Missing Data. On Missing Data and Interactions in SNP Association Studies. Missing Data - Approaches. Missing Data - Approaches. Multiple Imputation

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1 October, U of British Columbia Missing Data On Missing Data Interactions in SNP Association Studies 5 SNPs Ingo Ruczinski Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health 5 Subject Missing Data - Approaches Missing Data - Approaches The most common approach f dealing with missing data is to omit the observations that have missing recds in the model s covariates. This approach can have several shtcomings, including: Loss of power. Bias in the parameter estimates. A good reference on this topic is Greenl Finkle (995). Some other used approaches are: To impute a value from the marginal distribution of the covariate. To create an extra level indicating missingness, if the covariate is a fact. These choices tend to be not so great either. Multiple imputation can be used to draw valid statistical inference from data with missing values when the data are missing at rom (Little Rubin 987, Schafer 997). In essence, multiple imputation acknowledges the uncertainty due to missing data, instead of simply igning it: several complete data sets are generated, the uncertainty in the model parameter estimates incpates the stard errs of the parameter estimates as well as the variability between the parameter estimates from the replicate data sets. While the hypothesis of missing at rom cannot fmally be tested, it is a lot less stringent than the requirement of missing completely at rom, which is the underlying assumption made when observations are omitted. References: Greenl S, Finkle WD (995). A Critical Look at Methods f Hling Missing Covariates in Epidemiologic Regression Analyses. American Journal of Epidemiology, (): Little RJ, Rubin DB (987). Statistical Analysis with Missing Data. John Wiley Sons, NewYk. Schafer JL (997). Analysis of Incomplete Multivariate Data. Chapman & Hall. Not Missing at Rom Multiple Imputation parameter estimates From the white paper, updates/brlmm algithm.affx estr Snp7CG Snp7GG SnpTC SnpTT SnpGC SnpGG SnpGT SnpTT SnpAG SnpGG Snp6CT Snp6TT SnpGA SnpGG SnpGA SnpGG SnpAG SnpGG SnpAG SnpGG

2 Multiple Imputation Multiple Imputation parameter estimates estr Snp7CG Snp7GG SnpTC SnpTT SnpGC SnpGG SnpGT SnpTT SnpAG SnpGG Snp6CT Snp6TT SnpGA SnpGG SnpGA SnpGG SnpAG SnpGG SnpAG SnpGG estr Snp7CG Snp7GG confidence intervals SnpTC SnpTT SnpGC SnpGG SnpGT SnpTT SnpAG SnpGG Snp6CT Snp6TT SnpGA SnpGG SnpGA SnpGG SnpAG SnpGG SnpAG SnpGG Example Example Number of Pairs Odds Ratio Confidence Interval Family Histy not complete Family Histy complete AA AC CC na AA AC CC na XPD Lys75Gln iginal data set.9 (.. ) multiple imputations.5 (.. ) raw numbers case control XPD Gln75Gln iginal data set.8 (.8. ) multiple imputations. (.7. ) percents case control Positive Family Histy iginal data set.5 (..5 ) multiple imputations.5 (.58. ) Brewster AM, Jgensen TJ, Ruczinski I, Huang HY, Hoffman S, Thuita L, Newschaffer C, Lunn RM, Bell D, Helzlsouer KJ (6). Polymphisms of the DNA Repair Genes XPD (Lys75Gln) XRCC (Arg99Gln Arg9Trp): Relationship to Breast Cancer Risk Familial Predisposition to Breast Cancer. Breast Cancer Research Treatment, 95(): 7-8. Example Example XPD Lys75Gln 6 5 XPD Gln75Gln hazards ratio Family Histy combined iginal SNP SNP SNP SNP SNP 5 SNP 6 SNP 7 SNP 8 SNP 9 SNP SNP SNP SNP The missing data were imputed using decision trees. In a minute... Unpublished data.

