An affine linear model for the relation between two sets of frequency counts.
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Animals↗
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Results for “Linear Models”
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The use of linear models for EEG investigations.
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Brain↗
Comment on linear models article.
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Chemistry, Clinical↗
A linear models application of competing risks to multiple causes of death.
An analysis is performed to ascertain the joint incidence of two causes of death, acute myocardial infarct and stroke, for the deaths of residents of Massachusetts and North Carolina in 1969. To assay their association an explicit biological model of the nature of the relation is posited. It is shown that, under this model, Chiang's (1968) theory of competing risks may be extended to the case in which an individual's death may have multiple causes. Furthermore, techniques are developed which allow us to model the survival parameters derived under the model by categorical data procedures of the type introduced by Grizzle, Starmer and Koch (1969). The study shows that there is a greater incidence of the joint occurrence of stroke and myocardial infarct on death certificates in North Carolina than in Massachusetts, a pattern consistent with the generally higher stroke mortality in North Carolina. Furthermore, the incidence of the joint occurrence of the two diseases shows a clear age "gradient" increasing through the age range of the analysis. Males and females show somewhat different patterns of age variation in that state-by-age interaction terms are more prominent in the model fitted for females than for males.
Age Factors↗
[Linear model of biactive electrocoagulation].
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Electrocoagulation↗
Log-linear modeling with the negative multinomial distribution.
We develop a negative multinomial sampling plan in which observed cell counts are positively correlated. We show that maximum likelihood estimates of cell means are the same as those found under independent Poisson sampling. There is no maximum likelihood estimate for the shape parameter in general. We propose an estimate of the shape parameter based on the mean and quantiles of Pearson's chi-squared statistic. These techniques are applied to models of cancer incidence for three cities in Ohio and longitudinal health care utilization by a group of senior citizens.
Aged↗
Proceedings: Non-linear modelling of colorectal electrical potentials.
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Action Potentials↗
Factoring vs linear modeling in rate estimation: a simulation study of relative accuracy.
A common strategy for modeling dose-response in epidemiology is to transform ordered exposures and covariates into sets of dichotomous indicator variables (that is, to factor the variables). Factoring tends to increase estimation variance, but it also tends to decrease bias and thus may increase or decrease total accuracy. We conducted a simulation study to examine the impact of factoring on the accuracy of rate estimation. Factored and unfactored Poisson regression models were fit to follow-up study datasets that were randomly generated from 37,500 population model forms that ranged from subadditive to supramultiplicative. In the situations we examined, factoring sometimes substantially improved accuracy relative to fitting the corresponding unfactored model, sometimes substantially decreased accuracy, and sometimes made little difference. The difference in accuracy between factored and unfactored models depended in a complicated fashion on the difference between the true and fitted model forms, the strength of exposure and covariate effects in the population, and the study size. It may be difficult in practice to predict when factoring is increasing or decreasing accuracy. We recommend, therefore, that the strategy of factoring variables be supplemented with other strategies for modeling dose-response.
Bias↗
An investigation of fetal growth using serial ultrasound data.
Serial ultrasonic measurements were performed in 67 Caucasian fetuses from 20 weeks' gestation until term to derive reference standards for abdominal circumference (AC) and estimated fetal weight (EFW). The variances of both AC and EFW increased with gestational age. Four mathematical models (linear, quadratic, Gompertz and Rossavik) were fitted to the log(10)(AC) and log(10)(EFW) data from each fetus using least squares regression analysis. The standard deviations of the residual error were greatest for the linear model. The linear model also overestimated the final log(10)(AC) while the linear, Gompertz and Rossavik models all overestimated the final log(10)(EFW) when these data were omitted from the fitting process. The individual quadratic constants were, therefore, used to produce 67 individual growth curves. Values of log(10)(AC) and log(10)(EFW) for each fetus at exact gestational ages between 20 and 40 weeks were derived by interpolation; the mean and standard deviation values were then used to derive centile ranges for abdominal circumference and estimated fetal weight. These standards should prove useful in the assessment of fetal growth.
