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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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5 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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30 views

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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25 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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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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1answer
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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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27 views

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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17 views

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
39 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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22 views

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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48 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
51 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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81 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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68 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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162 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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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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26 views

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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1answer
710 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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34 views

The link between logistic regression and logistic sigmoid

My professor once stated that the logistic sigmoid function is the inverse link function to logistic regression. I cannot find mathematical deductions showing this. Can anyone confirm his statement ...
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36 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
631 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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Set different link functions in Generalized Mixed Effects Model in R

Suppose I have a dataset of fish, some are salmon some are trout. I have a bivariate regression model specified roughly like below: prob(caught) = 0.5 + 0.5 * logit_inv(diet + fish_type) for salmon ...
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90 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
294 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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132 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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242 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
90 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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36 views

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
119 views

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

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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114 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
43 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 ...
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1answer
286 views

For glmer, is it appropriate to use the negative binomial distribution for non-binary response variable, and if so, which link should be used?

I have a dataset where the response variable fits a negative binomial distribution much better than the normal distribution (see image below). The variable is discrete but not binary. I am hoping to ...
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188 views

Selecting error distribution and link function for GLMMs on count data

I would like to find out how to select an appropriate error distribution and link function to model count data to determine a treatment effect using a generalized linear mixed effects model. I have ...
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1answer
494 views

Is there a mathematical definition of non-collapsibility?

When teaching GLMs, it's commonly taught that the linear and log link are collapsible but that the logit is not. The log link is collapsible because the curves are proportional and the means are ...
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330 views

Alternatives to ordinal regression when proportional odds assumption not met

I'm trying to fit an ordinal logistic regression using the ordinal package but the proportional odds assumption is not met. I have read all post here on this. My specific question is that, if you ...
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1answer
181 views

Strange variance weights for Poisson GLM for square root link

Is there a reason why all the variance weights from a Poisson GLM are equal when a square root link is used? That is (on R): ...
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3k views

Family in GLM - how to choose the right one?

When modeling data sampled in the field, I often come across the problem of determining the Family of the dependent variable for GLM (or GLMM). An example: in an ecological study, I have ~ 60 patches. ...
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291 views

What is the canonical link function for a multinomial logit?

For the multinomial logit model, what is the canonical link function and how does it relate to the definition of the canonical link as defined in this answer? I note from this link on glms, Agresti ...
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1answer
414 views

Variance Of Uniform Distribution After Log Transformation

Suppose I have a random variable $X \sim Unif(0.61, 0.79)$ and now I'm interested in the mean and variance of $Y \sim \log X$. I can easily calculate the mean using the equation, $$ E(g(X)) = \int ...
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Quasi-likelihood and complementary log-log link

I need help to understand how R computes the adjusted value when I use the quasi-likelihood distribution and complementary log-log link with variance function mu(1-mu). My data are rates from ...
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1answer
60 views

Link Function Help Needed!

Help needed with link functions! I'm attempting to do some linear models examining how fork lenghts of fish are dependent upon sex (0 or 1), hatchery mark (0 or 1) and age (continuous). When I try a ...
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1answer
127 views

Why do I get such a large divergence between Gaussian and Gamma family GLMs?

From what I understand the Gamma distribution is a good choice for positive right skew data. My data is a magnitude quantity with heavy outliers as you can see here from a histogram of the pooled set: ...
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1answer
85 views

Accounting for denominator in Poisson rate regression model with identity link and categorical predictors

This is a followup to this question: Offset in Poisson regression when using an identity link function? I am using a Poisson regression model with identity link to model risk based on a couple of ...
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560 views

Interpretation of cloglog model in layman's terms

I fitted a beta regression model via MCMC with a complementary log-log link function. Is there a way to interpret it in a layman's terms? The estimates of the model are: \begin{align} \beta_0 &=...
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76 views

Ratios, differences, and link functions?

I recently learned that with Poisson regression, you can model rate difference by using an identity link function, or rate ratio by using a log link function. Does that work the same way with other ...
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1answer
591 views

Offset in Poisson regression when using an identity link function?

It appears that in Poisson regression, using an identity link function means your betas will be rate differences, and using a log link function your (exponentiated) betas will be rate ratios. You can ...
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1answer
139 views

Why is the link function a function of the mean and not the linear predictor?

Generalized linear models are formulated so that the link function $g$ of the mean $\mu$ of a random variable $Y$ is equal to the linear predictor $\eta=x'\beta$, i.e. \begin{align} g(\mu)=\eta. \end{...
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131 views

Am I doing too many likelihood ratio tests to simplify the model?

I am building a GLM model on a large data set. The main variables thought to be relevant are X_1,...,X_7. I think the interactions X_1*X_3 and X_2 *X_4 might be significant, so I'll include them ...
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284 views

How do sufficiency statistics help in the interpretation of regression results?

One of the results why canonical link functions are widely used in GLMs is the existence of sufficiency statistics for the regression parameters, which in turn allow for: ... minimal sufficient ...