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I am fitting a continuous proportion as my response in a beta distribution model.( I am using rstanarm to implement this model). For context the continuous proportion is the amount of time out of 30 seconds an animal engaged in a particular behavior.

Would it still be appropriate to model this with a logit link? If so, why not a log link? I am having a hard time understanding why I would want to put this on a log odds scale, since it is a continuous proportion and not discrete success or failure.

If a logit link should be used in this instance, are there other examples using the beta distribution where a log link is appropriate instead?

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Nice question!

Page 2 of the STATA document on beta regression available at https://www.stata.com/manuals/rbetareg.pdf does a nice job of outlying the possibilities available to you in terms of link functions:

logit
probit
cloglog (complementary log-log)
loglog  (log-log) 

All of these link functions are designed to transform a proportion p (where 0 < p < 1) so that it "lives" on the real axis. (The real axis encompasses values that are negative, zero or positive.) That makes it easier to model the transformed proportion p as an intercept plus a linear combination of predictor variables.

The link function you had in mind - log - is not used because it would take a proportion p - be it discrete or continuous - (where 0 < p < 1) and make it "live" on part of the real axis (i.e., the negative part). The log link is typically used to transform quantities q for which q > 0. You will see this link used to model things like a scale parameter $\sigma$, rather than a mean parameter $\mu$.

Going back to the four admissible link functions stated above, I remember reading in a book by Joseph Hilbe that the logit link is preferable to use if we are interested in interpreting the effects of the various predictors in the model. The other links make that interpretation tenuous and are better used in a predictive context, where we care about how the predictors combine to predict p rather than their individual effects on p.

When you work with beta regression with a logit link, you are right that the interpretation of a beta regression model like $logit(E(y|x_1, ..., x_p)) = \beta_0 + \beta_1 x_1 + ... + \beta_p x_p$ becomes convoluted. What this model is saying is that the logit-transformed expected (or mean) value of your response variable is an intercept plus a linear function of your predictors. In other words, you couldn't model the expected value of your response directly because it "lived" in the restricted range (0,1) - you had to logit-transform it first.

When you interpret this model on the logit scale, you can say things like:

Using beta regression, the data provided evidence of a positive (or negative) linear effect of the predictor $x_1$ on the logit-transformed expected/mean/average value of the response variable y, adjusting for the effects of all other predictors included in the model. Each 1-unit increase in the value of $x_1$ was associated with $b_1$ units increase (or decrease) in the expected/mean/average value of the response variable $y$, all else being equal. (In your case, the response variable $y$ is the amount of time out of 30 seconds an animal engaged in a particular behaviour - in other words, the proportion of time an animal engaged in that behaviour over a span of 30 seconds).

You can then supplement a statement like the above with effect plots*, which will show the nonlinear effects of each predictor (or at least of the predictors of primary interest in the model) on the expected/mean/average value of the response variable y.

Note that in a beta regression you can model both the expected value and the variability of your response variable $y$ as a function of the predictor variables.

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