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In this post, OP asked the difference between log linear regression and logistic regression. Two answers in the post are very clear and directly address OP's question.

I understand Log-linear regression and logistic regression are quirete different but not understand what's the difference between log-linear regression and Poission regression?

I think AdamO and Gung's answer are not explain my question in detail.

From AdamO

the log-linear model is actually just a Poisson regression model

From Gung

"log-linear regression" is usually understood to be a Poisson GLiM applied to multi-way contingency tables.

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  • $\begingroup$ That's weird. I've always understood "log-linear regression" to mean linear regression where the dependent variable is the log-transformed variable of interest (something positive-valued). Which is obviously different from Poisson regression. But if there is some ambiguity, you have to say what you mean by log-linear regression before anyone can explain the difference from Poisson regression. $\endgroup$ – The Laconic Feb 14 '17 at 20:51
  • $\begingroup$ I find "log-linear regression" to be somewhat of a misnomer since the whole idea extends to much more than just plain regression (probably why wikipedia calls it log-linear analysis. Log-linear analysis is a form of categorical data analysis mostly used with three-way contingency tables. A common Poisson GLM would only explain a subset of the independence relations that a log-linear analysis can explain, though in some cases (e.g. two-way tables) both methods are exactly the same. $\endgroup$ – Digio Dec 4 '17 at 11:35
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A Poisson regression is a regression where the outcome variable consists of non-negative integers, and it is sensible to assume that the variance and mean of the model are the same.

A log-linear regression is usually a model estimated using linear regression, where the response variable is replaced by a new variable that is the natural logarithm of the of the original response variable. Or, if using a GLM, this is done via a logarithmic link function (essentially the same idea, but the mechanics of fitting the model are the different).

The Poisson regression and log-linear regression are not the same thing, but are often used for very similar problems, particularly among older statisticians (the Poisson regression model only became widely available in software in the 1980s).

Most people these days prefer a Poisson regression because it can deal with 0 values, whereas you will get an error using a log-linear regression.

It is possible to use a Poisson regression to model data from a contingency table, where the predictor variables are the dimensions (e.g., row and column labels) of a contingency table. This can be referred to as a log-linear model. Perhaps some people call it a log-linear regression (one of the challenges of statistics is that the language is used rather loosely, but many people act as if the language is precise).

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    $\begingroup$ I don't agree while agreeing with the quotes provided by OP. Poisson regression is called "log-linear model" while I can't recall anyone using this name for linear regression with log transformed response variable. $\endgroup$ – Tim Feb 16 '17 at 16:31

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