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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92 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 ...
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How to Interpret Parameter Estimates (B) in GLM With Different Link Functions (inverse, 1/mu^2)

I'm experimenting with different PDF (gamma, inverse gaussian, quasi binomial) as well as different link functions (inverse, 1/mu^2, log, logit) in order to find out which generalized linear ...
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20 views

Which type of distribution and link function for quasi-proportion and proportion data in a GLM with high variance in DV?

First of all, I hope my question is not too broad for Cross Validated. If it should be, I'm sorry, and it would be nice if someone could tell me another statistics forum, but this one seemed most ...
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25 views

Which type of distribution and link function for quasi-proportion and proportion data in a multilevel/longitudinal GLM?

First of all, I hope my question is not too broad for Cross Validated. If it should be, I'm sorry, and it would be nice if someone could tell me another statistics forum, but this one seemed most ...
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10 views

Is the following Bayesian GLM reasonable?

Let $Y\geq 0$ count dependent variable following a $Poisson$ distribution with parameter $\lambda$. Also, let $X_{1}\geq 0,...,X_{q}\geq 0$ counts treated as independent variables. I want to fit the ...
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27 views

Is log-link function important in this case?

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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17 views

Why the p-value of the interaction term changes when changing the link function of binomial Generalised Linear Model

I have fitted three binomial generalised linear models with three different link functions in order to investigate the relative risk/odds ratio/probability difference of the data. fiverelapse: an ...
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1answer
41 views

Finding a confidence interval for difference of proportions

Let two independent random variables, $Y_1$ and $Y_2$ that have binomial distribution have parameters $n_1 = n_2 = 100$, $p_1$ and $p_2$, respectively, be observed to be equal to $y_1 = 50$ and $y_2 = ...
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1answer
70 views

Help understand the virtue of generalized linear models

On page 4 of https://www.sagepub.com/sites/default/files/upm-binaries/21121_Chapter_15.pdf, the authors state the following strength of generalized models, which I don't quite understand. Indeed, one ...
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130 views

Which link function could be used for a glm where the response is per cent (0 - 100%)?

I am thinking about building a model (glm) where the response variable (y) is the cover (in per cent) of a plant species in a defined area, dependant of environmental variables. However, I don't think ...
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1answer
34 views

Conway-Maxwell-Poisson (CMP) - Coefficient interpretation (Log/IRR)

I'm using the Conway-Maxwell-Poisson (CMP) distribution to model the amount of nouns in a clause (data is under-dispersed). I've run the model using glmmTMB (family= "compois") but I'm ...
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Why are we entitled to use the link function we prefer the most?

For a project, we have been trying to fit different models. When we used a Poisson regression, so a glm with a Poisson family, initially our fit was quite bad. But once we used the identity link ...
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Definition of a link function (brms or multilevel model -associated)

I have one question related to a link function defined in BRM. According to the reference (Bürkner, Paul-Christian. "brms: An R package for Bayesian multilevel models using Stan." Journal of ...
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15 views

Why do we link the rate parameter of the Gamma distribution for a Gamma GLM?

I've seen several explanations of GLMs that link the linear combination of coefficients to the rate parameter, and assume the shape parameter is constant for all values of $y_i$ (for example, here: ...
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159 views

chi square GLM inference

Suppose at $m$ different positions on a line $a_1,....,a_m$, we sample from a i.i.d normal distribution $N(\mu_i,\sigma_i^2)$, $n_i$ times for each of the $1\le i\le m$ different points. Here of ...
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How do we do GEE with a dataset having a lot of zeroes? (Statistics doubts regarding exploring climate data)

I am working on relationship between climate variables available per zip code and a certain disease incidence over 3 years. I found that Gini index and Generalized estimating equations (GEE) are the ...
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2answers
98 views

Fit at zero inflated poisson GAM

I am trying to fit at zero inflated poisson GAM to my count data, and I want a log link. ziP() from the mgcv package does not support the log link. what can I do?
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In GLIM, how do I understand “the link maps mu to the entire real line, from −∞ to +∞”?

I always read in generalised linear model that the link function has to have a 1-1 correspondence from the range of mu to (-infinity, infinity). But, when we look at log link, for instance, it is not ...
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1answer
76 views

Appropriate link function for beta distribution

I am fitting a continuous proportion as my response in a beta distribution model.( I am using rstanarm to implement this model). For context the continuous proportion is the amount of time out of 30 ...
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17 views

Explaining the link-function in regression

I am writing a medical paper, which provides a methodological strategy for analysing non-transformed, non-normal (zero-inflated, extremely positively skewed) outcome variables. As I am not a ...
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43 views

How does link function work in GLM?

I have several questions regarding the link function of generalized linear regression. I know how link function changes range of the distribution function's mean to the complete real line. But is that ...
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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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188 views

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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1answer
82 views

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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2answers
29 views

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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22 views

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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1answer
148 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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1answer
592 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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1answer
202 views

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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1answer
51 views

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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1answer
59 views

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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2answers
225 views

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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63 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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2answers
1k 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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1answer
103 views

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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38 views

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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33 views

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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473 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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1answer
109 views

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 ...
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2answers
305 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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224 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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1answer
39 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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1answer
77 views

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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1answer
391 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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2answers
168 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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23 views

How to interpret the beta estimates of a generalized linear model with a square root power link?

I'm running a generalized linear model (GLM) in SAS with a gamma distribution (since my Y response variable is skewed to the right) and a specified square root power link (since I found that ...
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2answers
791 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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30 views

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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37 views

How to run a quadratic model with plateau with binary outcome?

I'm trying to build a generalised linear model in which the predicted probability of y follows a quadratic curve but hits a plateau at its maximum. I have found some solutions for a linear model ...