new heritability measure: median dissimilarity ratio

Median Dissimilarity RatioNew Scale Measure of Heritability in Binomial Trait Using Median

Odds Ratio

김진섭

GSPH, SNU

October 24, 2014

김진섭 (GSPH, SNU) Median Dissimilarity Ratio October 24, 2014 1 / 33

개요

1 Binomial Trait에서 heritabily를 가늠할 수 있는 새로운 scale의 지표를소개

2 Heritability + λsib + Median Odds Ratio(MOR)

3 Median Dissimilairty Ratio between Sibling pair and Unrelated pair(MDRSU)

4 HyperTG trait(Healthy Twin Study, Korea)에 적용

5 새로운 역학지표로서의 의의.


Previous Measure of Heritability

Contents

1 Previous Measure of HeritabilityHeritabilityλsib

2 Median Odds RatioIntroduction of MORMOR formula

3 New measure of HeritabilityIdeaMDRSU formula

4 Example of hyperTG(≥200): Healthy Twin Study, Korea

5 Discussion


Previous Measure of Heritability Heritability

Heritability

The portion of phenotypic variance in a population attributable toadditive genetic factors

Intraclass correlation coefficient(ICC) scale

Mixed model로 계산 - Covariate 보정가능.



문제점

Binomial trait: Logit scale VS probability Scale

해석 어렵다.



Different scale: ICC??

Var(Yi ) = pi (1− pi ) (1)

logit(pi ) = Xiβ + Groupi (2)

Proportional scale VS Logistic scale[1]



Example: binomial case

glmer(formula = hyperTG ~ age + sex + BMI + (1 | FID), data = a,

family = binomial)

Estimate Std. Error z value Pr(>|z|)

(Intercept) -6.65451749 1.48227814 -4.4893852 7.142904e-06

age 0.01052907 0.01206682 0.8725635 3.829010e-01

sex -1.48506920 0.60773433 -2.4436158 1.454090e-02

BMI 0.19131619 0.05022612 3.8090977 1.394749e-04

Groups Name Std.Dev.

FID (Intercept) 1.1163



Solution

1 Linearization : logit → proportion

2 Simulation : proportion → logit

3 Latent variable

Approximation of ICC, calculation issue[1, 5]


Previous Measure of Heritability λsib

λsib

1 Recurrence risk ratio

2 Sibling recurrence risk ratio(λsib)

3 일반 인구집단의 prevalence에 비해 affected people의 sibling집단에서의 prevalence가 몇배나 더 높은지를 나타내어 유전적인정도를 측정한다.


Previous Measure of Heritability λsib

문제

1 일반 인구 집단에서의 prevalence를 정확히 측정해야 한다는 부담

2 Ascertainment bias에 민감

3 Polygenic effect 없이 common environmental factor의 영향에의해서도 증가할 수 있다.


Median Odds Ratio

Contents





5 Discussion


Median Odds Ratio Introduction of MOR

Median Odds Ratio(MOR)

Larsen et al.(2000, 2005)임의의 두 group을 골랐을 때 (Odds가 큰 그룹: 작은 그룹) 의 OR이대충(median) 얼마나 되는가?[3, 2, 4]

MOR = exp (√

2VGroup × Φ−1(0.75)) ' exp (0.95√VGroup) (3)

1 1 ∼ inf : Group효과 없다, 엄청 크다.

2 VGroup만 있으면 계산가능: mixed model

3 OR scale로 해석: age, sex 해석하듯이 하면 된다.

4 Covariate 보정가능!!



Example: binomial case

glmer(formula = hyperTG ~ age + sex + BMI + (1 | FID), data = a,

family = binomial)

Groups Name Std.Dev.

FID (Intercept) 1.1163

MOR = exp(√

2× 1.11632 × 0.6745) = 3.67 (4)

: 임의의 두 가족을 뽑으면 대충(median) OR이 3.67이다.


Median Odds Ratio MOR formula

Multilevel logistic regression[3]

Logit[Pr(Yij = 1|Xij ,Gj)] = β0 + X ′ijβ1 + Gj (5)

(β0: intercept, β1: vector of fixed regression coefficients, Gj : randomintercept Gj ∼ N(0,Vg ))

Odds[Pr(Yij = 1|Xij ,Gj)] = exp (β0) exp (X ′ijβ1) exp (Gj) (6)

Odds[Pr(Yij = 1|X ,Gj)]

Odds[Pr(Yik = 1|X ,Gk)]= exp (Gj − Gk) (7)


Median Odds Ratio MOR formula

Odds가 큰그룹을 Odds가 작은 그룹과 비교!

OR = exp |Gj − Gk | (8)

(Gj − Gk) ∼ N(0, 2Vg ) (9)

결국 임의로 두 그룹을 뽑았을 때 Odds가 큰 그룹과 Odds가 작은 그룹을비교하여 OR의 median값을 계산하였을 때 그 결과는

MOR = exp (√

2Vg × Φ−1(0.75)) ' exp (0.95√

Vg) (10)

(Φ: probability density function(PDF) of standard normal distribution)


New measure of Heritability

Contents





5 Discussion


New measure of Heritability Idea

장점만 빼내자

Heritability : Covariate보정

λsib : 직관적인 해석(sib VS unrelated)

MOR의 비율로 만듦.

MDRSU : Ratio of MOR(sib VS unrelated)


New measure of Heritability Idea

단점은 극복

Heritability: 해석어려움.

λsib : 유병률 알아야.. Bias.. Covariate보정안됨.


