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PubMed2002

Reconsidering the use of the general linear model with single-case data.

Using a low point estimate of autocorrelation to justify analyzing single-case data with the general linear model (GLM) is questioned. Monte Carlo methods are used to examine the degree to which bias in the estimate of autocorrelation depends on the complexity of the linear model used to describe the data. A method is then illustrated for determining the range of autocorrelation parameters that could reasonably have led to the observed autocorrelation. The argument for using a GLM analysis can be strengthened when the GLM analysis functions appropriately across the range of plausible autocorrelations. For situations in which the GLM analysis does not function appropriately across this range, a method is provided for adjusting the confidence intervals to ensure adequate coverage probabilities for specified levels of autocorrelation.

Analysis of Variance↗

PubMed1992

[Mathematical simulation of intracranial condition--Part 1. Linear model stimulation].

We developed a linear mathematical model of the intracranial vessels, which reflects changes of the pulse wave (pulse pressure) of intracranial pressure after ligation of the internal jugular vein. The model composed of eight major variables: 1. resistance of arteries, 2. resistance of small arteries and capillary vessels, 3. resistance of veins, 4. resistance of internal jugular and vertebral veins, 5. compliance of arteries, 6. compliance of small arteries and capillary vessels, 7. compliance of veins and 8. intracranial compliance. All variables are presumed to have linear elements and replaced with electrical elements. The model of neck dissection is expressed as the change of resistance of the internal jugular and vertebral veins. Intracranial condition is expressed as the pulse wave (pulse pressure) of intracranial pressure and driving pressure. After unilateral ligation of the internal jugular vein, the pulse wave of intracranial pressure increased 24% and, after bilateral ligation of the internal jugular vein, it increased 55%. After unilateral ligation of the internal jugular vein, the pulse wave of intracranial pressure increased 27%, and, after bilateral ligation, it increased 79%. When intracranial compliance is normal, the respective ratios of pulse wave of intracranial pressure and driving pressure to cross-sectional area decreased, whereas those after increase of intracranial compliance increased.

Cerebrovascular Circulation↗

PubMed1998

Hospital- and patient-related characteristics determining maternity length of stay: a hierarchical linear model approach.

OBJECTIVES: The purpose of this study was to identify factors related to pregnancy and childbirth that might be predictive of a patient's length of stay after delivery and to model variations in length of stay. METHODS: California hospital discharge data on maternity patients (n = 499,912) were analyzed. Hierarchical linear modeling was used to adjust for patient case mix and hospital characteristics and to account for the dependence of outcome variables within hospitals. RESULTS: Substantial variation in length of stay among patients was observed. The variation was mainly attributed to delivery type (vaginal or cesarean section), the patient's clinical risk factors, and severity of complications (if any). Furthermore, hospitals differed significantly in maternity lengths of stay even after adjustment for patient case mix. CONCLUSIONS: Developing risk-adjusted models for length of stay is a complex process but is essential for understanding variation. The hierarchical linear model approach described here represents a more efficient and appropriate way of studying interhospital variations than the traditional regression approach.

Adult↗

PubMed2002

Linear modeling of steady-state behavioral dynamics.

The observed steady-state behavioral dynamics supported by unsignaled periods of reinforcement within repeating 2,000-s trials were modeled with a linear transfer function. These experiments employed improved schedule forms and analytical methods to improve the precision of the measured transfer function, compared to previous work. The refinements include both the use of multiple reinforcement periods that improve spectral coverage and averaging of independently determined transfer functions. A linear analysis was then used to predict behavior observed for three different test schedules. The fidelity of these predictions was determined.

Animals↗

PubMed1994

Diagnostics for the multivariate linear model analysis of 2 x 2 crossover designs.

