Logistic Regression, and an Introduction to Hierarchical Analysis

Question 1

Simplest Linear Model

Answer 1

Just a constant term
Returns the mean of our data
Postive mean = positive change
Beta divided by standard error

Question 3

One Continuous Regressor

Answer 3

Add one more term returning the intercept and slope of our graph
Explains variation in the score
Beta value divided by standard error - significance difference?

Question 4

Multiple Continuous Regressors

Answer 4

Add more terms
Returns the slope controlling for the effect of other regressors

Question 5

Multiple Categories

Answer 5

Add a ‘dummy variable’ for each additional category
Returns the mean for the reference category, and difference from this reference for each other category

Question 6

Multiple Independent Variables

Answer 6

Add extra sets of ‘dummy variables’ for each additional discrete factor, plus their interaction terms

Question 7

Logistic Regression

Answer 7

p(y=right) = 1/1+exp(=B0 + B1 x coherence))

Question 8

Probability to Odds

Answer 8

Odds = p/1-p

Question 9

Log Odds

Answer 9

Log odds of a binary outcome can be modelled with a straight line
Logarithm of odds
Continuous function
Log odds = log(p/1-p)

Question 10

2 Ways of Expressing the Same Idea

Answer 10

B1 = impact of coherence
B0 = intercept term
Biases choices

Question 11

Summary

Answer 11

When the outcome variable is binary, rather than continuous we use logistic regression to predict outcomes
Equivalent to a linear model predicting the log odds of the probability of the outcome
Log odds = link function
Generalises linear models to predict outcomes that are binary in nature

Question 12

Aggregate Information across Participants

Answer 12

Estimate models ‘hierarchically’
Advantages - interpretability, can apply to any ‘summary statistic’ from each participat, computational simplicity
Disadvantages - requires many observations at ‘first level’ (each participant), assumes each subject is equally reliable, doesn’t explicitly account for ‘correlated observations within participants

Question 13

Conclusion

Answer 13

Repeat experiment across many different individuals
Aggregate information across individuals by performing a ‘hierarchical’ analysis - estimate a GLM for each subject at first model, and then take the parameters of this model to the 2nd level performing inference across the population
Straightforward and usually valid
Some limitations dealt with by a more sophisticated method of aggregating information across participants, mixed effects model