Risk Prediction Modeling on Family-Based Sequencing Data Using a Random Field Method

| INVESTIGATION

Risk Prediction Modeling on Family-Based Sequencing

Data Using a Random Field Method

Yalu Wen,*,†,1_{Alexandra Burt,}‡_{and Qing Lu}§,1 *Institute of Cancer Stem Cell, Dalian Medical University, Liaoning, 116044, China,†Department of Statistics, University of Auckland, 1010, New Zealand,‡Department of Psychology, and§Department of Epidemiology and Biostatistics, Michigan State University, East Lansing, Michigan 48824

ABSTRACTFamily-based design is one of the most popular designs in genetic studies and has many unique features for risk-prediction research. It is robust against genetic heterogeneity, and the relatedness among family members can be informative for predicting an individual’s risk for disease with polygenic and shared environmental components of risk. Despite these strengths, family-based designs have been used infrequently in current risk-prediction studies, and their related statistical methods have not been well developed. In this article, we developed a generalized randomfield (GRF) method for family-based risk-prediction modeling on sequencing data. In GRF, subjects’phenotypes are viewed as stochastic realizations of a randomfield in a space, and a subject’s phenotype is predicted by adjacent subjects, where adjacencies between subjects are determined by their genetic and within-family similarities. Different from existing methods that adjust for familial correlations, the GRF uses this information to form surrogates to further improve prediction accuracy. It also uses within-family information to capture predictors (e.g., rare mutations) that are homogeneous in families. Through simulations, we have demonstrated that the GRF method attained better performance than an existing method by considering additional information from family members and accounting for genetic heterogeneity. We further provided practical recommenda-tions for designing family-based risk prediction studies. Finally, we illustrated the GRF method with an application to a whole-genome exome data set from the Michigan State University Twin Registry study.

KEYWORDSfamily-based studies; genetic heterogeneity; high-dimensional data; randomfield theory

T

designed rationally from a detailed understanding of the disease etiology and individual differences, has been widely accepted as the future goal of precision medicine (Collins and Varmus 2015). Toward that end, there is an expectation that

human genome discoveries and otherfindings (e.g.,

environ-mentalfindings) will revolutionize the current trial-and-error

practice of medicine by enabling more accurate disease pre-diction and precise prevention/treatment strategies (Collins

et al.2003; Collins and Varmus 2015). The use of emerging

genomicfindings and other existing knowledge to predict

disease risk is an essential step toward precision medicine

(Rogowskiet al.2009). The hope is that formed risk-prediction

models can successfully identify high-risk subpopulations so that appropriate prevention/treatment strategies can be used to reduce morbidity and mortality. While there has been suc-cess in predicting certain diseases, risk-prediction models for

most diseases lack sufficient accuracy for clinical use because

currently known predictors are insufficient for accurately

predicting disease risk (Kraft and Hunter 2009).

One possible explanation of the low accuracy of existing models is that complex diseases are likely driven by many yet-to-be discovered genetic variants, which include common genetic variants with small marginal effects and rare variants, but the current genetic studies are not powerful enough to detect all these variants (Goldstein 2009; Golan and Rosset 2014). In light of this theory, the traditional approach of

using only the significant common variants to form genetic

risk scores is expected to perform poorly because it overlooks the lion’s share of causal variants that fail to reach statistical

significance, either due to small effect sizes or minor allele

frequencies (Allen et al.2010; Yanget al.2010; Chatterjee

Copyright © 2017 by the Genetics Society of America doi:https://doi.org/10.1534/genetics.117.199752

Manuscript received January 9, 2017; accepted for publication June 27, 2017; published Early Online July 5, 2017.

Supplemental material is available online atwww.genetics.org/lookup/suppl/doi:10. 1534/genetics.117.199752/-/DC1.

et al.2013). Recent efforts in improving the prediction accu-racy have attempted to include a large number of genetic variants by applying more lenient inclusion criteria. Alterna-tive approaches have also been proposed, among which best linear unbiased prediction (BLUP) is one of the most widely used methods to predict complex traits in both animal

breed-ing and human genetics (Meuwissenet al.2001; VanRaden

2008; Yanget al.2011; de los Camposet al.2013). The basic

premise is that instead of estimating the effects for each in-dividual genetic variant, it attempts to estimate their cumu-lative effects. The genetic variance component is estimated based on a genetic relationship matrix, which gives high

val-ues for those sharing similar genetic profiles and low values

for those with different genetic profiles. In animal breeding,

the genetic variance component can be estimated using

avail-able pedigree information (Meuwissenet al.2001; VanRaden

2008), and in human genetic research the genetic variance component can be based on the genetic relationship matrices

from whole-genome data (Yanget al.2011; de los Campos

et al.2013). The implicit assumption for BLUP in human

ge-netic studies is that all common gege-netic variants have the same effect-size distribution, which can be subject to the sub-optimal performance when effects are heterogeneous (Speed and Balding 2014).

The advent of whole-genome sequencing studies has dem-onstrated that rare variants can play an important role in many common complex diseases, such as type 1 diabetes, coronary

heart disease, and drug dependence (Nejentsevet al.2009;

Cirulli and Goldstein 2010; Yanget al.2015; Zanoniet al.

