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PubMed2004

Functional data analysis in longitudinal settings using smoothing splines.

Data in many experiments arise as curves and therefore it is natural to use a curve as a basic unit in the analysis, which is termed functional data analysis (FDA). In longitudinal studies, recent developments in FDA have extended classical linear models and linear mixed effects models to functional linear models (also termed varying-coefficient models) and functional mixed effects models. In this paper we focus our review on the functional mixed effects models using smoothing splines, because functional linear models are special cases of this more general framework. Due to the connection between smoothing splines and linear mixed effects models, functional mixed effects models can be fitted using existing software such as SAS Proc Mixed. A case study is presented as an illustration.

Biomedical Research↗

PubMed1997

Basic concepts of pharmacokinetic/pharmacodynamic (PK/PD) modelling.

Pharmacokinetic (PK) and pharmacodynamic (PD) information from the scientific basis of modern pharmacotherapy. Pharmacokinetics describes the drug concentration-time courses in body fluids resulting from administration of a certain drug dose, pharmacodynamics the observed effect resulting from a certain drug concentration. The rationale for PK/PD-modelling is to link pharmacokinetics and pharmacodynamics in order to establish and evaluate dose-concentration-response relationships and subsequently describe and predict the effect-time courses resulting from a drug dose. Under pharmacokinetic steady-state conditions, concentration-effect relationships can be described by several relatively simple pharmacodynamic models, which comprise the fixed effect model, the linear model, the long-linear model, the Emax-model and the sigmoid Emax-model. Under non steady-state conditions, more complex integrated PK/PD-models are necessary to link and account for a possible temporal dissociation between the plasma concentration and the observed effect. Four basic attributes may be used to characterize PK/PD-models: First, the link between measured concentration and the pharmacologic response mechanism that mediates the observed effect, direct vs. indirect link; second, the response mechanism that mediates the observed effect, direct vs. indirect response; third, the information used to establish the link between measured concentration and observed effect, hard vs. soft link; and fourth, the time dependency of the involved pharmacodynamic parameters, time-variant vs. time-invariant. In general, PK/PD-modelling based on the underlying physiological process should be preferred whenever possible. The expanded use of PK/PD-modelling is assumed to be highly beneficial for drug development as well as applied pharmacotherapy and will most likely improve the current state of applied therapeutics.

Dose-Response Relationship, Drug↗

PubMed1996

Evaluation of methacholine dose-response curves by linear and exponential mathematical models: goodness-of-fit and validity of extrapolation.

Several models have been proposed to analyse dose-response curves recorded in bronchoprovocation challenge tests. The aims of the present work were: 1) to investigate which model (linear vs exponential) and which minimization method (trials and errors vs Levenberg-Marquardt) gives better results in terms of data interpolation (goodness-of-fit); and 2) to verify the validity of extrapolation by comparing forced expiratory volume in one second (FEV1) observed after 4 mg methacholine with values extrapolated after truncation of the curves at 2 mg. For these purposes, methacholine dose-response curves were obtained in 832 subjects from a random population sample, as part of the European Community Respiratory Health Survey (ECRHS) in Italy. Methacholine was inhaled up to a maximum dose of 6 mg by dosimeter technique. The coefficient of determination (r2) was significantly higher with the exponential model (0.81 +/- 0.22; mean +/- SD) than with the linear model (0.69 +/- 0.27). With both models, extrapolated values were usually lower than observed values. As a consequence, a 20% fall in FEV1 with respect to postsaline FEV1 was observed in only 24% and 21% of the tests, where a 20% fall had been predicted, respectively, according to the linear and exponential model. In conclusion, exponential models are better than linear models with respect to data interpolation of methacholine dose-response curves. However, they are worse with respect to extrapolation to higher doses. With any model, extrapolation of dose-response curves by one doubling-dose should be avoided.

Adult↗

PubMedDate not supplied

Fitting regression models to censored survival data.

A short review of regression models for the analysis of censored survival data is given. These include multiplicative hazard rate models, log-linear models (accelerated failure time models), linear models and polynomial models. An application of some of these models to the analysis of a large retrospective study on carcinomas of the oral cavity is described. The results obtained by parametric and semiparametric analyses are compared.

