Ascertainment bias for non-twin relatives in twin proband studies

When families are ascertained through affected twins, as for example when twin probands are selected from a registry and their non-twin relatives studied, a correction for ascertainment bias is needed. It is shown that probandwise counting (where relatives of doubly ascertained twin pairs are counted twice) is the appropriate method. The bias resulting from pairwise counting is given and depends on the genetic model and on the probability of selecting an affected twin as a proband. For the multifactorial and generalized single major locus models the bias is small, and the problems associated with nonindependent ascertainment are negligible in practice.

Hum. Hered. 32: 202-207 (1982) Ascertainment Bias for Non-Twin Relatives in Twin Proband Studies! John Ricea• b, Irving T. Gottesman" ", Brian K. Suarez" c, Dennis H. O'Rourkea, Theodore Reicha , c Departments of aPsychiatry, hBiostatistics and cGenetics, Washington University School of Medicine and the Jewish Hospital of St. Louis, St. Louis, Mo., USA Key Words. Ascertainment· Proband· Twin study· Probandwise concordance Abstract. When families are ascertained through affected twins, as for example when twin probands are selected from a registry and their non-twin relatives studied, a correction for ascertainment bias is needed. It is shown that probandwise counting (where relatives of doubly ascertained twin pairs are counted twice) is the appropriate method. The bias re- I sulting from pairwise counting is given and depends on the genetic model and on the probability of selecting an affected twin as a proband. For the multifactorial and generalized single major locus models the bias is small, and the problems associated with nonindepen­dent ascertainment are negligible in practice. .. Introduction Twin studies provide a unique data base for the investigation of human diseases, and existing twin registries have proven a valua­ble source of clinical data for disorders ranging from schizophrenia [Gottesman and Shields, 1972] to. diabetes [Harvald and Hauge, 1976; Pyke and Nelson, 1976] and breast cancer [Holm et aI., 1980]. However, if observations are limited solely to MZ and DZ pairs, restrictive environmental assump­tions must be made since only two unknown model parameters may be estimated from 1 Supported in part by USPHS grants AA-03539, GM-28067, MH-25430, MH-31302 and MH-14677. two observations. Recently, much attention has been given to genetic models which al­low for phenomena such as the transmission of environmental factors from parent to off­spring (cultural transmission) and effects due to a contemporaneous environment of rearing [Wright, 1931; Rao et aI., 1976; Eaves, 1976; Cloninger et aI., 1979; Rice et aI., 1980]. Accordingly, the range of hy­potheses testable from twin data can be broadened by including, for example, the parents and siblings of the twins. Essen-Moller and Fischer [1979] point out, however, that there is confusion in the literature on the appropriate procedure for taking the method of ascertainment into ac­count when families are sampled through J Ascertainment of Relatives through Affected Twins - affected twins. To estimate the prevalence of affected non-twin relatives in an ascer­tained sample, they discuss pairwise count-ing (where each relative is counted once), probandwise counting (where relatives of doubly ascertained twins are counted twice), and argue that both approaches may be in-adequate since affected non-twin relatives cannot themselves be probands. In report-ing rates in non-twin relatives of ascertained twins, Kring/en [1967] used pairwise rates, whereas other investigators [ef. Gottesman and Shields, 1972; Fischer, 1973] report probandwise rates, although without a for­mal justification [ef. Allen et aI., 1967]. In what follows we show that the proband- 203 Table I. Summary of notations used K R_lwin x y Prob ( Prob (asc) Prevalence of illness in twins Prevalence of il lness in relatives (e.g., parents, siblings, etc.) Probability that the co-twin of an affected twin is affected Probability that a relative of an affected twin is affected Probability that an affected twin is a proband Number of affected twins in a twin pair, with X = 0, I or 2 Number of probands in a twin pair, with Y ~ X Probability of the event in parentheses Probability o(ascertaining a twin pair Correlation in liability for the multifactorial model wise rate is indeed the correct one, and we q, fl , f1, f3 quantify the degree of bias resulting from Parameters of the single major locus model