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PubMed2004

Population response to climate change: linear vs. non-linear modeling approaches.

BACKGROUND: Research on the ecological consequences of global climate change has elicited a growing interest in the use of time series analysis to investigate population dynamics in a changing climate. Here, we compare linear and non-linear models describing the contribution of climate to the density fluctuations of the population of wolves on Isle Royale, Michigan from 1959 to 1999. RESULTS: The non-linear self excitatory threshold autoregressive (SETAR) model revealed that, due to differences in the strength and nature of density dependence, relatively small and large populations may be differentially affected by future changes in climate. Both linear and non-linear models predict a decrease in the population of wolves with predicted changes in climate. CONCLUSIONS: Because specific predictions differed between linear and non-linear models, our study highlights the importance of using non-linear methods that allow the detection of non-linearity in the strength and nature of density dependence. Failure to adopt a non-linear approach to modelling population response to climate change, either exclusively or in addition to linear approaches, may compromise efforts to quantify ecological consequences of future warming.

Age Factors↗

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↗

PubMed2005

Some limitations of a proposed linear model for antimicrobial risk management.

The FDA Center for Veterinary Medicine (CVM) (Bartholomew et al., 2005) recently proposed an approach to risk management based on the linear modeling framework: Risk = K x Exposure. They suggest that, once K has been estimated from historical data, it can be used to predict how limiting future exposure will reduce future risk. They illustrate the approach for fluoroquinolone-resistant campylobacter in chicken. However, despite its appealing simplicity, the proposed approach confuses a possibly meaningless descriptive statistical ratio with a valid predictive causal relation. In general, the historical ratio K = (Risk/Exposure) may not predict how changing future exposures will affect future risks, and hence it does not necessarily provide an appropriate guide to current risk management actions. We identify several limitations of the proposed framework, including omission of frequency and severity of human health harm in quantifying "Risk" and omission of microbial load from "Exposure." Finally, we show that an extended linear modeling approach that considers impacts of changing animal antibiotic use on susceptible as well as on resistant bacteria is consistent with the conclusion that reducing "Exposure" can greatly increase "Risk."

Animals↗

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↗

PubMed2004

Evaluating assumptions for least squares analysis using the general linear model: a guide for the pharmaceutical industry statistician.

A review of graphical and test based methods for evaluating assumptions underlying the use of least squares analysis with the general linear model is presented along with some discussion of robustness. Alternative analyses are described for situations where there is evidence that the assumptions are not reasonable. Evaluation of the assumptions is illustrated through the use of an example from a clinical trial used for US registration purposes. It is recommended that: (1) most assumptions required for the least squares analysis of data using the general linear model can be judged using residuals graphically without the need for formal testing, (2) it is more important to normalize data or to use nonparametric methods when there is heterogeneous variance between treatment groups, and (3) nonparametric analyses can be used to demonstrate robustness of results and that it is best to specify these analyses prior to unblinding.

Aging↗

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↗

PubMed2007

A practical approach to computing power for generalized linear models with nominal, count, or ordinal responses.

Data analysts facing study design questions on a regular basis could derive substantial benefit from a straightforward and unified approach to power calculations for generalized linear models. Many current proposals for dealing with binary, ordinal, or count outcomes are conceptually or computationally demanding, limited in terms of accommodating covariates, and/or have not been extensively assessed for accuracy assuming moderate sample sizes. Here, we present a simple method for estimating conditional power that requires only standard software for fitting the desired generalized linear model for a non-continuous outcome. The model is fit to an appropriate expanded data set using easily calculated weights that represent response probabilities given the assumed values of the parameters. The variance-covariance matrix resulting from this fit is then used in conjunction with an established non-central chi square approximation to the distribution of the Wald statistic. Alternatively, the model can be re-fit under the null hypothesis to approximate power based on the likelihood ratio statistic. We provide guidelines for constructing a representative expanded data set to allow close approximation of unconditional power based on the assumed joint distribution of the covariates. Relative to prior proposals, the approach proves particularly flexible for handling one or more continuous covariates without any need for discretizing. We illustrate the method for a variety of outcome types and covariate patterns, using simulations to demonstrate its accuracy for realistic sample sizes.

Computer Simulation↗

PubMed2006

Genetic evaluation of mastitis in dairy cattle using linear models, threshold models, and survival analysis: a simulation study.

The objective was to study, by simulation, whether survival analysis results in a more precise genetic evaluation for mastitis in dairy cattle than cross-sectional linear models and threshold models by using observation periods for mastitis of 2 lengths (the first 150 d of lactation, and the full lactation, respectively). True breeding values for mastitis liability on the underlying scale were simulated for daughters of 400 sires (average daughter group size, 60 or 150), and the possible event of a mastitis case within lactation for each cow was created. For the linear models and the threshold models, mastitis was defined as a binary trait within either the first 150 d of lactation or the full lactation. For the survival analysis, mastitis was defined as the number of days from calving to either the first case of mastitis (uncensored record) or to the day of censoring (i.e., day of culling, lactation d 150 or day of next calving; censored record). Cows could be culled early in lactation (within 10 d after calving) for calving-related reasons or later on because of infertility. The correlation between sire true breeding values for mastitis liability and sire predicted breeding values was greater when using the full lactation data (0.76) than when using data from the first 150 d (0.70) with an average of 150 daughters per sire. The corresponding results were 0.60 and 0.53, respectively, with an average of 60 daughters per sire. Under these simulated conditions, the method used had no effect on accuracy. The higher accuracy of sire breeding values can be translated into a greater genetic gain, unless counteracted by a longer generation interval.

Animals↗

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↗

PubMed2006

A simple linear model for the effect of changes in metabolic risk factors on coronary heart disease.

OBJECTIVES: The risk of having one or more metabolic risk factors in the development of coronary heart disease (CHD) is well estimated, but it still remains to be shown which influence any given change in the number of risk factors has on the overall risk of CHD. DESIGN: In this prospective cardiovascular population study, 10 194 participants were examined twice with a 5-year interval for metabolic risk factors such as obesity, hypertension, hypercholesterolaemia and diabetes mellitus. RESULTS: During 14 years of follow-up, 1724 incident cases of CHD were identified, 973 in men and 751 in women. The effect of an increase in the number of metabolic risk factors during a 5-year interval on the relative risk of CHD could be determined to be statistically significant for both genders. But only for men, the effect of a decrease in the number of metabolic risk factors was statistically significant during the same time interval. Hence, by statistical analysis, a simple linear model could be constructed for men with two linear trend parameters, one corresponding to the number of metabolic risk factors at the first examination and one corresponding to the change in the number of metabolic risk factors from the first to the second examination. The parameters were 1.50 (1.39-1.63); P < 0.001 and 1.29 (1.18-1.41); P < 0.001, respectively. CONCLUSIONS: This study suggests that it is possible to fit a simple linear model for the effect of changed number of metabolic risk factors on the risk of CHD.

Adult↗

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↗