# Log Linear Models

Can someone please explain why do we use Log Linear Models in very lay-man terms? I come from Engineering background, and this is really turning out to be a difficult subject for me, statistics that is. I will be grateful for a response.

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Are you talking about loglinear models for proportions (generally in tables) or loglinear models for something else? – Glen_b Nov 22 '12 at 0:15
Glen, I am talking about tables. – user1343318 Nov 22 '12 at 5:19
@user1343318 If some of these answers gave you what you're looking for then maybe you should consider picking one of them so we can move on with our lives. :) – Dr. Mike Mar 16 '15 at 11:12

Log linear models, like crosstabs and chi-square, are usually used when none of the variables can be classed as dependent or independent but, rather, the goal is to look at association among sets of variables. In particular, log linear models are useful for association among sets of categorical variables.

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Log-linear models are often used for proportions because independent effects on probability will act multiplicatively. After taking logs, this leads to linear effects.

In fact there are other reasons why you might use loglinear models (such as the fact that the log-link being the canonical link function for the Poisson), but I think the first reason probably suffices from a general modelling point of view.

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Here's a list of related reasons why $\ln$ (aka $\log_e$) transformation may be used. Since all logarithms are proportional to each other, many people tend to use base $e$, since it has some nice properties. To quote John D. Cook,

I don't always use logs, but when I do, they're natural logarithms.

This list is taken from Nick Cox's Intro To Transformations (with some added commentary):

• Reduce skewness - Gaussian distribution is regarded as ideal or necessary for many statistical methods (sometimes mistakenly). Taking logs helps.
• Equalize spreads - induce homoskedasticity when there's lots of variation in levels.
• Linearize relationships - For example, a plot of logarithms of a series against time has the property that periods with constant rates of change are straight lines
• The coefficients$\cdot$100 have a semi-elasticity interpretation: for a 1 unit change in $x$, you get b*100% change in $y$. For binary $x$ going from 0 to 1 effect, the effect is $100 \cdot(\exp\{\beta\}-1)$%. Some people find exponentiated coefficients easier to think about than elasticities. That gives a ratio of Y-values per unit change in X, assuming an exponential relationship (a kind of multiplier).
• "Additivize" relationships - Trying to get the parameters of a Cobb-Douglas production function is a whole lot easier without non-linear methods. Analysis of variance also requires additivity.
• Convenience/Theory - log scale may more natural for some phenomena.

Finally, logs aren't the only way to accomplish some of these goals.

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A common interpretation, and way of seeing the difference, between a normal linear model and a log linear model is if your problem is multiplicative or additive.

A normal linear model has the following form $Y=\sum_{i=1}^M \beta_i X_i+\beta_0$

A log linear model has a log transformation on the response variable which gives the following equation

$\ln Y=\sum_{i=1}^M \beta_i X_i+\beta_0$

which turns into

$Y=e^{\beta_0}\prod_{i=1}^M e^{\beta_i X_i}$