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What is the purpose of the link function as a component of the generalized linear model? Why do we need it?

Wikipedia states:

It can be convenient to match the domain of the link function to the range of the distribution function's mean

What's the advantage of doing this?

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A.J. Dobson pointed out the following things in her book:

  1. Linear regression assumes that the response variable is normally distributed. Generalized linear models can have response variables with distributions other than the Normal distribution– they may even be categorical rather than continuous. Thus they may not range from $-\infty$ to $+\infty$.

  2. Relationship between the response and explanatory variables need not be of the simple linear form.

This is why we need the link function as a component of the generalized linear model. It links the mean of the dependent variable $Y_i$, which is $E(Y_i)=\mu_i$ to the linear term $x_i^T\beta$ in such a way that the range of the non-linearly transformed mean $g(\mu_i)$ ranges from $-\infty$ to $+\infty$. Thus you can actually form a linear equation $g(\mu_i)$=$x_i^T\beta$ and use an iteratively reweighted least squares method for maximum likelihood estimation of the model parameters.

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  • $\begingroup$ Welcome @Chris. $\endgroup$ – Blain Waan Jan 27 '13 at 6:46
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It may help you to read my answer here: Difference between logit and probit models, which discusses GLiM links somewhat extensively.

The basic way of explaining this issue is laid out clearly by @BlainWaan, and Wikipedia: The actual parameter (e.g., $p$ for a binomial response--i.e., logistic regression) cannot range from negative infinity to positive infinity, but your predicted parameter will. The second big reason is that without a properly specified link, the variances of your residuals will not be constant (a required assumption for inference with an Ordinary Least Squares estimate) or handled correctly.

Another way to go at this issue is that by using the identity link (this is another way of saying / thinking about 'not using' a link function) means that you are thinking about your situation incorrectly in a way that necessarily distorts the picture of your situation that you derive from your analysis. For example, unless the true probabilities that you are trying to model (again for logistic regression situations) exist only in the middle of the range (where they are fairly linear), and the range of $X$ you are examining is centered on the point where $p=.5$, your betas will be biased and your predicted $\hat p_{x_i}$'s will be far from the true values. In addition, your inferences will be distorted as well (e.g., the type I error rate won't equal $\alpha$).

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  • $\begingroup$ Does link function make the residual distribution normal ? $\endgroup$ – ABC Jul 15 '15 at 12:20
  • $\begingroup$ @ABC, no the link function just connects the structural part of the model with (a transformation of) the predicted parameter. In a GLiM, you also need to specify the response distribution & the dispersion. $\endgroup$ – gung Jul 15 '15 at 14:17

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