2016). Recent rare variants could also be more deleterious than the common variants as they are under less selection pressures according to the evolutionary theory. While inte-grating rare mutations into a risk-prediction model can po-tentially lead to improved accuracy, few methods have been developed for risk prediction on sequencing data. The con-ventional statistical methods are not designed for sequencing data and could have been subject to low performance due to the high dimensionality of the sequencing data and low fre-quency of rare variants. Recently, gene-based approaches have been proposed for association analyses of sequencing

data, and they have led to promising findings (Neale and

Sham 2004; Wuet al.2011; Heet al.2014). Compared with

the single-variant method, gene-based methods have several advantages. The gene is the functional unit of the human genome, and jointly evaluating all genetic variants within a gene can aggregate the signal from each variant and increase the detection power. A similar idea can also be used in pre-diction analysis, where the contribution of rare variants to

disease risk can be accumulated at the gene level (Wenet al.

2016).

It is well known that family history can provide important information for disease risk prediction. Correlations among relatives can explain unmeasured polygenic and shared en-vironmental variations, and thus contribute additional infor-mation, beyond the measured genetic and environmental

data, for risk prediction (Ruderferet al.2010). Furthermore,

affected members from the same family tend to have homo-geneous genetic causes. Therefore, family-based genetic studies can provide robustness against genetic heterogeneity

(Mihaescu et al. 2013), which is important for predicting

diseases with heterogeneous underlying genetic causes

(McClellan and King 2010; Morriset al.2010). Despite these

advantages, family-based studies have been used infre-quently in current risk prediction research and their related statistical methods have not been well developed. With the emerging genetic data from family-based studies, there is a great need for new statistical methods for family-based risk prediction analysis. Existing family-based risk-prediction

methods, such as GEE-GS (Meigset al.2008), are designed

for risk prediction on a limited number of known predictors and thus are not suitable for high-dimensional genetic data. In this article, we describe a statistical approach for risk-prediction modeling using high-dimensional data from family-based genetic studies. The new method is robust against genetic heterogeneity and has the capacity of using relatedness among family members as a surrogate for unmeasured genetic and shared environmental predictors to further increase

predic-tion accuracy. Our method is built within the randomfield

framework (Cressie 1993; Adler and Taylor 2007). Moti-vated by the idea used in spatial statistics that adjacent points share similar outcomes (Cressie 1993; Cressie and

Johannesson 2008; Wu et al. 2011; He et al. 2014), we

assume genotypic similarity will lead to phenotypic

similar-ity and this assumption has been widely used in the field.

Instead of adjusting for correlations between family

mem-bers (Meigset al.2008; Chenet al.2013; Svishchevaet al.

2014), we use this information to construct within-family similarities, which is then used to form a surrogate for un-measured risk predictors. We estimate the effects of the selected genes and family information through solving gen-eralized estimating equations. Simulation studies were con-ducted to assess the effects of family information and genotypic heterogeneity on risk-prediction modeling, and further give practical recommendations for designing fam-ily-based risk-prediction studies. Finally, we illustrate the proposed method through an application to a whole-genome exome data set from the Michigan State University Twin Reg-istry (MSUTR) (Burt and Klump 2012).

Methods

A fundamental concept in quantitative genetics is to link

genetic profiles with disease outcomes through genetic

sim-ilarity among individuals (Wright 1921), and this concept has

been recently exploited using spatial models (Heet al.2014;

Wheeleret al.2014; Wenet al.2016). Spatial autocorrelation

is a common phenomenon in geospatial and ecological data, because observations from nearby locations tend to be more similar than would be expected on a random basis. The dis-ease outcomes of subjects can be viewed as observations from

specific locations in a space, and they form a stochastic

analogy with spatial statistics motivates the use of random

field models for genetic risk-prediction modeling. Specifi

-cally, we assume the outcome of a new subject cannot only be predicted by their own environmental and clinical risk factors but also adjacent subjects, where neighborhood struc-tures between subjects are determined by the genetic simi-larities of subjects and the closeness of subjects in the family. Because correlation between family members is informative for unmeasured genetic and shared environmental factors

(Ruderferet al.2010), this additional information is

incorpo-rated into generalized randomfield (GRF) to further improve

the model’s prediction accuracy. We form our GRF risk pre-diction model as

E

Yi_;jY2ði;jÞ

¼ m_i;jþ X

ði;jÞ6¼ðm;lÞ

l_{ði;jÞ;ðm;lÞ}Ym;l2mm;l

þXK_kg_k X

ði;jÞ6¼ðm;lÞ

sk_{ði;jÞ;ðm;lÞ}

Ym_;l2mm;l

;

whereY_2ði;jÞdenotes the phenotypic values for all

individ-uals excluding thejth subject in theith family i.e.,Yði;jÞ;

