# Tagged Questions

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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### 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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### 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 ...
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### Where in this stata code does it indicate that the link test failed?

The link test was ran for a possion model. This picture was taken out of the Negative binomial regression textbook by Joseph Hilbe.
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### Log binomial regression with a case-control sample

It is my understanding that log binomial regression involves a direct comparison of prevalence ratios ("% cases among the exposed" vs. "% cases among the unexposed"), rather than using prevalence odds ...
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### What are the error distribution and link functions of a model family in R?

When building models with the glm function in R, one needs to specify the family. A family specifies an error distribution (or variance) function and a link ...
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### Same Estimate and Confidence Intervals with Logistic and Link Functions

I fit a logistic regression and calculated the expected difference in probabilities of my outcome between two treatment levels holding all other variables constant. I obtained confidence intervals ...
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### Sigmoid type functions for logistic regression

I am trying to find sigmoid function alternatives for logistic regression. I am curious that if any cumulative distribution function can be replaced with sigmoid function and what will be the best?
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### Pros and Cons of Log Link Versus Identity Link for Poisson Regression

I am carrying out a Poisson regression with the end goal of comparing (and taking the difference of) the predicted mean counts between two factor levels in my model: $\hat{\mu}_1-\hat{\mu}_2$, while ...
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### Link function in a Gamma-distribution GLM

In a GLM, if the response variable has a Gamma distribution, why is the inverse used as the link function, i.e.: $\mu = -(X\beta)^{-1}$? In particular, why is the inverse the canonical link? Does it ...
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### What type of Generalized Linear Model can handle high-to-low-variance heteroscedasticity?

I am trying to model the relationship between a continuous response variable (sample-corrected species-diversity estimates) and a continuous predictor variable (geographic spread). I have log-...
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### Selecting Link Function for Negative Binomial GLM

I'm trying to model insect abundance data with a variety of vegetation/site related covariates. Because it is count data that is over-dispersed, I've decided to use the negative binomial distribution. ...
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### Is it possible to convert a Gaussian Mixture Model implementation into a Categorical Mixture Model?

I am modelling whether a customer will spend when given a voucher. I have a theory that a customer falls into one of two latent classes: call them spendthrift and miser. So I would like to fit a ...
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### Are there any reasons to use the identity link in logistic regression (or any other glm)?

From this answer, the following statement is posed: 'Though not "wrong", you'd want a good reason for using an identity link to model a Bernoulli probability.' I would like to know what good reasons ...
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### Inverse Gaussian with MCMCglmm in R

I am trying to specify a mixed model using the MCMCglmm package/function in R. My data follow an inverse gaussian distribution, so I want to use MCMCglmm as an alternative to using an inverse ...
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### modeling probability with the multinomial logit link

I am attempting to model probabilities using the multinomial logit link and I am confused about how the link works. To study the link function I have been attempting to use a deterministic system. ...
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### Canonical Link Function in Zero-Inflated Poisson Model

For a zero-inflated Poisson model, I understand that the Poisson Model uses the canonical link of log(mu). For the logistic portion of the model, I usually see the equation written as logit(phi), and ...
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### Interpreting GLM regression analysis result

I'm using the following code in R to predict votes (e.g. non-negative integer count data). ...
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### Selecting an appropriate link function for zero inflated negative binomial regression

I have count data distributed according to zero-inflated negative binomial RV. I have been able to find good sources for a lot of model diagnostic steps, but there are a few things that are eluding ...
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### In a GLM, does the link transform the estimated mean, or is the mean estimated from the transformed RHS?

Is a GLM with a log link function the same as estimating: $$y_i = \exp(\beta_0 + \beta_1 x_i)\ ?$$ A GLM is the following: $$g(\mu_i) = \eta_i = b_0 + b_1 X_i$$ where $g(x)$ is the link function. ...
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### GLM vs least squares with Gamma errors

To illustrate the usefulness of GLMs in comparison to the least square method I did a simple program in which I add random noise to a straight line (Y=m*x + b; red line in the attached plot). The ...
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I'm running a glm that estimates gaussian variable of production in kilogrames using different independent variables. i found a problem of heteroskedasticity so i tried different transfromations of my ...
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### How does the logit link handle binomial (1/0) data?

