Preferential Transmission of Type 1 Diabetes From Parents to Offspring: Fact or Artifact?

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1 Genetic Epidemiology 23: (2002) Preferential Transmission of Type 1 Diabetes From Parents to Offspring: Fact or Artifact? Sun-Wei Guo 1 3n and Jaakko Tuomilehto 4 1 Department of Pediatrics, Medical College of Wisconsin, Milwaukee, Wisconsin 2 Max McGee Center for Juvenile Diabetes, Medical College of Wisconsin, Milwaukee, Wisconsin 3 Division of Biostatistics, Medical College of Wisconsin, Milwaukee, Wisconsin 4 Department of Epidemiology and Health Promotion, National Public Health Institute and Department of Public Health, University of Helsinki, Helsinki, Finland It has been widely reported that men with type 1 diabetes (T1D) tend to be more likely to transmit the disease to their offspring than their female counterparts in Caucasoid populations. Several theories to explain this preferential transmission have been proposed, but so far none of them has been unequivocally proven. Whatever the mechanism, confirmation or refutation of this observation is nonetheless important and practical to the design of future genetic studies of T1D. We carried out some statistical modeling of the preferential transmission. The well-established fact that males have higher a prevalence of T1D than females, an apparent sex difference in fecundity, and a possible misclassification of gestational diabetes mellitus (GDM) as T1D in women have been considered. We demonstrated, first, that the ascertainment of study families through the affected offspring with T1D would generate a higher proportion of fathers than mothers having T1D, even though there was no preferential transmission at all. This can be explained by the male preponderance in T1D prevalence as compared with females, coupled with a greater likelihood of being selected and/or recruited for study in families with T1D fathers due to the fecundity difference. Second, when the study population is ascertained through affected parents, misclassification of mothers with GDM as T1D, and the existence of male/female difference in Grant sponsor: NIH; Grant numbers: GM 56515; Grant sponsor: Academy of Finland; Grant numbers: 38387, 46558, 51224; Grant sponsor: Juvenile Diabetes Research Foundation; Grant sponsor: Sigrid Juselius Foundation. n Correspondence to: Sun-Wei Guo, Ph.D., Department of Pediatrics, Medical College of Wisconsin, 8701 Watertown Plank Road, MS 756, Milwaukee, WI swguo@mcw.edu Received for publication 10 April 2002; Revision accepted 28 May 2002 Published online in Wiley InterScience ( DOI: /gepi r 2002 Wiley-Liss, Inc.

2 324 Guo and Tuomilehto fecundity in conjunction with a birth order effect, can contribute to the observed preferential transmission, even though there was none. In light of the plausibility of assumptions employed in the analysis and, in particular, an apparent failure to critically examine the effects of these causes of bias in earlier studies, it is perhaps prudent to say that the jury for the existence of preferential transmission in T1D is still out. Genet. Epidemiol. 23: , & 2002 Wiley-liss, Inc. Key words: ascertainment bias; birth order effect; epidemiology; fecundity; genetics; misclassification; preferential transmission; type 1 diabetes INTRODUCTION One intriguing conundrum in the research of insulin-dependent diabetes mellitus (T1D) is an apparent higher prevalence of T1D in offspring of fathers with T1D than in offspring of mothers with T1D, at least in Caucasoid populations. By direct estimation of the recurrence risks in offspring of parents with T1D, it has been shown that the offspring of affected fathers are more likely to develop T1D than those of affected mothers [Warram et al., 1984; Tuomilehto et al., 1995]. These findings seem to have been validated by family-history studies of children with T1D and their parents, which have demonstrated that the prevalence of paternal T1D in families with at least one child with T1D is significantly higher than the prevalence of maternal T1D [Degnbol et al., 1978; Wagener et al., 1982; Dahlquist et al., 1982; Jefferson et al., 1985; Gavard et al., 1989; O Leary et al., 1991; Tuomilehto et al., 1992; Metcalfe and Baum, 1992; Pociot et al., 1993; Lorenzen et al., 1994]. A summary of studies, by no means exhaustive, that support such a preferential transmission in T1D is given in Table I. Various hypotheses have been proposed to explain the observed preferential transmission of T1D susceptibility, but so far none of them has proven unequivocally true. First, haplotypic preservation in T1D susceptibility genes in males could be responsible for the sex difference in transmission. Males tend to have a lower recombination frequency than females between linked loci during gametogenesis, and multiple loci could be responsible for T1D susceptibility [Warram et al., 1984]. Testing of this hypothesis, however, has been difficult, in part because few genes responsible for T1D susceptibility have been positively and consistently identified yet, although it is known that such genes exist within the major histocompatibility complex (MHC). In addition, the mild sex difference in recombination fraction is unlikely to account for the substantial difference observed in the prevalence of T1D between fathers and mothers of children with T1D. Second, the selective loss of fetuses that carry susceptibility to T1D in women with the disease could result in a lower prevalence of T1D in the offspring of women with T1D than in men with T1D [Warram et al., 1984]. This hypothesis, however, was later dismissed by empirical data [Warram et al., 1988; Mills et al., 1988]. Third, fetuses exposed to maternal diabetes could be protected from subsequent manifestation of T1D and thus could also lead to the observed preferential transmission [Warram et al., 1984]. Fourth, the diabetogenic genes may be preferentially transmitted from fathers to offspring, regardless of whether the

