SNP-SNP Interactions in Case-Parent Trios

Transcription

1 Detection of SNP-SNP Interactions in Case-Parent Trios Department of Biostatistics Johns Hopkins Bloomberg School of Public Health June 2, 2009 Karyotypes

2 Single Nucleotide Polymphisms urgi.versailles.inra.fr Haplotype Blocks Kim Dionne (2007)

3 SNP-SNP Interactions Many statistical approaches have been used in the literature f the detection of SNP-SNP interactions: Regression with Higher Order Interactions Logic (Boolean) Regression Monte Carlo Logic Regression Multifact Dimensionality Reduction (MDR) Classification Regression Trees (CART) Rom Fests Boosting Multivariate Adaptive Regression Splines (MARS) Neural Netwks References: Heidema AG et al (2006). The Challenge f Genetic Epidemiologists: How to Analyze... BMC Genetics 7: 23. McKinney BA et al (2006). Machine Learning f Detecting Gene-Gene Interactions... Appl Bioinf 5(2): Musani SK (2007). Detection of Gene x Gene Interactions in... Hum Hered 63(2): Biological Statistical Interactions BB Bb bb AA Aa aa (SNPA D SNPB R ) (SNPB D SNPA R )

4 Biological Statistical Interactions BB Bb bb AA Aa aa (SNPA D SNPB R ) (SNPB D SNPA R ) Statistical interaction: Deviation from additivity in a linear statistical model. Epistasis: Masking of phenotype expressed by one gene by the effects of another gene. Reference: Moe JH (2005). A Global View of Epistasis. Nature Genetics, 37: Introduction Current methods f analyzing complex traits include analyzing localizing disease loci one at a time. However, complex traits can be caused by the interaction of many loci, each with varying effect....patterns of interactions between several loci, f example, disease phenotype caused by locus A locus B, A but not B, A (B C), clearly make identification of the involved loci me difficult. While the simultaneous analysis of every single two-way pair of markers can be feasible, it becomes overwhelmingly computationally burdensome to analyze all 3-way, 4-way to N-way patterns, patterns, combinations of loci. Reference: Lucek PR, Ott J (1997). Neural Netwk Analysis of Complex Traits. Genet Epi 14(6):

5 Introduction Assume we have a cidate gene study with 1000 cases of a certain disease, 1000 controls. We typed 100 SNPs of interest, we were lucky - the two causal SNPs were typed. Also assume that a double penetrance model describes the relationship between SNPs phenotype: subjects with two variant alleles of either SNP 24 SNP 80 at least one variant allele of the other have higher odds of disease: logit(p) =α + β Ind{(SNP24 D SNP80 R ) (SNP24 R SNP80 D )} How could we detect this signal? Introduction !2!4!6 SNP1 SNP2 SNP3 SNP4 SNP5 SNP6 SNP7 SNP8 SNP9 SNP10 SNP11 SNP12 SNP13 SNP14 SNP15 SNP16 SNP17 SNP18 SNP19 SNP20 SNP21 SNP22 SNP23 SNP24 SNP25 SNP26 SNP27 SNP28 SNP29 SNP30 SNP31 SNP32 SNP33 SNP34 SNP35 SNP36 SNP37 SNP38 SNP39 SNP40 SNP41 SNP42 SNP43 SNP44 SNP45 SNP46 SNP47 SNP48 SNP49 SNP50 SNP51 SNP52 SNP53 SNP54 SNP55 SNP56 SNP57 SNP58 SNP59 SNP60 SNP61 SNP62 SNP63 SNP64 SNP65 SNP66 SNP67 SNP68 SNP69 SNP70 SNP71 SNP72 SNP73 SNP74 SNP75 SNP76 SNP77 SNP78 SNP79 SNP80 SNP81 SNP82 SNP83 SNP84 SNP85 SNP86 SNP87 SNP88 SNP89 SNP90 SNP91 SNP92 SNP93 SNP94 SNP95 SNP96 SNP97 SNP98 SNP99 SNP100

