D. Gordon, M.A. Levenstien, S.J. Finch, J. Ott. Pacific Symposium on Biocomputing 8: (2003)

Transcription

1 Errors and Linkage Disequilibrium Interact Multilicatively Wen Comuting Samle Sizes for Genetic Case-Control Association Studies D. Gordon, M.A. Levenstien, S.J. Finc, J. Ott Pacific Symosium on Biocomuting 8:49-5(3)

2 ERRORS AD LIKAGE DISEQUILIBRIUM ITERACT MULTIPLICATIVELY WHE COMPUTIG SAMPLE SIZES FOR GEETIC CASE-COTROL ASSOCIATIO STUDIES D. GORDO, M. A. LEVESTIE, S. J. FICH, AD J. OTT Laboratory of Statistical Genetics, Rockefeller University 3 York Avenue, ew York, Y Deartment of Alied Matematics and Statistics, State University of ew York at Stony Brook, Stony Brook, Y 794 Single nucleotide olymorisms (SP) may be used in case-control designs to test for association between a SP marker and a disease. Suc designs may assume tat te genotye data are reorted witout error. Our goal is quantifying te effects tat errors ave on samle size for case-control studies wit alotyes formed by a disease locus and a SP marker locus in te resence of linkage disequilibrium (LD). We consider te effects of a recently ublised error model on 3 ci-square analysis. We study te joint relation of LD and errors wit samle size for tree secific genetic disease models and two settings eac of marker allele frequencies (total of 6 studies). Minimal samle size necessary for fixed asymtotic ower is estimated as a 4 t degree olynomial in te variables S (error) and D (LD measure) via a backward ste-wise regression. We find tat increased error rates lower ower. In all studies, we observe tat LD and errors interact in a non-linear fasion. In articular, regression analyses sows tat several iger order interaction terms ave coefficients significantly different from in eac study, wit fraction of variance exlained greater tan Finally, te increase in samle size necessary to maintain constant asymtotic ower and level of significance as a function of S is smallest wen D = (erfect LD). Te increase grows monotonically as D decreases to.5 for all studies. Introduction Single nucleotide olymorisms (SPs) may be used in case-control designs to test for genetic association between marker and disease. Suc designs usually assume tat genotye data are reorted witout error. In statistical genetics, errors in genotying or enotying (incorrectly assigning a case to be a control, or vice versa) can significantly affect linkage and genetic association studies. A number of autors ave studied suc effects -. Sobel et al. summarize results to date. Major findings are tat errors lead to inflation in genetic ma distances, an increase in tye I error or a decrease in ower for statistical metods designed for gene localization, and biased estimates of arameters suc as te recombination fraction among loci and te amount of linkage disequilibrium (LD) between two loci. For case-control studies of genetic association, researcers,3 ave found tat, for a articular error model (not resented ere), errors lead to a loss in ower to detect association between a disease and a locus. However, to our knowledge, tere as been no quantitative assessment of te relation between errors and LD in genetic case-control association studies for multile disease models, altoug oter

3 autors 6,4-7 ave develoed metods tat allow for errors in genetic linkage and/or association analyses. Te urose of tis work is terefore a quantitative assessment, in terms of increased samle size, of error rates in genetic case-control association studies. Te data we consider is alotye data for cases and controls from a SP marker locus tat is in LD wit a disease locus. Te SP marker is observed, and te disease locus is unobserved. Te test statistic considered is te standard χ on 3 tables. We comute asymtotic ower analytically by means of a non-centrality arameter. Errors affect te ower of suc statistics by deviating genotye frequencies in cases and controls away from teir true values. Furtermore, determining samle size for fixed ower level is equivalent to determining ower for a fixed samle size, and it is tis first question tat we study in tis work. For tree articular genetic disease models and two different settings of SP marker allele frequencies (a total of 6 studies), we comute genotye frequencies for cases and controls in te resence of errors, and comute te samle size necessary to maintain constant asymtotic ower and level of significance for different values of te error model arameters. Finally, we erform model fitting by regressing te minimal samle size necessary to maintain constant ower on a 4 t degree olynomial in te variables S (error arameter) and D ' (LD arameter). Materials and Metods. otation Te following notation is used troug te remainder of tis work: Count arameters: A = number of cases U = number of controls Frequency arameters: = allele frequency of SP marker allele = allele frequency of SP marker allele = - d = allele frequency of disease locus d allele + = allele frequency of disease wild-tye allele = - d Aij = frequency of SP marker genotye ij in te case oulation (ij {,, }) Uij = frequency of SP marker genotye ij in te control oulation (ij {,, }) Disequilibrium arameters:

