5. Sitzung

Signal Detection Theory

Andrew Ellis

Neurowissenschaft Computerlab FS 22

2022-03-22

1

Signal Detection Theory

Signal detection theory

• What happpens when a person must whether or not an event has occured, using information that insufficient to determine the answer?

• Signal detection theory is a widely-used method for analyzing this type of situation and of separating characteristics of the signal from those of the person who is detecting it.

• We can begin to investigate how information is used to guide a decision.

Signal detection theory

What is a decision?

• A doctor is examining a patient and trying to make a diagnosis.

• A witness to a crime is asked to identify a suspect. Was this person present at the time of the crime or not?

• Purchase decision, gambling decisions, perceptual decisions, etc.

Binary choice

	Signal
Response	Yes	No
——–	————–	———————
Yes	Hit	False alarm (FA)
No	Miss	Correct rejection (CR)

Signal detection theory

• Assumptions of SDT
• Graphical representation

The key assumptions of SDT

• Separate representation and decision-making.

• Signal and noise trials can be represented as values along a uni-dimensional “strength” construct.

• Both types trials produce Gaussian random variables that represent strength.

• Both variances are assumed to be the same.

• The decision-making assumption of is that yes and no responses are produced by comparing the strength of the current trial to a fixed criterion. If the strength exceeds the criterion a “yes” response is made, otherwise a “no” response is made. ]

Equal-variance Signal Detection Model

c and d'

• The mean of the signal distribution is d'. This makes d' a measure of the discriminability of the signal trials from the noise trials – it corresponds to the distance between the two distributions.

• The strength value d/2 is special, because it is the criterion value that maximizes the probability of a correct classification when signal and noise trials are equally likely to occur.

• c a measure of bias, because it corresponds to how different the actual criterion is from the unbiased one. Positive values of c correspond to a bias towards saying no, and so to an increase in correct rejections at the expense of an increase in misses. Negative values of c correspond to a bias towards saying yes, and so to an increase in hits at the expense of an increase in false alarms.

2

Computing Signal Detection Measures in R

Computing SDT Measures in R

• Distribution function: code
• Distribution function: figure

library(tidyverse)
tibble(x = seq(-3, 3, by = 0.1)) %>% 
  ggplot(aes(x)) +
  stat_function(fun = pnorm, colour = "steelblue3", 
                  args = list(mean = 0, sd = 1),
                  size = 1.5) +
  labs(y = "Probability", x = "Z Score") +
  scale_y_continuous(limits = c(0, 1)) +
  scale_x_continuous(limits = c(-3.5, 3.5), breaks = -3:3) + 
  ggtitle("Cumulative distribution function / pnorm()") +
  theme_linedraw()

Cumulative distribution function: pnorm()

• Quantile function: code
• Quantile function: figure

tibble(x = seq(0, 1, by = 0.01)) %>% 
  ggplot(aes(x)) +
  stat_function(fun = qnorm, colour = "steelblue3", 
                  args = list(mean = 0, sd = 1),
                  size = 1.5) +
  labs(x = "Probability", y = "Z Score") +
  scale_x_continuous(limits = c(0, 1)) +
  scale_y_continuous(limits = c(-3.5, 3.5), breaks = -3:3) + 
  ggtitle("Probit function / quantile function / inverse cdf / qnorm()") +
  theme_linedraw()

Probit function / quantile function / inverse cdf: qnorm()

3

Using Signal Detection to Analyze Data in R

Using Signal Detection to Analyze Data in R

We will look at an example using data from a recognition memory test.

• Subjects first learn words (study phase).

• In the test phase, they are presented with words that were on the list, and with new words.

• Subjects have to classify words into

  • old (yes, this was on the list)
  • new (no, this wasn’t on the list)

• These kind of data are usually analyzed using a Signal Detection model.