6 Introduction !2!4!6 SNP1 SNP2 SNP3 SNP4 SNP5 SNP6 SNP7 SNP8 SNP9 SNP10 SNP11 SNP12 SNP13 SNP14 SNP15 SNP16 SNP17 SNP18 SNP19 SNP20 SNP21 SNP22 SNP23 SNP24 SNP25 SNP26 SNP27 SNP28 SNP29 SNP30 SNP31 SNP32 SNP33 SNP34 SNP35 SNP36 SNP37 SNP38 SNP39 SNP40 SNP41 SNP42 SNP43 SNP44 SNP45 SNP46 SNP47 SNP48 SNP49 SNP50 SNP51 SNP52 SNP53 SNP54 SNP55 SNP56 SNP57 SNP58 SNP59 SNP60 SNP61 SNP62 SNP63 SNP64 SNP65 SNP66 SNP67 SNP68 SNP69 SNP70 SNP71 SNP72 SNP73 SNP74 SNP75 SNP76 SNP77 SNP78 SNP79 SNP80 SNP81 SNP82 SNP83 SNP84 SNP85 SNP86 SNP87 SNP88 SNP89 SNP90 SNP91 SNP92 SNP93 SNP94 SNP95 SNP96 SNP97 SNP98 SNP99 SNP100 Logic Regression The predicts are the SNPs in dominant recessive coding. SNP X X.R X.D AA 0 0 AT 0 1 TT 1 1 Let X 1,...,X k be binary (0/1) predicts, Y a response variable. Fit a model g(e(y)) = b 0 + t b j L j, j=1 where L j is a Boolean combination of the covariates, e.g. L j = (X 1 X 2 ) X c 4. Determine the logic terms L j estimate the b j simultaneously. Reference: Ruczinski et al (2003). Logic Regression. J of Comp Graph Stat, 12(3):

7 Logic Regression Alternate Leaf Alternate Operat Grow Branch Possible Moves Initial Tree Prune Branch Split Leaf Delete Leaf Logic Regression We try to fit the model g(e(y)) = b 0 + t j=1 b j L j. Select a scing function (RSS, log-likelihood,...). Pick the maximum number of Logic Trees. Pick the maximum number of leaves in a tree. Initialize the model with L j = 0 f all j. Carry out the Simulated Annealing Algithm: Propose a move. Accept reject the move depending on the sces the temperature.

8 Logic Regression Logic Regression SNP80.R

9 Logic Regression SNP80.R SNP24.R SNP80.R Logic Regression SNP80.R SNP24.R SNP80.R SNP80.R SNP24.R SNP80.D

10 Logic Regression SNP80.R SNP24.R SNP80.R SNP80.R SNP24.R SNP80.D SNP80.R SNP24.R SNP80.D SNP73.R Logic Regression SNP80.R SNP24.R SNP80.R SNP80.R SNP24.R SNP80.D SNP80.R SNP24.R SNP86.R SNP30.R SNP80.D SNP73.R SNP80.R SNP24.R SNP80.D

11 Logic Regression We implemented two flavs f the required model selection. Both approaches require a definition of model size. 1 Cross-validation This is most applicable when prediction is the main objective, i. e. not necessarily the best option in SNP association studies. 2 Permutation tests This is a test f association, i. e. the preferred test in SNP association studies. The model size is chosen via a sequence of hypothesis tests. Schizophrenia Diagnosis classification of Schizophrenia (SZ) depends on the observation of disease related symptoms: delusion, hallucination, bizarre behavi, disganized speech,... About 1% prevalence wldwide, equal in males females. Schizoaffective disder (SZA) is closely related to SZ. Over 2000 papers on linkage association studies. About 150 genes implicated, most on chromosomes 1, 6, 22. Most findings are somewhat inconclusive, though. Most studies are single marker analysis only, very few address SNP-SNP interactions.

12 Schizophrenia Study Case-parent trios of Ashkenazi Jewish descents. Diagnosis of SZ SZA based on DSM IV. Dense coverage of 64 cidate genes. 375 SNPs on 11 chromosomes genotyped f 312 trios. Original analysis through single marker TDT. Goal: Exple SNP-SNP interactions f association with SZ SZA. Reference: Fallin et al (2005). Bipolar I disder schizophrenia... Am J Hum Genet 77(6): TDT - Allelic The transmission disequilibrium test measures the over-transmission of an allele from parents to affected offsprings. F a set of n parents with alleles 1 2 at a genetic locus, each parent can be summarized by the transmitted the non-transmitted allele: Non-TA P a b a + b TA 2 c d c + d P a + c b + d 2n Under the null of no association, Only the heterozygous parents contribute infmation! (b c)2 b + c χ 2 1

