Bounded counts are pretty common

Binomial regression, our old friend

• We’ve already used binomial regression, for modeling categorical data.

• Every respondent had one “attempt” to succeed.

\[ logit(Y) \sim Binom(X\beta, 1) \]

How well do you EU?

• We administered a knowledge test on the EU. We want to model sum scores based on education, age and attitudes towards the EU.

Test questions

• “Members of which of the following institutions are elected by the people directly?”

• “Who is currently the President of the European Commission?”

• “Which two countries are not EU members?”

• “In which of the following areas does Czechia have the least freedom in setting its own policy as a result of its membership in the EU?”

Specifying binomial regression

• Almost identical to “classical” logistic regression, but we need to tell R what the number of trials, i.e. the maximum possible score, is.

• Every package is different, glm() requires to specify number of success and number of failures.

Marginal effects for binomial model

• Interpreting as usual through marginal effects.

Assumptions

• Validity and reliability

• Representativity

• Linearity between logit of dependent variable and predictors

• The probability of answering correctly is the same for each variable, e.g. difficulty is the same.

  • If violated, difficulty can be modeled directly using beta-binomial regression.

Single likert items

• We can treat a single likert item as a “score”.

• E.g. Attitudes towards EU

  • 0 Strongly eurosceptic

  • 1 Eurosceptic

  • 2 Pro-EU

  • 3 Strongly pro-EU

• Unlike linear regression, this approach naturaly accounts for item’s bound.

EU attitudes an binomial regression

Good old proportions

Beta distribution

Modeling mean scores

• Beta distribution assumes values between 0-1, we need to rescale the data.

\[ logit(Y) \sim Beta(X\beta) \]

• Logit link! Non-collapsibility, we meet again…

• Doesn’t work if there are 0s or 1s in our data. Solution?

Ordered beta

• Multiple ways to incorporate 0s and 1s into our model.

• A neat one is ordered beta (other option is zero-one-inflated beta)

• (Roughly speaking) We predict whether an observation is 0 or 1 using logistic regression. If not, predict the proportion using beta.

\[ \begin{align} logit(Y = 0) &\sim Binomial(X\beta) \\ logit(1> Y > 0) &\sim Beta(X\beta) \\ logit(Y = 1) &\sim Binomial(X\beta) \\ \end{align} \]

Modeling EU index

• Are EU attitudes associated with knowing how the EU works, after controlling for age and education?

\[ \begin{align} logit(EU\;score) \sim OrdBeta(&Age + Knowledge + \\ &Education + Knowledge \cdot Education) \end{align} \]

• Interpreted as usual trough marginal effects and predicted values plots.

Modeling EU index

Assumptions

• Validity and reliability

• Representative

• Linearity

• Conditional distribution is beta

GLMs wrap up

• By remembering few distribution, you can make modeling much easier.

• Proper Generalized linear model respects properties of data and serves as a base for more complicate models.

  • Item response theory

  • Conjoint analysis

  • Propensity score matching

  • and more…

• Don’t rely on regression coefficients, use marginal effects and predicted probabilities plot.

• Model fit can be checked using randomized residuals and posterior predictive plots.

GLMs in a table

GLMs and the R ecosystem

• Modelling functions are spread across number of packages.

  • Takes time to find the right package

  • Compatibility problems

  • Different syntax

• Solution?

The Beauty of the BRMS package

BRMS trade-offs

• brms uses simulations to estimate models (Stan language).

• Advantage:

  • Extremely flexible

  • Easy to work with/postprocesses

  • More robust

• Disadvantage:

  • It takes much longer to estimate models

  • Need to make sure the simulation went ok

BRMS trade-offs

• When Maximum likelihood based model takes 0.5 seconds to estimate, brms takes 1-5 minutes.

• When ML based model takes 3 minutes to estimate, brms takes 30-60 minutes.

• Is it worth it?

Time management, modeling style

• From my experience, yes.

• Waiting 5 minutes for a model > Spending 1 hour looking for packages and fixing compatibility issues.

• Waiting 1 hour for a model > Spending 4 hours making sure a model converges.

• Sometimes, you straight up don’t have a choice

• So give brms a try!