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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### Handling Non-normality of Reaction Time Data in Mixed Models

I am examining the effect of 'Phase' on reactions time (RT) data using a mixed model in lme4. However, as is common with RT data, the residuals are non-normal. This is the first model, which is a ...
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### GLM link function, variance Function, and dispersion for linear regression

we were recently introduced in class to GLM's and I am still trying to wrap my head around link functions, variance functions and dispersion paramters. In particular we were asked re-write the linear ...
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### Link function for exponential regression

I have a dataset with an exponential relationship that I'm linearizing before fitting with code like this: glm(log(y) ~ log(x), family = gaussian) I'm trying to ...
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### Custom link function for logistic regression model

I am trying to build a generalized linear model with a custom link function. This is a follow up to this question: Understand and specify a generalized logistic model in R In this paper A Generalized ...
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151 views

### Understand and specify a generalized logistic model in R

I am reading a paper in which the authors models tree survival (mortality). They go and remeasure tagged trees for decades to establish "survival functions" for the given tree species and ...
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### Residuals assumptions in glmm not verified! Help

I am trying out GLMMs models to test whether two categorical variables (species and sex) and their interaction (sex + species + sex*species= fixed factors) influence certain acoustic parameters (...
1 vote
61 views

### Predicted probabilities from logistic vs log-binomial model

I am giving a talk on logistic regression and I was going to mention log-binomial models to estimate risk-ratios. I understand the difference between odds and probabilities and that they only converge ...
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### Is there a link function $g$ which is a monotonic function with $g(0) = 0$? [closed]

I am working on a machine learning project (similar to this paper https://arxiv.org/abs/1912.04136). For this I need to make an assumption that a link function is monotonic and takes value $0$ at $0$. ...
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1 vote
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### Can expected variance of GLM be expressed through gradients under non-cannonical link functions?

In GLMs (generalized linear models), one can typically obtain an estimate of the expected variance of the response variable given the predictors as a transformation of the same parameter that defines ...
1 vote
26 views

### How to model suspected piecewise-linear data with a lasso GLM

My data consist ~130 observations. Each observation has several thousand features (including many collinear or otherwise useless features) and a position along a single spatial dimension. Some sets of ...
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### meaning of this GLM defintion

The answer key says "Different link functions have different shapes and can therefore fit to different nonlinear relationships between the predictors and the target variable." Shouldn't we ...
59 views

### Extremely small confidence intervals and standard deviations in a clmm-model

I have a data set with an effort rating (varying in an ordinal scale from 1 to 7) as the response variable and a time point of measure (seven consecutive measures) as a fixed factor. An individual is ...
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### Use a GLM/GAM model with Gamma family & identity link despite warning?

What I have tried and done so far: I am running GLMs and GAMs on my positive, continuous response variable. I have determined that the Gamma distribution would fit my data best by plotting various QQ ...
1 vote
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### Coding error? Why is the treatment not showing as a significant effect with GLM and standard error is so high?

I've been handed off some data to analyze and I'm looking for something simple and straightforward. The main study question is testing the efficacy of 7 herbicides, with one untreated control. The 1 ...
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### is a threshold model on ordinal data ~ a link function? SEM OpenMx

Are anyone familiar with OpenMx's capacity for handling ordinal data in SEM using a link function like ordered logit or probit (Stata gsem does this)? Some folks have highlighted issues with feeding ...
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### How to interpret coefficients of a GLM (Gamma family with identity link)

I'm trying to interpret a coefficient of a glm model with the gamma family and the identity link function. The outcome is continuous, positive and right-skewed. Transformation did not yield a normal ...
1 vote
80 views

### bootstrapped confidence interval for glmer model with identity link

In an experiment Subjects decide as fast as possible whether letter strings presented on screen are real Words or non-words and reaction time (RT) in milliseconds is recorded. We want to know for ...
67 views

### XGBoost custom objective/loss: when is a Reverse (inverse) Link Function required?

I'm trying to implement a custom objective function in XGBoost. I read the docs on this topic. I am not sure if I need to define a "reverse link function" (aka inverse link function) to ...
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### GLM: Differing results for Interaction Effects depending on the link-function

I want to test whether whether a dispositional risk factors moderates the relation between a situational risk factor and a negative outcome in a regression model (including several control variables). ...
1 vote
48 views

### Is this model linear function belongs to GLM models?

In GLM, we assume that $\mathbb{E}[Y|X]=\mu(\beta^\top X)$ and $Y|X$ follows exponential family distribution. I am going to assume that the probability of success in the Bernoulli distribution is ...
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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 ...
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### 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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### 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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1 vote
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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
22 views

### 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 ...
22 views

### 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 ...
536 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 ...
361 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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Consider a model $$Y = inv.logit(\beta_0 + \beta_1X_1 + \dots, +\beta_dX_d) + \varepsilon,$$ where $inv.logit(x) = \frac{e^x}{1+e^x}$, and $\varepsilon\sim Gumbel$ is a centered Gumbel noice. How can ...
793 views

### 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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1 vote
214 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 ...
603 views

### Confidence intervals for binomial generalised linear model with cauchit link function

The correct way to calculate a confidence interval (CI) for a generalised linear model (GLM) and avoid the problems of normal approximation intervals has been adequately discussed by Gavin Simpson ...
3k views

### 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
39 views

### 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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