Journal Article↗
Ordinal regression model and the linear regression model were superior to the logistic regression models.
OBJECTIVE: Ordinal scales often generate scores with skewed data distributions. The optimal method of analyzing such data is not entirely clear. The objective was to compare four statistical multivariable strategies for analyzing skewed health-related quality of life (HRQOL) outcome data. HRQOL data were collected at 1 year following catheterization using the Seattle Angina Questionnaire (SAQ), a disease-specific quality of life and symptom rating scale. STUDY DESIGN AND SETTING: In this methodological study, four regression models were constructed. The first model used linear regression. The second and third models used logistic regression with two different cutpoints and the fourth model used ordinal regression. To compare the results of these four models, odds ratios, 95% confidence intervals, and 95% confidence interval widths (i.e., ratios of upper to lower confidence interval endpoints) were assessed. RESULTS: Relative to the two logistic regression analysis, the linear regression model and the ordinal regression model produced more stable parameter estimates with smaller confidence interval widths. CONCLUSION: A combination of analysis results from both of these models (adjusted SAQ scores and odds ratios) provides the most comprehensive interpretation of the data.
Adolescent↗
Confidence and prediction intervals for generalised linear accident models.
Generalised linear models, with "log" link and either Poisson or negative binomial errors, are commonly used for relating accident rates to explanatory variables. This paper adds to the toolkit for such models. It describes how confidence intervals (for example, for the true accident rate at given flows) and prediction intervals (for example, for the number of accidents at a new site with given flows) can be produced using spreadsheet technology.
Accident Prevention↗
A semi-parametric Bayesian approach to generalized linear mixed models.
The linear mixed effects model with normal errors is a popular model for the analysis of repeated measures and longitudinal data. The generalized linear model is useful for data that have non-normal errors but where the errors are uncorrelated. A descendant of these two models generates a model for correlated data with non-normal errors, called the generalized linear mixed model (GLMM). Frequentist attempts to fit these models generally rely on approximate results and inference relies on asymptotic assumptions. Recent advances in computing technology have made Bayesian approaches to this class of models computationally feasible. Markov chain Monte Carlo methods can be used to obtain 'exact' inference for these models, as demonstrated by Zeger and Karim. In the linear or generalized linear mixed model, the random effects are typically taken to have a fully parametric distribution, such as the normal distribution. In this paper, we extend the GLMM by allowing the random effects to have a non-parametric prior distribution. We do this using a Dirichlet process prior for the general distribution of the random effects. The approach easily extends to more general population models. We perform computations for the models using the Gibbs sampler.
Bayes Theorem↗
Experimental determination of the theophylline diffusion coefficient in swollen sodium-alginate membranes.
In this paper attention is focused on the determination of the drug diffusion coefficient in a swollen polymeric membrane referring to a recent mathematical model (linear model). The main advantage deriving from its use is that, despite its analytical nature and its ability to account for the most important aspects characterising a permeation experiment, it can also be applied in the case of thick membranes. To check the model reliability, a comparison is made with a more complex numerical model and with a largely employed model in terms of data fitting quality. To this purpose, particular care is devoted to the experimental and theoretical tools employed to calculate the auxiliary parameters required by the three models, and with the aim of getting a drug diffusion coefficient value as accurate as possible. Theophylline was chosen as model drug owing to its wide employment in the pharmaceutical field. Membranes were prepared with sodium alginates hydrogels at three different polymer concentrations. The present analysis demonstrates the reliability of the linear model and reveals that the theophylline diffusion coefficient is not significantly affected by the polymer concentration. Indeed, such a parameter is reflected in different membrane thicknesses rather than in different mesh sizes of the polymeric network.
Alginates↗
Model comparison of generalized linear mixed models.
Generalized linear mixed models (GLMMs) have been widely appreciated in biological and medical research. Maximum likelihood estimation has received a great deal of attention. Comparatively, not much has been done on model comparison or hypotheses testing. In this article, we propose a path sampling procedure to compute the observed-data log-likelihood function, so that the Bayesian information criterion (BIC) can be applied to model comparison or hypothesis testing. Advantages of the proposed path sampling procedure are discussed. Two medical data sets are analysed for providing illustrative examples of the proposed methodology.
