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Disease Models and Association StatisticsDisease Models and Association Statistics
Nicolas Widman
CS 224- Computational GeneticsNicolas Widman
CS 224- Computational Genetics
IntroductionIntroduction
Certain SNPs within genes may be associated with a disease phenotype
Statistical model used in class only considers inheritance of a single copy of an SNP location: Single Chromosome Model Expand the statistic to a diploid model and
take into account different expression patterns of a SNP
Certain SNPs within genes may be associated with a disease phenotype
Statistical model used in class only considers inheritance of a single copy of an SNP location: Single Chromosome Model Expand the statistic to a diploid model and
take into account different expression patterns of a SNP
Basic Statistic- Haploid ModelBasic Statistic- Haploid Model
: Relative Risk pA: Probability of disease-associated allele F: Disease prevalence
For this project, F is assumed to be very small +/-: Disease StateDerivation of case (p+) and control (p-) frequencies:
P(A)=pA p+A=P(A|+) p-
A=P(A|-) F=P(+)
P(A|+)=P(+|A)P(A)/P(+)
P(+|A)= P(+|¬A)
: Relative Risk pA: Probability of disease-associated allele F: Disease prevalence
For this project, F is assumed to be very small +/-: Disease StateDerivation of case (p+) and control (p-) frequencies:
P(A)=pA p+A=P(A|+) p-
A=P(A|-) F=P(+)
P(A|+)=P(+|A)P(A)/P(+)
P(+|A)= P(+|¬A)
Derivation- ContinuedDerivation- Continued
P(+)=F=pAP(+|A)+(1-pA)P(+|¬A)
P(+)=F= pAP(+|A)+(1-pA)P(+|A)/
P(+)=F=P(+|A)(pA+(1-pA)/)=P(+|A)(pA(-1)+1)/
P(+|A)= F/(pA(-1)+1)
P(A|+)=P(+|A)P(A)/P(+)=P(+|A)pA/F=pA/(pA(-1)+1)
P(-|A)=1-P(+|A)=1- F/(pA(-1)+1)
P(A|-)=P(-|A)P(A)/P(-)
If F is small, then 1-F ≈ 1 and P(-|A) ≈ 1
then, P(A|-) ≈ P(A) = pA
P(+)=F=pAP(+|A)+(1-pA)P(+|¬A)
P(+)=F= pAP(+|A)+(1-pA)P(+|A)/
P(+)=F=P(+|A)(pA+(1-pA)/)=P(+|A)(pA(-1)+1)/
P(+|A)= F/(pA(-1)+1)
P(A|+)=P(+|A)P(A)/P(+)=P(+|A)pA/F=pA/(pA(-1)+1)
P(-|A)=1-P(+|A)=1- F/(pA(-1)+1)
P(A|-)=P(-|A)P(A)/P(-)
If F is small, then 1-F ≈ 1 and P(-|A) ≈ 1
then, P(A|-) ≈ P(A) = pA
Haploid ModelHaploid Model
The relative risk formula:
Association Power:
The relative risk formula:
Association Power:
1)1( +−=+
A
AA p
pp
)1()/2(
)(
AA
AAA
ppN
ppN
−−
=−+
λ
))2/((1
))2/((1
1
N
N
A
A
λα
λα
+Φ−Φ−+
+ΦΦ−
−
AssumptionsAssumptions
Low disease prevalence F ≈ 0: Allows p-
A ≈ pA
Uses Hardy-Weinberg Principle A-Major Allele a-Minor Allele
P(AA)=P(A)^2
P(Aa)=2*P(A)*(1-P(A))
P(aa)=(1-P(A))^2
Uses a balanced case-control study
Low disease prevalence F ≈ 0: Allows p-
A ≈ pA
Uses Hardy-Weinberg Principle A-Major Allele a-Minor Allele
P(AA)=P(A)^2
P(Aa)=2*P(A)*(1-P(A))
P(aa)=(1-P(A))^2
Uses a balanced case-control study
Diploid Disease ModelsDiploid Disease Models
When inheriting two copies of a SNP site, there are three common relationships between major and minor SNPs Dominant
Particular phenotype requires one major allele
Recessive Particular phenotype requires both minor alleles
Additive Particular phenotype varies based whether there are one or
two major alleles
When inheriting two copies of a SNP site, there are three common relationships between major and minor SNPs Dominant
Particular phenotype requires one major allele
Recessive Particular phenotype requires both minor alleles
Additive Particular phenotype varies based whether there are one or
two major alleles
Diploid Disease ModelsDiploid Disease Models
AA- Homozygous major Aa, aA- Heterozygous aa- Homozygous minor
AA- Homozygous major Aa, aA- Heterozygous aa- Homozygous minor
Modifying the Calculation for Relative Risk
Modifying the Calculation for Relative Risk
Previous relative risk formula only considered the haploid case of having a SNP or not having a SNP.