m_i;j¼fðXT

i;jbÞ denotes the mean function as defined in a

generalized linear model,bis a vector of regression

coef-ficients for covariatesX, fðxÞ is a link function he.g., for

binary outcomes, we use logit link,fðxÞ ¼_1þexp_expðx_ðÞ_xÞi;gk

mea-sures the genetic effect of thekth gene on the phenotype,K

is the total number of genes,sk

ði;jÞ;ðm;lÞmeasures the genetic

similarity of thekth gene between subjectjin familyiand

subjectlin familym, andlði;jÞ;ðm;lÞmeasures the

contribu-tion of family informacontribu-tion to predictYi;jfrom subjectlin

family m. The genetic similarity between two

sub-jects based on the kth gene

h

i.e., sk_ði;jÞ;ðm;lÞ

i

is defined as

sk

ði;jÞ;ðm;lÞ¼

Pnk

h vkh

22gki;;jh2g k;h m;l

;wheregki;;jhdenotes the

genotype of thehth marker on thekth gene for thejth

sub-ject in familyi,nkis the total number of markers on thekth

gene, andvk

his the weight used to reflect the effects of rare

variants and is commonly defined based on minor allele

frequency. The genetic similarity for the same subject is

set at zero [i.e.,sk

ði;iÞ;ðl;lÞ¼0], and thus the phenotype of a

subject does not contribute to his/her own phenotype pre-diction. When two subjects come from different families,

we assume there is no family information and thuslði;jÞ;ðm;lÞ

is set at zero. For subjects from the same family, we assume that the correlation between family members are mainly due to the shared environmental factors and genetic

relat-edness. Therefore, we modell_ði;jÞ;ðm;lÞas a linear

combina-tion of shared environmental effects and the theoretical correlation between family members due to genetic relatedness,

l_{ði;jÞ;ðm;lÞ}¼g_wF_ði;jÞ;ðm;lÞþgsIði¼mandj6¼lÞ;

whereFði;jÞ;ðm;lÞis the kinship coefficient betweenjth

individ-ual in family i and lth individual in family m, and

Iði¼mandj6¼lÞ is an indicator function with

Iði¼mandj6¼lÞ ¼1 if two different subjects are from the

same family (i.e., i¼m and j6¼l), and 0 otherwise. The

F_ði;jÞ;ðm;lÞ for the same subject is set at zero [i.e.,

F_ði;jÞ;ði;jÞ¼0], and thus the phenotype of a subject does not

contribute to his/her own phenotype prediction.gwandgs

respectively measure predictive effects of unmeasured genet-ic variants and shared environmental factors. While here we use a kinship matrix and compound symmetric matrix to model genetic relatedness and shared environmental factors,

the model can be easily modified by using other types of

matrices and considering other factors (e.g., spouse

correlation).

For the purpose of derivation, we write the GRF model in matrix format:

EðYjY2Þ ¼mþ ðgwFþgsIeÞðY2mÞ þ

XK

k

g_kS_kðY2mÞ;

(1)

where Y2¼ ðY92ð1;1Þ;Y92ð1;2Þ;. . .;Y2ðM;NMÞ9Þ9; M is the total

number of families,Niis the number of individuals in family

i,Ntis the total number of subjects (i.e.,Nt¼PMi Ni), andSk

is an Nt3Nt genetic similarity matrix with sk

ði;jÞ;ðm;lÞ as its

element and zeros on the diagonal.F andI_eare two block

diagonal matrices where each block represents a family. The

ðj;lÞelements of blockiforFandI_eare kinship coefficient

between subjectjand subjectlin familyi[i.e.,F_ði;jÞ;ði;lÞ] and 1,

respectively. Theðj;jÞelements of blockiforFandIeare set at

zero, so that the same individual does not contribute to his/ her own phenotype prediction.

LetG¼ ðg1;g2;. . .;gKÞ9denote the genetic effects

associ-ated with K genes. The parameters ðG;gw;gsÞ in Equation

1 can be estimated by solving the following unbiased gener-alized estimating equations:

Ugkðb;G;gw;gsÞ ¼

@EðYjY2Þ9

@g_k

½Y2EðYjY₂Þ

¼ ðY2mÞ9Sk

I2PK_kgkSk2gwF2gsIe

ðY2mÞ ¼0

Ug_wðb;G;gw;gsÞ ¼

@EðYjY₂Þ9 @g_w

½Y2EðYjY2Þ

¼ ðY2mÞ9FI2P_kKgkSk2gwF2gsIe

ðY2mÞ ¼0

Ugsðb;G;gw;gsÞ ¼

@EðYjY2Þ9

@g_s

½Y2EðYjY₂Þ

¼ ðY2mÞ9Ie

I2PK_kgkSk2gwF2gsIe

ðY2mÞ ¼0:

0 B B B B B B B B B B B B B B B B B B B B B B B B @

The regression coefficients (b) are usually unknown, and can

assumption of G¼0 andgw¼gs¼0: With the estimated parameters, the predicted value of a new subject’s phenotype is

^

yf;p¼m^f;pþ

X

ðm;lÞ h

^

g_wF_ðf;pÞ;ðm;lÞþg^sIðf ¼mÞ

i

Ym_;l2m^m;l

þX

K

k ^

g_kX ðm;lÞ

sk_{ðf;pÞ;ðm;lÞ}

Ym_;l2m^m;l

:

(2)

