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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↗

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↗

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↗

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↗

PubMed1996

Interval censored survival data: a generalized linear modelling approach.

A method is described for weak parametric modelling of arbitrarily interval censored survival data using generalized linear models. The method makes use of an associated Bernoulli model, with standard errors based on the observed information matrix. Three types of models are discussed: additive and multiplicative hazard models with piecewise constant baseline hazard, and a proportional hazards model with discrete baseline survivor function. These models may be fitted in the statistical package GLIM.

Breast Neoplasms↗

PubMed1993

Application of a hierarchical linear model to the study of adolescent deviance in an overlapping cohort design.

Hierarchical linear models provide a conceptual orientation and a flexible set of analytic techniques for studying psychological change in repeated measures studies. The researcher first formulates a model for individual change over time, with each individual's development characterized by a unique set of parameters. These parameters are then viewed as varying randomly over the population of persons. We illustrate this approach with data on attitudes toward deviance during adolescence (Raudenbush & Chan, 1992), indicating how one may assess the psychometric properties of an instrument for studying change, compare the adequacy of linear and curvilinear growth models, control for time invariant and time-varying covariates, and link overlapping cohorts of data. The results suggest that prodeviant attitudes characteristically increase during early adolescence, achieving a peak between 17 and 18 years of age. The typical trajectories for male and female adolescents have the same shape, although female adolescents tend to be less deviant than male adolescents at each age. We briefly consider the statistical power of tests of cohort differences at the points where they overlap.

Adolescent↗

PubMed1985

Confidence intervals for a variance ratio, or for heritability, in an unbalanced mixed linear model.

A procedure is presented for constructing an exact confidence interval for the ratio of the two variance components in a possibly unbalanced mixed linear model that contains a single set of m random effects. This procedure can be used in animal and plant breeding problems to obtain an exact confidence interval for a heritability. The confidence interval can be defined in terms of the output of a least squares analysis. It can be computed by a graphical or iterative technique requiring the diagonalization of an m X m matrix or, alternatively, the inversion of a number of m X m matrices. Confidence intervals that are approximate can be obtained with much less computational burden, using either of two approaches. The various confidence interval procedures can be extended to some problems in which the mixed linear model contains more than one set of random effects. Corresponding to each interval procedure is a significance test and one or more estimators.

Analysis of Variance↗

PubMed1992

Estimating prevalence by group testing using generalized linear models.

A method is described for estimating prevalence by group testing using generalized linear models. This provides a simple way of analysing such data using widely available software. Existing methodology to correct for overdispersion using quasi-likelihoods is applied to the group testing model. The methods are illustrated by an estimation of salmonella contamination in eggs, and of yellow fever virus infection in a mosquito population.

Animals↗

PubMed1994

Directional selectivity in a nonspiking interneuron of the crayfish optic lobe: evaluation of a linear model.

1. Intracellular recordings, sine wave gratings, and paired flashes were used to characterize the directional selectivity (DS) of the peripheral neurons of the crayfish visual pathway. DS was observed in nonspiking tangential (Tan1) neurons of the distal medulla externa and it is expressed by the amplitude of the modulated synaptic potential elicited with drifting gratings. 2. The directional mechanism was characterized by variations in the grating contrast, spatial frequency, and temporal frequency. DS is both contrast and velocity dependent. 3. The velocity dependence of DS for fixed stimulus contrast can be described by a linear model including a delay and subtractive compare operation. This mechanism operates over the entire useful range of spatial and temporal frequencies. 4. The parameters of the linear model can be estimated from the spatiotemporal structure of the Tan1 cell receptive field. The receptive field exhibits a spatially asymmetric inhibitory subfield that is offset from the excitatory subfield by 3-5 degrees (1-2 ommatidia). The inhibition is delayed relative to excitation by 50-100 ms. 5. The contrast dependence of DS reflects an apparent nonlinearity in the mechanism that determines the null response amplitude. The preferred response magnitude is approximately linear with variations in contrast. 6. The nonlinearity observed in the null direction can in principle be attributed to either a tonic excitation at 0 contrast or a threshold for inhibition. There is evidence for both processes in the Tan1 cell visual response.

Acetylcholine↗

PubMed1986

Bivariate linear models in neurobiology: problems of concept and methodology.

Bivariate linear models, used to describe morphological and functional characteristics between two sets of observations, are examined both in concept and in application. This paper focuses on the underlying assumptions and statistics of the method most frequently used: ordinary linear regression, principal axis and standard major axis. It is shown how the choice of method should depend on: the purpose of the analysis and the a priori assumptions regarding the residual variance. It appears that none of the methods has a universal application. Differences among the models discussed are illustrated by a bivariate morphometric analysis of cerebrocortical regions in primates.

Animals↗

PubMed1997

Predictive variable selection for the multivariate linear model.

We develop a predictive Bayesian approach to variable selection in the multivariate linear model. A criterion derived from the Bayesian predictive density is proposed and a calibration is provided for it. Reference and informative priors are discussed, and an automated method that focuses on the response variable is proposed for specifying informative priors for the regression parameters. Relationships between the proposed criterion and other several well-known criteria are examined. Illustrative examples involving real data are given to demonstrate the methodology.

Asbestos↗