New measure of Heritability MDRSU formula

MDRSU

Yi : ith individual의 health status(case: 1, control: 0, 1 ≤ i ≤ n)

Xi : vector of covariates

Gi : ith individual의 polygenic effect.

Logit[Pr(Yi = 1|Xi ,Gi )] = β0 + X ′i β1 + Gi (11)

(β0: intercept, β1: vector of fixed regression coefficients, Gi : polygeniceffect of ith individual



Gi들의 vector: G = (G1,G2, · · · ,Gn)′

G ∼ N(0,Vp∑

). (∑

: genetic relationship matrix, Vp: variance ofpolygenic effect).

Odds[Pr(Yi = 1|Xi ,Gi )] = exp (β0) exp (X ′i β1) exp (Gi ) (12)

로 표현할 수 있으며 임의로 뽑은 ith individual과 jth individual의 Oddsratio는

Odds[Pr(Yi = 1|X ,Gi )]

Odds[Pr(Yj = 1|X ,Gj)]= exp (Gi − Gj) (13)



unrelated individuals

i와 j를 unrelated individuals에서 뽑았을 경우

(Gi − Gj) ∼ N(0, 2Vg )

MORunrelated = exp (√

2Vg × Φ−1(0.75)) ' exp (0.95√

Vg ) (14)



i ,j를 sibling pair에서 뽑았다면

i와 j가 항상 sibling이라면 Cov(Gi ,Gj) = 12Vg

(Gi − Gj) ∼ N(0,Vg )

MORsibling = exp (√

Vg × Φ−1(0.75)) ' exp (0.67√Vg ) (15)



MOR은 대략적인 pair의 polygenic dissimilarity를 의미

Unrelated pair, sibling pair에서의 MOR의 비인 Median dissimilarityratio(MDR)를 정의.

MDRSU =MORunrelated

MORsibling' exp(0.28

√Vp) (16)



Sibling pair의 Dissimilarity vs Unrelated의 Dissimilarity

MDRSU 크다.→ unrelated에서 sibling pair보다 dissimilarity가 크다.→ 유전적인 부분이 크다.

Polygenic effect의 variance만 이용하여 계산 → 간단히 계산GLMM이용 → covariate 보정가능!!


Example of hyperTG(≥200): Healthy Twin Study, Korea

Contents





5 Discussion



R의 hglm 패키지 이용

Variable: Mean (SD) or N (%) Male Female P-value P-value:NP*Age 44.6 (13.6) 43.94 (12.73) 0.2 0.285

Smoking status < 0.001 < 0.0011 278 (26.18) 1505 (90.28)2 290 (27.31) 53 (3.18)3 494 (46.52) 109 (6.54)

TG 140.34 (92.34) 98.8 (62.82) < 0.001 < 0.001HyperTG* < 0.001 < 0.001

Control 862 (81.02) 1550 (92.65)Case 202 (18.98) 123 (7.35)

Table: Descriptive statistics of study population

(NP: non parametric: Wilcox rank sum test or Fisher exact test, HyperTG: I(TG≥200))



Parameter(95% CI) Model 1 Model 2 Model 3

Vp 0.92(0.76˜1.11) 0.96(0.79˜1.16) 0.94(0.78˜1.14)MORunrelated 2.49(2.29˜2.72) 2.54(2.33˜2.78) 2.51(2.31˜2.76)

MORsibling 1.9(1.8˜2.03) 1.93(1.82˜2.06) 1.92(1.8˜2.05)MDRsu 1.31(1.28˜1.34) 1.32(1.28˜1.35) 1.31(1.28˜1.35)

Table: Variance parameter and MOR related measures of Polygenic effects

(Model 1: no covariate, Model 2: age & sex as covariates, Model 3: age,sex and smoking status as covariates)


Discussion

Contents





5 Discussion


Discussion

1 New measure of Heritability

2 h2, λsib의 장점만 취하고 단점은 버린다.

3 본 지표가 binomial trait의 polygenic effect를 직관적으로 설명하는간단한 지표가 될 것이라 생각한다.


Discussion

Reference I

[1] Browne, W. J., Subramanian, S. V., Jones, K., and Goldstein, H. (2005). Variance partitioning in multilevel logistic modelsthat exhibit overdispersion. Journal of the Royal Statistical Society: Series A (Statistics in Society), 168(3):599–613.

[2] Larsen, K. and Merlo, J. (2005). Appropriate assessment of neighborhood effects on individual health: integrating randomand fixed effects in multilevel logistic regression. American journal of epidemiology, 161(1):81–88.

[3] Larsen, K., Petersen, J. H., Budtz-Jørgensen, E., and Endahl, L. (2000). Interpreting parameters in the logistic regressionmodel with random effects. Biometrics, 56(3):909–914.

[4] Merlo, J., Chaix, B., Ohlsson, H., Beckman, A., Johnell, K., Hjerpe, P., Rastam, L., and Larsen, K. (2006). A briefconceptual tutorial of multilevel analysis in social epidemiology: using measures of clustering in multilevel logistic regressionto investigate contextual phenomena. Journal of Epidemiology and Community Health, 60(4):290–297.

[5] Vigre, H., Dohoo, I., Stryhn, H., and Busch, M. (2004). Intra-unit correlations in seroconversion to actinobacilluspleuropneumoniae and mycoplasma hyopneumoniae at different levels in danish multi-site pig production facilities. Preventiveveterinary medicine, 63(1-2):9–28.


Discussion

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new heritability measure: median dissimilarity ratio

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