It is worth noting that when the new subject is not from the families in the training data, ^gwFðf;pÞ;ðm;lÞþg^sIðf ¼mÞ ¼0 and its predicted value only relies on the demographic and

genetic predictors (i.e., the family information does not

con-tribute to the prediction). When the new individual comes from the families in the training data set, the GRF method not only uses the demographic and genetic predictors, but also uses the additional information provided by family members to capture the unmeasured genetic and shared environmen-tal predictors, and thus further improves prediction accuracy. While different measurements can be used to evaluate the discriminative ability of the model, we here use the Pearson correlation and the area under the receiver operating

character-istic curve (AUC) to evaluate the model’s performance for

contin-uous and binary outcomes, respectively. TheAUCis calculated as

AUC¼X

ND t

i¼1 XND

t

j¼1 cyib;yjb

N_tDND_t

;

whereND

t andNtDare the total number of cases and controls,

respectively. The kernel functioncis defined as

cðyi;yjÞ ¼

8 < :

1 if yi.yj

0:5 if yi¼yj

0 if yi,yj

:

Data availability

The R codes for the algorithm can be found athttps://github.

com/YaluWen/FRF.

Results

Simulations

Simulation studies were conducted to evaluate the perfor-mance of the new method by considering additional informa-tion from family members and predictors that are homogeneous in families but heterogeneous across families. To illustrate, we compared the performance of the proposed method (GRF) with

another randomfield-based risk-prediction method (RF) in which

family correlation was adjusted for and only genetic information

was used to predict phenotypes (Wen et al.2016). In all the

simulations described below, the genotypes of the founders were drawn from the 1000 Genome Project and the genotypes of off-spring were generated using the gene-dropping method. In par-ticular, we randomly selected a 2-Mb region from the genome

(i.e., chromosome 1: 9,411,243–11,411,242) and randomly

chose 20-kb segments, which have200 single nucleotide

variants (SNVs). The Pearson correlation/AUC of the two

methods were compared based on 1000 replicates under various disease models with different contributions of shared environmental and unmeasured genetic factors.

We further evaluated the methods’ performance when

causal variants have heterogeneous effects on the pheno-types. Finally, we evaluated the effects of pedigree struc-tures on family-based risk prediction and provided practical recommendations for designing a family-based risk-prediction study.

Scenario I:In this set of simulations, we varied the effects of unmeasured shared environmental and genetic risk factors,

and evaluated their impacts on methods’performance. We

considered two pedigree structures, two generations with two offspring (Supplemental Material, Figure S1a in File S1) and three generations with two offspring per

gen-eration (Figure S1b inFile S1). We used 1092 individuals

from the 1000 Genome Project as the founders. Therefore, they formed 546 families and 273 families for the

two-generation scenario (i.e., two founders per family) and

three-generation scenario (i.e., four founders),

respec-tively. The genotypes for all offspring in each generation were generated using the gene-dropping method. We

sim-ulated two disease-associated genes (i.e., we selected two

nonoverlapped 20-kb segments from the genome), on which 33% of SNVs were causal. The phenotypes were simulated based on causal variants under the following additive models:

wherebk;his the genetic effect for thehth causal variant on

genek(k¼1 or 2),nkis the total number of causal variants

for thekth gene,hiNð0;f

2_Þ

is the shared environmental

effect, andei;j

iid

Nð0;s2_{Þis the random error. For}

quantita-tive phenotypes, the intercept awas set to be zero. For

binary phenotypes, a was adjusted to ensure that the

case/control ratio was1:1:

We setb1;1¼b1;2¼. . .¼b1;n1¼b1¼0:2 for all the

sim-ulations we considered. To mimic the situation where some

(_Yi

;j¼aþPn_h1b1;hg1i;;jhþ

Pn2

hb2;hgi2;;jhþhiþei;j for a quantative outcome

logitpYi;j¼1

¼aþPn1

h b1;hg1i;;jhþ

P

h n2

of the genetic predictors were not measured, we dropped one

of causal genes (i.e., gene 2 with effectsb2;h) from ourfinal

data set to be analyzed. For simplicity and without loss of generality, we set the effects of all the causal variants on the

dropped gene the same (b2;1¼b2;2¼. . .¼b2;n2¼b2). We

varied the effects of the dropped gene (i.e.,b2 varied from

zero to two) and shared environmental factors (i.e.,uranges

from zero to four) to evaluate the effects of unmeasured ge-netic and environmental risk factors on the performance of the method. The proportions of variations explained by mea-sured genetic factors, unmeamea-sured genetic factors, and shared environmental factors are summarized in Table S1 inFile S1. We randomly chose two-thirds of the data from each family to serve as the training data set and used the remaining data as the testing data set. The performance of the two models was assessed based on the testing samples

using the Pearson correlation andAUCs for quantitative and

binary outcomes, respectively.

The results of scenario I are summarized in Table 1. Given a

fixed level of unmeasured genetic effects, the performance of

GRF increased with the increase of the variation of the shared environmental effects, whereas the performance of RF de-creased. This indicates that when shared environmental fac-tors contribute to the disease risk, considering correlation between family members can increase the prediction accu-racy. Because correlations between family members are ad-justed, the RF model cannot account for the risk due to shared environmental factors, and this thus reduces the

pre-diction accuracy. Similarly, given afixed level of shared

envi-ronmental effects, the performance of GRF increased with the increase of unmeasured genetic effects, whereas the

perfor-mance of RF decreased. Intuitively, an individual from a fam-ily with many affected individuals tends to have a higher chance of getting the disease, and this could be either due to the shared environmental and/or shared genetic attributes.