3 TABLE I. A Summary of Published Studies Reporting Preferential Transmission and Their Characteristics a Parent with TIDM Study Ascertainment Father (%) Mother (%) Ratio GDM excluded Comments Degnbol and Green [1978] [recalculated by Warrem et al., 1984] Wagener et al. [1982] [recalculated by Warrem et al., 1984] Dahlquist et al. [1982] [recalculated by Warrem et al., 1984] P ? O ? O ? Warrem et al. [1984] P ? Dahlquist et al. [1985] O ? Jefferson et al. [1985] O ? Sex difference in incidence Tilli and Kobberling [1987] O ? P Gavard et al. [1989] O ? O Leary et al. [1991] O?? 4.0? Tuomilehto et al. [1992] O ? Differential fertility Metcalfe and Baum [1992] O ? Pociot et al. [1993] O ? Lorenzen et al. [1994] O ? Verge et al. [1994] O ? Dahlquist and Mustonen [1995] O?? 2.5? Tuomileho et al. [1995] P ? Birth order effect Eurodiab ACE Study Group [1998] O ? Gillespie et al. [2002] O Yes a P, from parent; O, from offspring.

4 326 Guo and Tuomilehto fathers are themselves diabetic or not [Vadheim et al., 1986]. Data from 107 nuclear families appeared to support this hypothesis, which was confirmed by Field [1989, 1994]. However, an analysis of 172 multiplex diabetic pedigrees from the United Kingdom found no such evidence [Bain et al., 1994]. The last hypothesis is genomic imprinting, referring to the differential expression of a disease depending on the sex of the parent transmitting the susceptibility allele [McCarthy et al., 1991]. Data collected from 1,774 families with at least one child with T1D did not support this hypothesis [McCarthy et al., 1991]. The resolution of this conundrum obviously has important and practical implications in understanding the etiology of T1D, which has so far remained elusive. First, preferential transmission, if proven true and measured precisely, can be incorporated into genetic counseling to provide more accurate prediction of the probability of T1D. Second, since linkage/association studies based on transmission/ disequilibrium tests can be a powerful tool in identifying T1D susceptibility genes [Spielman et al., 1993; Risch, 2000], more powerful genetic tests can be devised if preferential transmission is quantified and its mechanisms understood. Third, preferential transmission, if confirmed, could point to new directions for further investigations of T1D etiology. The evidence for the preferential transmission of T1D has so far come from two types of epidemiological studies [Podar et al., 1994]. One is to ascertain children with T1D first, through either cross-sectional data or retrospective data, and then examine the affection status of their parents. The other approach is to ascertain parents with T1D and estimate directly the recurrence risk of T1D in their offspring. It should be noted that caution needs to be exercised in making inferences from these studies, since both types of studies are subject to biases that may be different in different types of studies. In the former, the higher prevalence of paternal diabetes in families with a T1D child than that of maternal diabetes cannot be translated directly into preferential transmission. A crude but adequate analogy would be the inference from the statement, A rat is a mammal to A mammal is a rat. In the latter, other factors, such as differential fecundity between the two sexes, should be ruled out before reaching the conclusion of the true existence of the preferential transmission of T1D. To further understand the conundrum, we carried out some simple statistical modeling of the preferential transmission and came up with several alternative explanations. It is a well-established fact that males have a higher prevalence of T1D than females in many populations [Karvonen et al., 1997]. Thus, we propose that, when an epidemiologic study ascertains the affected offspring first and then, through them, ascertains the affection status of their parents, the gender difference in T1D prevalence in the background population alone could result in the observation that children with T1D are more likely to have a diabetic father than a diabetic mother. This bias can be further magnified by ascertainment bias due to a sex difference in fecundity. We also propose that an epidemiologic study that ascertains parents with T1D first and then estimates directly the prevalence in their offspring could also arrive at a spurious conclusion of preferential transmission if differential fitness in fecundity is not accounted for. Lastly, a misclassification of mothers with gestational diabetes as T1D also could contribute to apparent preferential transmission, since

