scholarly journals An Exploration of Link Functions Used in Ordinal Regression

2020 ◽  
Vol 18 (1) ◽  
pp. 2-15
Author(s):  
Thomas J. Smith ◽  
David A. Walker ◽  
Cornelius M. McKenna
Keyword(s):  
Survey Data ◽  
Generalized Linear Models ◽  
Linear Models ◽  
Simulated Data ◽  
Ordinal Regression ◽  
Research Center ◽  
Link Function ◽  
25Th Anniversary ◽  
Link Functions ◽  
The Web

The purpose of this study is to examine issues involved with choice of a link function in generalized linear models with ordinal outcomes, including distributional appropriateness, link specificity, and palindromic invariance are discussed and an exemplar analysis provided using the Pew Research Center 25th anniversary of the Web Omnibus Survey data. Simulated data are used to compare the relative palindromic invariance of four distinct indices of determination/discrimination, including a newly proposed index by Smith et al. (2017).

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.

Summary goodness-of-fit statistics for binary generalized linear models with noncanonical link functions

2015 ◽  
Vol 58 (3) ◽  
pp. 674-690 ◽  
Author(s):  
Jana D. Canary ◽  
Leigh Blizzard ◽  
Ronald P. Barry ◽  
David W. Hosmer ◽  
Stephen J. Quinn
Keyword(s):  
Generalized Linear Models ◽  
Goodness Of Fit ◽  
Linear Models ◽  
Fit Statistics ◽  
Link Functions

On the Approximation Rate of Hierarchical Mixtures-of-Experts for Generalized Linear Models

1999 ◽  
Vol 11 (5) ◽  
pp. 1183-1198 ◽  
Author(s):  
Wenxin Jiang ◽  
Martin A. Tanner
Keyword(s):  
Generalized Linear Models ◽  
Upper Bound ◽  
Linear Models ◽  
Sobolev Class ◽  
Link Function ◽  
Approximation Rate ◽  
Smooth Functions ◽  
Mixtures Of Experts ◽  
The Family

We investigate a class of hierarchical mixtures-of-experts (HME) models where generalized linear models with nonlinear mean functions of the form ψ(α + xTβ) are mixed. Here ψ(·) is the inverse link function. It is shown that mixtures of such mean functions can approximate a class of smooth functions of the form ψ(h(x)), where h(·) ε W∞2;k (a Sobolev class over [0, 1]s, as the number of experts m in the network increases. An upper bound of the approximation rate is given as O(m−2/s) in Lp norm. This rate can be achieved within the family of HME structures with no more than s-layers, where s is the dimension of the predictor x.

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.

scholarly journals Instantaneous Geometric Rates via Generalized Linear Models

2017 ◽  
Vol 17 (2) ◽  
pp. 358-371 ◽  
Author(s):  
Andrea Discacciati ◽  
Matteo Bottai
Keyword(s):  
Clinical Trial ◽  
Randomized Clinical Trial ◽  
Linear Model ◽  
Generalized Linear Models ◽  
Generalized Linear Model ◽  
Renal Carcinoma ◽  
Linear Models ◽  
Rate Model ◽  
Model Framework ◽  
Link Functions

The instantaneous geometric rate represents the instantaneous probability of an event of interest per unit of time. In this article, we propose a method to model the effect of covariates on the instantaneous geometric rate with two models: the proportional instantaneous geometric rate model and the proportional instantaneous geometric odds model. We show that these models can be fit within the generalized linear model framework by using two nonstandard link functions that we implement in the user-defined link programs log_igr and logit_igr. We illustrate how to fit these models and how to interpret the results with an example from a randomized clinical trial on survival in patients with metastatic renal carcinoma.

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