Air Pollution↗
A method for fitting regression splines with varying polynomial order in the linear mixed model.
The linear mixed model has become a widely used tool for longitudinal analysis of continuous variables. The use of regression splines in these models offers the analyst additional flexibility in the formulation of descriptive analyses, exploratory analyses and hypothesis-driven confirmatory analyses. We propose a method for fitting piecewise polynomial regression splines with varying polynomial order in the fixed effects and/or random effects of the linear mixed model. The polynomial segments are explicitly constrained by side conditions for continuity and some smoothness at the points where they join. By using a reparameterization of this explicitly constrained linear mixed model, an implicitly constrained linear mixed model is constructed that simplifies implementation of fixed-knot regression splines. The proposed approach is relatively simple, handles splines in one variable or multiple variables, and can be easily programmed using existing commercial software such as SAS or S-plus. The method is illustrated using two examples: an analysis of longitudinal viral load data from a study of subjects with acute HIV-1 infection and an analysis of 24-hour ambulatory blood pressure profiles.
Blood Pressure Monitoring, Ambulatory↗
Consideration in selecting crops for the human-rated life support system: a Linear Programming model.
A Linear Programming model has been constructed which aids in selecting appropriate crops for CELSS (Controlled Environment Life Support System) food production. A team of Controlled Environment Agriculture (CEA) faculty, staff, graduate students and invited experts representing more than a dozen disciplines, provided a wide range of expertise in developing the model and the crop production program. The model incorporates nutritional content and controlled-environment based production yields of carefully chosen crops into a framework where a crop mix can be constructed to suit the astronauts' needs. The crew's nutritional requirements can be adequately satisfied with only a few crops (assuming vitamin mineral supplements are provided) but this will not be satisfactory from a culinary standpoint. This model is flexible enough that taste and variety driven food choices can be built into the model.
Crops, Agricultural↗
Smooth random effects distribution in a linear mixed model.
A linear mixed model with a smooth random effects density is proposed. A similar approach to P-spline smoothing of Eilers and Marx (1996, Statistical Science 11, 89-121) is applied to yield a more flexible estimate of the random effects density. Our approach differs from theirs in that the B-spline basis functions are replaced by approximating Gaussian densities. Fitting the model involves maximizing a penalized marginal likelihood. The best penalty parameters minimize Akaike's Information Criterion employing Gray's (1992, Journal of the American Statistical Association 87, 942-951) results. Although our method is applicable to any dimensions of the random effects structure, in this article the two-dimensional case is explored. Our methodology is conceptually simple, and it is relatively easy to fit in practice and is applied to the cholesterol data first analyzed by Zhang and Davidian (2001, Biometrics 57, 795-802). A simulation study shows that our approach yields almost unbiased estimates of the regression and the smoothing parameters in small sample settings. Consistency of the estimates is shown in a particular case.
Biometry↗
Mathematical modelling of drug permeation through a swollen membrane.
This work proposes two different mathematical models (linear and numerical) able to simulate the drug permeation through a swollen membrane sandwiched by two external layers (trilaminate system). Moreover, a solid drug dissolution phenomenon in the donor compartment may be accounted for. Indeed, this is a situation that may often occur in permeation experiments. An insufficient stirring of the donor and of the receiver volume may give rise to two sandwiching layers and the target of a constant drug concentration in the donor compartment may be accomplished by putting a solid drug amount in the saturated donor solution. The linear model shows the advantage of having an analytical expression which extremely simplifies the calculation of the drug diffusion coefficient D inside the membrane. Its main drawback lies in the fact that it works only for thin trilaminate systems. The numerical model is more general than the linear one, as it works for all kind of trilaminate thickness and it may account for a solid powder dissolution in the donor compartment. Of course, it does not have an analytical solution and, thus, the D determination is less easy to perform as the numerical model is more time consuming than the linear one. These two models are then compared with the classical approach developed by Flynn and Barrie in order to better define its validity limits.
Diffusion↗