4 D= disequilibrium (non-standardized as defined in Hartl and Clark 8 ) [ote: max (- +, - d ) D min ( d, + )] D = min ( d, + ) (we assume in tis work tat disequilibrium is ositive) max D = roortion of total disequilibrium (or standardized disequilibrium 9 ) = D/ Dmax Penetrances: f = Pr(affected + + at disease locus) f f = Pr(affected + d at disease locus) = Pr(affected dd at disease locus) Conditional robabilities: A = Pr( genotye at SP locus affected) A = Pr( genotye at SP locus affected) U = Pr( genotye at SP locus unaffected) U = Pr( genotye at SP locus unaffected) Prevalence and oter arameters: φ = disease revalence= ( d ) f + ( d )( d ) f + d f (ote: We assume Hardy-Weinberg equilibrium (HWE) at te disease locus; no suc assumtion is made for te marker locus) ij = alotye frequency of i allele at disease locus (i = + or d) and j allele at marker locus (j = or ) (see Metods) Error model arameters: ε = Pr(true eterozygote incorrectly coded as a omozygote), ε = Pr(true eterozygote as one allele misread), ε = Pr(jointly misreading bot alleles of a genotye), 3 ε 4 = Pr(falsely adding an allele to a true omozygote), ε = Pr(re-gel error). 5 Sobel et al. describe tese arameters more comletely. It sould be noted tat, for a di-allelic locus, te arameter ε =, since it is not ossible for one eterozygote to be incorrectly read as anoter eterozygote for a di-allelic locus. Wen considering te χ statistic on 3 tables, te samle size determination for fixed asymtotic ower and significance level is comletely determined by te non-centrality arameter λ, wic is a function of te genotye

5 frequencies in te case and control oulations and te ratio of cases to controls. In section., we demonstrate ow to comute genotye frequencies in eac oulation as a function of te genetic model arameters (enetrance values, disease allele frequency), an LD arameter and te SP marker allele frequency. In section.3, we resent an error model and comute recisely ow genotye frequencies determined in section. are altered for general settings of te error model arameters. Comutation of genotye frequencies We assume tat we know te following six arameter values: te enetrance values f, f, f, te SP marker allele frequency, te disease allele frequency d, and te standardized disequilibrium D. Using te definition of conditional robability, we calculate all suc values Pr(ab at SP marker locus affection status),. For examle, we ave te following case genotye frequency exressions: = Pr( affected) = [/( φ)]{( ) f + ( )( ) f + ( ) f }, A + + d d A = Pr( affected) = [/( φ)]{( + )( + ) f + ( + d + d + ) f + ( d )( d ) f }, A = Pr( affected) = [/ φ]{( + ) f + ( + )( d ) f + ( d ) f }. To comute te corresonding genotye frequencies for controls, relace φ by -φ and eac f by f in eac exression. Te alotye frequencies are functions of i i te arameters,, +,, and D. Using te notation defined above, we ave: d = d + = + D' D + max, + = D' D + max, d D' Dmax, d = d + D' Dmax. To obtain te genotye frequency exressions as functions of LD, substitute te alotye relations above in te genotye frequency exressions..3 Error model Recently, Sobel, Pa, and Lange roosed a model to describe ow errors affect genotyes, in terms of te robabilities Pr(observed genotye is ab true genotye is cd) (were{ ab, cd} {,, } ). We call tese robabilities error enetrances.

