Linear regression as a probability model
Slides are licensed under CC BY-NC-SA 4.0
Note: Canadian Income Survey (CIS) uses the Labour Force Survey (LFS) sex variable, which asks respondents for their sex “assigned at birth” and requires respondents to answer either “male” or “female.” While the LFS includes a gender item, this is not available in the CIS.
Entire population has one mean and one standard deviation
Standard linear regression allows mean to vary depending on respondent
Each observation () can have a different value for
for female respondents for male respondents
Prior for each of the parameters
Stochastic relationship
Deterministic relationship
* at least three
When we estimate this model, we get a single joint posterior distribution for all three parameters:
What can we do with a joint posterior?
Data: Sample of 3,181 working adults in Canada
Describe the marginal posterior distributions
Describe posterior probability of theoretically relevant scenarios
In general: if the outcome variable is on a log-scale, then exponentiating coefficient estimates () gives multiplicative factors
These results suggest that men make about 22.3% more than women on average
From here, we can add covariates to model income however we like
Compact notation:
Figures by Peter McMahan (source code)
"Distributional models" allow varying sigma