The multivariate linear model Y = X beta + epsilon is used to analyze data in 2 x 2 crossover designs with either univariate or multivariate response. Diagnostics are performed on estimating the effect of interest formulated as C beta U and on testing the general linear hypothesis C beta U = k. The multivariate Cook's distance is extended to be the influence measure by incorporating the contrast matrix C and the transformation matrix U, and magnitude of F(1)-F, the difference between the F approximation of a multivariate test statistic, is proposed as a measure to detect influential observations for testing the hypothesis C beta U = k. Both measures prove to be very useful because the diagnostics are now associated with estimating and testing effects of interest in the context of the experimental design.

Biopharmaceutics↗

PubMed1984

Covariate analysis of competing-risks data with log-linear models.

A general system of log-linear modeling is proposed for analysis of competing-risks data with discrete covariates. The instantaneous cause-specific failure rates, approximated by step-functions, are analyzed by techniques for multidimensional contingency tables. Censored observations are accommodated. Counts of failures of each type, and the amount of follow-up, are summarized in two arrays in which each cell denotes a distinct combination of failure type, time interval and covariate value. Maximum likelihood estimators for the parameters of the model are derived by iterative proportional fitting; the resulting estimates of the number of failures in each cell are used for goodness-of-fit tests. The principal advantages of this approach are its simple display of data, its computational ease for the fitting and comparison of models and its provision of explicit goodness-of-fit tests. Interpretation of the models is facilitated by reference to several alternative models for survivorship and competing risks. The basic model is extended to incorporate stochastic covariates whose values change during follow-up, and to accommodate quantitative covariates.

Analysis of Variance↗

PubMed1992

Methods for estimating the parameters of a linear model for ordered categorical data.

In many empirical analyses, the response of interest is categorical with an ordinal scale attached. Many investigators prefer to formulate a linear model, assigning scores to each category of the ordinal response and treating it as continuous. When the covariates are categorical, Haber (1985, Computational Statistics and Data Analysis 3, 1-10) has developed a method to obtain maximum likelihood (ML) estimates of the parameters of the linear model using Lagrange multipliers. However, when the covariates are continuous, the only method we found in the literature is ordinary least squares (OLS), performed under the assumption of homogeneous variance. The OLS estimates are unbiased and consistent but, since variance homogeneity is violated, the OLS estimates of variance can be biased and may not be consistent. We discuss a variance estimate (White, 1980, Econometrica 48, 817-838) that is consistent for the true variance of the OLS parameter estimates. The possible bias encountered by using the naive OLS variance estimate is discussed. An estimated generalized least squares (EGLS) estimator is proposed and its efficiency relative to OLS is discussed. Finally, an empirical comparison of OLS, EGLS, and ML estimators is made.

Abnormalities, Drug-Induced↗

PubMed1994

Integration of shading and texture cues: testing the linear model.

One of the first attempts to develop a formal model of depth cue integration is to be found in Maloney and Landy's [(1989) Proceedings of the SPIE: Visual communications and image processing, Part 2 (pp. 1154-1163)] "human depth combination rule". They advocate that the combination of depth cues by the visual system is best described by a weighted linear model. The present experiments tested whether the linear combination rule applies to the integration of texture and shading. As would be predicted by a linear combination rule, the weight assigned to the shading cue did not vary as a function of its curvature value. However, the weight assigned to the texture cue varied systematically as a function of the curvature values of both cues. Here we describe a non-linear model which provides a better fit to the data. Redescribing the stimuli in terms of depth rather than curvature reduced the goodness of fit for all models tested. These results support the hypothesis that the locus of cue integration is a curvature map, rather than a depth map. We conclude that the linear combination rule does not generalize to the integration of shading and texture, and that for these cues it is likely that integration occurs after the recovery of surface curvature.

Cues↗

PubMed1992

Categorical data analysis in primary care research: log-linear models.

Primary care researchers often wish to perform multiple variable analyses using variables measured at a nominal or ordinal level. This paper provides a step-by-step description of log-linear modeling, an approach uniquely well suited to explore and describe interactions among three or more nominal or ordinal variables. The method of log-linear analysis is illustrated with the use of an example from a primary care research project in which the relationships among hypertension, diet, and sodium were examined. The advantages and disadvantages of log-linear models and logistic regression are compared and available computer software programs discussed.