13 TDT - Genotypic Assume that at a certain locus the father has alleles 11 the mother has alleles 12. The four Mendelian children thus have alleles 11, 12, 11, 12. Assume the affected prob has genotype 11. The three Pseudo controls then have the genotypes 11, 12, 12. Y X Affected prob 1 11 Pseudo control # Pseudo control # Pseudo control # We can use coonditional logistic regression to analyze the data! Outline 1 We extended the logic regression methodology to analyze trios with affected probs. 2 We developed an R package to a) impute missing genotypes, using a haplotype-based approach that employs mating tables, to b) simulate case-parent trios with disease risk dependent on SNP-SNP interactions. 3 We carried out simulation studies to verify the validity of the methodology the software. 4 We analyzed the data from the study of Schizophrenia among Ashkenazi Jewish families.

14 Trio Logic Regression The rough idea is as follows: Pseudo controls are generated from the trio data, taking the LD block structure into account. Missing data are hled using haplotype-based imputation. The conditional logistic likelihood is used in logic regression to assess differences in cases pseudo controls. Trio Logic Regression

15 Trio Logic Regression The steps in me detail: 1 Estimate the haplotype blocks the haplotype frequencies using the parents genotypes. 2 F each block each trio, sample haplotype pairs f the parents the offspring consistent with the observed genotypes in the trio, allowing f missing data. 3 Generate the probs genotype data from the haplotypes that were passed from the parents. 4 F each block each trio, generate genotypes f three pseudo-controls (PC1, PC2, PC3) using the parents haplotypes that were not passed to the prob. The assignment to PC1, PC2, PC3 is rom. 5 Assemble three pseudo-controls f each trio by augmenting the genotypes from the blocks. 6 F each locus, translate the genotype data into two binary variables in dominant recessive coding. Trio Logic Regression The conditional logistic likelihood is L(β) = ( ) N exp(x i0 β) i=1 exp(x i0 β)+σ 3 m=1 exp(x imβ) In this setting, i refers to a trio, X i0 refers to the exposure of the prob in trio i, (X i1, X i2, X i3 ) are the exposures of the 3 pseudo-controls in trio i. Does the likelihood always have a maximum?

16 Trio Logic Regression Number of trios where k pseudo-controls match the prob k X i0 = 0 a 0 b 0 c 0 d 0 X i0 = 1 a 1 b 1 c 1 d 1 Likelihood contributions f X i0 = 0 a 0 b 0 c 0 d 0 X i X i X i X i L i (β) exp(β) exp(β) 1 3+exp(β) Likelihood contributions f X i0 = 1 a 1 b 1 c 1 d 1 X i X i X i X i L i (β) exp(β) exp(β)+3 exp(β) 2 exp(β)+2 exp(β) 3 exp(β) Trio Logic Regression The log-likelihood is: log(l(β)) = a 1 {β log(exp(β)+3)} + b 1 {β log(2 exp(β)+2)} + c 1 {β log(3 exp(β)+1)} + d 1 4 a 0 log{1 + 3 exp(β)} b 0 log{2 + 2 exp(β)} c 0 log{3 + exp(β)} d 0 4

17 Trio Logic Regression The first derivative of the log-likelihood is: log(l(β)) β = a 1 (a 1 + c 0 ) exp(β) exp(β)+3 + b 1 (b 1 + b 0 ) c 1 (c 1 + a 0 ) 2 exp(β) 2 exp(β) exp(β) 3 exp(β)+1 The log-likelihood is monotonically decreasing in β. Further: lim β log(l(β)) β = a 1 + b 1 + c 1 0 lim β + log(l(β)) β = (a 0 + b 0 + c 0 ) 0 Trio Logic Regression Since the derivative of the log likelihood function is a continuous function in β, it follows that the likelihood function has a maximum unless a 1 + b 1 + c 1 = 0 a 0 + b 0 + c 0 = 0. In other wds, f either the exposure the non-exposure groups, all pseudo-controls are equal to the respective probs. This might occur in a particular sample where the number of trios is small the ( some of the) SNPs contributing to the exposure definition have extremely low min allele frequency, however, it is not plausible that such a separation would exist in the population in truth.

