Solved – R – (why) does fitting a binomial glm to a 3-level factor work

binomial distributiongeneralized linear modellme4-nlmer

I am trying to determine which of several independent variables {A,B,C} best predicts a subject's response R, which can be one of 3 choices, {high, medium, low}.

As the dependent variable is multinomial i.e. not binomial, a binomial logistic regression is not suitable. However, using the lme4 package to fit a glm with family = 'binomial' to the data runs without error or warning, and correctly identifies that independent variable B is the best predictor.

Is this a lucky coincidence, or is the call to glm() automatically fitting multiple binomial regressions to the data (one for each binary combination), or is it something else entirely? Any comments are welcome.

Best Answer

Here's what's happening:

For binomial and quasibinomial families the response can also be specified as a factor (when the first level denotes failure and all others success) or as a two-column matrix with the columns giving the numbers of successes and failures.

Taken from the glm.fit docs.

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Solved – Why does a binomial glm give negative predictions

Assuming that you are using the predict.glm() from the stats package.

A quote from the manual, under the entry explaining the type parameter:

Thus for a default binomial model the default predictions are of log-odds (probabilities on logit scale) and ‘type = "response"’ gives the predicted probabilities.

So instead, try the following:

predict(glm(cbind(suc,fail)~c(1:10),family=binomial), type="response")

Solved – Arm package: BayesGLM prior

As described in Bayesian logit model - intuitive explanation? thread, logistic regression can be described as a linear combination

$$ \eta = \beta_0 + \beta_1 X_1 + ... + \beta_k X_k $$

that is passed through the link function $g$:

$$ g(E(Y)) = \eta $$

where the link function is a logit function

$$ E(Y|X,\beta) = p = \text{logit}^{-1}( \eta ) $$

where $X_1,\dots,X_k$ are the predictor variables and $Y$ is a target variable. What we want to estimate is $\beta_0,\beta_1,\dots,\beta_k$ parameters. When using Bayesian approach, for estimating some parameter of interest we start with defining a prior that describes out out-of-data knowledge about the parameter, and then use the prior combined with the likelihood function (that tells us what the data says about the parameter) to obtain a posterior (i.e. estimate of distribution of the parameter). So we choose priors for parameters and normal, $t$, Cauchy etc. are perfectly fine priors for parameters of regression model or generalized linear model. Beta distribution is inappropriate choice for usual regression model since it would imply that estimated parameters must be bounded in $[0,1]$ interval (you are probably confusing it with beta-binomial model).

The parameters for priors are different story. You need to choose them based on your out-of-data knowledge about the parameters, so this is a problem-specific choice. If you have no a priori knowledge, you can use a weekly informative prior, e.g. normal distribution centered at zero with large variance.

Best Answer

Related Solutions

Solved – Why does a binomial glm give negative predictions

Solved – Arm package: BayesGLM prior

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