Humans↗

PubMed1999

A simulation study of confounding in generalized linear models for air pollution epidemiology.

Confounding between the model covariates and causal variables (which may or may not be included as model covariates) is a well-known problem in regression models used in air pollution epidemiology. This problem is usually acknowledged but hardly ever investigated, especially in the context of generalized linear models. Using synthetic data sets, the present study shows how model overfit, underfit, and misfit in the presence of correlated causal variables in a Poisson regression model affect the estimated coefficients of the covariates and their confidence levels. The study also shows how this effect changes with the ranges of the covariates and the sample size. There is qualitative agreement between these study results and the corresponding expressions in the large-sample limit for the ordinary linear models. Confounding of covariates in an overfitted model (with covariates encompassing more than just the causal variables) does not bias the estimated coefficients but reduces their significance. The effect of model underfit (with some causal variables excluded as covariates) or misfit (with covariates encompassing only noncausal variables), on the other hand, leads to not only erroneous estimated coefficients, but a misguided confidence, represented by large t-values, that the estimated coefficients are significant. The results of this study indicate that models which use only one or two air quality variables, such as particulate matter [less than and equal to] 10 microm and sulfur dioxide, are probably unreliable, and that models containing several correlated and toxic or potentially toxic air quality variables should also be investigated in order to minimize the situation of model underfit or misfit.

Air Pollution↗

PubMed1991

A log-linear modeling framework for selective mixing.

Nonrandom mixing can significantly alter the diffusion path of an infectious disease such as AIDS that requires intimate contact. Recent attempts to model this effect have sought a general framework capable of representing both simple and arbitrarily complicated mixing structures, and of solving the balancing problem in a nonequilibrium multigroup population. Log-linear models are proposed here as a general framework for solving the first problem. This approach offers several additional benefits: The parameters used to govern the mixing have a simple, intuitive interpretation, the framework provides a statistically sound basis for the estimation of these parameters from mixing-matrix data, and the resulting estimates are easily integrated into compartmental models for diffusion. A modified selection model is proposed to solve the second problem of generalizing the selection process to nonequilibrium populations. The distribution of contacts under this model is derived and is found to satisfy the assumptions of statistical inference for log-linear models. Together these techniques provide an integrated and flexible framework for modeling the role of selective mixing in the spread of disease.

Acquired Immunodeficiency Syndrome↗

PubMed2003

Wavelet statistics of functional MRI data and the general linear model.

PURPOSE: To improve the signal-to-noise ratio (SNR) of functional magnetic resonance imaging (fMRI) data, an approach is developed that combines wavelet-based methods with the general linear model. MATERIALS AND METHODS: Ruttimann et al. (1) developed a wavelet-based statistical procedure to test wavelet-space partitions for significant wavelet coefficients. Their method is applicable for the detection of differences between images acquired under two experimental conditions using long blocks of stimulation. However, many neuropsychological questions require more complicated event-related paradigms and more experimental conditions. Therefore, in order to apply wavelet-based methods to a wide range of experiments, we present a new approach that is based on the general linear model and wavelet thresholding. RESULTS: In contrast to a monoresolution filter, the application of the wavelet method increased the SNR and showed a set of clearly dissociable activations. Furthermore, no relevant decrease of the local maxima was observed. CONCLUSION: Wavelet-based methods can increase the SNR without diminishing the signal amplitude, while preserving the spatial resolution of the image. The anatomical localization is strongly improved.

Adult↗

PubMed1998

Sensitivity of parametric link functions in generalized linear models.

A common method of choosing the link function in generalized linear models is to specify a parametric link family indexed by unknown parameters. The maximum likelihood estimates of such link parameters, however, may often depend on one or several extreme observations. Diagnostics are derived to assess the sensitivity of the parametric link analysis. Two examples demonstrate that the proposed diagnostics can identify jointly influential observations on the link even when masking is present.

