Examples of link functions

toffa et al. x psicostat

Linear Predictor

Consider a linear equation:

\(Y =\) \(\beta_0 + \beta_1X_1 + \beta_2X_2\)\(+\) \(\epsilon\)

  • the linear predictor (often called \(\eta\), but in the following slides simply called \(X\)) is the “\(\beta_0 + \beta_1X_1 + \beta_2X_2\)” part.

  • the job of the link function in GLMs is to transform (re-map) the linear predictor \(X\), which may span in \((-\infty, + \infty)\), to the appropriate range of the response variable \(Y\) (e.g., times in \((0, +\infty)\), probabilities in \((0, 1)\))

link="identity"

link="log"

link="inverse"

link="logit"

link="probit"

link=mafc.probit(3)

Probit or Logit?

  • Logit: “thinking in terms of odds ratios makes sense”; a linear increase in predictors multiplies the odds of an event happening; e.g., chance of diagnosis (true categorical); ???

  • Probit: “thinking about an underlying Gaussian distribution makes sense”; when a linear increase in predictor reflects a linear increase in an underlying normally-distributed trait; e.g., chance of: signal detection (normally-distributed noise in sensory processes), diagnosis (underlying dimensional), correctly answering a question / a math problem / passing an exam