Questions tagged [link-function]

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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15 views

When AIC and pseudo-R2 give opposite conclusions in beta regression models

I conducted an experiment to quantify the effect of two factors on a response variable: the response variable (Y) is a proportion (percentage cover) factor A is represented by the continuous ...
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182 views

Generalized Linear Model and Identity link, what's its benefit?

I found a paper saying that a Generalized linear model with an identity link function was used. They standardize some continuous independent variable as well as the continuous dependent variable and ...
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Generalised Linear Model help

Kind of new to coding using Rstudio here. I have data for a survey for 600 individuals, over 4 years (150 p/year), for 30 categories being an absence or presence (0/1) resulting in a total score /30 ...
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Is the following a GLM?

We know that the standard linear model is a partial case of the GLM scenario by taking the identity link function, i.e. $$g(μ)=μ=η=x_i^Tβ$$ However, in one of our past papers we are asked to ...
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Why does the canonical parameter give a link function? Why does this relate $E[Y]$ to $x^T \beta$?

If I have a pdf in the form $f(y|\theta,\phi)=\text{exp}\bigg(\frac{y\theta-b(\theta)}{a(\phi)}+c(y,\phi)\bigg)$, then $\theta$ is called the canonical parameter. I'm told we can get a link function $...
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66 views

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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23 views

Poisson/NegBinom model with identity or logarithmic link

I am wondering about the implications of using (1) a logarithmic, and (2) an identity link within a Poisson model for count data. I have read through related posts here on CV: Pros and Cons of Log ...
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21 views

is there a difference in fitted values $\mu_i$ depending on the link function chosen for a poisson GLM

I'm new to stats/R, and have just started learning about generalized linear models and am a little lost. Does choice of link function (in this case identity link vs log link) affect the fitted values $...
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Why should link functions be differentiable?

I'm beginner in stats. I do know that link functions should be continuous, but I do not understand that why should they be differentiable.
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252 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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How to choose $b(\theta)$ and $g(\mu)$ for the identity link case for Poisson GLM?

I am trying to understand Poisson GLMs for the canonical and non-canonical link functions. I am having difficulty understanding the non-canonical case. I am considering the canonical exponential ...
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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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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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28 views

How to interpret coefficients of parametric terms in comp.risk?

I am trying to fit a flexible competing risks semiparametric regression model with the timereg package. My primary goal is to estimate the effect of Z on the cumulative incidence of the event of ...
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36 views

Log-likelihood using the link identity for poisson?

I understood the Log-likelihood using the link “log” for poisson, λ=exp(α+βx). But I can’t get the Log-likelihood in the case of “identity”, λ=α+βx. How do I get it?. The example is the following data....
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How do I interpret generalized linear regression with continues independent variable with Gaussian family and log link

I am running generalized linear regression Gaussian family and log link. Independent variable is Time (continues variable). Dependent variables: years of practice (continues variable). ...
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102 views

Valid GLM with square root link

I came across the following answer to a problem, and I couldn't reconcile the answer with what I found. I'm sure I did something wrong, but I'm not sure where my mistake is. The model is of the ...
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120 views

Understanding glm and link functions: how to generate data?

I'm trying to take the approach for understanding how certain concepts work, by trying to generate data for them and checking how the output behaves. Currently, I thus realized I don't quite get what'...
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Link function $g(\mu)=\mu$ and what does it mean to the parameters $\beta_1,..,\beta_p$

I am currently taking a course about generalized linear models and I am confused about the concept of link functions. Specifically, suppose we have $y_i$ which is $Bin(1,\mu_i)$ distributed and let $...
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726 views

Why are Poisson regression coefficients biased?

Suppose I run a simple Poisson regression, where $$Y \sim \text{Pois} (5X) $$ If I run a Poisson regression of $Y$ on $X$, I am expecting to get back $5$. Instead I get numbers much higher. Why is ...
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22 views

GEV distribution: nonstationary location sign changes when adding nonstationary scale parameter in MLE

I am trying to estimate the location, scale and shape for a nonstationary GEV distribution with block maxima. HRindex is the yearly maximum daily rainfall. I used following code in R to achieve ...
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Link function of Gamma distribution?

So if I understand correctly, the link function $g(\mu)$ is supposed to transform the range of $\mu$ to $(-\infty, \infty)$ and $g(\mu) = \mu^{-1}$ is a link for the Gamma distribution. But doesn't ...
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Poisson regression with custom offset and link

I have the following model $$Y \sim \operatorname{Poisson}\left(\frac{1}{1+\exp(\beta X)} E\right)$$ In other words, I have count data for Poisson process with exposure E and rate given by the ...
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Interprete GLMM Estimates with log link

i am relatively new to this field and this is my first time using Generalized Linear Mixed-Effects Model. my response variable is Reaction Time (RT) and i have two fixed effects: prime and type. both ...
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34 views

Model/link function to deal with dependent variable in range [-1,1]?

My dependent variable, $Y$, contains values anywhere from -1 to 1 (i.e. it is bounded continuously on the range $[-1,1]$). I know that a regular OLS regression on such a variable would sometimes ...
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Log transformation vs. log link function: Analysing proportional differences

I'm analysing the effect of rhizobial inoculation, fertilisation, and species identity on plant shoot weight. The two species I'm studying have different mean weights, so I'm mostly interested in ...
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Marginal log likelihood for Tobit model heteroskedasticity link function

I am using a Tobit estimator for a demand model left censored at 0. To account for heteroskedasticity, I specify the standard error as follows (using the crch function in the R package crch): $log(\...
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1answer
60 views

Do you specify priors according to the link function's transformed space?