5 such women may not posses susceptibility genes for T1D and do not transmit the T1D susceptibility to their offspring. STATISTICAL METHODS Ascertainment Through Offspring In the majority of Caucasian populations, males have a higher age-specific incidence or risk of T1D than their female counterparts, whereas the trend is reversed in African or Asian populations, at least in children [Karvonen et al., 1997]. In Europe, the sex-difference in incidence becomes even larger in the age group years [Nystrom et al., 1992]. Since most studies that reported preferential transmission were done in Caucasians, and since the T1D prevalence in Asian populations is in general low, we can reasonably assume that males have a higher risk of T1D than their female counterparts. With this assumption, we can show that the gender difference in prevalence alone would lead to an apparently higher prevalence of paternal T1D than maternal T1D in Caucasian families with at least one child with T1D. To show this, let P, M, ando be the event that a father, mother, and offspring are affected with T1D, respectively. Under the assumption of a higher prevalence of T1D in males than in females, we have P(P)4P(M). We assume further that there is no preferential transmission. In probabilistic terms, we have P(O P) ¼ P(O M). Under these assumptions, we have PðPjOÞ ¼ PðOjPÞPðPÞ PðOÞ Preferential Transmission of T1D 327 ¼ PðOjMÞPðPÞ 4 PðOjMÞPðMÞ ¼ PðMjOÞ: PðOÞ PðOÞ ð0:1þ That is, the gender difference alone can explain why affected children are more likely to have affected fathers than affected mothers, if the affection status of parents is examined among the offspring with T1D. Under the same assumptions, we obtain PðPjOÞ PðMjOÞ ¼ PðOjPÞPðPÞ PðOjMÞPðMÞ ¼ PðPÞ PðMÞ : That is, conditional on an affected offspring, the ratio of prevalence of paternal T1D vs. that of maternal T1D is identical to the ratio of male/female prevalences of T1D in the population. If, in addition, given that a parent is affected with T1D, his/her daughters and sons are equally likely to be affected, i.e., P(S M) ¼ P(D M), and P(S P) ¼ P(D M), where S and (D) denote the event that the son (or daughter) is affected, then PðPjSÞ ¼ PðSjPÞPðPÞ o PðDjPÞPðPÞ ¼ PðPjDÞ ð0:2þ PðSÞ PðDÞ simply because P(S)4P(D) due to gender difference. In other words, the fathers of affected daughters would appear more likely to be affected with T1D than fathers of affected sons. Because the T1D prevalence in males is only slightly higher than that in females in many populations, P(P)/P(M) may not deviate greatly from 1. Therefore, factors other than sex difference in prevalence may also contribute to the apparently higher prevalence of diabetic fathers than mothers in the offspring afflicted with T1D. One such possibility is ascertainment bias due to a sex difference in fecundity.

