Polynomial Growth Curve Research Articles

Test for mean matrix in GMANOVA model under heteroscedasticity and non-normality for high-dimensional data

This paper develops a unified testing methodology for high-dimensional generalized multivariate analysis of variance (GMANOVA) models. We derive a test of the bilateral linear hypothesis on the mean matrix in a general scenario where the dimensions of the observed vector may exceed the sample size, design may be unbalanced, the population distribution may be non-normal and the underlying group covariance matrices may be unequal. The suggested methodology is suitable for many inferential problems, such as the one-way MANOVA test and the test for multivariate linear hypothesis on the mean in the polynomial growth curve model. As a key component of our test procedure, we propose a bias-corrected estimator of the Frobenius norm of the mean matrix. We derive null and non-null asymptotic distributions of the test statistic under a general high-dimensional asymptotic framework that allows the dimensionality to arbitrarily exceed the sample size of a group. The accuracy of the proposed test in a finite sample setting is investigated through simulations conducted for several high-dimensional scenarios and various underlying population distributions in combination with different within-group covariance structures. For a practical demonstration we consider a daily Canadian temperature dataset that exhibits group structure, and conclude that the interaction of latitude and longitude has no effect to predict the temperature.

Body mass index across adulthood and the development of airflow obstruction and emphysema.

Low body mass index (BMI) is associated with COPD, but temporal relationships between airflow obstruction (AO) development and emphysematous change are unclear. We investigated longitudinal changes in BMI, AO, and lung density throughout adulthood using data from the Framingham Offspring Cohort (FOC). BMI trajectories were modelled throughout adulthood in 4587 FOC participants from Exam 2 (mean age = 44), through Exam 9 (mean age = 71), in AO participants and non-AO participants (AO n = 1036), determined by spirometry, using fractional polynomial growth curves. This process was repeated for low lung density (LLD) and non LLD participants (LLD n = 225) determined by Computed Tomography. Spirometry decline was compared separately between tertiles of BMI in those aged <40years and associations between fat and lean mass (measured using Dual Energy X-ray Absorptiometry, DEXA) and development of AO and LLD were also assessed. Additional analyses were performed with adjustment for smoking volume. The BMI trajectory from 30years of age was visually lower in the AO group than both non-AO smokers (non-<AO-S) and non-AO non-smokers (non-AO-N). Similarly, BMI trajectories were visually lower in participants with LLD throughout adulthood compared to normal lung density smokers and non-smokers. Differences remained after adjustment for smoking volume. The lowest BMI tertile in ages <40years was associated with the steepest subsequent decline in FEV1/FVC ratio in both sexes. Mean BMI is lower throughout adulthood in AO and LLD participants. Lower BMI is associated with a steeper decline in the ratio of FEV1/FVC. These findings suggest body mass may precede and potentially have a role in the development of COPD lung pathophysiology.

Statistical inference in a growth curve quantile regression model for longitudinal data.

This article describes a polynomial growth curve quantile regression model that provides a comprehensive assessment about the treatment effects on the changes of the distribution of outcomes over time. The proposed model has the flexibility, as it allows the degree of a polynomial to vary across quantiles. A high degree polynomial model fits the data adequately, yet it is not desirable due to the complexity of the model. We propose the model selection criterion based on an empirical loglikelihood that consistently identifies the optimal degree of a polynomial at each quantile. After the parsimonious model is fitted to the data, the hypothesis test is further developed to evaluate the treatment effects by comparing the growth curves. It is shown that the proposed empirical loglikelihood ratio test statistic follows a chi-square distribution asymptotically under the null hypothesis. Various simulation studies confirm that the proposed test successfully detects the difference between the curves across quantiles. When the empirical loglikelihood is employed, we incorporate the within-subject correlation commonly existing in longitudinal data and gain estimation efficiency of the quantile regression parameters in the growth curve model. The proposed process is illustrated through the analysis of randomized controlled longitudinal depression data.

Evaluation of Weight Loss Over Time in Cats with Chronic Kidney Disease.

