Link functions- Probit compared to logit

Generalized Linear Models in R

Richard Erickson

Instructor

Why link functions?

  • Understand and simulate GLMs
  • Probit vs logit as example
Generalized Linear Models in R

Why probit?

  • Demonstrate link function
  • Used in some fields (e.g., toxicology)
  • Preferred by some people
Generalized Linear Models in R

What is a probit?

  • Probability unit
  • Toxicology by Chester Bliss in 1934
  • Computationally easier than logit
  • Model know as probit analysis, probit regression, or probit model
Generalized Linear Models in R

Probit equation

Model of binary data

$Y = \text{Binomial}(p)$

Linked to linear equation

$\Phi^{-1} (p) = \beta_0 + \beta_1 x + \epsilon $

Generalized Linear Models in R

Probit function

Based upon cumulative normal

$\Phi(z) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^z e^{- \frac{1}{2} z^2} \, dz$

Generalized Linear Models in R

Cumulative probability of probit and logit

Generalized Linear Models in R

Fitting a probit in R

  • family option for glm()
    • Character: glm(..., family = "binomial")
    • Function: glm(..., family = binomial())
  • Default: binomial(link = "logit")
  • Probit: binomial(link = "probit")
  • Match instructions for DataCamp
Generalized Linear Models in R

Simulate with probit

Convert from probit scale to probability scale:

p = pnorm(-0.2)

Use probability with binomial distribution

rbinom(n = 10, size = 1, prob = p)
Generalized Linear Models in R

Simulate with logit

Convert from logit scale to probability scale:

p = plogis(-.2)

Use probability with a binomial distribution

rbinom(n = 10, size = 1, prob = p)
Generalized Linear Models in R

When to use probit vs logit?

  • Largely domain specific
  • Thicker tails of logit
  • Either is tenable
Generalized Linear Models in R

Let's practice!

Generalized Linear Models in R

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