6 Wile teir model generalizes to a marker locus wit any number of alleles, we resent in table te error enetrances for a di-allelic locus. Table Error enetrances for a SP marker locus using te Sobel-Pa-Lange error model True Genotye Observed Genotye ( ε 3 + ε4 + ε5 ) ( ε + ε5 ) / ε 3 + ε 5 / ε 4 ε 5 / ( ε + ε5 ) ε 4 + ε 5 / ε 3 ε 5 / ( ε + ε5 ) / ( ε 3 + ε4 + ε5 ) Using table, we comute te observed genotyes for eiter cases or controls wen errors are resent. If table is tougt of as a 3 3 matrix M, we can comute te vector of observed case genotye frequencies in te resence of errors, A = ( ) T A, A,, (ere, T is te transose oerator) by erforming te A matrix multilication M A. For examle, = ( ε + ε + ε )] + [( ε + ε ) / ] + [ ε + ε / ]. A [ A 5 A 3 5 A ote tat te observed genotye frequencies are a function of bot te error rates and te LD arameter. Wile te Sobel-Pa-Lange error model assumes 5 arameters, in order for us to resent 3-dimensional lots of te interaction between LD and errors, we must reduce it to a single arameter. Terefore, we use fixed multiles of te settings: ε =.5, ε =, ε =.5, ε =., ε. 5 from = u to 6 (increments of.5) from tis oint forward. Sobel et al. give tese settings as te default settings for teir error model arameters wen considering a di-allelic locus. Te notation S reresents te sum k ε, were k =.,.5,,,, i= i.4 on-centrality arameter Using te notation above and a general result roved by Mitra, Gordon et al. 3 found tat tat te non-centrality arameter λ for te test of genotye frequency differences among cases and controls is given by: ( A U ) ( A U ) ( A U ) λ = A U [ A A + U + U A A + U + U A A + U ]. () U Tis formula rovides us wit te samle size for a fixed value of te non-centrality arameter. Assuming a fixed ower and significance level, te non-centrality is

7 known. It is ten ossible to solve equation () for samle sizes. We comute tis solution for all genetic models resented in te next section..5 Genetic models Here we resent values for te arameters in section.. Eac set of genetic model arameters (enetrances + disease allele frequency) comes from a genetic disease model in wic te disease revalence is.3 and te disease allele frequency is.. In all studies, te non-centrality arameter is set to 5.448, wic corresonds to a fixed asymtotic ower of.95 at te.5 level of significance for a χ distribution wit degrees of freedom. Also, te LD arameter D ' is varied between.5 and. in increments of.5. Finally, te SP marker -allele frequency is set at bot. and.5 in all studies. Te genetic model arameter values are: (Dominant model) f =.4, f =.7, f =.7, d =. (Additive model) f =.4, f =.8, f =.4, d =. (Multilicative model) f., f =.8, f =.7,. = d =.6 Regression analysis As a furter means of describing te quantitative relationsi among samle size, LD, and errors, we erform a backward ste-wise regression analysis. For eac setting of error arameter S and te LD arameter D ', te value of te deendent variable is te samle size necessary for asymtotic ower.95 at level of significance.5. Te general form of te fitted regression equation (i.e., te uer model) is: Yˆ = βˆ + βˆ 4 4 i= i j + = j 4 i, j i S D' j, wereŷ is te fitted samle size (in terms of case individuals) corresonding to a given setting of S and D ', and te terms ˆβ are te arameters of te regression (regression coefficients) tat minimize te sum of squares of differences between te fitted values for settings of S and D ' (using equation ) and te observed values for te same settings. Te regression coefficients are determined using te S-PLUS 6. software (see Electronic Database Information). i, j

8 3 Results We ave tree main results. Our first is tat, for te genetic models considered in section.5, tere is multilicative interaction between te error arameter S and te standardized LD D '. Tis interaction is documented graically in figures and and quantitatively in our regression analysis results (Table ). Table Regression coefficients for all genetic model studies and SP allele frequency settings Exonent air (i,j) for term Genetic Model a /SP allele frequency S i D j Dom/.5 Dom/. Mult/.5 Mult/. Add/.5 Add/. (,)(intercet) (,) (,) (,) (,) (,) (,3) (3,) (,) (,) (,4) (,) (3,) (,3) a (Dom = Dominant, Mult = Multilicative, Add = Additive) Figures and resent te minimal samle size necessary to maintain constant asymtotic ower of.95 at te.5 significance level for our dominant model wit SP -allele frequency of.5 and our additive model wit SP -allele frequency of., resectively. Te samle size, as indicated above, is a function of S and D '. We comment tat in table, te non-zero coefficients, wen tested (using te t-test) for being non-zero, are all significant at te. level (data not sown). Te observations tat several interaction terms in table are significantly non-zero and tat te fraction of variance (multile R value) for eac regression is at least.9999 (data not sown) indicate tat, for tese error models, samle size is well fit by a ig degree olynomial in te variables S and D ', and ence tere is significant interaction between tese two variables in exlaining te samle size increase.