BackgroundThin body condition and weight loss are common in cats with chronic kidney disease (CKD). However, the time course and progression of weight loss before and after diagnosis have not been thoroughly evaluated.Hypothesis/ObjectivesTo describe weight loss in cats with CKD before and after diagnosis and its relationship to survival.AnimalsA total of 569 cats (55.5% females and 44.5% males) with CKD from 6 US veterinary practices for which International Renal Interest Society (IRIS) stage, age, date of CKD diagnosis, and at least two body weight measurements were available.MethodsBody weight measurements were analyzed by time windows and polynomial growth curve analysis. Survival analysis was performed by Kaplan–Meier curves and log‐rank tests.ResultsMedian age at diagnosis was 14.9 years (range, 5.0–22.8 years). Cats were categorized at diagnosis as IRIS stage 1 (n = 34 [6%]), stage 2 (n = 345 [61%]), stage 3 (n = 141 [25%]), and stage 4 (n = 49 [9%]). Median body weight at diagnosis was 4.2 kg (range, 1.6–9.9 kg). Cats lost a median of 8.9% of body weight in the 12 months before diagnosis, but weight loss was already present 3 years before diagnosis and accelerated after diagnosis of CKD. Cats <4.2 kg at the time of diagnosis had significantly shorter survival time compared to cats ≥ 4.2 kg at diagnosis (P < .0001).Conclusions and Clinical ImportanceWeight loss can be detected in cats before diagnosis of CKD, accelerates after diagnosis, and is associated with shorter survival. Tracking body weight may help clinicians in earlier diagnosis of CKD.

High-dimensional [formula omitted] in the growth curve model

The AIC and its modifications have been proposed for selecting the degree in a polynomial growth curve model under a large-sample framework when the sample size n is large, but the dimension p is fixed. In this paper, first we propose a high-dimensional AIC (denoted by HAIC) which is an asymptotic unbiased estimator of the AIC-type risk function defined by the expected log-predictive likelihood or equivalently the Kullback–Leibler information, under a high-dimensional framework such that p/n→c∈[0,1). It is noted that our new criterion gives an estimator with small biases in a wide range of p and n. Next we derive asymptotic distributions of AIC and HAIC under the high-dimensional framework. Through a Monte Carlo simulation, we note that these new approximations are more accurate than the approximations based on a large-sample framework.

Consistency of AIC and its modification in the growth curve model under a large-(q,n) framework

The AIC and its modifications have been proposed for selecting the degree in a polynomial growth curve model under a large-sample framework and a high-dimensional framework by Satoh, Kobayashi and Fujikoshi [9] and Fujikoshi, Enomoto and Sakurai [4], respectively. They note that the AIC and its modifications have no consistency property. In this paper we consider asymptotic properties of the AIC and its modification when the number q of groups or explanatory variables and the sample size n are large. First we show that the AIC has a consistency property under a large-(q,n) framework such that q/n→d∈[0,1), under a condition on the noncentrality matrix, but the dimension p is fixed. Next we propose a modification of the AIC (denoted by MAIC) which is an asymptotic unbiased estimator of the risk under the asymptotic framework. It is shown that MAIC has a consistency property under a condition on the noncentrality matrix. Our results are checked numerically by conducting a Mote Carlo simulation.

Developmental models for children's temperament: alternatives to chronometric polynomial curves

Partridge and Lerner (2007), in a secondary analysis of the New York Longitudinal Study, employed a chronometric polynomial growth curve model to argue that the developmental course of difficult temperament follows a non-linear trajectory over the first 5 years of life. The free curve slope intercept (FCSI) growth curve model of Meredith and Tisak (1990) is presented as a preferable conceptual alternative because it contains a number of currently popular statistical models, including repeated measures multivariate analysis of variance, factor mean, linear growth, linear factor analysis, and hierarchical linear models as special cases. As such, researchers can compare the fit of each of these models relative to the FCSI model, and, at times, to each other. The present paper conducts a re-analysis of the data, and establishes that fit of the FCSI model is arguably better than other statistical alternatives. The FCSI model is also used as the basis for identifying subgroups of individuals with their qualitatively distinct growth patterns within a growth mixture modeling framework. Copyright © 2010 John Wiley & Sons, Ltd.