6 328 Guo and Tuomilehto Having T1D can be debilitating, especially when the quality of diabetes care is inadequate or there are difficulties in accessing healthcare. Although the management of T1D has advanced much since the clinical use of insulin, childbearing and parenting can still be challenging for individuals with T1D [Dorman et al., 1999]. Since the physical demand of parenting is much higher for females than for males, it seems plausible that women with T1D may have less progeny than their male counterparts, a decision made consciously or otherwise. Since the family size of fathers with T1D tends to be larger than that of mothers with T1D, one would expect more T1D offspring born to T1D fathers than to T1D mothers, assuming everything else to be equal. Therefore, sibship with at least one T1D born to a father with T1D is more likely to be ascertained than those born to mothers with T1D. Hence, we have PðPjO; SÞ PðMjO; SÞ ¼ PðPÞPðOjPÞPðSjO;PÞ PðO;SÞ ¼ PðMÞPðOjMÞPðSjO;MÞ PðO;SÞ PðPÞPðSjO; PÞ PðMÞPðSjO; MÞ ð0:3þ where S denotes the event that the sibship has been ascertained, and PðOjPÞ and PðOjMÞ cancel out under the assumption that there is no transmission preference. To compute PðSjO; PÞ and PðSjO; MÞ, we assume that the number of offspring has a Poisson distribution with mean l. Thus, the probability that a parent has k children is l k e l =k!, where k = 0, 1, 2, y. Suppose, for simplicity, that each offspring is equally likely and independently to be affected with T1D with probability P. Obviously, families without any child will never be recruited for study and are thus excluded. Therefore, we are only interested in sibships with at least one child, and thus we have a truncated Poisson distribution l k e l = ð1 e l Þk!, where k ¼ 1; 2; ::::. For ascertainment, we consider the multiple ascertainment scheme, in which case there is a constant probability p that any offspring affected with T1D becomes an index case, and the ascertainment of different affected individuals in the same family is assumed to be independent [Bailey, 1951], a scheme often used in human genetics [Ewens, 1991]. Under this scheme, the probability that a sibship with r affected individuals contains at least one index case and thus enters the sample is 1 ð1 pþ r. Assuming average family sizes for men and women with T1D are l 1 and l 2, respectively. Then PðSjO; PÞ ¼ X1 Pðthe sibship size is kþpðsjthe sibship size is kþ k¼1 ¼ X1 l k 1 e l 1 X k k!ð1 e l 1 k¼1 Þ j¼1 ¼ X1 l k 1 e l 1 X k k!ð1 e l 1 k¼1 Þ j¼1 ¼ X1 k¼1 l k 1 e l 1 k!ð1 e l 1 ð1 ppþ k 1 Þ ¼ 1 e ppl 1 1 e l 1 : Pðthere are j affectedþpðsjthere are j affectedþ k! j!ðk jþ! p j ð1 pþ k j 1 ð1 pþ j

7 Therefore, we have PðPjO; SÞ PðPÞPðSjO; PÞ ¼ PðMjO; SÞ PðMÞPðSjO; MÞ ¼ PðPÞ PðMÞ 1 e ppl 1 1 e l 1 1 e ppl 2 1 e l 2 ¼ PðPÞ ð1 e ppl 1 Þð1 e l 2 Þ PðMÞ ð1 e ppl : 2 Þð1 e l 1Þ Thus, the ratio P(P)/P(M) will be further amplified by the ratio PðSjO; PÞ=PðSjO; MÞ if fathers with T1D have a higher fecundity than mothers with T1D. Clearly, the result would be more pronounced if the affection statuses in a sibship are correlated, which seem to be the case. It should be noted that the above result still holds approximately under the single ascertainment scheme. Ascertainment Through Parents The offspring of females with T1D could be less likely to be inflicted with T1D than those of T1D males if the risk of T1D increases with birth order and/or maternal age in conjunction with difference in fecundity. In other words, the higher prevalence of T1D in the offspring of fathers with T1D than in the offspring with mothers with T1D could be due to fewer progeny that T1D mothers have, if coupled with a birth order effect. Suppose, as before, the number of offspring has a Poisson distribution with mean l. Suppose also that each offspring is equally likely and independently to be affected with T1D with probability P. Therefore, if men with T1D and women with T1D have on average l 1 and l 2 offspring, respectively, then the male/female ratio of the number of offspring with T1D would be l 1 /l 2.Ifin addition we assume that a birth order effect exists, i.e., the later the birth order, the higher risk of having T1D, such that the risk of developing T1D is pg i 1 for the ith child, where g41 and i 1, and g is the risk ratio multiplier for offspring with later birth order as compared with the first-born, then the expected number of offspring affected with T1D born to a parent drawn at random from the population is 1 X 1 l k e l 1 e l p þ pg þþpg k 1 pe lðg 1Þ 1 ¼ k! ðg 1Þð1 e l Þ : k¼1 Therefore, the expected prevalence of T1D among the offspring born to a parent with T1D who has on average l children is p½e ðg 1Þl 1Š ðg 1Þð1 e l Þ l 1 e l Preferential Transmission of T1D 329 ¼ p½eðg 1Þl 1Š ; lðg 1Þ and the ratio of T1D prevalence in offspring born to T1D fathers and mothers would be PðOjPÞ PðOjMÞ ¼ ðel1ðg 1Þ 1Þl 2 ðe l 41: 2ðg 1Þ 1Þl 1 Misclassification Another potential source for generating apparent transmission bias is misclassification. Specifically, if one misclassifies gestational diabetes mellitus (GDM) as T1D, then one would also observe apparent transmission bias. To see this, let M 0 denote the event that the ascertained mother is diabetic, without

