What's a power link used for in generalized linear models (GLMs)?

Question

The documentation for the power link function in R says it "Creates a link object based on the link function η = μ ^ λ."

and: "If lambda is non-positive, it is taken as zero, and the log link is obtained. The default lambda = 1 gives the identity link."

What is a power link function used for?

Trying to learn GLMs by reading the R help will be a very slow and painful process -- it might help to post and ask for recommended reading that will cover things like power functions and quasi models to get you up to speed on GLMs. — Glen_b, Nov 15 '17 at 04:53
By making it look like a coding question you make it more likely that someone will close your question. Since your question isn't really about R (it's a question about how glms work) it's probably better to frame it more generally. I will attempt an edit that would make your question a slightly better fit; you can roll it back if you really wish. — Glen_b, Nov 15 '17 at 04:55
@glen_b My prof assigned the R doc as reading for this assignment. :-( What book(s) do you recommend? — GuyMatz, Nov 16 '17 at 14:50
That's a new question (tag `references`); other people may have better suggestions than I do. There are also a number of questions here explaining some of the basics of GLMs. If you're new to GLMs, use some of the distribution families before tackling issues like quasi models and power link functions. Many people use the binomial family (especially for logistic and probit regression) but the Poisson and Gamma are very useful in a wide range of situations (and both have power variance functions). — Glen_b, Nov 16 '17 at 22:40

Glen_b · Answer 1 · 2017-11-16T23:30:15.777

GLMs model the conditional distribution of a response given a set of predictors $[Y|\underline{x}]$,
where $\underline{x}=(x_1,x_2,...,x_p)'$.

One of the main components of the model is the link function, which relates the conditional expectation $E(Y|\,\underline{x}\,)=\mu(\,\underline x\,)$ to the linear predictor ($\eta=\underline{x}'\beta$), through the link $g$; that is, it specifies that the model for the conditional mean is $g(\mu_i)=\underline{x}_i'\beta$ (here the $i$ subscript is denoting the random variable for the $i$th observation).

For example, a Poisson family has a default log-link $\log(\mu) = \eta = \underline{x}_i'\beta$, while the default link for the Gaussian is the identity link $\mu= \eta =\underline{x}_i'\beta$

In the case of the power link, some power of the conditional mean is linear in the predictors:

$$\mu^\lambda=\underline{x}_i'\beta\,.$$

The Poisson family in R includes the square root link, which is an example of a power link. With the power-link function you can specify some power that's not otherwise available.

In short, the power link is used so you can have a mean-model like $E(Y|x)^\lambda = \beta_0+\beta_1x_1+...+\beta_px_p$ for whichever value of $\lambda$ you choose.

An example:

   glm(mpg~disp,mtcars,family=Gamma(link=power(1/3)))

Here $Y$ is is the variable mpg and $x$ is disp. This model says that $E(Y|x)=\mu(x)$, where $\mu(x)^\frac13=\beta_0+\beta_1x$ and that the conditional distribution of $Y$ is gamma with mean $\mu$ (so the variance function is $\mu^2$). Here's the resulting fit:

(this isn't a suitable model for several reasons, but it will do to demonstrate the point)

The Tweedie family (which is not implemented in the vanilla glm function in R) has for its canonical link a form of power link, ($\eta=\frac{\mu^{1-p}}{1-p}$ -- as well as a power variance function $V(\mu)=\mu^p$).

What's a power link used for in generalized linear models (GLMs)?

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