Developmental changes in cerebral grey and white matter volume from infancy to adulthood

In order to quantify human brain development in vivo, high resolution magnetic resonance images of 158 normal subjects from infancy to young adulthood were studied (age range 3 months–30 years, 71 males, 87 females). Data were analysed using algorithms based on voxel-based morphometry (VBM) (an objective whole brain processing technique) to generate global volume measures of whole brain, grey matter (GM) and white matter (GM). Gender-specific development of WM and GM volumes is characterised using a piecewise polynomial growth curve model to account for the non-linear nature of human brain development, implemented using Markov chain Monte Carlo simulation.The statistical method employed in this study proved to be successful and robust in the characterisation of brain development. The resulting growth curve parameter estimates lead to the following observations: total brain volume is demonstrated to undergo an initial rapid spurt. The total GM volume peaks during childhood and decreases thereafter, whereas total WM volume increases up to young adulthood. Relative to brain size, GM decreases and WM increases markedly over this age range in a non-linear manner, resulting in an increasing WM-to-GM ratio over much of the observed age range. In addition, significant gender differences are found. In general, brain volume and total white and grey matter volume are larger in males than in females, with a time-dependent difference over the age range studied. Over part of the observed age range females tend to have more GM volume relative to brain size and lower WM-to-GM ratio than males.The presented findings should be taken into account when investigating physiological and pathological changes during brain development.

Goodness-of-Fit Testing for Exponential Polynomial Growth Curves

A goodness-of-fit test is proposed for the family of exponential polynomial growth curve models (EPGCM; Heinen, 1999), which has wide applications in different areas of science. The exponential growth curve model (EGCM), the most prominent member of the EPGCM family, is a simple and biologically meaningful growth model. Other members of the EPGCM family also cover many realistic growth processes. Thus, a goodness-of-fit test for the EPGCM class has substantial practical value. The goodness-of-fit test developed here is based on the properties of finite differences. The performance of the theory developed is illustrated through simulation and analysis of real data.

A Natural Goodness-of-Fit Testing Procedure for the Logistic Growth Curve Model

Growth curve models are frequently used in a wide range of disciplines such as biology, ecology, demography, population dynamics etc. Living organisms exhibit different types of growth patterns. To analyze these curves investigators need adequate parametric models. A proper identification of the growth model is very important for the appropriateness of the subsequent analysis. In this paper we develop a natural goodness of fit test for the logistic growth curve model. Bhattacharya et al. (2004, 2008) have provided some interesting approaches based on Hill's method of finite differences (Hill 1968) in case of the exponential and exponential polynomial growth curve models. But in case of the logistic model their approach leads to very complicated and cumbersome mathematics. Basu and Bhattacharj6ee (2006) have presented an alternative method to test the goodness of fit for the exponential growth curve model by directly modeling the relative growth rate rather than the size variable itself. In this paper we extend that approach for testing goodness of fit in case of the logistic growth curve model.

A Natural Goodness­of­Fit Test for the Exponential Growth Curve Model Based on the Relative Growth Rate

This paper develops a natural goodness-of-fit test for the exponential growth curve model (EGCM) based on the property of finite differences. The EGCM has wide applications in various scientific fields such as biology, ecology, demography, population dynamics, finance, econometrics etc. Hill (1968, Biometrics, 4, 192-196) proposed a test for the polynomial growth curve model (PGCM) based on finite differences of the size variable. However, the PGCM has limited practical use and does not have any biologically interpretable para.ffieters. In comparison, the exponential model is more realistic and biologically meaningful; a goodness-of-fit test for the EGCM has substantial practical value. Bhattacharya et. al. (2004) generalized Hill's approach to the case of the EGCM based on appropriate finite differences of functions of the size variable. In this paper we present an alternative test for the EGCM by directly modeling the relative growth rate (rather than the size variable itself) and hence avoid the assumptions necessary in Bhattacharya et al (2004). The performance of the theory developed is illustrated with several sets of real data and through simulation. AMS {2000} Subject Classification: 62F03, 62P10.

Latent Curve Analyses of Longitudinal Twin Data Using a Mixed-Effects Biometric Approach

In a recent article McArdle and Prescott (2005) showed how simultaneous estimation of the biometric parameters can be easily programmed using current mixed-effects modeling programs (e.g., SAS PROC MIXED). This article extends these concepts to deal with mixed-effect modeling of longitudinal twin data. The biometric basis of a polynomial growth curve model was used by Vandenberg and Falkner (1965) and this general class of longitudinal models was represented in structural equation form as a latent curve model by McArdle (1986). The new mixed-effects modeling approach presented here makes it easy to analyze longitudinal growth-decline models with biometric components based on standard maximum likelihood estimation and standard indices of goodness-of-fit (i.e., chi(2), df, epsilon(a)). The validity of this approach is first checked by the creation of simulated longitudinal twin data followed by numerical analysis using different computer programs (i.e., Mplus, Mx, MIXED, NLMIXED). The practical utility of this approach is examined through the application of these techniques to real longitudinal data from the Swedish Adoption/Twin Study of Aging (Pedersen et al., 2002). This approach generally allows researchers to explore the genetic and nongenetic basis of the latent status and latent changes in longitudinal scores in the absence of measurement error. These results show the mixed-effects approach easily accounts for complex patterns of incomplete longitudinal or twin pair data. The results also show this approach easily allows a variety of complex latent basis curves, such as the use of age-at-testing instead of wave-of-testing. Natural extensions of this mixed-effects longitudinal approach include more intensive studies of the available data, the analysis of categorical longitudinal data, and mixtures of latent growth-survival/frailty models.