I have a data set that contains a continuous explanatory variable and a set of responses as binary success and failures. For example, ...
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### Log vs square root link for Poisson data in R

I am currently working to model deaths from AIDS over time using a GLM in R. I know that there are two possible options for the link function for Poisson data, log and square root. I know that square ...
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### link fx and distribution in GEE model

I am working on my dissertation using a GEE approach. The reason is that my outcome is nonnormal, nonlinear, heteroscedastic, and clustered. Additionally, it is also truncated (a cutoff score) and ...
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### Closed form function relating $\mu$ to the natural parameter for the logarithmic series distribution?

While answering another question here, I mentioned the logarithmic series distribution as a possible model for species per genus. In the course of looking at the pmf while answering that I realized ...
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### Can we write the likelihood of a GLM in generality?

So I know we can explicitly write down the likelihood of any specified GLM model, for example the likelihood for the logistic regression model would be L(\mathbf{\beta};y,x)=\prod_{i=1}^np(x_i)^{...
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### Slope testing for regression lines between non-normal explanatory and response variables

I would like to estimate a regression line between my explanatory and response variables. The explanatory and the response variables are (paired) instrumental measures of the same thing under ...
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### What is the difference between lm(log(y) ~ x) and glm(y ~ x, family = gaussian(link = “log”))? [duplicate]

Is all in the title. I would like to know if there is any difference in terms of coefficients, residuals, p-values, but also conceptually.
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### Why are the fixed effects of a panel probit regression inconsistent?

I was taught that a probit with fixed effects would not be consistent because the estimates of a non-linear model with a link function other than the canonical (in this case the logit) are not ...
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### GLM: verifying a choice of distribution and link function

I have a generalized linear model that adopts a Gaussian distribution and log link function. After fitting the model, I check the residuals: QQ plot, residuals vs predicted values, histogram of ...
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### Adding a square root link function to an overdispersed negative binomial GLM

I'm analyzing nematode count data (80 data points) from a randomized block design in which I have two factors with both four levels (Plant and Inoc). The data show heavy overdispersion when analyzed ...
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### R binomial family with identity link

I want to fit a linear model by R with family=binomial(link="identity"), however, binomial family do not have identity link. What should I do?
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### Exponential distribution: How to avoid negative predictor of $\lambda$?

I have a joint distribution $P(X, Y_{1}, Y_{2}, ....)$ which contains one univariate exponential distribution ($X$) and several univariate gaussian distributions ($Y_{1}, ...$). For details regarding ...
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### Is it possible to model the conditional expectation of a binary outcome using an additively-separable link function?

Logit and probit link functions aren't additively separable. So, fitting a model using these link functions implies that the effect of one predictor in determining the outcome is not independent of ...
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### simulate GLM with square root link in R

I'm trying to simulate a fitted GLM using basic functions, not using the simulate() and predict() functions that are widely questioned and answered. I get different results when I compare my math ...
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### quantile regression with e.g. gamma distribution and log link

I have a basic question about quantile regression (I'm new to it): Why doesn't it seem possible to do a quantile regression with a specified family (e.g. gamma) and link function (e.g. log), as in a ...
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### Nonlinear vs. generalized linear model: How do you refer to logistic, Poisson, etc. regression?

I have a question about semantics that I would like fellow statisticians' opinions on. We know models such as logistic, Poisson, etc. fall under the umbrella of generalized linear models. The model ...
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### GLM link function for bimodal probit fitting?

I am trying to model a set of data I have physical reason to believe can be represented by a bimodal normal cumulative distribution function (Technically it is a bimodal log-normal CDF, but I think I ...
583 views

### selecting a link function for GLM's

If you don't care about using GLM model parameters to predict anything, but simply want to select the best-fitting model for your data, is it necessary to get into the theoretical debate as to which ...
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### How to customize a link function to perform a logistic regression?

My data was collected using Randomized Response Technique. So I have additional variability into the data. I have a binary response variable. Should I customize a logit link function to incorporate ...
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### Sine link with binary regression

I have used the SIN link to estimate probabilities, mostly with Program MARK. However, I am not sure how the SIN link works. I know the SIN link enables parameter ...
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