2019-03-27

# Contents

• What is a GLM?
• Signal detection theory basics
• Example: recognition memory experiment
• Example: perceptual experiment

# Main points

• Equal-variance signal detection models can formulated as a subclass of generalized linear models (GLM) (DeCarlo 1998; Knoblauch and Maloney 2012)
• SDT models can be estimated using multilevel GLMs
• Multilevel models provide regularizarion (shrinkage) of group-level estimates
• More generally, we can formulate any variation of SDT as a probabilistic model, e.g. unequal variance SDT, extreme value SDT
• Parameters can be estimated using Bayesian inference, resulting in posterior distributions.
• flexibility
• access to full statistical workflow, i.e. model checking, model comparison, etc.

# GLM

• What is a GLM? A quick reminder

The GLM generalizes a regression model by allowing the linear predictor to be related to the response variable via a link function.

We have to the following components:

• Response (outcome): Observable behaviour. E.g. subjects’ choices in an experiments, number of cars driving by per hour, etc.
• Linear model: We use known variables (stimulus identity) to predict the expected value of the response. E.g. when a previously seen image is presented, the subject is more likely to respond “I’ve seen that before”.
• Link function: This is a function that relates the output of the linear model to the response. E.g. in a 2AFC experiment, the linear predictor gives a value on the real line. The link function is chosen to map this real number to a probability (in this case the inverse of a sigmoidal function mapping the interval $$(0, 1)$$ to $$(-\infty, \infty)$$).

# Probit model

The most common model for binary responses is the logistic model. The probit model is related, but uses a different link function. The probit function is the quantile function associated with the normal distribution. The quantile function is also the inverse cumulative distribution function:

$probit(x) = \Phi^{-1}(x)$

and looks like this: