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Learning bayesian data analysis with Statistical Rethinking

27 Lecture 18 - 2019

27.1 Centering/non-centered

  • not always one or the other that is better in all situations
  • use the reparameterized version if modeling is inefficient
  • check if the number of effective parameters increases after

27.2 Multilevel horoscopes

  1. Think about causal model first
  2. Draw the DAG
  3. Begin with “empty” model with varying intercepts on relevant clusters
  4. Standardize predictors
  5. Use regularizing priors + prior predictive simulation
  6. Add in predictors and vary their slopes
  7. Possibly drop varying effects with tiny sigmas
  8. Consider two kinds of posterior prediction
    • same units: “what happened in this data?”
    • new units: “what might we expect in new units?”

27.3 Other covariance structures

  • Instrumental variables
  • Social relations model (covariance in behaviour among nodes)
  • Factor analysis
  • Animal model, heritability of phenotype
  • Phylogenetic regressions
  • Spatial autocorrelation

27.4 Instrumental variables

Recall: adding variables can introduce confounds, so we use the back door criterion to determine which to include and which may introduce confounds.

Sometimes the back door says there is no way to shut it.

For example, E->W, U->E, U->W, where U is some unobserved confound

U is a confound, but U is unmeasured

Solution: an instrument that influences the exposure but not the outcome

Q->E->W, U->E, U->W

Where Q in this case, is the quarter of the year someone is born in. E is education, W is wage. Quarter influences 1) when you start school and 2) when you are allowed to quit school, since Q is the calendar year and not biological age.

In this example, the confounds are not likely related to quarter, and therefore quarter is a valid instrument.

  • this makes E into a collider
  • U generates a correlation between E, W
  • Q tells us something about deviation in E, separate from the E, W correlation from U

Instrumental variables are also called “natural experiments” in biology.

They can be limited. They depend greatly on the DAG, and it is hard to find a plausible instrument. Instruments with weak effects are not very useful.

27.5 Social relationoso models

Example: giving and receiving rates

How to disentangle dyadic offsets, general giving, general receiving rates, etc.

Use multiple covariance matrices. 2x2 giving + receiving, 2x2 dij + dji (paired dyad offsets). dij != dji because one may give more than the other.