18 Lecture 09 - 2019
18.1 Conditioning
Interaction of variables on each other
Dependence on the state
eg. Influence of genes on phenotype depends on environment
Approaches
- Use interacting terms (simplest)
- Generalized linear models
- Multilevel models
Interactions arise wherever there is a boundary in the outcome space. All GLMs have interactions.
In a DAG, an interaction looks like
gene → phenotype ← environment
But DAGs can’t fully tell you if it’s an interaction
Before interaction terms, all variables are simply independent additive terms.
18.1.1 Example: ruggedness
“Ruggedness is bad for the economy outside of Africa, but within Africa is it good”
Reminder - constrain priors to possible outcome space
- Scale ruggedness between 0, 1
- Constrain change in GDP bc evidently eg GDP x 2 would be a huge effect
Keep it reasonable
Options
- Split the data?
Run two linear regressions. This means there is no statistical criteria to measure the split.
We are interested in the contrast in slope, but to do that we need to use the same model.
- Add a categorical variable for Africa?
Use alpha[id] and different estimates for each
This means the slope is forced to be the same, but difference intercepts.
Relationship is held constant across groups, not what we want.
- Interaction
\(\mu_{i} = \alpha_{CID[i]} + \beta_{CID[i]}(r_{i} - \bar{r})\)
Slope and intercept are allowed to vary for each
18.2 Interpreting interactions
- Interpreting interactions is hard
- The influence of predictors depends upon multiple parameters and their co variation
- Interactions are symmetric within the data.
- Eg. effect of continent depends on ruggedness is the same as effect of ruggedness on continent
- Statistically the same
- We need to apply our outside knowledge and causal information