Quadratic Growth Model Research Articles

Education and trajectories of cognitive decline over 9 years in very old people: methods and risk analysis

the investigation of cognitive decline in the older population has been hampered by analytical considerations. Most studies of older people over prolonged periods suffer from loss to follow-up, yet this has seldom been investigated fully to date. Such considerations limit our understanding of how basic variables such as education can affect cognitive trajectories. we examined cognitive trajectories in a population-based cohort study in Cambridge, UK, of people aged 75 and over in whom multiple interviews were conducted over time. Cognitive function was assessed using the Mini-Mental State Examination (MMSE). Socio-demographic variables were measured, including educational level and social class. An age-based quadratic latent growth model was fitted to cognitive scores. The effect of socio-demographic variables was examined on all latent variables and the probability of death and dropout. at baseline, age, education, social class and mobility were associated with cognitive performance. Education and social class were not related to decline or its rate of change. In contrast, poor mobility was associated with lower cognitive performance, increased cognitive decline and increased rate of change of cognitive decline. Gender, age, mobility and cognitive ability predicted death and dropout contrary to much of the current literature, education was not related to rate of cognitive decline or change in this rate as measured by MMSE. Higher levels of education do not appear to protect against cognitive decline, though if the MMSE is used in the diagnostic process, individuals with less education may be diagnosed as having dementia somewhat earlier.

Longitudinal predictors of attitudes toward aging among women with multiple sclerosis.

The purpose of the study was to explore the impact of change in functional limitation (FL), controlling for social support (SS), on attitudes toward aging using longitudinal survey data collected over a 7-year period. The 503 women with multiple sclerosis (MS; age, M=57 years, SD=10.25) were mostly Anglo (93%) and married (69%). First, the authors identified a quadratic growth model as the best for describing change in FL. Next, they considered SS as a time-varying covariate of FL to assess both within- and between-individual effects of SS on FL over time. Within individuals, higher FL levels were associated with lower SS levels. Between individuals, level of but not change in FL was associated with average SS level. Finally, average SS and response level and change in FL were studied as predictors of attitudes toward aging, accounting for 38% of the variance. Women with higher FL and lower SS scores had more negative views of aging. Negative views of aging among women with MS may have long-term consequences for their health outcomes as well as for the quality of their later years.

Attention-deficit/hyperactivity disorder is characterized by a delay in cortical maturation

There is controversy over the nature of the disturbance in brain development that underpins attention-deficit/hyperactivity disorder (ADHD). In particular, it is unclear whether the disorder results from a delay in brain maturation or whether it represents a complete deviation from the template of typical development. Using computational neuroanatomic techniques, we estimated cortical thickness at >40,000 cerebral points from 824 magnetic resonance scans acquired prospectively on 223 children with ADHD and 223 typically developing controls. With this sample size, we could define the growth trajectory of each cortical point, delineating a phase of childhood increase followed by adolescent decrease in cortical thickness (a quadratic growth model). From these trajectories, the age of attaining peak cortical thickness was derived and used as an index of cortical maturation. We found maturation to progress in a similar manner regionally in both children with and without ADHD, with primary sensory areas attaining peak cortical thickness before polymodal, high-order association areas. However, there was a marked delay in ADHD in attaining peak thickness throughout most of the cerebrum: the median age by which 50% of the cortical points attained peak thickness for this group was 10.5 years (SE 0.01), which was significantly later than the median age of 7.5 years (SE 0.02) for typically developing controls (log rank test chi(1)(2) = 5,609, P < 1.0 x 10(-20)). The delay was most prominent in prefrontal regions important for control of cognitive processes including attention and motor planning. Neuroanatomic documentation of a delay in regional cortical maturation in ADHD has not been previously reported.