The kinship coefficient measures the genetic relationship

be-tween two individuals, and thus could be used as a surrogate for unmeasured genetic risk factors to improve the accuracy of the prediction. As shown in Table 1, for diseases with polygenic and shared environmental components of risk, the genotypes and phenotypes of family members can be informative for an individual’s disease risk. Ignoring such information and predict-ing an individual’s disease risk solely based on his/her own genotypes may be subject to low prediction accuracy.

Scenario II:Converging evidence suggests the inherited pre-disposition to most cancers and spectrum disorders are

het-erogeneous in nature because biological pathways can be modulated by various cellular components, and any nonsy-nonymous mutations could ultimately lead to disease

devel-opment (McClellan and King 2010; Warde-Farleyet al.2012;

Wen and Lu 2013, 2016). However, most existing risk-prediction methods assume that the disease under study has homogeneous genetic/environmental causes, which can attenuate the predictive accuracy if genetic etiology

of a disease is heterogeneous (Warde-Farleyet al.2012;

Wen and Lu 2013). Compared with population-based stud-ies, family-based genetic studies offer an advantage of being robust against genetic heterogeneity. Individuals within the same family tend to share similar environmental risk factors and be more genetically related, and thus the affected individ-uals within the same family tend to have the same underlying causes. Therefore, a model built on within-family informa-tion can be formed to capture predictors that are homoge-neous in families, and provides robustness against genetic heterogeneity.

In this set of simulations, we evaluated the performance of the proposed method when the phenotypes were affected by different sets of genetic variants across families. We started with the case where the disease was caused by the same set of

genetic variants for all individuals (i.e., homogeneous

causes). We then gradually increased the number of hetero-geneous groups, with each group having its own underlying genetic causes. For simplicity, we did not consider the effects of shared environment, and the disease models were simu-lated according to Equation 4, which involves two disease-associated genes with 33% of SNVs on each gene being causal:

where bi

1;h is the genetic effect of a causal variant on gene

1 for familyi, which could vary across different families. To

be specific, let H represent the number of heterogeneous

groups, Mrepresent the total number of families, and the

function½xrepresent the integer part ofx. Thebi1;h was set

according to the following criterion:

bi

1;h¼

2; if   i21

M=H

* n1

H

,h#

i21

M=H

þ1

!

* n1

H

0; otherwise

:

8 > > < > > :

For example, when two heterogeneous underlying causes were considered, the simulated families were split into two

groups, with thefirstn1=2 variants on gene 1 being causal to

thefirst family group, and the remaining variants on gene

(

Yi_;j¼aþ

Pn1 hb

i

1;hg

1;h i;j þ

Pn2

h b2;hgi2;;jhþei;j for a quantative outcome logitpYi_;j¼1

¼aþPn1

hb i

1;hg

1;h i;j þ

Pn2

hb2;hg2i;;jh for a binary outcome

1 being causal to the second family group. To mimic the situation where some of the causal variants were not

mea-sured, we dropped the second gene (i.e., gene 2 with effects

b2;h¼0:01) from the data set to be analyzed. Similarly to

scenario I, we also considered two pedigree structures

(Fig-ure S1, a and b, inFile S1). Individuals from the 1000

Ge-nome Project served as the founders and the genotypes of the offspring were generated based on the gene-dropping permu-tation. We randomly chose two-thirds of the data from each family to train the model and used the remaining data to test the performance of the GRF and RF methods. Pearson

corre-lations andAUCs were calculated for continuous and binary

outcomes, respectively.

The results of scenario II are summarized in Table 2. As expected, the performance of both methods decreased as the number of heterogeneous groups increased. However, the prediction accuracy of the GRF method was much more ro-bust against the genetic heterogeneity than the RF method. For example, for continuous outcomes with the three-generation family structure, the Pearson correlation of GRF dropped from 0.7650 to 0.5932, whereas that of RF dropped from 0.7396 to 0.0896. This trend was preserved regardless of pedigree struc-tures and types of phenotypes. In the presence of genetic

het-erogeneity, the effects of risk predictors associated with a specific

group of individuals are attenuated in the entire sample and thus results in a low accuracy model. GRF uses within-family

information to capture genetic variants that are homogeneous in subgroups, and thus it can partially capture the residual effects

of the risk factors contributing specificity to individuals within a

family. This makes the GRF method more robust against genetic heterogeneity.

Scenario III: In this set of simulations, we evaluated the effects of pedigree structures on the performance of the method and provided practical recommendations for

design-ing a family-based risk-prediction study. Wefirst considered

the situation where the total number of individuals isfixed,

and then considered the case where the total number of families is predetermined. For all the above settings, we

considered the sibling design (Figure S1c in File S1), the

two-generation pedigree design (Figure S1d in File S1),

and the three-generation pedigree design (Figure S1b inFile

S1). The details of the settings are summarized in Table S2 in

File S1. Genotypes of the founders were drawn from the 1000 Genome Project, and the genotypes of the offspring were generated using the gene-dropping method. The phe-notypes were simulated using Equation 3. We considered