9 Our second result is tat te general trend of samle size increase as a function of te two variables S and D ' is robust to genetic model secification for te models we consider ere. Tis result may be observed quantitatively by noting tat, for eac monomial term in table, te sign of te regression coefficient for te non-zero coefficients is te same across genetic models and SP allele frequency secifications, and may be observed graically by studying figures and. We comment te sae of te surfaces in figures and is identical to te sae of te surfaces for tose figures determined by all oter genetic model and SP allele frequency secifications (data not sown). Te tird result is tat, for all values of S, samle size increase as a function of S is smallest wen D ' =, and is largest wen D ' =.5 (table ; figures and ). Tis result suggests tat ig levels of LD, in addition to increasing ower for genetic case-control studies, may ave te additional benefit of mitigating te effects of errors in data in te sense of requiring te smallest ossible increase in samle size for a given error setting. 4 Summary and Discussion In tis work, we ave demonstrated tat it is ossible to comute analytically samle size requirements for genetic case-control studies in te resence of errors. In sections.-.5, we ave described ow tese comutations are done for te test of genotyic association using te 3 contingency table. Furter, we ave sown tat, for our genetic model, error model, and LD arameter settings, samle size is accurately redicted by a olynomial of ig degree in te variables S and D '. From te viewoint of marker selection, we ave documented tat ig levels of LD ave te smallest cost, in terms of increased samle size, for a given setting of error arameters. Tis result sould be reassuring to researcers wo are lanning association studies and wo are concerned about errors in teir data. Tis work generalizes to an analytic metod for samle-size calculations in te resence of errors wen te observed data are alotyes or multi-locus genotyes. One only needs to secify multi-locus error models. Peras te simlest and most reasonable model is one in wic errors in individual marker loci are indeendent of errors in oter marker loci. Also, tis work is not restricted to just di-allelic loci; it can also be extended to markers loci wit any number of alleles. Te analytic rice is tat one as to secify multile LD arameters and multile allele or alotye frequency arameters for te marker loci. We ave considered te question of interaction between errors and LD over a larger set of values for te genetic model arameters secified in section.; our observation is tat te interaction between S and D is robust to genetic model secifications. Tat is, te sae of figures and is reeated for every set of genetic model arameters considered (data not sown).