with two alleles A, a, with q the frequency of a and fl , f1, f3 the probability that an individual with genotype AA, Aa, aa , respectively, is affected the use of the pairwise rate. Methods Estimation of the Prevalence of Affected Co-Twins Let us first review the need for the use of pro­bandwise concordance when calculating the prev­alence of illness in co-twins of affected twins sam­pled using the proband method, and then general­ize to the prevalence of illness in the relatives of the twins. The correction for ascertainment bias was developed by Weinberg [1928] and has been discussed in detail by Morton [1959], Crow [1965], AI/en et al. [1967], Gottesman and Shields [1972] and Smith [1974] . Consider first the case when the twins are or­dered (TI, T2), say by birth order. Let A and U de­note affected and unaffected, respectively, and PAA, PAU, PUA, PUU denote the probabilities of the various affectational possibilities. The preval­ence of the disease in twins, assuming PAU = PUA is KP.twin = PAA + PAU and we wish to es­timate KR.twin = PAA/(PAA + PAU). Let n denote the probability that an affected twin will be a proband in the study. Then, using the notation summarized in table I, Prob (asclX = 2) = n 2 + 2n (l -n) = n(2-n), (1) Prob (asclX = 1) = n, Prob (asclX = 0) = 0, Prob (asc) = n(2-n)pAA + 2npA u. (2) (3) (4) That is, the probability of ascertaining a pair conditional on having both members affected is the probability that they are both pro bands (n 2) plus the probability that exactly one is a proband; the probability of ascertaining a pair conditional on having exactly one affected member is n; the probability of ascertaining a pair both of whom are unaffected is O. Therefore, the probability of ascertaining a pair is the weighted sum given in equation 4. Let Pt denote the probability in the ascer­tained sample that both twins will be affected and one is a proband, P2. the probability that both will be affected and both are probands, and P3 the probability that one is affected and a proband. 204 Then PI = Prob (X=2, Y=l\asc) = 2n(1-n )PA ,\/Prob(asc), P2 = Prob (X=2, Y=2!asc) = n2pAA/ (5) Prob(asc), (6) Pa = Prob (X=1, Y=1Iasc) = 2nP.\l;! Prob(asc), (7) The Pi's are the expected proportions for the quantities actually observed in the sample, and we note that PAA PAA + PAU KR-twin, (8) so that an asymptotically unbiased estimate of KR-twin may be obtained by counting each doubly ascertained pair twice. This is remarkable both in the fact that this is true independently of the gene­tic/ environmental model and independently of the value of n· Tn contrast the pairwise rate is given by which depends both on the values of PA,\ and PAU and on n. Note, however, that as :t be­comes small, then the term in brackets approaches 1. Estimation of the Prevalence of Affected Relatives Now consider the case where twin pairs are sampled using the proband method and that a par­ticular type of relative, e.g., parent or sibling, is studied. Let (TI, T2, S) denote an ordered trio consisting of twin 1, twin 2, and relative, and let Pijk denote the probability in a random sample that TI, T2 and S have phenotypes i, j, k, respec­tively. Thus, PAAA is the probability that all three are affected, P.c\,U- the probability that both twins are affected and the relative unaffected, etc. We use a '.' to indicate a marginal distribution, so that PA" = KP-twin, P-'A = KP-reI, PA_\' = P:\A (as above), and PA-A the probability that twin 1 and the relative are affected. Since ascertainment is limited to the twins T 1 or T2, the probability of ascertainment, Prob(asc), is the same as that given in equation 4, Rice!Gottesman/Suarez/O'Rourke/Reich Prob(asc) = n(2-n)p,u., + 2nPAU' (10.1)4 ~1O.2) 1 (10.3) The probabilities in the ascertained sample are Qt = Prob[(A,A,A), Y = 2!asc] =n2pAAA/ Prob(asc) (11) t Q2 = Prob[(A,A,U), Y = 2\asc] = n 2pAA1./ Prob(asc) (12) Qa = Prob[(A,A,A), Y = l!asc] -~ h(1-n)PAAAi Prob(asc) (13) Q4 = Prob[(A,A,U), Y = l!asc] = hC1-n)PAAul Prob(asc) (14) Q5 = Prob[CA,U,A), Y = l~asc] = nPAUA/ Prob(asc) Q. = Prob[(A,U,U), Y = llasc] = nPAuui Prob(asc) Qr = Prob[(U,A,A), Y = l\asc] = nPlJAAI Prob(asc) Qs = Prob[(U,A,U), Y = 1lasc] = npL\U/ Prob(asc), (15) (16) (17) (18) so that the probandwise rate in the relatives is giv­en by PA-A -- = K R -re1, PA" (19) Equation 19 follows in a straightforward way us­ing the formulas P.\UA = PA-A - P.