three situations: (1) both measured (b1;h) and unmeasured

(b2;h) genetic causes contributed to disease risk (i.e.,

b1;h¼0:3;b2;h¼0:4;hi¼0); (2) only measured genetic

and shared environmental factors affected the outcomes

[i.e.,b1;h¼0:3;hiNð0;1Þ;b2;h¼0]; and (3) both measured

Table 1 The accuracy performance of GRF and RF with different unmeasured genetic and shared environmental effects for two-generation and three-two-generation families

f2a

b₂¼0b b2¼0:4 b2¼0:8 b2¼1:2 b2¼2

GRF RF GRF RF GRF RF GRF RF GRF RF

Continuous outcomes with the two-generation pedigreec

0 0.4716 0.5711 0.6336 0.1805 0.6449 0.0840 0.6482 0.0440 0.6496 0.0125

1 0.6555 0.4715 0.6487 0.1661 0.6495 0.0781 0.6498 0.0409 0.6499 0.0133

2 0.7426 0.4128 0.6648 0.1678 0.6550 0.0803 0.6526 0.0445 0.6512 0.0105

4 0.8272 0.3419 0.6896 0.1523 0.6635 0.0742 0.6567 0.0411 0.6521 0.0143

16 0.9408 0.2021 0.7753 0.1227 0.7062 0.0668 0.6803 0.0376 0.6626 0.0109

Binary outcomes with the two-generation pedigreed

0 0.6850 0.6900 0.7624 0.5790 0.7866 0.5492 0.7943 0.5378 0.7979 0.5297

1 0.7079 0.6673 0.7717 0.5768 0.7920 0.5484 0.7980 0.5388 0.8018 0.5303

2 0.7394 0.6522 0.7776 0.5757 0.7924 0.5480 0.7984 0.5380 0.8012 0.5300

4 0.7848 0.6322 0.7912 0.5724 0.7984 0.5481 0.8005 0.5378 0.8029 0.5302

16 0.8818 0.5859 0.8331 0.5614 0.8155 0.5444 0.8098 0.5367 0.8064 0.5299

Continuous outcomes with the three-generation pedigreec

0 0.5293 0.5544 0.6903 0.1825 0.7045 0.0915 0.7089 0.0538 0.7114 0.0248

1 0.7053 0.4553 0.7017 0.1710 0.7072 0.0882 0.7094 0.0544 0.7105 0.0265

2 0.7864 0.3967 0.7163 0.1708 0.7131 0.0880 0.7129 0.0537 0.7128 0.0251

4 0.8606 0.3268 0.7349 0.1567 0.7179 0.0848 0.7139 0.0536 0.7116 0.0261

16 0.9538 0.1920 0.8071 0.1283 0.7524 0.0775 0.7333 0.0500 0.7209 0.0246

Binary outcomes with the three-generation pedigreed

0 0.6820 0.6838 0.7933 0.5775 0.8249 0.5503 0.8340 0.5415 0.8397 0.5342

1 0.7309 0.6620 0.8031 0.5774 0.8270 0.5516 0.8352 0.5418 0.8402 0.5349

2 0.7739 0.6470 0.8108 0.5739 0.8298 0.5496 0.8362 0.5416 0.8409 0.5342

4 0.8264 0.6274 0.8245 0.5718 0.8336 0.5505 0.8384 0.5416 0.8412 0.5345

16 0.9203 0.5819 0.8701 0.5603 0.8546 0.5459 0.849 0.5392 0.8458 0.5341

a_{The effects of shared environmental factors.} b

The effects of causal variants from the dropped gene.

and unmeasured genetic causes, as well as shared

envi-ronmental factors contributed to disease risk [i.e.,

b_1;h¼0:3;h_iNð0;1Þ;b2;h¼0:4]. We randomly chose

two-thirds of the data from each family to serve as the train-ing data set and used the remaintrain-ing data as the testtrain-ing data set. The performance of the GRF model was assessed based on the testing samples using the Pearson correlations and

AUCs for continuous and binary outcomes, respectively.

The results of scenario III are summarized in Table 3. With

the total sample size fixed, the sibling design achieved the

highest prediction accuracy while the three-generation de-sign had the lowest accuracy when unmeasured genetic fac-tors contributed to disease risk. This could be explained by the fact that kinship correlation between adjacent genera-tions tended to be larger as compared to that for generagenera-tions that are far apart. Similarly, the average kinship correlation in the sibling design is larger than that in the two-generation designs with unrelated founders. When only shared environ-mental factors contributed to disease risk, the three designs tended to perform similarly, among which the three-generation design attained the highest accuracy. This is mainly due to the fact that correlation between family members is treated the same as when only shared environmental factors contributed

to disease risk. Given thefixed sample size, larger pedigrees

have more related individuals, and thus contribute more infor-mation to disease-risk prediction.

When the total number of families wasfixed, the

conclu-sions were similar to those when the total sample sizes were

fixed. One exception is that the two-generation design tended

to perform worse than the three-generation design when unmeasured genetic factors contributed to disease risk. With

afixed number of families, a three-generation design has more

samples than a two-generation design, and thus attained better performance. It is worth noting that the sibling design tends to achieve a higher prediction accuracy than the three-generation design when unmeasured genetic factors contrib-uted to disease risk, even when the total sample size in sibling design is smaller than that in the three-generation design. This is mainly due to the fact that, on average, the genetic corre-lation between siblings within a family is larger than that in a three-generation design. Therefore, the sibling design is more informative to capture unmeasured genetic factors.

