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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 quite different but do not understand what's the difference between log-linear regression and Poisson regression?

I think AdamO and Gung's answer do 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.

Update: I am reading some source code fro R0 package in R. The author was trying to estimate the exponential growth rate using different methods:

 ##details<< method "poisson" uses Poisson regression of incidence.
 ## method "linear" uses linear regression of log(incidence)
 if (reg.met == "linear") {
 tmp <-lm((log(incid)) ~ t, data=epid)
 ...
 } 

 # Method 2 == Poisson regression
 else if (reg.met == "poisson") {
   tmp <- glm(incid ~ t, family=poisson(), data=epid)
   ...
 }

Is there any relationship between linear regression on log scale and poisson regression? what is the reason to use different methods?

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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
    Commented Feb 14, 2017 at 20:51
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    $\begingroup$ 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
    Commented Dec 4, 2017 at 11:35
  • $\begingroup$ Agree the terminology is confusing, but as I understand, loglinear analysis (e.g. MASS::loglm) is identical to a Poisson GLM on frequencies, with statistical significance assessed via likelihood ratio tests), see stats.stackexchange.com/questions/124581/… for an example... Loglinear analysis is typically not understood as linear regression with a log transformed outcome variable... $\endgroup$
    – Tom Wenseleers
    Commented Jan 24, 2023 at 13:36

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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
    Commented Feb 16, 2017 at 16:31
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    $\begingroup$ While there is some qualitative similarity between a GLM with log link and a linear model applied to log transformed data, they are substantially very different regressions with different implied mean variance relationships, and will often give completely different answers. $\endgroup$
    – Paul
    Commented Dec 21, 2019 at 21:45
  • $\begingroup$ The other SE answer (linked by the OP) agrees with Tim and Paul, too: "a log transformed outcome variable in a linear regression model is not a log-linear model, (neither is an exponentiated outcome variable, as log-linear would suggest)" $\endgroup$
    – jdcrossval
    Commented Apr 22, 2022 at 0:54
  • $\begingroup$ Agree the terminology is confusing, but as I understand, loglinear analysis (e.g. MASS::loglm) is identical to a Poisson GLM on frequencies, with statistical significance assessed via likelihood ratio tests), see stats.stackexchange.com/questions/124581/… for an example... Loglinear analysis is typically not understood as linear regression with a log transformed outcome variable... $\endgroup$
    – Tom Wenseleers
    Commented Jan 26, 2023 at 8:50

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