Moving away from Linear-Gaussian assumptions Cons: Some things become much harder. No baked-in test of global fit Non-recursive models Error correlations and Latent variables harder to deal with How do we label an arrow? Pros: Flexibility to model nodes with whatever statistical assumption we want to make. Better inference Better predictions
Moving away from Linear-Gaussian assumptions. Pros: Flexibility to model nodes with whatever statistical assumption we want to make. Better inference Better predictions . Cons: Some things become much harder. No baked-in test of global fit - PowerPoint PPT Presentation
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Moving away from Linear-Gaussian assumptions
Cons:Some things become much harder.
No baked-in test of global fitNon-recursive models Error correlations and Latent
variables harder to deal withHow do we label an arrow?
Pros:Flexibility to model nodes with whatever statistical assumption we want to make.
Better inferenceBetter predictions
Causal Effects in Non-linear models: How big is the effect?
firesev
age
The Logic of Graphs: Conditional Independences, Missing link & Testable implications
How do we test structure of the model without Var-Cov matrix?
x
y1 y2
y3
For directed, acyclic models where all nodes are observed,
Vi Non-Child(V⏊ j)|Pa(Vi,Vj)
The residuals of each pair of nodes not connected by a link should be independent.
Each missing link represents a local test of the model structure
Individual test results can be combined using Fisher’s C to give a global test of structure.
k
iipC
1
)ln(2
The Logic of Graphs: Conditional Independences, Missing link & Testable implications
How do we test structure of the model without Var-Cov matrix?
x
y1 y2
y3
How many implied CI there?
N(N-1)/2-L
Where N= number of nodesL=number of links
Strategy for local estimation analysis
1.Create a causal graph
2.Model all nodes as functions of variables given by graph (using model selection of pick functional form)
3.Evaluate all conditional independences implied by graph using model residuals
4. If conditional independence test fails modify graph and goto 2
Generalized Linear Models – 3 components
A probability distribution from the exponential familyNormal, Log-Normal, Gamma, beta, binomial, Poisson, geometric
A Linear predictor
A Link function g such thatIdentity, Log, Logit, Inverse
firesev dist | (age)⏊cover dist | (firesev)⏊cover age | (firesev)⏊rich age | (cover,dist)⏊rich firesev | (cover,dist)⏊
Method for testing conditional indepedences:For each implied conditional independence statement:1. Hypothesize that a link between the variables exists
2. Quantify the evidence that the link explains residual variation in the variable chosen as the response.
C. Testing the conditional independences
C. Testing the conditional independences
C. Testing the conditional independences
What we need:1. List of all implied conditional independences2. Residuals for all fitted nodes>source(‘glmsem.r')
>fits=c("a1.q","f.sat","c.lin","r.q")
>stuff<-get.stuff.glm(fits,dat)
get.stuff.glm returns:1. R^2 for each node ($R.sq)2. Estimated Causal Effect*(over obs. range) ($est.causal.effects)3. Graph implied condition independences ($miss.links)4. Predicted values for each node ($predictions)5. Residuals for each node ($residuals)6. Matrix of links in the graph ($links)7. Matrix of prediction equations ($pred.eqns)