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

7

California wildfires example

age firesev cover

distance

abio

hetero

rich

8

California wildfires example

age firesev cover

distance

abio

hetero

rich

Causal Assumptions:

dist ageage firesevfiresev covercover richdist rich

Implied Conditional Independences:

firesev dist | (age)⏊cover dist | (firesev)⏊cover age | (firesev)⏊rich age | (cover,dist)⏊rich firesev | (cover,dist)⏊

A. Submodel – it’s causal assumptions and testable implications.

A. Functional Specification I – Models of Uncertainty

Variable Potential values Prob. Dist.

age {0,1,2,3,…} Negative Binom

rich {0,1,2,3,…} Negative Binom

firesev (0, ∞) Gamma

cover (0, ∞) Gamma

A. Functional Specification II – Models for Expected Values

B. Modeling the Nodes - Age

age

dist

>library(MASS)>a1.lin<-glm.nb(age~distance,data=dat)>a1.q<-glm.nb(age~distance+I(distance^2),…)

> AICtab(a1.lin,a1.q,weights=T)

dAIC df weight a1.q 0.0 4 0.99662a1.lin 11.4 3 0.00338

>curve(exp(p.l[1]+p.1[2]*x),from=0,to=100,add=T)>curve(exp(p.q[1]+p.q[2]*x+p.q[3]*x^2),from=0,to=100,add=T,lty=2)

firesev

age

>f.lin<-glm(firesev~age,family=Gamma(link="log"),…)

B. Modeling the Nodes - Firesev

>curve(exp(p.f.lin[1]+p.f.lin[2]*x),from=0,to=100,add=T)

1axybax

Aside- Linearization of a saturating function

1 1 baxy ax

firesev

age

>f.sat<-glm(firesev~I(1/age),family=Gamma(link="inverse"),…)

>curve(1/p.f.sat[2]*x/(1+1/p.f.sat[2]*p.f.sat[1]*x),from=0, to=65,add=T,lty=2)

B. Modeling the Nodes - Firesev

firesev

age

B. Modeling the Nodes - Firesev

> AICtab(f.lin,f.sat,weights=T)

dAIC df weight f.sat 0.0 3 1f.lin 16.2 3 <0.001

B. Modeling the Nodes - Cover

cover

firesev

>c.lin<-glm(cover~firesev,family=Gamma(link=log),…)

>curve(exp(p.c[1]+p.c[2]*x),from=0,to=9,add=T,lwd=2)

B. Modeling the Nodes - Richness

cover

firesev

dist

>r.lin<-glm.nb(rich~distance+cover,data=dat)

>r.q<-glm.nb(rich~distance+I(distance^2)+cover,…)

> AICtab(r.lin,r.q,weights=T)

dAIC df weight r.q 0.0 5 0.99767r.lin 12.1 4 0.00233

C. Testing the conditional independences

Implied Conditional Independences:

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)

C. Testing the conditional independences

>nl.detect3(dat,stuff$residuals,stuff$miss.links)

$p.valsdistance-firesev distance-cover age-cover age-rich firesev-rich 0.058 0.252 0.523 0.872 0.134

$fisher.c [1] 14.04139

$d.f[1] 10

$fisher.c.p.val[1] 0.1711122

D. Check Model - Residuals

>pairs(stuff$residuals)

D. Check Model- Parameter Estimates

>sapply(fits,function(x)summary(get(x))$coefficients)$a1.q Estimate Std. Error z value Pr(>|z|)(Intercept) 3.4600063194 8.944635e-02 38.682476 0.0000000000distance -0.0228871119 5.925116e-03 -3.862728 0.0001121277I(distance^2) 0.0002595776 6.729042e-05 3.857571 0.0001145194 $f.sat Estimate Std. Error t value Pr(>|t|)(Intercept) 0.150971 0.01325182 11.39247 5.264449e-19I(1/age) 1.427400 0.26099889 5.46899 4.189435e-07 $c.lin Estimate Std. Error t value Pr(>|t|)(Intercept) 0.213267 0.1382210 1.542942 1.264334e-01firesev -0.132441 0.0284891 -4.648832 1.166142e-05 $r.q Estimate Std. Error z value Pr(>|z|)(Intercept) 3.4603244955 7.030880e-02 49.216093 0.000000e+00distance 0.0164087246 3.150035e-03 5.209060 1.897993e-07I(distance^2)-0.0001408172 3.540241e-05 -3.977617 6.960945e-05cover 0.2361592759 8.581527e-02 2.751949 5.924170e-03

D. Check Model- Print Resulting Graph

#requires graphviz and {PNG}>glmsem.graph(stuff)

E. Run a Query (intervention)

new.dat<-datnew.dat[,'age']<-2dat.int<-calc.intervention.glm(fits,stuff$links,"age",new.dat)

Discussion

Get glmsem.r and these slides and R code for exmpl at:www.msu.edu/~schoolm4/Code_and_More.html