\AA and PAU' = PA' - - PAA' to simplify the expressio_~ in the equation. The pairwise rate a is seen to be a= Prob(asc) (20.0 2Kp-twin KR-rel - rcPAAA (20.2) 2Kp -twin - rcKp-twin KR-rel which depends on the parameter n and on the joint probability that all three individuals are af­fected. The Bias Resulting from Use of Pairwise Rates From equation 20.2, we see that the bias from the use of the pairwise rate approaches 0 as :J( approaches O. This is clear since as n becomes Ascertainment of Relatives through Affected Twins Table II. Value of the pairwise rate (1 as com­pared to K R_,,: for the multifactorial model K R_rc1 0.30 0.7 0.637 0.5 0.523 0.10 0.7 0.470 0.5 0.320 0.01 0.7 0.266 0.5 0.129 J[ 1.00 0.75 0.50 0.25 1.00 0.75 0.50 0.25 1.00 0.75 0.50 0.25 1.00 0.75 0.50 0.25 1.00 0.75 0.50 0.25 1.00 0.75 0.50 0.25 O.5S3 0.600 0.615 0.627 0.480 0.494 0.505 0.515 0.418 0.434 0.448 0.459 0.286 0.295 0.304 0.313 0.233 0.244 0.251 0.259 0.118 0.121 0.124 0.126 small, there will be few twin pairs with two pro­bands, with the ratio of doubly ascertained to singly ascertained pairs being n/(2-2:r) in pairs where both twins are affected. To assess the bias when :n is large, it is necessary to consider partic­ular models of disease transmission in order to ob­tain the joint probabilities needed for computation of the expected value of the pairwise rate. We will consider the multifactorial model and the general­ized single major locus model. For the multifactorial model, KR-rel depends On the prevalences in twins and in the relatives and on the correlation in liability between the 205 Table III. Value of the pairwise rate a as com­pared to KR_rel for the generalized single major locus model q J[ 0.0513 0.0 1.0 1.0 0.10 0.544 1.00 0.533 0.75 0.537 0.50 0.540 0.25 0.542 0.0050 0.0 1.0 1.0 0.0 I 0.504 1.00 0.503 0.75 0.504 0.50 0.504 0.25 0.504 0.3162 0.0 0.0 1.0 0.10 0.433 1.00 0.40 I 0.75 0.411 0.50 0.419 0.25 0.423 0.1000 0.0 0.0 1.0 0.0 I 0.303 1.00 0.294 0.75 0.297 0.50 0.299 0.25 0.301 0.1000 0.0 0.5 1.0 0.10 0.325 1.00 0.315 0.75 0.318 0.50 0.320 0.25 0.323 0.0 I 00 0.0 0.5 1.0 0.0 I 0.258 1.00 0.257 0.75 0.257 0.50 0.257 0.25 0.257 twins and between a twin and his relative [Falco­ner, 1965; Smith, 1970; Reich et a!., 1972; Curnow and Smith, 1975]. Tn table II we assume for sim­plicity that KP-twin = KP-rel and that the twin­twin and twin-relative correlations are equal to a common r, and show the value of KR-twin along with the a of formula 20. The bias decreases as r decreases, and we display results for r of 0.7 and 0.5. In the case of polygenic transmission and ran­dom mating, a parent-offspring correlation of 0.5 would correspond to a trait with 100% heritabili­ty. The probabilities were computed using the methods described in Rice et al. [1980]. 206 For the generalized single major locus model [Suarez et aI., 1976], we consider a locus with two alleles A,a and genotypes AA, Aa, aa. The model is parameterized in terms of q, the gene frequency of allele a, and the penetrances ft, f2, fa, the probabilities that individuals with geno­types AA, Aa, aa, respectively, are affected. Table III indicates the bias for dominant, recessive, and additive models, where the relative is a sibling and where the twins arc DZ. The above analysis shows that probandwise rates are the correct ones when the relativies of ascertained twins are studied. However, tables II and III indicate that in practice the pairwise and probandwise rates will be close to one another, even with large Jr, and even at the 'extremes' of the two models. This is consistent with the various studies which have calculated both types of rates and report very little difference. Discussion The above analysis was motivated by the twin data on schizophrenia where the rates of affected parents and siblings of twins have been reported to be lower than those reported in studies where probands are not twins [Essen-Moller and Fischer, 1979]. We have shown that this difference is not due to problems associated with ascertainment. These authors point out that these differ­ences may also be due to less thorough stu­dy of the non-twin relatives or be simply due to 'chance' as suggested by Gottesman and Shields [1976]. Although environmental ef­fects (prenatal and postnatal) are no doubt more similar for DZ siblings than for single­ton siblings, it is not yet clear whether or not they are relevant to the pathogenesis of schizophrenia. The corrections here for ascertainment are simply those one would use if sibships were sampled but probands were restricted, say, to the first- or second-born. However, Rice/Gottesman/Suarez/O'Rourke/Reich they are in general more important in twin research. For example, Fischer [1973] dre\v her sample from a Danish central register of psychiatric