Blood Sedimentation↗

PubMedDate not supplied

Non-ignorable missing covariates in generalized linear models.

We propose a likelihood method for estimating parameters in generalized linear models with missing covariates and a non-ignorable missing data mechanism. In this paper, we focus on one missing covariate. We use a logistic model for the probability that the covariate is missing, and allow this probability to depend on the incomplete covariate. We allow the covariates, including the incomplete covariate, to be either categorical or continuous. We propose an EM algorithm in this case. For a missing categorical covariate, we derive a closed form expression for the E- and M-steps of the EM algorithm for obtaining the maximum likelihood estimates (MLEs). For a missing continuous covariate, we use a Monte Carlo version of the EM algorithm to obtain the MLEs via the Gibbs sampler. The methodology is illustrated using an example from a breast cancer clinical trial in which time to disease progression is the outcome, and the incomplete covariate is a quality of life physical well-being score taken after the start of therapy. This score may be missing because the patients are sicker, so this covariate could be non-ignorably missing.

Algorithms↗

PubMed1996

Fitting limiting dilution experiments with generalized linear models results in a test of the single-hit Poisson assumption.

Limiting dilution analysis is a common technique that is used in immunology to estimate accurately the frequency of cells possessing a wide variety of functional activities such as growth, cytotoxicity and production of lymphokines. The reliability of the estimated frequency is usually checked by a standard chi-square (x2) test validating the goodness-of-fit to the single-hit Poisson model (SHPM). We present evidence that modelling limiting dilution data according to a generalized linear model offers an alternative to the standard x2 test for detecting departures from the SHPM, with a considerable increase in power compared to the x2 test.

Immunologic Techniques↗

PubMed1998

Choosing among generalized linear models applied to medical data.

When testing for a treatment effect or a difference among groups, the distributional assumptions made about the response variable can have a critical impact on the conclusions drawn. For example, controversy has arisen over transformations of the response (Keene). An alternative approach is to use some member of the family of generalized linear models. However, this raises the issue of selecting the appropriate member, a problem of testing non-nested hypotheses. Standard model selection criteria, such as the Akaike information criterion (AIC), can be used to resolve problems. These procedures for comparing generalized linear models are applied to checking for difference in T4 cell counts between two disease groups. We conclude that appropriate model selection criteria should be specified in the protocol for any study, including clinical trials, in order that optimal inferences can be drawn about treatment differences.

Clinical Trials as Topic↗

PubMedDate not supplied

Divide-and-conquer approach for brain machine interfaces: nonlinear mixture of competitive linear models.

This paper proposes a divide-and-conquer strategy for designing brain machine interfaces. A nonlinear combination of competitively trained local linear models (experts) is used to identify the mapping from neuronal activity in cortical areas associated with arm movement to the hand position of a primate. The proposed architecture and the training algorithm are described in detail and numerical performance comparisons with alternative linear and nonlinear modeling approaches, including time-delay neural networks and recursive multilayer perceptrons, are presented. This new strategy allows training the local linear models using normalized LMS and using a relatively smaller nonlinear network to efficiently combine the predictions of the linear experts. This leads to savings in computational requirements, while the performance is still similar to a large fully nonlinear network.

Artificial Intelligence↗

PubMed1991

The evaluation of agreement by means of log-linear models: proxy interviews on reproductive history among floriculture workers in Colombia.

We used data from Colombia to compare responses from husbands and wives concerning the reproductive history of the women. We analyzed agreement in two ways: First, we compared kappa statistics between independent subgroups. Second, we evaluated agreement by means of log-linear models. Men underreported the total number of pregnancies and the number of abortions of their wives. Agreement on the term of the pregnancy was dependent on the ages of the partners. Log-linear modeling provides an attractive alternative to more usual ways of studying agreement.

Adolescent↗