18 Missing Data Simulation G 1 SNP R SNP R 4 2 SNPR (SNP R 1 1 SNPD 6 2 ) SNPR (SNP R 2 3 SNPD 5 2 ) SNPR (SNP D 4 1 SNPR 8 2 ) (SNPR 5 1 SNPD 6 1 ) 6 ((SNP R 4 2 SNPR 7 2 ) SNPR 8 3 ) (SNPR 9 2 SNPR 6 1 ) 7 (SNP R 3 11 SNPR 12 2 ) (SNPR 5 4 SNPR 15 1 ) (SNPR 9 2 SNPR 8 3 ) P(G) Haplotypes / block # of haplotypes # mating table rows , , 3, , 5, , 5, 5, , 8, 5, 3, 3 2, , 4, 5, 5, 3, 5 4,

19 Simulation L1/11 L2/22 L1/11 Haplotype Frequencies population parents prob Haplotype Frequencies Simulation When using conditional logistic regression to compare cases pseudo-controls, the expected value of the parameter estimates is not the logs odds ratio β, but the log relative risk (Schaid 1996). RR = P(D I G = 1) P(D I G = 0) = exp(α + β)/(1 + exp(α + β)) exp(α)/(1 + exp(α)) = exp(β) ( ) exp(α + β) 1 + exp(α) Therefe log(rr) = β log ( ) 1 + exp(α + β) 1 + exp(α)

20 Simulation log(rr) =β log! 1 + exp(α + β) 1 + exp(α) 4 3 2!^ 1 0 "!!5!3!1!5!3!1!5!3!1!5!3!1!5!3! Simulation log(rr) =β log! 1 + exp(α + β) 1 + exp(α) 4 3 2!^ 1 0 "!!5!3!1!5!3!1!5!3!1!5!3!1!5!3!

21 Software 1 The trio logic regression methods are implemented as an augmentation in the logic regression R package LogicReg. 2 The R package trio contains functions to generate logic regression input from pedigree genotype files, to check f Mendelian errs, to impute missing data, to simulate case-parent trios. 3 A software vignette is also available. Results Logic model exp( ˆβ) I n o 302 D I n o 302 D 166 D I n o 302 D 166 D 148 D I n o 302 D 166 D 148 D 368 R 3.65 SNP 302 Chromosome 12 NOS SNP 166 Chromosome 8 CHRNB SNP 148 Chromosome 8 PNOC SNP 368 Chromosome 22 COMT

22 Results Logic model exp( ˆβ) I n o 302 D I n o 302 D 166 D I n o 302 D 166 D 148 D I n o 302 D 166 D 148 D 368 R 3.65 SNP 302 Chromosome 12 NOS SNP 166 Chromosome 8 CHRNB SNP 148 Chromosome 8 PNOC SNP 368 Chromosome 22 COMT Results exp( ˆβ) ˆβ (se) z p Marginal 302 D (0.16) e D (0.22) Logic 302 D 166 D (0.18) e-06 Additive 302 D (0.16) e D (0.22) D (0.19) e-06 Additive 166 D (0.33) D : 166 D (0.34)

23 Results exp( ˆβ) ˆβ (se) z p Marginal 302 D (0.16) e D (0.22) Logic 302 D 166 D (0.18) e-06 Additive 302 D (0.16) e D (0.22) D (0.19) e-06 Additive 166 D (0.33) D : 166 D (0.34) Results exp( ˆβ) ˆβ (se) z p Marginal 302 D (0.16) e D (0.22) Logic 302 D 166 D (0.18) e-06 Additive 302 D (0.16) e D (0.22) D (0.19) e-06 Additive 166 D (0.33) D : 166 D (0.34)

24 Results 0.67 I n 302 Do I {166D } 302 D 302 D D main effects only Results 0.90 I n 302 Do I {166D } 1.09 In 302 D :166 Do 302 D 302 D D main effects + interaction

25 Results 0.89 I n 302 D 166 Do 302 D 302 D D logic regression Results 302 D 302 D 302 D 302 D 302 D 302 D D 166 D 166 D main effects only main effects + interaction logic regression

26 Results exp( ˆβ) ˆβ(se) z p 302 D (0.16) e-05 Marginal 166 D (0.22) D (0.25) Logic 302 D 166 D 148 D (0.20) e-08 Acknowledgments Methods: Dani Fallin, Qing Li, Tom Louis. Data: Ann Pulver. Computing suppt: Marvin Newhouse, Jiong Yang. Funding: NIH R01 DK061662, GM083084, HL090577, a CTSA grant to the Johns Hopkins Medical Institutions.

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