Approach:
Create a virtual SNP which replaces pA in the formula.
Previous relative risk formula only considered the haploid case of having a SNP or not having a SNP.
Approach:
Create a virtual SNP which replaces pA in the formula.
1)1( +−=+
A
AA p
pp
Virtual SNPsVirtual SNPs
Use Hardy-Weinberg Principle to calculate a new pA - the virtual SNP using the characteristics of diploid disease models. Recessive
pA=pd*pd
DominantpA=pd*pd+2*pd*(1-pd)
AdditivepA=pd*pd+c*pd*(1-pd)
Pd: Probability of disease-associated allele. In the calculations used to determine the association power, c was set to sqrt(2).
Use Hardy-Weinberg Principle to calculate a new pA - the virtual SNP using the characteristics of diploid disease models. Recessive
pA=pd*pd
DominantpA=pd*pd+2*pd*(1-pd)
AdditivepA=pd*pd+c*pd*(1-pd)
Pd: Probability of disease-associated allele. In the calculations used to determine the association power, c was set to sqrt(2).
Diploid Disease Models: =1.5Diploid Disease Models: =1.5
Diploid Disease Models: =1.5Diploid Disease Models: =1.5
Diploid Disease Models: =1.5Diploid Disease Models: =1.5
Diploid Disease Models: =2Diploid Disease Models: =2
Diploid Disease Models: =2Diploid Disease Models: =2
Diploid Disease Models: =2Diploid Disease Models: =2
Diploid Disease Models: =3Diploid Disease Models: =3
ResultsResults
Achieving significant association power with low relative risk SNPs (=1.5) Minimum of 200 cases and 200 controls required to
reach 80% power within strongest pd intervals for each type of SNP
At a sample size of 1000 cases and 1000 controls, dominant and additive SNPs show very significant power for almost all SNP probabilities below 50%
Difficult to obtain significant association for low probability recessive SNPs regardless of sample size
Achieving significant association power with low relative risk SNPs (=1.5) Minimum of 200 cases and 200 controls required to
reach 80% power within strongest pd intervals for each type of SNP
At a sample size of 1000 cases and 1000 controls, dominant and additive SNPs show very significant power for almost all SNP probabilities below 50%
Difficult to obtain significant association for low probability recessive SNPs regardless of sample size
ResultsResults
SNP probability ranges for greatest association power Dominant: .10 - .30 Recessive: .45 - .70 Additive: .15 - .40
Higher relative risk SNPs require fewer cases and controls to achieve the same power.
As approaches 1, the association power to detect a recessive allele with probability p is the same as the power to detect dominant allele with probability 1-p.
SNP probability ranges for greatest association power Dominant: .10 - .30 Recessive: .45 - .70 Additive: .15 - .40
Higher relative risk SNPs require fewer cases and controls to achieve the same power.
As approaches 1, the association power to detect a recessive allele with probability p is the same as the power to detect dominant allele with probability 1-p.
ResultsResults
Diseases with higher relative risk have their range of highest association power skewed toward lower probability SNPs.
Challenges in obtaining high association power: Low probability recessive SNPs Low relative risk diseases, especially with small
sample sizes High probability dominant SNPs, however these are
unlikely due natural selection and that the majority of the population would be affected by such diseases.
Diseases with higher relative risk have their range of highest association power skewed toward lower probability SNPs.
Challenges in obtaining high association power: Low probability recessive SNPs Low relative risk diseases, especially with small
sample sizes High probability dominant SNPs, however these are
unlikely due natural selection and that the majority of the population would be affected by such diseases.