In summary, when a substantial amount of disease risk is due to unmeasured genetic factors, a study design that aims at recruiting individuals that are highly genetically related can help to build an accurate risk-prediction model. On the other hand, when shared environmental factors play key roles in the disease development, the risk-prediction model built from a study with large pedigrees tends to have higher prediction accuracy.

Real data application

We applied the GRF method to whole-genome exome data from the Twin Study of Behavioral and Emotional Develop-ment in Children (TBED-C), a study within the MSUTR (Burt and Klump 2012). The TBED-C is a population-based twin

study investigating genetic–environmental interactions that

contribute to conduct problems in children. A total of 1000 twins aged between 6 and 10 years old (500 twin fam-ilies and 50% monozygous twins) in Michigan were recruited. DNA samples were collected from each pair of twins, and they were genotyped using the Illumina Human Core Exome chip that includes common variants, rare vari-ants, mitochondrial DNA, and indels. We applied relatively stringent quality-control criteria, where samples with call

rate ,97% or missing rate larger than 3% were excluded.

The exclusion criteria for SNVs were (1) call rate,98%, (2) a

P-value for Hardy–Weinberg equilibrium test of,1025_;and

(3) a missing rate .2%. After the quality assessment,

957 samples and 513,886 SNVs remained for the analysis. Parents completed the child behavior checklist (CBCL) separately for each twin, and they rated the extent to which a series of statements describing their children’s behavior (three point scale with 0 = never and 2 = mostly true) (Achenbach and Rescorla 2003; Burt and Klump 2012). Teacher(s) of each twin also completed the teacher report form, whose scales were fully comparable to those on the

CBCL. The well-known aggressive scales (e.g.,“bullies,” “gets

infights,” “attacks people”; 18 items) were used (Burt and

Klump 2012). Children’s aggressive behavior was summarized based on the recommended approach that the data were

aver-aged across parental informant and the teachers’reports and

the analyses were conducted on the raw scale scores

(Achenbachet al.1987; Achenbach and Rescorla 2003).

Table 2 Accuracy performance of GRF and RF with different underlying heterogeneous causes for two-generation and three-generation families

Heterogeneous groups

Two-generation pedigree Three-generation pedigree

Continuousa Binaryb Continuousa Binaryb

GRF RF GRF RF GRF RF GRF RF

1 0.7776 0.7577 0.8262 0.8328 0.7650 0.7396 0.8186 0.8179

2 0.7587 0.4412 0.8218 0.6747 0.7379 0.4315 0.8189 0.6662

4 0.7503 0.2877 0.7825 0.6097 0.7181 0.2839 0.7852 0.6091

8 0.6892 0.1304 0.7221 0.5665 0.6823 0.1898 0.7182 0.5635

16 0.6051 20.0049 0.6364 0.5350 0.5932 0.0896 0.6426 0.5345

Before the risk-prediction analysis, wefirst conducted the Fisher’s exact test on each of the common variants (minor

allele frequencies .5%) and only included those variants

with a P-value less than a relatively lenient prespecified

threshold (i.e.,P= 0.1) for the risk-prediction analysis. The

selected markers that were within 50,000 bp were grouped into one group. We applied both the GRF and RF methods to the data. To account for the contribution of rare variants, we

specified the weight for each variant asvkh¼ ½1=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

fk hð12fhkÞ

q

;

wherefk

h is the minor allele frequency for thehth marker on

thekth gene. The Pearson correlation for the risk-prediction

models built by GRF and RF are 0.594 and 0.186,

respec-tively. To avoid overfitting and the chancefinding problems,

we randomly split the data into training and testing sets with 100 individuals in the testing set. We repeated this process 500 times. For each replicate, we used the training data to train the model and calculated the Pearson correlations for both the training data and the testing data. The mean Pear-son correlations in the training data sets for GRF and RF are 0.582 and 0.212, respectively. The mean Pearson correlations in the testing data sets for GRF and RF are 0.485 and 0.067,

respectively. The RF method had significantly lower accuracy

than the GRF method (Figure 1), which indicates that incor-porating family information into the risk-prediction model can potentially improve the model’s performance.

Discussion

The recent successes made through large-scale association studies have stimulated interest in the genomic prediction of disease risk, which may potentially enable risk prediction at the individual level and ultimately facilitate precision medi-cine. Although risk-prediction methods use similar genetic data to those from association studies, their main goal is different. While association studies focus on detecting re-producible disease-associated signals, the main aim of risk-prediction research is to maximize the predictive power. Validity and robustness are the two key concerns for a pre-diction model, and a causal interpretation is not absolutely necessary for a good predictive model (Abraham and Inouye

2015). Therefore, family-based genetic studies are appealing for risk-prediction research, because family information can be used to form surrogates to capture genetic heterogeneity as well as unmeasured polygenic and/or shared environmental factors so that prediction accuracy can be further improved.