hospital admissions, matched against a national twin birth cohort register [Hauge, 1981] so that n is nearly 1. This result indirectly addresses the prob­lems associated with nonindependent ascer­tainment of pro bands. The formulas derived above assume that in a doubly affected twin pair, the probability that an affected twin will be a proband conditional on his co-twin being a proband is n. It is possible, of course, that this conditional probability is higher due to a greater awareness of the ill­ness once the first twin case falls ill or due to referral to the same familiar treatment fa­cility. The problems with secondary ascer­tainment have been considered by Smith [1974] and Allen and Hrubec [1979]. Nonetheless, the mathematically precise corrections should, in general, lie be­tween the probandwise and pairwise rates, and the above results indicate that these dif­ferences are trivial when considering the morbid risks in the relatives of twins. References Allen, G.; Harvald, B.; Shields, J.: Measures of twin concordance. Acta genet. 17: 475-481 (1967). Allen, G.; Hrubec, Z.: Twin concordance; a more general model. Acta Genet. med. Gemell. 28: 3-13 (1979). Cloninger, C.R.; Rice, I.; Reich, T.: Multifactorial inheritance with cultural transmission and as­sortative mating. II. A general model of com­bined polygenic and cultural inheritance. Am. I. hum. Genet. 31: 176-198 (1979). Crow, I.F.: Problems of ascertainment in the analysis of family data: In Neel, Shaw, Schull, Ascertainment of Relatives through Affected Twins Genetics and the epidemiology of chronic di­seases (Washington 1965). Curnow, R.N.; Smith, c.: Multifactorial models for familial diseases in man. J. R. Stat. Soc. A. 138: 131-169 (1975). Eaves, L.J.: The effect of cultural transmission on continuous variation. Heredity 37: 41-57 (1976). Essen-Moller, E.; Fischer, M.: Do the partners of dizygotic schizophrenic twins run a greater risk of schizophrenia than ordinary siblings? Hum. Hered. 29: 161-165 (1979). Falconer, D.S.: The inheritance of liability to cer­tain diseases, estimated from the incidence among relatives. Ann. hum. Genet. 29: 51-76 (1965). Fischer, M.: A Danish twin study of schizopre­nia. Acta psychiat. scand., suppl. 238 (1973). Gottesman, I.I.; Shields, J.: Schizophrenia and ge­netics. A twin study vantage point (Academic Press, New York 1972). Gottesman, 1.1.; Shields, J.: A critical review of re­cent adoption, twin, and family studies of schi­zophrenia: behavioral genetics perspectives. Schizo Bull. 2: 360-401 (1976). Harvald, B.; Hauge, M.: Diabetes in twins. Nord. Council Arct. Med. Res. Rep. 13: 13-15 (1976). Hauge, M.: The Danish twin register; in Mednick, Baert, Prospective longitudinal research in Eu­rope: an empirical basis for the primary pre­vention of psychosocial disorders (Oxford Uni­versity Press, Oxford 1981). Holm, N.V.; Hauge, M.; Harvald, B.: Etiologic factors of breast cancer elucidated by a study of unselected twins. J. natn. Cancer Inst. 65: 285-298 (1980). Kringlen, E.: Heredity and environment in the functional psychoses (Heinemann, London 1967). Morton, N.E.: Genetic tests under incomplete as­certainment. Am. J. hum. Genet. 11: 1-16 (1959). 207 Pyke, D.A.; Nelson, P.O.: Diabetes mellitus in identical twins; in Creutzfe1dt, Kobberling, Nee1, The genetics of diabetes mellitus (Sprin­ger, New York 1976). Rao, D.C.; Morton, N.E.; Yee, S.: Resolution of cultural and biological inheritance by path analysis. Am. J. hum. Genet. 28: 228-242 (1976). Reich, T.; James, J.W.; Morris, C.A.: The use of multiple thresholds in determining the mode of transmission of semi-continuous traits. Ann. hum. Genet. 36: 163-184 (1972). Rice, J.; Cloninger, C.R.; Reich, T.: Analysis of behavioral traits in the presence of cultural transmission and assortative mating: applica­tions to IQ and SES. Behav. Genet. 10: 73-92 (1980). Smith, c.: Heritability of liability and concord­ance in monozygous twins. Ann. hum. Genet. 34: 85-91 (1970). Smith, c.: Concordance in twins: methods and in­terpretation. Am. J. hum. Genet. 26: 454--466 (1974). Suarez, B.K.; Reich, T.; Trost, J.: Limits of the general two-allele single locus model with in­complete penetrance. Ann. hum. Genet. 40: 231-244 (1976). Weinberg, W.: Mathematische Grundlage der Pro­bandenmethod. Z. indukt. Abstamm.-Vererb.­Lehre 48: 179-338 (1928). Wright, S.: Statistical methods in biology. Proc. Am. Stat. Ass. 26: 155-163 (1931). John P. Rice, PhD, Department of Psychiatry, Washington University School of Medicine, and Jewish Hospital of St. Louis, 216 South Kingshighway, St. Louis, MO 63110 (USA)