10 An imortant question for tis work regards arameter estimation. We are currently working on metods to determine genotying error rates. Also, LD arameters can be estimated using inter-marker LD atterns. Wit traits for wic te genetic model arameters are difficult to estimate, one can secify genetic model-free arameters 3 rater tan te genetic model-based arameters we ave secified in tis work. Software erforming tese calculations will be available from our website tt://linkage.rockefeller.edu/awe/ by January 3. Te rogram is called PAWE (Power of Association Tests Wit Errors). Acknowledgments Te autors gratefully acknowledge grants K-HG55 and MH5949 from te te ational Institutes of Healt. Electronic Database Information S-PLUS 6. Academic Site Edition Release. Coyrigt 988- Insigtful Cor. References. Douglas, J.A., Boenke, M. & Lange, K. A multioint metod for detecting genotying errors and mutations in sibling-air linkage data. Am J Hum Genet 66, ().. Sields, D.C., Collins, A., Buetow, K.H. & Morton,.E. Error filtration, interference, and te uman linkage ma. Proc atl Acad Sci 88, 65-5 (99). 3. Buetow, K.H. Influence of aberrant observations on ig-resolution linkage analysis outcomes. Am J Hum Genet 49, (99). 4. Terwilliger, J.D., Weeks, D.E. & Ott, J. Laboratory errors in te reading of marker alleles cause massive reductions in lod score and lead to gross overestimates of te recombination fraction. Am J Hum Genet 47, A (99). 5. Gordon, D., Matise, T.C., Heat, S.C. & Ott, J. Power loss for multiallelic transmission/disequilibrium test wen errors introduced: GAW simulated data. Genet Eidemiol 7 Sul, S587-9 (999). 6. Gordon, D., Heat, S.C., Liu, X. & Ott, J. A transmission/disequilibrium test tat allows for genotying errors in te analysis of single-nucleotide olymorism data. Am J Hum Genet 69, 37-8 (). 7. Goldstein, D.R., Zao, H. & Seed, T.P. Te effects of genotying errors and interference on estimation of genetic distance. Hum Hered 47, 86- (997). 8. Cerny, S.S., Abecasis, G.R., Cookson, W.O., Sam, P. & Cardon, L.R. Te effect of genotye and edigree error on linkage analysis: analysis of tree astma genome scans. Genet Eidemiol, S7- (). 9. Abecasis, G.R., Cerny, S.S. & Cardon, L.R. Te imact of genotying error on family-based analysis of quantitative traits. Eur J Hum Genet 9, 3-4 ().

11 .Akey, J.M., Zang, K., Xiong, M., Doris, P. & Jin, L. Te effect tat genotying errors ave on te robustness of common linkage-disequilibrium measures. Am J Hum Genet 68, ()..Sobel, E., Pa, J.C. & Lange, K. Detection and integration of genotying errors in statistical genetics. Am J Hum Genet 7, ()..Bross, I. Misclassification in x tables. Biometrics, (954). 3.Gordon, D. & Ott, J. Assessment and management of single nucleotide olymorism genotye errors in genetic association analysis. Pac Sym Biocomut, 8-9 (). 4.Goring, H.H. & Terwilliger, J.D. Linkage analysis in te resence of errors II: marker-locus genotying errors modeled wit yercomlex recombination fractions. Am J Hum Genet 66, 7-8 (). 5.Goring, H.H. & Terwilliger, J.D. Linkage analysis in te resence of errors I: comlex-valued recombination fractions and comlex enotyes. Am J Hum Genet 66, 95-6 (). 6.Goring, H.H. & Terwilliger, J.D. Linkage analysis in te resence of errors III: marker loci and teir ma as nuisance arameters. Am J Hum Genet 66, (). 7.Goring, H.H. & Terwilliger, J.D. Linkage analysis in te resence of errors IV: joint seudomarker analysis of linkage and/or linkage disequilibrium on a mixture of edigrees and singletnos wen te mode of ineritance cannot be accurately secified. Am J Hum Genet 66, 3-7 (). 8.Hartl, D.L. & Clark, A.G. Princiles of oulation genetics, (Sinauer Associates, Sunderland, 989). 9.Lewontin, R.C. Te interaction of selection and linkage. I. General considerations; eterotic models. Genetics 49, (964)..Risc,. A general model for disease-marker association. Ann Hum Genet 47, 45-5 (983)..Sam, P. Statistics in Human Genetics, (J. Wiley and Sons, Inc., ew York, 998)..Mitra, S.K. On te limiting ower function of te frequency ci-square test. Ann Mat Stat 9, -33 (958). 3.Gordon, D., Finc, S.J., otnagel, M. & Ott, J. Power and samle size calculations for case-control genetic association tests wen errors are resent: alication to single nucleotide olymorisms. Hum Hered (in ress) ().

12 Figure. Minimum samle size necessary to maintain.95 ower at.5 significance level for dominant genetic model, SP allele frequency =.5 Samle Size (Cases+Controls) S D' Minimum Samle Size: 46, for D'=, S= Maximum Samle Size: 56, for D'=.5, S=.8

13 Figure. Minimum samle size necessary to maintain.95 ower at.5 significance level for additive genetic model, SP allele frequency =. 5 Samle Size (Cases+Controls) S D' Minimum Samle Size: 46, for D'=, S= Maximum Samle Size: 376, for D'=.5, S=.8