Wissenschaftswachstum in wichtigen naturwissenschaftlichen Disziplinen vom 17. bis zum 21. Jahrhundert

AbstractThe Growth of Science within Important Scientific Disciplines from 17th to 21st century. – In this paper the growth of knowledge is investigated for the disciplines of Astrophysics / Astronomy, of Physics and of Chemistry in the last centuries. In this context the main emphasis is devoted to the discussion of the growth of literature. Besides, this is also illustrated by the growth of indicators like for example important books, new discoveries in science, the chemical elements, the known chemical compounds and the number of known inorganic chemical structures. Compared to numerous earlier studies a systematic evaluation of all presently available data sources is carried out. The data are fitted by different mathematical growth models (linear, quadratic, exponential, hyperbolic). These are discussed and compared. As far as the exponential growth model is applied the doubling periods obtained have values between 11 and 32 years for the cumulative total of publications, between 24 and 98 years for that of important books, between 79 and 163 years for that of new discoveries and between 10 and 14 years for that of chemical substances. From an order of magnitude point of view this is in agreement with the results of other authors. It is however also demonstrated that the exponential growth model which is normally used is not necessarily the best to describe the growth of the data. Other models are often better suitable to explain the growth. It is shown that linear growth of the yearly publications describes the data better than exponential growth in certain epochs. In this latter case it is demonstrated that the parameter (linear slope) for the rate of increase has a constant values in certain epochs, but rises in steps from epoch to epoch by a factor of 3 to 40. After the second world war the linear growth model for the yearly publications or the quadratic growth model for the cumulative total of publications respectively delivers the best results. This is therefore used for forecasting the future.

Triplet Ultrasound Growth Parameters

To create ultrasound growth curves for normal growth of fetal triplets using statistical methodology that properly accounts for similarities of growth of fetuses within a mother as well as repeated measurements over time for each fetus. In this longitudinal study, all triplet pregnancies managed at a single tertiary center from 1992-2004 were reviewed. Fetuses with major anomalies, prior selective reduction, or fetal demise were excluded. Data from early and late gestation in which there were fewer than 30 fetal measurements available for analysis were excluded. We used multilevel models to account for variation in growth within a single fetus over time, variations in growth between multiple fetuses within a single mother, and variations in fetal growth between mothers. Medians (50th), 10th, and 90th percentiles were estimated by the creation of multiple quadratic growth models from bootstrap samples adapting a previously published method to compute prediction intervals. Estimated fetal weight was derived from Hadlock's formula. One hundred fifty triplet pregnancies were identified. Twenty-seven pregnancies were excluded for the following reasons: missing records (23), fetal demise (3), and fetal anomaly (1). The study group consisted of 123 pregnancies. The gestational age range was restricted to 14-34 weeks. Figures and tables were developed showing medians, 10th and 90th percentiles for estimated fetal weight, femur length, biparietal diameter, abdominal circumference, and head circumference. Growth curves for triplet pregnancies were derived. These may be useful for identification of abnormal growth in triplet fetuses. III.

Robustness properties of A-, D-, and E-optimal designs for polynomial growth models with autocorrelated errors

Optimal designs for polynomial growth models for longitudinal data with autocorrelated errors determine the optimal allocation and number of time points. A-, D-, and E-optimal designs for linear, quadratic and cubic growth models with four or less time points on the time interval [0,2] are given and four robustness properties of these designs are examined. The first considers the robustness against using too many time points. It is shown that the design with the number of time points equal to the number of regression coefficients is optimal, and that the efficiency of a design decreases when the number of time points increases. The second robustness property deals with the consequences of using an incorrect order of the polynomial. The efficiency of a design is shown to be generally higher if the assumed order of the polynomial is closer to the true order. The third robustness property deals with the robustness against an incorrect value of the autocorrelation coefficient. Results show that the optimal designs are very robust. The fourth robustness property considers the robustness of optimal designs with respect to other optimality criteria. The optimal designs are shown to be very robust to the other two optimality criteria.

Multilevel model applications to the analysis of longitudinal data

This work is an introduction to repeated measurement analysis for longitudinal studies. It uses a two stage modelling framework, using hierarchical linear models with two levels. The first level pertains to the repeated measures, the second level pertains to the individual. For the last 25 years, hierarchical linear models have been used in the Social Sciences to analyse data coming from organizations with multiple levels. Their applications have been extended to the study of change in populations, both to describe the average change in an outcome variable in a population and to analyse the factors associated with variability in the individual trajectories of change. In this article, the basic concepts are introduced: between subjects and within subjects variability, the person-specific model for the individual trajectory and the between person model to describe how individuals vary in their trajectories, fixed and random effects, linear and quadratic growth models. At the end of each section, an illustration is given for the study of cognitive function of the older people cohort "Aging in Leganés", followed in four occasions between 1993 and 1999. Results from fitting the models to answer the most frequently asked research questions in the descriptions and analysis of individual change are presented. Lastly, we present possible generalizations of these linear models to non linear situations which arise when outcomes are dichotomous, nominal or ordinal.

