1. Association Analysis, Logistic Regression, R and S-PLUS
Richard Mott

2. Logistic Regression in Statistical Genetics
Applicable to Association Studies
Data:
Binary outcomes (eg disease status)
Dependent on genotypes [+ sex, environment]
Aim is to identify which factors influence the outcome
Rigorous tests of statistical significance
Flexible modelling language
Generalisation of Chi-Squared Test

3. What is R ? Statistical analysis package Free
Similar to commercial package S-PLUS
Runs on Unix, Windows, Mac
Many packages for statistical genetics, microarray analysis available in R
Easily Programmable

4. Modelling in R Data for individual labelled i=1…n: Response yi
Genotypes gij for markers j=1..m

5. Coding Unphased Genotypes
Several possibilities:
AA, AG, GG original genotypes
12, 21, 22
1, 2, 3
0, 1, 2 # of G alleles
Missing Data
NA default in R

6. Using R Load genetic logistic regression tools
> source(‘logistic.R’)
Read data table from file
> t <- read.table(‘geno.dat’, header=TRUE)
Column names
names(t)
t$y response (0,1)
t$m1, t$m2, …. Genotypes for each marker

7. Contigency Tables in R ftable(t$y,t$m31) prints the contingency table
>

8. Chi-Squared Test in R > chisq.test(t$y,t$m31)
Pearson's Chi-squared test
data: t$y and t$m31
X-squared = , df = 2, p-value =
Warning message:
Chi-squared approximation may be incorrect in: chisq.test(t$y, t$m31)
>

9. The Logistic Model Prob(Yi=0) = exp(hi)/(1+exp(hi))
hi = Sj xij bj - Linear Predictor
xij – Design Matrix (genotypes etc)
bj – Model Parameters (to be estimated)
Model is investigated by
estimating the bj’s by maximum likelihood
testing if the estimates are different from 0

10. The Logistic Function Prob(Yi=0) = exp(hi)/(1+exp(hi))

11. Types of genetic effect at a single locusAA AG GG
Recessive 1
Dominant
Additive 2
Genotype a b

12. Additive Genotype Model
Code genotypes as
AA x=0,
AG x=1,
GG x=2
Linear Predictor
h = b0 + xb1
P(Y=0|x) = exp(b0 + xb1)/(1+exp(b0 + xb1))
PAA = P(Y=0|x=0) = exp(b0)/(1+exp(b0))
PAG = P(Y=0|x=1) = exp(b0 + b1)/(1+exp(b0 + b1))
PGG = P(Y=0|x=2) = exp(b0 + 2b1)/(1+exp(b0 + 2b1))

13. Additive Model: b0 = -2 b1 = 2 PAA = 0.12 PAG = 0.50 PGG = 0.88
Prob(Y=0) h

14. Additive Model: b0 = 0 b1 = 2 PAA = 0.50 PAG = 0.88 PGG = 0.98
Prob(Y=0) h

15. Recessive Model Code genotypes as Linear Predictor
AA x=0,
AG x=0,
GG x=1
Linear Predictor
h = b0 + xb1
P(Y=0|x) = exp(b0 + xb1)/(1+exp(b0 + xb1))
PAA = PAG = P(Y=0|x=0) = exp(b0)/(1+exp(b0))
PGG = P(Y=0|x=1) = exp(b0 + b1)/(1+exp(b0 + b1))

16. Recessive Model: b0 = 0 b1 = 2 PAA = PAG = 0.50 PGG = 0.88
Prob(Y=0) h

17. Genotype Model Each genotype has an independent probability
Code genotypes as (for example)
AA x=0, y=0
AG x=1, y=0
GG x=0, y=1
Linear Predictor
h = b0 + xb1+yb2 two parameters
P(Y=0|x) = exp(b0 + xb1+yb2)/(1+exp(b0 + xb1+yb2))
PAA = P(Y=0|x=0,y=0) = exp(b0)/(1+exp(b0))
PAG = P(Y=0|x=1,y=0) = exp(b0 + b1)/(1+exp(b0 + b1))
PGG = P(Y=0|x=0,y=1) = exp(b0 + b2)/(1+exp(b0 + b2))

