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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Interpretetion of linear predictor of a random variable that follows a gamma distribution

Assuming that: $0 < \nu, \alpha, y < \infty$ $$f_Y(y; \nu, \alpha) = \frac{y^{\nu-1}{\alpha}^{\nu}e^{-y\alpha}}{\Gamma (\nu)} \mathbb{1}_{Y \in (0, \infty)}$$ $$ = \exp \{ -y\alpha + \nu \log \...
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Expected Fisher Information Matrix for Gamma Distribution using canonical link

How to find the fisher information matrix for a random variable $Y \sim $ Gamma$(\nu,\alpha)$? $0 < \nu, \alpha, y < \infty$ I have written: $$f_Y(y; \nu, \alpha) = \frac{y^{\nu-1}{\alpha}^{\nu}...
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Modeling both mean and variance in a linear model

I have a variable $X$ that decays log-normally with time, and I have estimated the mean and the SD of that log-linear relationship. I also have a (categorical) variable $Y$ which—I hypothesize—will ...
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Link functions and interpreting credible intervals

I am pretty new to statistics, and was trying to interpret credible intervals from a bayesian analysis I had preformed. Some of my models are glms, and so have a link function. I know that to ...
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SAS proc genmod regression equation

I am using proc genmod with tweedie distribution and log link to analyze positively skewed outcome. Trying to figure out the actual regression equation to include ...
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61 views

Log transformation in GLM and model fit

For a negative binomial GLM, are we allowed to write the log transformation in the following way? ...
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37 views

glm with log link in binomial family

I was reading this post https://r-posts.com/simulations-comparing-interaction-for-adjusted-risk-ratios-versus-adjusted-odds-ratios/ and found that the author adjusted a glm with binomial family and ...
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Gamma GLM: why log-link is more common than canonical link

"The canonical link of Gamma GLM is $g(x)=1/x$ is often not very practical. Log-link is more appropriated in most cases." One reason I can think of is that log-link makes sure $\mu$, the ...
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GLM: effect of link function on choice of transformation of covariate

It struck me that if I have data of the form below, ...
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Should you standardize when using a Log link?

If I use a model with a log link function should I still standardize independent variables (since they differ in the scales range) or the log transformation is enough?
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Why do we choose exponential function as the nonlinearity in Possion GLM

In Poisson GLM, the response variable $Y$ follows the Poisson distribution $$P(Y=y)=\lambda^y\exp(-\lambda)/y!$$ and: $$\lambda=\exp(\bf \theta^Tx)$$ My question is why do we use exponential as the ...
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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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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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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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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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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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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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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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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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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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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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745 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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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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39 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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1answer
66 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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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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82 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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139 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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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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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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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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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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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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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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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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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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139 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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633 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 ...