We’ve been doing a lot of work recently with multinomial logit choice models; that is, trying to predict how choices are made between multiple options. It’s a fairly new topic for me so I’m trying to get the hang of both the theory and the practical bits. But learning multinomial modelling before binomial modelling (the choice between two options) is like trying to run before you can walk. (Normal linear regression would be crawling.)

Binomial models are easy to do in R. Just feed your independent and response variables into the `glm`

function and specify the “binomial” regression family.

Without getting into the theory, this model estimates the logit *z* as a linear function of the independent variables.

[latex size=1]

z = \beta_0 + \beta_1 x_1 + \ldots + \beta_i x_i

[/latex]

This value can then be used to calculate the probability of a given outcome via the logistic function:

[latex size=1]

f(z) = \frac{1}{1 + e^{-z}}

[/latex]

Normally with a regression model in R, you can simply predict new values using the `predict`

function. The problem with a binomial model is that the model estimates the *probability* of success or failure. So, for a given set of data points, if the probability of success was 0.5, you would expect the `predict`

function to give `TRUE`

half the time and `FALSE`

the other half.

This is kind of what happens in R but it doesn’t predict these `TRUE`

/`FALSE`

outcomes directly, so you need to know how to convert the prediction outcome into the actual binary response variable. If you use the default `predict`

arguments, the model gives you the *z* values. One (wrong) approach is to interpret *z* less than 0 as success or failure depending on how you’ve coded your model. But what you really want to do is estimate the probability, then draw a random number and see if that’s above or below the threshold value. R’s `predict`

function can do this but it will take a couple lines of code:

# Create some new data n2 <- 100 new.df <- data.frame(x1 = runif(n2, 0, 100), x2 = runif(n2,0,100)) # Predict the probabilities # predict(model, new.df) will just give you the logit (z) values # We want the probabilities so use the "response" type probs <- predict(model, new.df, "response") # Draw some random values between 0 and 1 draws <- runif(n2) # Check if draw value is less than threshold probability results <- (draws < probs)

Created by Pretty R at inside-R.org

So if you want to predict the results again, simply rerun the last two lines. New random draws will mean new outcomes.

To illustrate all this, I’ve put together a quick ggplot picture to show the original data (small points) and the modelled results (larger points). The colours represent the success/failure outcomes. (ggplot doesn’t want to add the legends for some reason.)

Next time I’ll look at the more complicated multinomial case.

JohnHi James,

Thanks for the post. Do you have any references for this approach to draw a random number in order to determine the binary response?

James KeirsteadPost authorNot off the top of my head, no. But if any good book on microsimulation should describe this general approach.

NataliaHello, I have a doubt about binomial models: I did a binomial glm based on an experiment, and my response variable es female bird agression; females in my experimental group were agressive and females in the control group were not. when I run the model, theres some mistake, because all the 0’s are on the control group and all the 1’s are in the experimental group….I think it is a quasicomplete separation problem…..what can I do to run a binomial glm with data like this??

thanx a lot

Dalton HanceI’ve been playing around with a binomial regression model (though I’m still not clear on how this is different from logistic regression, if at all). I just discovered that the output of ‘predict(model, new.df, “response”) ‘ yields the

oddsnot the probability. Check it out yourself. The output of ‘predict(model, new.df, “response”) ‘ is the same as if you were to simply take the exponent of the logit (log-odds) values that is: ‘predict(model, new.df) = exp( predict(model, new.df, “response”) )’. This is not the probability. Probability is equal to (odds/1+odds). So you’ve got to take one extra step to get the probability. My issue is dealing with getting a CI in terms of probability not odds. Any help out there?James KeirsteadPost authorI’m not sure I follow your comment Dalton. Here’s the relevant section of ?predict.glm:

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FernandoYou could also use the

samplefunction, that enables you to set the probabilities.sarahHello , I am a student and do not understand the difference between logstic and regrresion binomial Do not understand Can you help me?

James KeirsteadPost authorHi Sarah,

A binomial regression is a statistical model that explains a binary outcome from covariates (e.g. wear a coat or not as a function of outdoor temperature, activity, length of time spent outside, etc). To do this, one needs to specify a link function that defines the probability of success. This could be a logistic curve, or it could be a normal or Cauchy cumulative distribution function. See

`?family`

for more information.