18. Genotype Model: b0 = 0 b1 = 2 b2 = -1 PAA = 0.5 PAG = 0.88 PGG = 0.27
Prob(Y=0) h

19. Models in R response y genotype gAA AG GG
model DF
Recessive 1
y ~ dominant(g)
Dominant
y ~ recessive(g)
Additive 2
y ~ additive(g)
Genotype a b
y ~ genotype(g)

20. Data Transformation g <- t$m1
use these functions to treat a genotype vector in a certain way:
a <- additive(g)
r <- recessive(g)
d <- dominant(g)
g <- genotype(g)

21. Fitting the Model Equivalent models: genotype = dominant + recessive
afit <- glm( t$y ~ additive(g),family=‘binomial’)
rfit <- glm( t$y ~ recessive(g),family=‘binomial’)
dfit <- glm( t$y ~ dominant(g),family=‘binomial’)
gfit <- glm( t$y ~ genotype(g),family=‘binomial’)
Equivalent models:
genotype = dominant + recessive
genotype = additive + recessive
genotype = additive + dominant
genotype ~ standard chi-squared test of genotype association

22. Parameter Estimates
> summary(glm( t$y ~ genotype(t$m31), family='binomial'))
Coefficients:
Estimate Std. Error z value Pr(>|z|)
b0 (Intercept) <2e-16 ***
b1 genotype(t$m31)
b2 genotype(t$m31)
---
Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
>

23. Analysis of Deviance Chi-Squared Test
> anova(glm( t$y ~ genotype(t$m31), family='binomial'))
Analysis of Deviance Table
Model: binomial, link: logit
Response: t$y
Terms added sequentially (first to last)
Df Deviance Resid. Df Resid. Dev
NULL
genotype(t$m31)

24. Model Comparison
Compare general model with additive, dominant or recessive models:
> afit <- glm(t$y ~ additive(t$m20))
> gfit <- glm(t$y ~ genotype(t$m20))
> anova(afit,gfit)
Analysis of Deviance Table
Model 1: t$y ~ additive(t$m20)
Model 2: t$y ~ genotype(t$m20)
Resid. Df Resid. Dev Df Deviance
>

25. Scanning all Markers > logscan(t,model=‘additive’)
Deviance DF Pval LogPval
m e e
m e e
m e e
m e e
m e e
m e e
m e e
m e e
m e e
m e e
m e e …

26. Multilocus Models
Can test the effects of fitting two or more markers simultaneously
Several multilocus models are possible
Interaction Model assumes that each combination of genotypes has a different effect
eg t$y ~ t$m10 * t$m15

27. Multi-Locus Models
> f <- glm( t$y ~ genotype(t$m13) * genotype(t$m26) , family='binomial')
> anova(f)
Analysis of Deviance Table
Model: binomial, link: logit
Response: t$y
Terms added sequentially (first to last)
Df Deviance Resid. Df Resid. Dev
NULL
genotype(t$m13)
genotype(t$m26)
genotype(t$m13):genotype(t$m26)
> pchisq(6.03,2,lower.tail=F) calculate p-value
[1]

28. Adding the effects of Sex and other Covariates
Read in sex and other covariate data, eg. age from a file into variables, say a$sex, a$age
Fit models of the form
fit1 <- glm(t$y ~ additive(t$m10) + a$sex + a$age, family=‘binomial’)
fit2 <- glm(t$y ~ a$sex + a$age, family=‘binomial’)

29. Adding the effects of Sex and other Covariates
Compare models using anova – test if the effect of the marker m10 is significant after taking into account sex and age
anova(fit1,fit2)

30. Multiple Testing
Take care interpreting significance levels when performing multiple tests
Linkage disequilibrium can reduce the effective number of independent tests
Permutation is a safe procedure to determine significance
Repeat j=1..N times:
Permute disease status y between individuals
Fit all markers
Record maximum deviance maxdev[j] over all markers
Permutation p-value for a marker is the proportion of times the permuted maximum deviance across all markers exceeds the observed deviance for the marker
logscan(t,permute=1000) slow!

31. Haplotype Association
Different from multiple genotype models
Phase taken into account
Haplotype association can be modelled in a similar logistic framework
Treat haplotypes as extended alleles
Fit additive, recessive, dominant & genotype models as before
Eg haplotypes are h = AAGCAT, ATGCTT, etc
y ~ additive(h)
y ~ dominant(h) etc