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## Moving away from Linear-Gaussian assumptions

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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

Non-recursive models Error correlations and Latent variables harder to deal with

How do we label an arrow?

The Logic of Graphs: Conditional Independences, Missing link & Testable implications

How do we test structure of the model without Var-Cov matrix?

For directed, acyclic models where all nodes are observed,

Vi⏊Non-Child(Vj)|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.

x

y1

y2

y3

The Logic of Graphs: Conditional Independences, Missing link & Testable implications

How do we test structure of the model without Var-Cov matrix?

How many implied CI there?

N(N-1)/2-L

Where N= number of nodes

L=number of links

x

y1

y2

y3

Strategy for local estimation analysis

Create a causal graph

Model all nodes as functions of variables given by graph (using model selection of pick functional form)

Evaluate all conditional independences implied by graph using model residuals

If conditional independence test fails modify graph and goto 2

Generalized Linear Models – 3 components

A probability distribution from the exponential family

Normal, Log-Normal, Gamma, beta, binomial, Poisson, geometric

A Linear predictor

A Link function g such that

Identity, Log, Logit, Inverse

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

Causal Assumptions:

dist age

age firesev

firesev cover

cover rich

dist rich

Implied Conditional Independences:

firesev⏊ dist | (age)

cover ⏊ dist | (firesev)

cover ⏊ age | (firesev)

rich ⏊ age | (cover,dist)

rich ⏊ firesev | (cover,dist)

A. Functional Specification I – Models of Uncertainty

VariablePotential valuesProb. Dist.

age {0,1,2,3,…} Negative Binom

rich {0,1,2,3,…} Negative Binom

firesev (0, ∞) Gamma

cover (0, ∞) Gamma

dist

>library(MASS)

>a1.lin<-glm.nb(age~distance,data=dat)

>a1.q<-glm.nb(age~distance+I(distance^2),…)

age

>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)

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

dAICdf weight

a1.q 0.0 4 0.99662

a1.lin 11.4 3 0.00338

B. Modeling the Nodes - Firesev

age

firesev

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

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

B. Modeling the Nodes - Firesev

age

firesev

>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

age

firesev

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

dAICdf weight

f.sat 0.0 3 1

f.lin 16.2 3 <0.001

firesev

cover

>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

dist

firesev

cover

>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)

dAICdf weight

r.q 0.0 5 0.99767

r.lin12.1 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

Quantify the evidence that the link explains residual variation in the variable chosen as the response.

C. Testing the conditional independences

What we need:

List of all implied conditional independences

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:

R^2 for each node ($R.sq)

Estimated Causal Effect*(over obs. range) ($est.causal.effects)

Graph implied condition independences ($miss.links)

Predicted values for each node ($predictions)

Residuals for each node ($residuals)

Matrix of links in the graph ($links)

Matrix of prediction equations ($pred.eqns)

C. Testing the conditional independences

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

$p.vals

distance-firesevdistance-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

>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 0.0000000000

distance -0.0228871119 5.925116e-03 -3.862728 0.0001121277

I(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-19

I(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-01

firesev-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+00

distance 0.0164087246 3.150035e-03 5.209060 1.897993e-07

I(distance^2)-0.0001408172 3.540241e-05 -3.977617 6.960945e-05

cover 0.2361592759 8.581527e-02 2.751949 5.924170e-03

new.dat<-dat

new.dat[,'age']<-2

dat.int<-calc.intervention.glm(fits,stuff $links,"age",new.dat)

Discussion

Get glmsem.r and these slides and R code for exmplat:

www.msu.edu/~schoolm4/Code_and_More.html

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