Latent Curve Analyses of Longitudinal Twin Data Using a Mixed-Effects Biometric Approach

AbstractIn a recent article McArdle and Prescott (2005) showed how simultaneous estimation of the bio-metric parameters can be easily programmed using current mixed-effects modeling programs (e.g., SAS PROC MIXED). This article extends these concepts to deal with mixed-effect modeling of longitudinal twin data. The biometric basis of a polynomial growth curve model was used by Vandenberg and Falkner (1965) and this general class of longitudinal models was represented in structural equation form as a latent curve model by McArdle (1986). The new mixed-effects modeling approach presented here makes it easy to analyze longitudinal growth-decline models with biometric components based on standard maximum likelihood estimation and standard indices of goodness-of-fit (i.e., χ2, df, εa). The validity of this approach is first checked by the creation of simulated longitudinal twin data followed by numerical analysis using different computer programs (i.e., Mplus, Mx, MIXED, NLMIXED). The practical utility of this approach is examined through the application of these techniques to real longitudinal data from the Swedish Adoption/Twin Study of Aging (Pedersen et al., 2002). This approach generally allows researchers to explore the genetic and nongenetic basis of the latent status and latent changes in longitudinal scores in the absence of measurement error. These results show the mixed-effects approach easily accounts for complex patterns of incomplete longitudinal or twin pair data. The results also show this approach easily allows a variety of complex latent basis curves, such as the use of age-at-testing instead of wave-of-testing. Natural extensions of this mixed-effects longitudinal approach include more intensive studies of the available data, the analysis of categorical longitudinal data, and mixtures of latent growth-survival/ frailty models.

LR Tests for some Linear Hypotheses in an Extended Growth Curve Model

SYNOPTIC ABSTRACTIn this paper we consider the problem of testing some linear hypotheses in an extended growth curve model which is described for the data set of k clusters. The model includes the one whose mean structure consists of polynomial growth curves with k different degrees. First we derive likelihood ratio tests for adequacy of the extended growth curve model. Next we consider to test a general linear hypothesis of the parameter matrix for each of k clusters. In general, we propose to use Wald type test. However, likelihood ratio tests are derived for two important special cases. It is shown that the null distributions of our likelihood ratio statistics can be expressed as products of independent lambda statistics. We also give a numerical example.

Efficiency of the MLE in a multivariate parallel profile model with random effects

In this paper we consider a multivariate parallel profile model with polynomial growth curves. The covariance structure based on a random e¤ects model is assumed. The maximum likelihood estimators (MLE’s) are obtained under the random e¤ects covariance structure. The efficiency of the MLE is discussed.

Multivariate distributions with correlation matrices for nonlinear repeated measurements

The polynomial growth curve model based on the multivariate normal distribution has dominated the analysis of continuous longitudinal repeated measurements for the last 50 years. The main reasons include the ease of modelling dependence because of the availability of the correlation matrix and the linearity of the regression coefficients. However, a variety of other useful distributions also involve a correlation matrix: the multivariate Student's t, multivariate power exponential, and multivariate skew Laplace distributions as well as Gaussian copulas with arbitrarily chosen marginal distributions. With modern computing power and software, nonlinear regression functions can be fitted as easily as linear ones. By a number of examples, we show that these distributions, combined with nonlinear regression functions, generally yield an improved fit, as compared to the standard polynomial growth curve model, and can provide different conclusions.