Aged↗

PubMed2003

Multivariate sib-pair linkage analysis of longitudinal phenotypes by three step-wise analysis approaches.

BACKGROUND: Current statistical methods for sib-pair linkage analysis of complex diseases include linear models, generalized linear models, and novel data mining techniques. The purpose of this study was to further investigate the utility and properties of a novel pattern recognition technique (step-wise discriminant analysis) using the chromosome 10 linkage data from the Framingham Heart Study and by comparing it with step-wise logistic regression and linear regression. RESULTS: The three step-wise approaches were compared in terms of statistical significance and gene localization. Step-wise discriminant linkage analysis approach performed best; next was step-wise logistic regression; and step-wise linear regression was the least efficient because it ignored the categorical nature of disease phenotypes. Nevertheless, all three methods successfully identified the previously reported chromosomal region linked to human hypertension, marker GATA64A09. We also explored the possibility of using the discriminant analysis to detect gene x gene and gene x environment interactions. There was evidence to suggest the existence of gene x environment interactions between markers GATA64A09 or GATA115E01 and hypertension treatment and gene x gene interactions between markers GATA64A09 and GATA115E01. Finally, we answered the theoretical question "Is a trichotomous phenotype more efficient than a binary?" Unlike logistic regression, discriminant sib-pair linkage analysis might have more power to detect linkage to a binary phenotype than a trichotomous one. CONCLUSION: We confirmed our previous speculation that step-wise discriminant analysis is useful for genetic mapping of complex diseases. This analysis also supported the possibility of the pattern recognition technique for investigating gene x gene or gene x environment interactions.

Adult Children↗

PubMed1993

Radiobiological assessment of non-standard and novel radiotherapy treatments using the linear-quadratic model.

The linear-quadratic (LQ) model is useful in the radiobiological assessment of a wide variety of radiotherapy treatment techniques, not being confined to analysis of fractionated treatments alone. The model uses parameters that must be separately specified for tumours and dose-limiting normal tissues, and may therefore be used to help identify treatments that are most likely to maximise tumour cell kill while minimising the risk of severe normal-tissue damage. Additionally, the model is capable of making tentative allowance for the tumour repopulation that can occur during extended treatments. Intercomparisons between different types of treatment are made through the concept of the Extrapolated Response Dose (ERD). The ERD is calculated for each critical tissue and takes account of both the radiobiological parameters and the dose/time pattern of radiation delivery. Known tolerance doses for specified organs may be expressed as an ERDtolerance value, and, if a proposed 'new' treatment is to be successful, its associated ERD value must not exceed ERDtolerance. Examples of this procedure are given in this paper. It is particularly important that medical physicists fully appreciate the scope and limitations of LQ equations, as the analysis of radiobiology problems using the model often requires a degree of mathematical understanding that clinicians may not possess.

Dose-Response Relationship, Radiation↗

PubMed2006

Analysis of non-invasive ventilation effects on gastric inflation using a non-linear mathematical model.

A non-linear mathematical model of the oesophagus was developed to study the effects of non-invasive ventilation variables on the severity of gastric inflation. The model was based on the non-linear physical characteristics of biological tissue. The model simulated oesophageal mechanical function during non-invasive ventilation in cardiac arrest (2:30 ventilations/chest compressions cycles) and respiratory arrest (1:5 ventilations/s) as recommended by the European Resuscitation Council (ERC) in its 2005 guidelines for adult basic and advanced life support. Model predictions establish a strong correlation between the expiratory time and the occurrence of gastric inflation. For cardiac arrest, when using ventilation pressure lower than 12 cmH2O, expiratory time between consequent ventilations and time until the occurrence of gastric inflation were linearly dependent (r = 0.98). This linear correlation changed abruptly when airway pressure exceeded the threshold pressure of 12 cmH2O, indicating that air had entered the stomach during the first ventilation. The interval at which the pressure at the distal section of the oesophagus was above the lower oesophageal sphincter (LES) opening pressure was significantly prolonged in the model of cardiac arrest (approximately 5.5 s compared to 3 s in respiratory arrest), thus allowing a greater amount of air to enter the stomach at relatively low airway pressures. During cardiac arrest, the mean pressure at the distal section of the oesophagus and the amplitude of air backflow were higher compared to the mean pressure and amplitude during respiratory arrest. This is also due to the shorter expiratory intervals in the 2:30 ventilations/chest compressions technique. The model indicates that the time required for the air trapped in the oesophagus to completely deflate is approximately 2 s. This may be longer than the expiratory time recommended by the 2005 guidelines. Model predictions support the 2005 guidelines regarding the decrease in the tidal volume and in the inspiratory pressure in an effort to minimise gastric inflation.

