A transformation of a parameter governing a response distribution that is used as a crucial part of the generalized linear model to map that parameter's range (which may be from 0 to 1, or only positive values, e.g.) to the real number line $(-\infty, +\infty)$.

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### How to prove the Poisson link function is a canonical link function?

So I'm a 3rd year undergraduate doing my thesisin football score models right now. In my thesis I want to include a proof of what the link function for the Poisson distribution is and why it relates ...
14 views

### Difference between specifying link function in GLM and transforming y data in advance? [duplicate]

I'm a beginner in stats and I have a very basis question on GLM. Suppose Gaussian distribution. GLM tries to do this: $$g(E(y|X))=X\beta$$ where $g$ is a link function. But if I transform $y$ before ...
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### Zero Inflated Beta Regression (ZOIB)

I'm using R to model zero inflated beta regression. Upon reading some documentation for the ZOIB package, I came across some functionality for adding random effects to one or many of the link ...
1 vote
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### Is it possible to build residuals that are uncorrelated with predictors from regressions with a binary outcome, without an identity link?

Here: What do the residuals in a logistic regression mean? is a description of the different ways how one can generate residuals from a logistic regression. My question concerns not the logistic ...
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### reasonable distributions for non-negative real-valued data with many zeroes, GAMs

Lots of data is real-valued, non-negative, and replete with 0s... for example any time count data are normalized by something: e.g., perhaps it makes sense normalize disease case counts taken over ...
1 vote
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### Is it possible to average models with different link functions?

While I can compare models that have different link functions in terms of AIC/BIC weights, I think it's impossible to use those weights to create an averaged model. Am I right in believing this?
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### Under what conditions do different choices of link function for GLMs result in considerably different models?

The statistical folklore I have heard is that the choice of link function usually does not considerably affect the resulting fit of a GLM. For example, usually probit regression and logistic ...
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### Different results with different link functions

I have used the R package eventglm to construct pseudo observations, and want to estimate the relative risk and the risk difference for exposure with age adjustment....
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On pg. 125 in Agresti's Categorical Data Analysis, it's suggested by a plot of the dependent variable (a count) vs an independent variable (categorized version of continuous width variable) that the ...
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### Purpose for the conditions of Link Function

I am studying GLM at the moment and have a few questions regarding link functions. Why are the conditions of the link function to be smooth monotonic function? What properties are preserved by having ...
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### Split linear predictors with link function

Is it possible to split linear predictors contribution up when talking glm of non-normal distributions? If: $$µ_i = g^{-1}(η_i)$$ and $$µ_i = g^{-1}(β_0 + β_1X_{i1} + β_2X_{i2} +···+β_kX_{ik})$$ Is it ...
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### GLM with Gamma distribution: Choosing between two link functions

I need to perform a GLM based analysis on a purely positive, continuous, and highliy right skewed (inflated around low values) outcome variable. I tested several combinations of distributions and link ...
262 views

### Back transform predict.gam() from nb link log model run?

I have model with 1 covariate. I would like to run y values from gam in another model. I used nb(log=link) in gam model. Because I used nb and link log in gam, do I need to back transform to use ...
145 views

### Back transforming standard errors in a GLMM with a log-link

I'm using lme4 and have a GLMM with a log link and gaussian variance structure. I would like to report my fixed effect estimates with their standard errors, as well as the standard deviation of my ...
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### Which link function in binomial regression is better?

Concerning the choice of the link function in binomial regression (e.g. logit versus probit or cauchit), I wonder what the recommended comparison criterion might be. Note that I am not interested in ...
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### How can I use the link(g, lam) function from "psyphy" package to adjuste asymptotes?

I need to customize the asymptotes of the model, and I am trying with psyphy package which provides parameters for adjusting asymptotes in its ...
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173 views

### Transforming the expected value of $Y_i$ in binomial regression

Currently, I'm learning generalized linear regression (GLM). There is something troubling me concerning binomial regression. In this text, in the part about the structure of a GLM, the random ...
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### Mean finger volume: Is a GLM with log link function appropriate?

I have a model where the volume ($V$) of a finger is normally distributed, with mean $\mu = \beta_0 L^{\beta_1}D^{\beta_2}$ (where $L=$ length, $D=$ diameter and $\beta_i \in \Bbb R$ for $i=0,1,2$) ...
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### Is this the correct way to compute confidence intervals on the original scale for GLM(M)s?

Suppose I have fitted a GLM and want to produce a confidence interval (or a prediction interval) on the original scale of the outcome. What I would do is estimate it on the link scale and then inverse ...
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### Why is the canonical parameter linearly related to the input x in GLMs and why does it give the link function?

In Andrew Ng's CS229 notes, one of the three assumptions he makes for constructing GLM models is: The natural parameter $\eta$ and the inputs x are related linearly: $\eta=\theta^Tx$ He goes on to ...
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### Link functions in poisson regression

I've recently started studying statistics and a question came up to my mind while reading about poisson regression: If we have to exponentiate all terms in order to have only positive values, why do ...
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### Interpretation difference between log link and log transformation

I have a question about the interpretation difference between log link of GLM and log transformation of LM. I know that log transformation is for target variable but log link is for mean .But related ...
1 vote
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### Is log-link function important in this case? [closed]

I have a positive count response $Y$ (number of times that a particular pattern was observed within a single day) and positive count independent variables $X_{1}, X_{2}$ (that are associated with the ...
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### Geometric distribution: finding canonical link and proving it is part of the natural exponential family?

Looking for some help on my statistics homework question! The background to the question is: suppose that you toss a biased coin repeatedly (and independently) until you get a head. Let Y denote the ...
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### The identity link not used for binary response

Questions: The identity link is the standard one with normal responses but is not often used with binary or count responses. Why do you think this is? My idea: The range for a linear predictor, and ...
747 views

### Logistic regression with actual probabilities $\in(a,b)$ where $0<a<b<1$

When modelling probabilities with a logistic regression$^1$, the range of fitted probabilities is $(0,1)$. The logit function$^2$ asymptotes at $0$ and $1$, so this is a good match. However, in some ...
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### Would this modification accelerate convergence of generalized linear model, or break it?

This page describes the following iteratively reweighted linear least-squares (IRLS) method for solving a generalized linear model (GLM): let $x_1=0$ for $j=1,2,...$ do linear ...
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473 views

### Negative Value Gamma GLM Inverse Link

I want to calculate a log-likelihood score for a gamma glm with inverse link function. The score will be used to find optimal parameters ($\beta$) for the model. I'm fixing the shape parameter and ...
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