Generalized Linear Models

2021 ◽  
pp. 195-208
Author(s):  
Andy Hector
Keyword(s):  
Regression Analysis ◽  
Maximum Likelihood ◽  
Generalized Linear Models ◽  
Linear Models ◽  
Square Root ◽  
Link Function ◽  
Data Transformations ◽  
Normal Distributions ◽  
Equal Variance ◽  
The Mean

This chapter revisits a regression analysis to explore the normal least squares assumption of approximately equal variance. It also considers some of the data transformations that can be used to achieve this. A linear regression of transformed data is compared with a generalized linear-model equivalent that avoids transformation by using a link function and non-normal distributions. Generalized linear models based on maximum likelihood use a link function to model the mean (in this case a square-root link) and a variance function to model the variability (in this case the gamma distribution, where the variance increases as the square of the mean). The Box–Cox family of transformations is explained in detail.

Covariance matrix of the bias-corrected maximum likelihood estimator in generalized linear models

2013 ◽  
Vol 55 (3) ◽  
pp. 643-652
Author(s):  
Gauss M. Cordeiro ◽  
Denise A. Botter ◽  
Alexsandro B. Cavalcanti ◽  
Lúcia P. Barroso
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Covariance Matrix ◽  
Generalized Linear Models ◽  
Linear Models ◽  
Likelihood Estimator

Posterior contraction in sparse generalized linear models

Biometrika ◽  
2020 ◽  
Author(s):  
Seonghyun Jeong ◽  
Subhashis Ghosal
Keyword(s):  
Generalized Linear Models ◽  
Bayesian Methods ◽  
Generalized Linear Model ◽  
Linear Models ◽  
Continuous Distribution ◽  
Fractional Power ◽  
Regression Coefficients ◽  
Link Function ◽  
Convergence Properties ◽  
Set Up

Summary We study posterior contraction rates in sparse high-dimensional generalized linear models using priors incorporating sparsity. A mixture of a point mass at zero and a continuous distribution is used as the prior distribution on regression coefficients. In addition to the usual posterior, the fractional posterior, which is obtained by applying Bayes theorem with a fractional power of the likelihood, is also considered. The latter allows uniformity in posterior contraction over a larger subset of the parameter space. In our set-up, the link function of the generalized linear model need not be canonical. We show that Bayesian methods achieve convergence properties analogous to lasso-type procedures. Our results can be used to derive posterior contraction rates in many generalized linear models including logistic, Poisson regression and others.

scholarly journals Association Between Pulsatile Components of Blood Pressure and Severe Tooth Loss in Rural Ecuador: The Three Villages Study

2020 ◽  
Vol 11 ◽  
pp. 215013272092867 ◽  
Author(s):  
Oscar H. Del Brutto ◽  
Robertino M. Mera ◽  
Bettsy Y. Recalde ◽  
Andrew P. Torpey ◽  
John P. Hill ◽  
...  
Keyword(s):  
Blood Pressure ◽  
Arterial Hypertension ◽  
Generalized Linear Models ◽  
Tooth Loss ◽  
Linear Models ◽  
Community Dwelling ◽  
Multivariate Models ◽  
Mediation Analyses ◽  
The Mean ◽  
Relevant Covariates

Background: Arterial hypertension has been associated with severe tooth loss, but differential associations with individual components of blood pressure (BP) have scarcely been investigated. We assessed the independent associations between pulsatile/steady components of BP and severe tooth loss in community-dwelling adults residing in 3 rural Ecuadorian villages. Methods: Individuals aged ≥40 years living in Atahualpa, El Tambo, and Prosperidad were identified during door-to-door surveys. Data collection focused on the number of remaining teeth and measurements of pulsatile/steady components of BP. Multivariate models were fitted to assess independent associations between pulsatile/steady BP components and severe tooth loss, after adjusting for relevant covariates. Results: A total of 1543 individuals were included. Oral exams identified 426 (28%) individuals with severe tooth loss. BP levels ≥140/90 mm Hg were determined in 481 (31%) individuals. The mean pulse pressure (PP) level was 55.3 ± 19 mm Hg. For systolic BP (SBP), the mean level was 133.1 ± 23.5 mm Hg, and for diastolic BP (DBP) it was 77.8 ± 11.5 mm Hg. Univariate models showed significant associations between severe tooth loss and SBP and PP, but not with DBP. However, the significance was taken away in fully adjusted generalized linear models. Age remained as an independent significant covariate in models using SBP and PP. Causal mediation analyses disclosed that percentages of the effect of severe tooth loss mediated by age were 99.5% for SBP and 98.9% for PP. Conclusion: This study shows that age captures most of the effect of the association between pulsatile components of BP and severe tooth loss.

scholarly journals The opposite effects of routine metabolic rate and metabolic rate during startle responses on variation in the predictability of behaviour in hermit crabs

Behaviour ◽  
2016 ◽  
Vol 153 (13-14) ◽  
pp. 1545-1566 ◽  
Author(s):  
Mariana Velasque ◽  
Mark Briffa
Keyword(s):  
Metabolic Rate ◽  
Generalized Linear Models ◽  
Startle Response ◽  
Linear Models ◽  
Elevated Temperatures ◽  
Hermit Crabs ◽  
Routine Metabolic Rate ◽  
Intraindividual Variation ◽  
The Mean ◽  
Pagurus Bernhardus

Studies on animal behaviour have suggested a link between personality and energy expenditure. However, most models assume constant variation within individuals, even though individuals vary between observations. Such variation is called intraindividual variation in behaviour (IIV). We investigate if IIV in the duration of the startle response is associated with metabolic rates (MR) in the hermit crabPagurus bernhardus. We repeatedly measured startle response durations and MR during each observation. We used double hierarchical generalized linear models to ask whether among and IIV in behaviour was underpinned by MR. We found no association between the mean duration of the startle responses and either routine MR or MR during startle response. Nevertheless, we found that IIV increased with MR during startle responses and decreased with routine MR. These results indicate that crabs with higher MR during startle responses behave less predictably, and that predictability is reduced during exposure to elevated temperatures.

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