The β 2-adrenergic receptor Arg 16-gly polymorphism and interactions involving β 2- and β 3-adrenergic receptor polymorphisms are associated with variations in longitudinal serum lipid profiles: The Bogalusa Heart Study

We examined the effects of combined genotypes of the β 2-adrenergic receptor (AR) Arg 16-Gly and β 3-AR Trp 64-Arg polymorphisms on longitudinal serum total (T-C) and low-density lipoprotein cholesterol (LDL-C) profiles in 1,198 subjects examined multiple times (6,488 observations) from 1973 to 1996 in the Bogalusa Heart Study, at ages from 4.5 to 38 years. Within 5-year age groups, T-C was significantly ( P < .05) higher in β 2-AR Arg 16/Arg 16 homozygotes than in Gly 16 carriers among those 4 to 8 (171.4 ± 30.0 v 161.5 ± 27.7 mg/dL), 9 to 13 (167.7 ± 28.6 v 162.4 ± 27.4 mg/dL), and 14 to 18 (158.8 ± 29.6 v 154.7 ± 27.5 mg/dL) years of age, but not in those 19 to 23, 24 to 28, 29 to 33, or 34 to 38 years of age. The β 3-AR polymorphism was not associated with variation in either T-C or LDL-C. In multilevel polynomial growth curve models, the combination of the β 2-AR Arg 16/Arg 16 genotype with either the β 3-AR Arg 64/Arg 64 or Trp 64/Arg 64 genotypes, denoted AA/AX, was associated with variation in longitudinal T-C ( P < .01) and LDL-C ( P < .01) profiles. The association between combined β 2/β 3-AR genotype and lipid profiles differed among race/sex groups, being most marked in black females, in whom the AA/AX combination was associated with higher T-C and LDL-C profiles across all ages. In White males, the AA/AX combination was most strongly associated with higher lipids in adults. In black males and white females, lipid profiles differed little between genotype groups. Our findings suggest that the β 2-AR Arg 16-Gly genotype influences T-C and LDL-C levels in an age-specific manner, that it may interact with β 3-AR Trp 64-Arg genotypes to influence longitudinal T-C and LDL-C profiles, and that the effect of combined β 2/β 3-AR genotypes on T-C and LDL-C profiles may differ among race/sex groups.

The impact of controlled breeding on milk production in pastoral goats in northern Kenya: an application of polynomial growth curve fitting

The paper investigates the effect of controlled seasonal breeding on milk production in a herd of Small East African (SEA) goats. Polynomial growth curve models were fitted to both daily and cumulative milk yield data obtained from an experiment conducted over a period of 4 years (1984–88) under simulated pastoral herd management in Isiolo District, northern Kenya. The experimental treatment consisted of six different mating seasons per year, which were replicated three times over the course of the experiment.Milk yields in the first 2 weeks of lactation were negatively affected (&lt;400 g/day) when kidding took place between June and September, whereas maximum initial yields of about 450 to 550 g/day were achieved at the onset and during the long rainy season. Multiple peaks in milk yield curves were observed when a rainy season occurred after about the first half of the lactation period. In terms of total amount of milk produced until 28 weeks of lactation, the production system could benefit from the introduction of a restricted breeding management allowing does to be bred in the period from June to November, with total milk yields being estimated at approximately 60 kg of milk. Maximum milk production until weaning can be expected to be achieved by does mated between October and January (between 46 and 48 kg of milk). The present experiment has revealed that mating just prior to or during the long rainy season leads to low milk yields until weaning and significantly increases the incidence of early kid deaths.It is concluded that evaluating milk production in goat herds exposed to strong seasonal changes in forage supply is perhaps best carried out in terms of cumulative milk yields, instead of average daily yields, which are subject to large fluctuations. Furthermore, under these conditions fitting polynomial growth curves to longitudinal milk yield data using the general linear mixed model appears to be more appropriate than the estimation of non-linear algebraic lactation curves.

SIMULTANEOUS CONFIDENCE REGIONS IN AN EXTENDED GROWTH CURVE MODEL WITH k HIERARCHICAL WITHIN-INDIVIDUALS DESIGN MATRICES

ABSTRACT We consider an extended growth curve model with k hierarchical within-individuals design matrices. The model includes the one whose mean structure consists of polynomial growth curves with k different degrees. First we propose certain simple estimators of the mean and covariance parameters which are closely related to the MLE's. Using these estimators we construct simultaneous confidence regions for each or all of k growth curves which is an extension of Fujikoshi.[2] A numerical example with k = 3 is also given.

Estimation in a random effects model with parallel polynomial growth curves

In this paper we consider an extended growth curve model with two hierarchical within-individuals design matrices, which is useful in analyzing mean profiles of several groups with parallel polynomial growth curves. The covariance structure based on a random e¤ects model is assumed. The maximum likelihood estimators (MLE’s) are obtained under the random e¤ects covariance structure. The efficiency of the MLE is discussed. A numerical example is also given.