8 330 Guo and Tuomilehto specifying whether she has GDM or T1D. Suppose that in the sample of interest, a is the misclassification rate from GDM to T1D. There is no consensus that the offspring born to mothers with GDM have a higher risk of T1D than those born to fathers with T1D, but the risk is very likely lower. Therefore, we can assume that PðOjMÞ4PðOjM GDM Þ, where M GDM denotes the event that the ascertained mother actually has GDM, not T1D. Therefore, even there is no preferential transmission, i.e., PðOjMÞ ¼PðOjPÞ, we still have PðOjM 0 Þ¼ð1 aþpðojmþþapðojm GDM ÞoPðOjPÞ for a40. NUMERICAL RESULTS In many published results, it seems that women with T1D do have fewer offspring than their male counterparts, especially if the sample is taken from earlier birth cohorts. In fact, there are numerous reports that document the sex difference in fecundity. For example, the cohort studied by Warram et al. [1984] reported that among a group of T1D men and women diagnosed with T1D at the Joslin Diabetes Center during the period of , 88 men produced 244 offspring, 99 women produced 175 offspring, and the remaining 30 men and 41 women either did not attempt to have children or no infants born alive survived for 6 weeks [Warram et al., 1984, p. 149]. In this case, women with T1D have on average 175/140 ¼ 1.25 children, while their male counterparts have 244/118 ¼ 2.07 children. In other words, the fecundity of women with T1D, as measured by the number of offspring, was only 60% of that in their male counterparts. Tuomilehto et al. [1992] also reported that, in a more recent cohort, male patients with T1D had on average 2.3 offspring, while diabetic female patients had 1.8 offspring, a reduction of more than 20% in fecundity in females as compared with males. This variation in fecundity difference between diabetic males and females may well vary from population to population, but also may be attributable to the major improvements in diabetes care that have occurred in the last few decades [Dorman et al., 1999]. From Warram et al. [1984], we have l 1 ¼ 2.07 and l 2 ¼ Now, assuming p ¼ 1/100,000 and the ascertainment probability p ¼ 0:2, we have PðSjO; PÞ= PðSjO; MÞ 1:35. If p ¼ 0:05, PðSjO; PÞ=PðSjO; MÞ is still roughly In other words, if males with T1D have on average 2.07 offspring while their female counterparts have about 1.25 offspring, the sibships born to fathers with T1D are about 35% more likely to be ascertained under the multiple ascertainment scheme. Varying p from 1/100,000 to 50/100,000 does not materially alter the result at all. To determine the male/female ratio of T1D prevalence, we used T1D incidence data in Finland. One reason to use Finnish data is that, thanks to an array of national registries and record linkage systems, the Finnish T1D incidence data are highly reliable. The annual age- and sex-adjusted T1D incidence rate (per 100,000 per year) and resultant cumulative risks (per 100) are listed in Table II. From Table II, we can calculate that P(M)/P(P)=1.37. Therefore, we would observe, theoretically PðPjO; SÞ ¼ 1:37 1:35 ¼ 1:85: PðMjO; SÞ