Cardiopulmonary Resuscitation↗

PubMed2000

Models of health-related quality of life in a population of community-dwelling Dutch elderly.

OBJECTIVE: Though health-related quality of life (HRQoL) is now commonly measured as an outcome in clinical trials, the relationships between its components remain unclear. The relation of physical symptoms, physical function, and psychological symptoms to each other and to overall quality of life is of special interest. METHOD: Cross-sectional data from 5,279 community-dwelling elders who participated in the Groningen Longitudinal Aging Study were analyzed using structural equation modeling techniques. Three models were examined. One "Linear" model included: number of chronic medical conditions, physical symptoms, physical functioning, activity interference, social function, perceived health and overall quality of life in a simple linear progression. Another 'non-linear' model included these variables, but allowed effects between non-adjacent variables. A third 'non-linear' model included these variables plus anxiety and depressive symptoms. RESULTS: The Linear Model did not satisfactorily account for the observed data [X2(15df) = 2946.96], so the saturated Non-Linear Model, incorporating paths between non-adjacent components, is described. When anxiety and depressive symptoms were added to this Non-Linear Model, they fit best in a position mediating the relation between perceived health and overall quality of life [X2(5df) = 136.78]. CONCLUSIONS: Overall quality of life appears to be related to symptom status as directly as it is related to functional status. Anxiety and depressive symptoms appear to mediate the relation between general health perceptions and overall quality of life. Quality of life measures should therefore include assessments of physical and psychological symptom severity as well as functional status if they are to truly reflect what matters to patients. The disability-adjusted life year (DALY) measure used by the WHO may inadequately reflect the effect of symptoms on patient's quality of life.

Aged↗

PubMed1994

Non-linear regression models to estimate the size of DNA fragments.

The least-squares, hyperbolic regression model is frequently used to estimate the size of unknown DNA fragments. This model avoids problems associated with semilog-plot interpolation, is computationally easy to use and provides an excellent fit to many experimental data sets. However, the methods commonly used to solve the hyperbolic regression model perform an inappropriate linearization of the original non-linear model. In this note, we describe advantages offered by standard, non-linear regression techniques, and provide computer code for a common statistical package to do these analyses.

Algorithms↗

PubMed1996

Evaluation of non-linear and linear mathematical models for creatinine and glucose fitting in peritoneal dialysis.

Four non-linear and five linear models for predicting the creatinine dialysate/ plasma ratio (CRD/P) and the glucose dialysate/initial concentration ratio (GLD/D0) were evaluated in a group of 31 patients on peritoneal dialysis and subjected to the peritoneal equilibration test (PET 3.86%, 240'). PET results and classification were compared to obtain a definition of patient peritoneal transport characteristics. The monomolecular and rectangular hyperbola non-linear models, the Lineweaver-Burk, Hanes-Woolf and Dadone linear transformations were considered for the CRD/P fitting. A monoexponential and two-exponential decay plus the semilogarithmic transformations were considered for the GLD/ D0. These models are simple, accurate and functionally homogeneous. Further studies are advisable however on the individual peritoneal transport classification, since approximately 30% of the patients were in different categories for CRD/P and GLD/D0 and the fittings do not give better classification results.

Aged↗

PubMed1995

Classifying and counting linear phylogenetic invariants for the Jukes-Cantor model.