3 Multiple Imputation Tree-based Imputation We looked into two approaches:. Haplotype-based imputation The idea here is to reconstruct the haplotypes (f example via the EM algithm), impute the missing values from the estimated haplotype frequencies.. Tree-based imputation The idea here is to use decision trees to impute the genotype data, browing infmation from neighbing SNPs other variables. F each individual i, let M i =(M i,m i,...,m ip) be the vect of p variables consisting of the covariates X i =(x i,...,x ir) the unphased SNP data G i =(g i,...,g ik) which have missing entries ( p r + k). Let C i be the vect of the remaining covariates unphased SNP data f which all data are available. We assume that the outcome D i is always observed. The joint probability distribution of the missing data f individual i given the observed data, Pr(M i,m i,...,m ip C i, D i), is difficult to get. An obvious problem is that the sets of missing data M i complete data C i, respectively, are different f each individual i. Dai J, Ruczinski I, LeBlanc M, Kooperberg C (6). A Comparison of Haplotype-based Tree-based SNP Imputation in Association Studies. Genetic Epidemiology, (in press). Instead of modeling the joint distribution, we use the Gibbs sampler, a Markov chain Monte Carlo technique that uses conditional (low-dimensional) distributions to draw samples from a high-dimensional distribution. Tree-based Imputation Simulation Specifically, we consider iteratively sampling from the following sequence of the full conditional distributions in the (n +) th iteration: M (n+) Pr(M M (n),m (n),...,m (n) p, C, D) M (n+) Pr(M M (n+),m (n),...,m (n) p, C, D) M (n+) p. Pr(M p M (n+),m (n+),...,m (n+) p where each full conditional distribution is modeled by CART., C, D). mean percent crect.7 percent crect.7.5 trees five sweeps.5 A convenient property of surrogate splits in CART is that we do not have to guess the initial values of the missing data in M. As a result only a very sht burn-in of the above sampler is required. trees one sweep marginal extra fraction missing [%] extra fraction missing [%] Simulation Motivation Mean imputation errs in the simulated data of four SNPs on the PGR gene f four imputation approaches: β = β = % missing data % missing data SNP SNP SNP SNP Approach SNP SNP SNP SNP Naive EM WEM Tree Naive EM 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 -way, -way to N-way patterns, patterns, combinations of loci. WEM Tree β = Naive EM WEM Tree Lucek PR, Ott J (997). Neural netwk analysis of complex traits. Genetic Epidemiology, (6): -6.

4 Double Penetrance Model Logic Regression AA Aa BB Bb bb X,...,X k are / (False/True) predicts. Y is a response variable. Fit a model t g(e(y)) = b + b j L j, j= where L j is a Boolean combination of the covariates, e.g. L j =(X X ) X c. Determine the logic terms L j estimate the b j simultaneously. SNP X X.R X.D aa (SNPaa SNPBB c ) (SNPbb SNPAA c ) SNPs are coded as dominant recessive: AA AT TT Ruczinski I, Kooperberg C, LeBlanc M (). Logic Regression. Journal of Computational Graphical Statistics, (): The Move Set f Logic Regression Simulated Annealing f Logic Regression Alternate Leaf Alternate Operat Grow Branch We try to fit the model g(e(y)) = b + t j= bj L j. Select a scing function (RSS, log-likelihood,...). Possible Moves 5 Pick the maximum number of Logic Trees. Pick the maximum number of leaves in a tree. Initial Tree Prune Branch Split Leaf Delete Leaf Initialize the model with L j = f all j. Carry out the Simulated Annealing Algithm: Propose a move. 6 Accept reject the move, depending on the sces the temperature. Growing Logic Models Model Selection X.7 We implemented two flavs f the required model selection. Both approaches require a definition of model size. X.7 X. X. Cross-validation: X.7 X. This is most applicable when prediction is the main objective, i. e. not in SNP association studies. 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. so on... X. X. X.7 X. X.9 X. X. X. X.7 Ruczinski I, Kooperberg C, LeBlanc M (). Logic Regression. Journal of Computational Graphical Statistics, (): 75-5.

5 Multiple Models : Monte Carlo LR Multiple Models : Metropolis-Hastings Goal: identify all models combinations of covariates that are potentially associated with the outcome. 7 6 Use reversible jumps to implement an MCMC algithm with pris on models model size. 5 The pri on model size does influence the total number of SNPs selected. sce The pri on model size has virtually no influence on the relative dering of the SNPs combinations thereof. 5. Kooperberg C, Ruczinski I (5). Identifying Interacting SNPs using Monte Carlo Logic Regression, Genetic Epidemiol., 8(): temperature Multiple Models : Metropolis-Hastings Multiple Models : Metropolis-Hastings Let γ S be the sce of a certain state S. Example: Simulate binary predicts X,...,X. We use the acceptance function α(γ old,γ new, t) = min{,exp([γ old γ new]/t)} If we keep the temperature constant, this defines a homogeneous Markov chain. Let Y =5+ L(X, X, X, X ) + ɛ, ɛ N(,). Run a homogeneous Markov chain during crunch time f two separate cases: We constructed the move set to be irreducible aperiodic, therefe each homogeneous Markov chain has a limiting distribution π t(s). If we know the model size where the signal ends the noise starts, we can read off the cresponding temperature from the diagnostic plot! Case Case All X are independent. All X are independent, except X (in the signal) X 5 (not in the signal), which are heavily crelated. Multiple Models : Metropolis-Hastings Multiple Models : Metropolis-Hastings X X X X X5 X6 X7 X8 X9 X X X X X X5 X6 X7 X8 X9 X predicts predicts SNPs