9 Preferential Transmission of T1D 331 TABLE II. Annual Age- and Sex-Adjusted T1D Incidence Rates (per 100,000 per Year) and Cumulative Risks for T1D in Finland in Incidence rate (100,000/year) Cumulative risk (per 100) Age group Male Female Total Male Female That is, fathers of T1D children would be about 85% more likely to be T1D than mothers of T1D children, despite the fact there is no preferential transmission at all. Even if sex difference in fecundity is not this pronounced, e.g., l 1 =2.3 and l 2 =1.8, we would observe PðPjO; SÞ 1:37 1:19 1:63: PðMjO; SÞ These numbers are close to what has been published (Table I). If ascertainment is through the parents, then for fecundity data calculated from Warram et al. [1984], PðOjPÞ=PðOjMÞ would be approximately 1.14, 1.20, 1.26, and 1.68, for g=1.3, 1.4, 1.5, and 2, respectively. For fecundity data calculated from Tuomilheto et al. [1995], PðOjPÞ=PðOjMÞ would be 1.09, 1.12, 1.16, and 1.39, respectively. This is not far off from what we have observed (Table I), especially if we take the possibility of misclassification into account. DISCUSSION Since a genetic component undoubtedly plays an important role in the etiology of T1D, the preferential transmission of T1D is a phenomenon that urgently needs to be confirmed or refuted in order to further facilitate genetic studies of T1D and the mode of inheritance of this disease. In this study, we demonstrated theoretically other possible explanations for the observed sex-specific preferential transmission of the disease. First, when the study population is ascertained through affected offspring, the male preponderance in T1D prevalence as compared with females would generate a higher proportion of fathers than mothers having T1D of T1D offspring, even though there is no preferential transmission at all. This higher prevalence is further amplified by the greater likelihood of being selected and/or recruited for study in families with T1D fathers due to a fecundity difference. Second, when the study population is ascertained through affected parents, misclassification of mothers with GDM as T1D, and the existence of male/female difference in fecundity in conjunction with birth order effect, can each and/or in combination contribute to the observed preferential transmission, even though there is none.

10 332 Guo and Tuomilehto How likely are the assumptions used in the above theoretical reasoning? Male preponderance of T1D prevalence appears to be true in many European populations [Karvonen et al., 1997], and this sex difference appears to be more pronounced when men and women enter into a reproductively active period, which, in itself, is a mystery yet to be solved. While several studies [e.g., Warram et al., 1984; Tuomilehto et al., 1992] provide evidence for a fecundity difference between T1D males and females, one recent study by Dorman et al. [1999] clearly demonstrates that women with T1D have higher spontaneous abortion rates than those without T1D, especially when older cohort of women are studied. This suggests that, at least for older cohorts, due to a higher likelihood of adverse pregnancy outcomes than in T1D-free women, the fecundity of women with T1D may be diminished as compared with their male counterparts. Women typically and traditionally have greater parental investment in bearing and raising children than men do. Several studies reported that older maternal age is a risk factor for T1D in the offspring [Flood et al., 1982; Wagener et al., 1983; Warram et al., 1991]. Since birth order correlates highly with maternal age, it is possible that increasing birth order is also a risk factor. This seems to be supported by some studies [e.g., Tuomilehto et al., 1995] but not by others [Wagener et al., 1982; Bingley et al., 2000; but see criticism of Bingley et al., 2000 in Patterson et al., 2001]. The assumption employed in the statistical modeling that the T1D affection statuses of siblings are independent is obviously not realistic. Given ample evidence of the involvement of both genetic and environmental factors, affection statuses are most likely to be dependent, since siblings obviously share genes and some environmental risk factors. This assumption is only for mathematical convenience. Since the dependence of affection statuses among siblings generally results in familial aggregation of the disease, the results of statistical modeling should be considered conservative, in the sense that they would be more pronounced for real data. It also should be noted that, while this paper only considered the effect of a single factor on transmission patterns of T1D from parents to offspring, it is possible that more than one factor can be involved to produce the apparent preferential transmission. For example, misclassification of GDM as T1D, coupled with a fecundity difference and birth order effect, could possibly produce the apparent preferential transmission. It should be noted that in all published papers that reported preferential transmission of T1D, none, to our knowledge, ever examined rigorously the effect of ascertainment bias due to differential fecundity and birth order effect, and the sex difference in T1D prevalence as a possible explanation for observed preferential transmission. The misclassification issue has been properly addressed in very few studies. In light of the analysis presented here, the plausibility of assumptions employed in the analysis, and, in particular, an apparent failure to critically examine the effects of these bias-causing factors in previous studies, it is perhaps prudent to say that the jury for the existence of preferential transmission in T1D is still out. In order to resolve the issue of preferential transmission, it is preferable to ascertain study population through affected parents, and it is necessary to eliminate or minimize misclassification, and to examine differential fecundity between the two sexes and birth order effect. If both fecundity difference and birth order effect exist, certain adjustments or corrections should be made.

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