Linear invariants are useful tools for testing phylogenetic hypotheses from aligned DNA/RNA sequences, particularly when the sites evolve at different rates. Here we give a simple, graph theoretic classification for each phylogenetic tree T, of its associated vector space I(T) of linear invariants under the Jukes-Cantor one-parameter model of nucleotide substitution. We also provide an easily described basis for I(T), and show that if I is a binary (fully resolved) phylogenetic tree with n sequences at its leaves then: dim[I(T)] = 4n-F2n-2 where Fn is the nth Fibonacci number. Our method applies a recently developed Hadamard matrix-based technique to describe elements of I(T) in terms of edge-disjoint packings of subtrees in T, and thereby complements earlier more algebraic treatments.

Base Sequence↗

PubMed2001

Factor analytic models of clustered multivariate data with informative censoring.

This article describes a general class of factor analytic models for the analysis of clustered multivariate data in the presence of informative missingness. We assume that there are distinct sets of cluster-level latent variables related to the primary outcomes and to the censoring process, and we account for dependency between these latent variables through a hierarchical model. A linear model is used to relate covariates and latent variables to the primary outcomes for each subunit. A generalized linear model accounts for covariate and latent variable effects on the probability of censoring for subunits within each cluster. The model accounts for correlation within clusters and within subunits through a flexible factor analytic framework that allows multiple latent variables and covariate effects on the latent variables. The structure of the model facilitates implementation of Markov chain Monte Carlo methods for posterior estimation. Data from a spermatotoxicity study are analyzed to illustrate the proposed approach.

Animals↗

PubMed2006

[Estimation on gene-environment interaction in the partial case-control study].

OBJECTIVE: To introduce the approaches for estimating gene-environment interaction based on partial case-control studies. METHODS: The effects of logistic model and log-linear model for estimating the main effects and gene-environment interaction effect were estimated by means of maximum likelihood methods in traditional case-control studies, case-only studies and partial case-control studies, respectively. An example was also illustrated. RESULTS: In traditional case-control study with complete data, the results of logistic model and log-linear model were equivalent. In case-only study without any information about controls, the logistic model can also efficiently estimate gene-environment interaction. In partial case-control study, environmental information was collected from all of the cases and controls, while genetic information was only collected from cases. For this case-control study with incomplete data, a suitable parameterized log-linear model could simultaneously and efficiently estimate the main effect of environment and gene-environment interaction, whereas the logistic model could not. CONCLUSION: For a partial case-control study, log-linear model could estimate not only the main effect of environment but also gene-environment interaction. If genotype and exposure were independent, estimators from partial case-control were as precisely as those from complete-data case-control studies.

Case-Control Studies↗

PubMed2006

Anomalous scaling exponents in nonlinear models of turbulence.

We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence. We construct, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. The statistics of the auxiliary linear model are dominated by "statistically preserved structures" which are associated with exact conservation laws. The latter can be used, for example, to determine the value of the anomalous scaling exponent of the second order structure function. The approach is equally applicable to shell models and to the Navier-Stokes equations.

Journal Article↗

PubMed1998

Heart rate dynamics in low risk human fetuses.

Evaluation of nonlinear heart rate (HR) dynamics has received considerable attention in the pediatric literature because such analyses not only provide insight into underlying control mechanisms, but may also help to differentiate between normal and abnormal infants. The purpose of this study was to determine, in eight low risk human fetuses, if nonlinear HR dynamics could be identified by analyzing the dispersion of interbeat intervals at slow (Ds) and fast (Df) HRs. The fetal cardiac electrical signal was captured transabdominally at a resolution of +/- 1 ms. To test the null hypothesis, that the time series is the result of a linear stochastic process, Ds and Df for the original time series were compared with the values calculated for three linear models. The linear models were constructed to preserve the major statistical properties of the original time series, including the mean, SD, and the Fourier power spectrum. For each fetus, there was no evidence of nonlinear cardiac dynamics based on analyses of Ds and Df. In contrast, the distribution of adjacent R-R intervals and the pattern of change across three successive interbeat intervals both revealed significant nonlinearities in HR control in each fetus. If the difference between normal and abnormal infants is the result of aberrant control of nonlinear processes, then our findings indicate that parameters which describe the nonlinearity may be more useful then Ds and Df in assigning a risk status.

Apgar Score↗

PubMed1994

Nonlinear parameter estimation by linear association: application to a five-parameter passive neuron model.

