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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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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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45 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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23 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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26 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
329 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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25 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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31 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
236 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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25 views

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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52 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
165 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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100 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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228 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
64 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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63 views

Can I fit a GLMM to near-ceiling binary accuracy data?

tl;dr Can I draw inferences from this binary response data with a ceiling effect, or is it too skewed? How do I check if the model is acceptable? I would like to analyse some binary response ...
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35 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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96 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
41 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
232 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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126 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
431 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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284 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
153 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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2k 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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261 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
361 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
56 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
118 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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440 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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0answers
57 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
506 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
122 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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113 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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247 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 ...
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2answers
86 views

How are fitted Poisson means constrained to be positive when the identity link is used in Poisson regression?

I apologise for the imprecise notation that follows, but hopefully I have conveyed the idea sufficiently. In Poisson regression of $Y \sim \mathbf{x}$ , the canonical link function $\ln$ constrains ...
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590 views

Why does the exponential distribution have a canonical link that can yield negative values?

The canonical link for the binomial is the logit. The linear predictor can be anything so it is usable for the probability after the logit transform is used. The case is analogous for the Poisson ...
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207 views

glm link functions for multinomial and ordered probit regression?

Here's what I understand, could someone please tell me if I'm wrong, and how? For a categorical variable $Y$, the expected value $\text{E}(Y)=\mu=\sum_{y}i\cdot\text{P}(Y=i)$. Using the descriptions ...
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1answer
217 views

Relationship between binomial regression link function and goodness-of-fit tests [now with link to R code]

Some background: A number of papers in the literature (various ones by Hosmer and Lemeshow; Copas; le Cessie and van Houwelingen; Cressie and Read; Osius and Rojek; J. R. Dale) discuss a family of ...
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255 views

Log link vs logging response variable in GAM model

hope someone might be able to help: My aim is to build a value predictor of an individual’s pension pot using customer data (based in the UK). It is to be used for reporting purposes and ...
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1answer
2k views

Can you give a simple intuitive explanation of IRLS method to find the MLE of a GLM?

Background: I'm trying to follow Princeton's review of MLE estimation for GLM. I understand the basics of MLE estimation: likelihood, ...
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137 views

Interpetation of estimates in Gamma-Regression with reciprocal link

I´m new to CrossValidated and this is my first question here. In the case of a generalized linear regression model where I assume that my data follow a gamma distribution, I would like to know how to ...
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0answers
268 views

Negative binomial GLM: model checking in R

I'm using a negative binomial GLM in R, but I could not solve the following issues (using McCullagh and Nelder (1989), textbooks and Google): Residual plots. McCullagh and Nelder (1989, p.398) ...
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1answer
677 views

Are canonical link functions and link functions the same thing? [duplicate]

Are Canonical link functions and the link functions the same thing? If not, can anyone tell the difference between them? I know a link function is a function that links a linear predictor to the ...
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1answer
414 views

Basic question on link function in GLM

I am a stats & R beginner and am trying to understand GLMs. I have a very basic question on the link function which is the following If I understand correctly the mean of the response variable Y ...
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1answer
67 views

In GLM do we try to model E(T(y)) or E(y)?

I'm trying to follow Andrew NG cs course on supervised learning. He defines the exponential family as: $$ p(y;\eta) = b(y)exp(\eta T(y) -a(\eta)) $$ and then continues to say that "our goal is to ...
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1answer
4k views

Generalized Additive Model interpretation with ordered categorical family in R

I am having a difficult time interpreting the gam.plots produced by the plot() function in the package mgcv in R—specifically, ...
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1answer
319 views

Reasoning behind reciprocal link function for exponential regression

The reciprocal link function in exponential regression doesn't constrain to positive values, whilst on the other hand it does thoroughly space out points close to zero when fitting the linear ...
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1answer
600 views

Link functions for Binomial Regression

So I have a dataset of presence (1) and absence (0) data, but it mainly consists of 0's (~80% of the 5200 observations). Now while constructing my binomial logistic model I am reading (Zuurt et al. ...
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560 views

Is it possible to calculate a pseudo-R squared for a binomial GLMM with a cauchit link?

I'm modeling some repeated-measures presence-absence data using a binomial GLMM in lme4. I've been using the method suggested by Nakagawa and Schielzeth (2013) to calculate a marginal and conditional ...