculation procedure can be recommended to mitigate risks arising from misspecified nuisance parameters at the planning stage.     

The analysis of longitudinal data using mixed model L-splines

Summary .   L-splines are a large family of smoothing splines defined in terms of a linear differential operator. This article develops L-splines within the context of linear mixed models and uses the resulting mixed model L-spline to analyze longitudinal data from a grassland experiment. In the spirit of time-series analysis, a periodic mixed model L-spline is developed, which partitions data into a smooth periodic component plus smooth long-term trend.     

Local estimation of age-dependent variance components from longitudinal twin data

In the study of longitudinal twin and family data, interest is often in the covariance structure of the data and the decomposition of this covariance structure into genetic and environmental components rather than in estimating the mean function. Various parametric models for covariance structures have been proposed but, e.g., in studies of children where growth spurts occur at various ages, it is difficult to a priori determine an appropriate parametric model for the covariance structure. In particular, there is a general lack of the visualization procedures, such as lowess, that are invaluable in the initial stages of constructing a parametric model for a mean function. Here we use kernel smoothing to modify a cross-sectional approach based on the sample covariance matrices to obtain smoothed estimates of the genetic and environmental variances and correlations for longitudinal twin data. The methods are proposed to be exploratory as an aid to parametric modeling rather than inferential, although approximate asymptotic standard errors are derived in the Appendix.     

Semiparametric analysis of zero-inflated count data

Medical and public health research often involve the analysis of count data that exhibit a substantially large proportion of zeros, such as the number of heart attacks and the number of days of missed primary activities in a given period. A zero-inflated Poisson regression model, which hypothesizes a two-point heterogeneity in the population characterized by a binary random effect, is generally used to model such data. Subjects are broadly categorized into the low-risk group leading to structural zero counts and high-risk (or normal) group so that the counts can be modeled by a Poisson regression model. The main aim is to identify the explanatory variables that have significant effects on (i) the probability that the subject is from the low-risk group by means of a logistic regression formulation; and (ii) the magnitude of the counts, given that the subject is from the high-risk group by means of a Poisson regression where the effects of the covariates are assumed to be linearly related to the natural logarithm of the mean of the counts. In this article we consider a semiparametric zero-inflated Poisson regression model that postulates a possibly nonlinear relationship between the natural logarithm of the mean of the counts and a particular covariate. A sieve maximum likelihood estimation method is proposed. Asymptotic properties of the proposed sieve maximum likelihood estimators are discussed. Under some mild conditions, the estimators are shown to be asymptotically efficient and normally distributed. Simulation studies were carried out to investigate the performance of the proposed method. For illustration purpose, the method is applied to a data set from a public health survey conducted in Indonesia where the variable of interest is the number of days of missed primary activities due to illness in a 4-week period.     

Joint modeling of longitudinal and survival data via a common frailty

We develop a joint model for the analysis of longitudinal and survival data in the presence of data clustering. We use a mixed effects model for the repeated measures that incorporates both subject- and cluster-level random effects, with subjects nested within clusters. A Cox frailty model is used for the survival model in order to accommodate the clustering. We then link the two responses via the common cluster-level random effects, or frailties. This model allows us to simultaneously evaluate the effect of covariates on the two types of responses, while accounting for both the relationship between the responses and data clustering. The model was motivated by a study of end-stage renal disease patients undergoing hemodialysis, where we wished to evaluate the effect of iron treatment on both the patients' hemoglobin levels and survival times, with the patients clustered by enrollment site.