In this article, we proposed a GRF method within the

random field framework for risk prediction research using

high-dimensional genetic data from family-based studies, where the correlations between family members have been explicitly modeled and incorporated into the prediction model to improve the model’s performance. Through simulation studies and an application to real data, we demonstrated that familial correlation can serve as a surrogate for unmeasured genetic and environmental risk factors, and can be used to improve prediction accuracy. Indeed, it has long been recog-nized that family history alone can help identify people at high risk. Family information remains an important factor in the prediction of many complex diseases with a substantial amount of heritability due to a combination of genetic fac-tors, shared familial environmental conditions, and lifestyle choices. Despite its clinical importance, few methods fully use this information when building a risk-prediction model from

high-dimensional genomic data. A common practice is tofirst

estimate the genetic effects with family correlation being ad-justed, and then build the risk-prediction model based on the

estimated genetic effects (Meigset al.2008). Although this

strategy allows researchers to evaluate the role of disease-associated variants in family-based risk-prediction studies, ignoring family information can lead to a less accurate model if there are unmeasured genetic and/or shared environmen-tal factors. There is a connection between GRF and the com-mon practice. If an individual is not related to anyone in the training data, then the prediction is solely determined by his/ her genotypes, which is exactly the same as the common practice. However, for individuals that are from the families in the training set, their predictions are determined not only

based on their own genotypes, but also risk profiles from their

family members. By using extra information from the family, the model’s performance can be further improved.

Converging evidence suggests that diseases with the same or similar clinical manifestations may have different Table 3 Accuracy performance of GRF given different pedigree structures and underlying causes

Designs

Disease model 1a Disease model 2b Disease model 3c

Continuous Binary Continuous Binary Continuous Binary

Total number of individuals arefixed

Sibling 0.8866 0.8660 0.7421 0.7412 0.8946 0.8709

Two generation 0.7327 0.8098 0.7422 0.7573 0.7438 0.8159

Three generation 0.7020 0.7982 0.7480 0.7652 0.7123 0.8066

Total number of families arefixed

Sibling 0.8661 0.8363 0.7171 0.7243 0.8821 0.8456

Two generation 0.6511 0.7639 0.7231 0.7471 0.6637 0.7736

Three generation 0.7024 0.7980 0.7480 0.7652 0.7123 0.8066

a_{Both measured and unmeasured genetic causes contributed to disease risk.}

b_{Measured genetic factors and shared environmental factors contributed to disease risk.}

underlying causes (McClellan and King 2010; Morris et al.

2010). Treating the disease under study as one unified

phe-notype with homogeneous genetic and environmental causes will likely attenuate the predictive power and results in a low-accuracy prediction model if heterogeneity presents. Com-pared with population-based studies, family-based studies offer a unique opportunity for considering genetic heteroge-neity. Family members tend to share similar environmental risk factors and they are genetically more related than the general population. Therefore, the affected individuals in the family tend to have more homogeneous underlying causes. Indeed, the idea of using the family to model genetic hetero-geneity can be traced back to the era of linkage analyses. In our proposed GRF method, we used the within-family simi-larities to capture the potential genetic heterogeneity. Through simulation studies, we demonstrated that such a strategy can substantially improve the model’s accuracy.

Random field models have long been used for mining

massive geospatial and imaging data (Cressie and Johannes-son 2008). Spatial autocorrelation is a common phenomenon in these data because observations from nearby locations tend to be more similar than would be expected on a random basis. Motivated by this idea, in our proposed GRF, we view

phenotypes of subjects as a randomfield in a space spanned

by their genotypes and family information, and thus the out-come of a new subject can be predicted by adjacent subjects in this space, where adjacencies between subjects are deter-mined by their genetic and family relatedness. As evidenced by association studies, rare variants can play an important role in the underlying mechanism of human diseases, and thus they hold great promise in improving prediction accu-racy. In our proposed method, we captured the effects of rare

variants by the weight function that is defined according

BLUP-type methods, in our proposed GRF we allowed different genes to have different directions and magnitudes of effect

sizes (gk). In this article, we also provide practical

recommen-dations for designing a family-based risk-prediction study. When a substantial amount of disease risk is due to unmea-sured genetic factors, the study that aims at recruiting more genetically correlated individuals tends to have greater pre-dictive power. On the other hand, when shared environmen-tal factors play key roles in the disease etiology, recruiting a large pedigree could lead to increased accuracy.

In the real data application, we built risk-prediction models for aggressive behavior using whole-genome exome data from TBED-C. On average, the Pearson correlations for GRF and RF are 0.485 and 0.067, respectively. This sug-gests that incorporating family information into the risk-prediction model can substantially improve risk-prediction

accuracy. The formed risk-prediction model is one of thefirst

prediction models for aggressive behavior, which explains

24% of the variations in the children’s aggressive behavior.

Further studies from independent samples are also needed to fully test the validity and robustness of our proposed model.

Acknowledgments

The authors thank all participating twins and their families. The Project was supported by the National Natural Science

Foundation of China (award no. 81502887); the Scientific

Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry; the National Institute of Dental and Craniofacial Research under award number R03 DE-022379; and the National Institute on Drug Abuse under award number K01 DA-033346. The Twin Study of Behav-ioral and Emotional Development in Children data collec-tion was supported by R01 MH-081813 from the Nacollec-tional Institute of Mental Health. The content is solely the re-sponsibility of the authors and does not necessarily

repre-sent the official views of the National Institute of Mental

Health or the National Institutes of Health.

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