The role of coding time in estimating and interpreting growth curve models.

The coding of time in growth curve models has important implications for the interpretation of the resulting model that are sometimes not transparent. The authors develop a general framework that includes predictors of growth curve components to illustrate how parameter estimates and their standard errors are exactly determined as a function of receding time in growth curve models. Linear and quadratic growth model examples are provided, and the interpretation of estimates given a particular coding of time is illustrated. How and why the precision and statistical power of predictors of lower order growth curve components changes over time is illustrated and discussed. Recommendations include coding time to produce readily interpretable estimates and graphing lower order effects across time with appropriate confidence intervals to help illustrate and understand the growth process.

Cluster Analysis as a Method of Recovering Types of Intraindividual Growth Trajectories: A Monte Carlo Study

There has been increased interest in and application of cluster analysis in longitudinal applications to identify distinctive developmental patterns of intraindividual change. This article used Monte Carlo experiments to evaluate the adequacy of cluster analysis to recover group membership based on simulated latent growth curve (LGC) models. The simulated LGC models were manipulated by varying growth parameters (e.g., elevation, dispersion, and shape) for subpopulation growth curves (e.g., linear and quadratic growth models). The evaluation of cluster analysis to recover individual membership in these growth curve subpopulations was completed via the Kappa statistics. Cluster analysis failed to recover adequately growth subtypes when the difference between growth curves was shape only. It was much more successful when the distance between initial mean levels was large (e.g., difference of two standard deviations), independent of difference in the shape of the growth curves. Tentative guidelines were proposed to facilitate the evaluation of the adequacy of a cluster analytic solution to recover subtype heterogeneity in longitudinal (intraindividual) growth curves.

Logarithmic Relation between the Initial Error and Predictability for the Barotropic Component of the Atmosphere.

In this study, predictability for the barotropic component of the atmosphere is examined based on analog weather maps in the historical data. The limit of predictability P is defined as the time taken for the initial error to reach the climate noise level which is defined by one standard deviation from the long term mean of the fluctuation in the observed atmosphere. According to the quadratic error growth model by Lorenz (1982), the predictability P is expected to obey a logarithmic function rather than a linear function of the initial error. Although we searched 15, 667, 760 combinations of weather maps, there are no good analog pairs to investigate the error growth for a sufficiently small initial error. For this reason, model experiments were conducted to demonstrate that the quadratic error growth model is applicable to infer the behavior of a small error from the distribution of a large error. From the results of the model experiments, and the best analog pairs in the historical data, we estimated that the predictability for the real atmosphere increases about 6.3 days when the initial error energy is reduced to 1/10. Hence, we may extend the predictability for the barotropic component of the atmosphere if we can reduce the initial error in the vertical mean of the atmosphere.

Elementary mathematical models: order aplenty and a glimpse of chaos

1. Overview 2. Sequences and difference equations 3. Arithmetic growth 4. Linear graphs, functions and equations 5. Quadratic growth models 6. Quadratic graphs, functions and equations 7. Polynomial and rational functions 8. Fitting a line to data 9. Geometric growth 10. Exponential functions 11. More on logarithms 12. Geometric sums and mixed models 13. Logistic growth 14. Chaos in logistic models.

Optimal designs in growth curve models — II Correlated model for quadratic growth: optimal designs for parameter estimation and growth prediction

This is a follow up of a recent paper on the study of the optimality aspects of linear growth models with correlated errors. In this paper, we examine optimality aspects of quadratic growth models with correlated errors and provide optimal designs for parameter estimation and growth prediction. In the former case we examine A-, D- and E-optimal designs and in the latter case we examine A-optimal designs, under two different correlation structures. In the process, we also examine the robustness of optimal designs with respect to the values of the correlation coefficient.