Suppose I'm developing a model where the response variable is weight measured in pounds and is Gamma distributed. I would like to specify a prior on my intercept coefficient using other information ...
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61 views

Trying to slightly alter logistic GLM - Link function seems unstable [closed]

So the data I have is whether a subject has performed a test correctly, or incorrectly. They have to match choose which of a pair of stimuli matches one they have memorized, and this gets harder and ...
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GLM: Empirical cloglog transformation for exploratory data analysis

Prior to fitting a GLM to an ordered categorical response $Y$ (6 levels), I would like to check the linearity assumption between the one (and only) continuous covariate $x$ in my linear predictor and ...
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66 views

What kind of regression to use with heavily skewed data?

I have data with an explanatory variable $X$ (I think I can treat this as continuous, as scores 1-100 on a certain test) and a response variable $Y$ (continuous variable, never lower than 0). Both ...
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1answer
116 views

Is this GLM for the Poisson distribution correct?

I'm currently taking a machine learning class, and one of the problem set questions is to construct a GLM that models the Poisson distribution, defined as $$P(y;\lambda) = e^{-\lambda}\frac{\lambda^y}...
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251 views

Attributes, Pros and Cons of GLM link functions [closed]

I would like some help understanding when the sqrt, 1 / mu^2, and inverse link functions would be useful. Thanks! My notes are shown below - . sqrt - mean of predicted value must be positive ...
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101 views

What is the difference between Gamma GLM on log output and Gamma GLM with log link function?

Here are two models (with R code to provide some context): Model 1: Take the log of the output variable $y$, then apply a Gamma GLM using the default identity link function: ...
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368 views

What useful properties does the canonical link function have?

So here I am studying generalized linear models. I know this question is quite naive and simple, but I do not exactly know why the link canonical function is so useful. Could someone provide me an ...
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57 views

Which model should I use? (genomic problem)

I have problems with choosing which model / link function should I use for my analysis. My response: numbers from -100% to +500% (increase of tumor after therapy, may switch to ratios or log-ratios, ...
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In gradient boosting, what is the thing being boosted called?

Quoting from this answer that explains how to do boosting when a 'link' function is involved: "Instead, we can re-express this as a function of $L_i$, (in this case also known as the log odds) $$ \...
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133 views

Conditions for link functions in glm

While studying the glm, I found the conditions for the link function to be confusing for me. Lecture notes I'm studying with say the link function should be monotone and differentiable. I understand ...
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1answer
2k views

Gamma glm log link - what does predicted values mean

Does the predict function in R for gamma glm with log link predict the actual values or the mean value? There is a gamma glm model in R with log link. Using predict(model,data,type = 'response') to ...
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53 views

Canonical Location Parameter

What does "Canonical Location Parameter" mean? I'm looking for a general meaning of this term and then separately within the context of generalized linear models.z UPDATE Given how often this term ...
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1answer
2k views

Canonical link of Gamma Distribution [duplicate]

I wonder why my professor said that Gamma's canonical link is $\frac{1}{\mu}$. My thoughts are: EDIT: $\theta$ is the canonical parameter. Since $$\mathbb{E}_\theta(Y)=b^{'}(\theta)=-\frac{1}{\theta}...
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134 views

In a gamma regression, how can i interpret coefficients?

My question is pretty simple, i have done a bayesian gamma regression with an inverse link, so: $\eta_i$=$\beta_0+\beta_1x_{i1}+\dots+\beta_px_{ip}$ < using an inverse link, mu is the ...
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2answers
541 views

Beta regression - interpret coefficients using loglog link

Although a number of similar questions (some of them duplicates) have been asked around the interpretation of the coefficients from a beta regression, these seem to be focused on models that have used ...
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190 views

Use of sqrt link with negative binomial glmer

It doesn't seem to be directly possible to use a sqrt link function in lme4::glmer.nb. This is a pity because on my specific data, with the fix effect model, the sqrt link does improve the ...
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2answers
282 views

GLMM for count data using square root link in lme4

I have data from a field survey. The objective of the study is to relate number of seedling (respond variable, count data), landform (exploratory variable, categorical variable with 3 levels) and ...
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1answer
119 views

Compare link function: generalized linear mixed binomial

How does one compare the Goodness of Fit for different models of some given binomial data using Generalized Linear Mixed Models. Specifically we want to know whether a model with a logit link gives a ...
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Bagging of models with link functions

I'm trying to predict proportion data, and I've got a small dataset (~4000), so holding out a test and validation set isn't practical. However, bagging is practical because the cost of training isn't ...
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1answer
230 views

Why does the natural logarithm function pop up as default link function when doing Poisson regression? [duplicate]

What is the philosophy behind the logarithm as link function for Poisson regression? What is it about count variables that modelling them as a GLM with the natural logarithm function is appropriate? ...
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141 views

Link functions and fitting models to data with mle2 in bbmle

I'm fitting models to data with the mle2() function in the bbmle package. In an example problem (reproducible code below) I am fitting a count response variable ...
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1answer
56 views

GLM difference between the links [duplicate]

I'd be very grateful if someone could help me understand the idea behind link functions: I know that the idea is that we want to map the mean to the $\eta$-vector. Also, I know that the canonical ...