Linear associative memories (LAM) have been intensely used in the areas of pattern recognition and parallel processing for the past two decades. Application of LAM to nonlinear parameter estimation, however, has only been recently attempted. The process consists in converting the nonlinear function in the parameters into a set of linear algebraic equations. The nature of the linearized system and the factors influencing the accuracy of the parameter estimates have not yet been fully investigated. In this paper, LAM is applied to a nonlinear five-parameter model of the neuron. Ill-conditioning, which is often exhibited in LAM, is treated with the method of regularization as well as by the singular value decomposition (SVD). Simulation results indicate that the parameters estimated by LAM exhibit a remarkable robustness against additive white noise in comparison with the classical gradient optimization technique. Moreover, it is shown that regularization can be superior to SVD under certain conditions. Our results suggest that LAM can be used both as a noise reduction technique and as a stand-alone nonlinear parameter estimation algorithm. The comparison between LAM and a gradient technique show that, for this estimation problem, the LAM method can give more reliable estimates. Further improvements in estimation quality may still be achieved by the use of other forms of regularizing functions.

Linear Models↗

PubMed2000

Empirical assessments of social networks, fertility and family planning programs: nonlinearities and their implications.

Empirical studies of the diffusion of modern methods of family planning have increasingly incorporated social interaction within nonlinear models such as logits. But they have not considered the full implications of these nonlinear specifications. This paper considers the implications of using nonlinear models in empirical analyses of the impact of family programs, modulated by social interaction, on reproductive behavior. Three implications of nonlinear models, in comparison with linear models, are developed. 1) With nonlinear models, there may be both low and high contraceptive-use equilibria (i.e., the ultimate level of use of modern family planning that a population can be expected to reach after the effects of a sustained change in a family planning program have worked through the population) rather than just one equilibrium as in linear models. If there are multiple equilibria, then one striking and important result is that a transitory large program effort may move a community from sustained low- to high-level contraceptive use. 2) With nonlinear models, the extent to which a social interaction multiplies program efforts depends on whether the community is at a low or high level of contraceptive use rather than being independent of the level of contraceptive use as in linear models. 3) With nonlinear models, intensified social interaction can retard or enhance the diffusion of family planning, in contrast to only enhancing diffusion as within linear models. To clarify these implications, for comparison a simple and more transparent linear model is also discussed. Illustrative estimates are presented of simple linear and nonlinear models for rural Kenya that demonstrate that some of these effects may be considerable.

Contraception↗

PubMed1977

Distribution pharmacokinetics of warfarin in the rat, a non-linear multicompartment model.

Preliminary analysis and linear two-compartment solutions of warfarin plasma concentrations recorded in the rat after intravenous bolus injections of 1, 2, 8 and 40 mg/kg of sodium warfarin revealed marked non-linearities. The half-life of total warfarin concentration in the plasma from 1-12h remained unchanged with all the doses used, but that of free warfarin was shorter with 40 mg/kg, possibly as the result of an increase in the binding of the drug to plasma proteins as the high total warfarin concentration decreased. The apparent volume of distribution generally increased with increasing dose, and differed according to the method used for its calculation. Liver warfarin data could be solved with Langmuir type saturation kinetics, but the saturation phenomena were slight in the concentration range studied. A non-linear multicompartment model was constructed, the physiological spaces of which were plasma, interstitial fluid and tissue. The binding of free warfarin to plasma proteins, interstitial fluid proteins and tissue structures was assumed to occur instantaneously, with saturable binding to plasma and interstitial fluid proteins, and a constant binding to tissues. The fluxes between the free warfarin pools of plasma and interstitial fluid as well as elimination were assumed to be linear. Following parameters were simulated simultaneously, using an analog hybrid computer: two for the above-mentioned fluxes, four for zero time drug mass distribution between plasma and interstitial fluid, and one for tissue binding. According to the best fits, warfarin is preferentially distributed into plasma, interstitial fluid and highly perfused tissues. The solution suggests that non-linearities in the pharmacokinetics of warfarin, a highly plasma protein-bound drug, first occur in plasma and interstitial fluid. Therefore, it is believed that the quantitative non-linear multicompartment approach presented in this paper might be useful in studying the kinetic behaviour of other highly plasma protein-bound drugs, too.

Animals↗