Genetic Epidemiology 9:123-139 (1992)
Linkage Studies of Schizophrenia: A Simulation Study of Statistical Power Wei J. Chen, Stephen V. Faraone, and Ming T. Tsuang Department of Epidemiology, Harvard School of Public Health, Boston, Massachusetts (W.J.C., M.T.TJ, Section of Psychiatric Epidemjology and Genetics, Department of Psychiatry, Harvard Medical School, Boston, Massachusetts (S.V.F., M. T.T.), and Psychiatry Service, Brockton-West Roxbury Veterans Affairs Medical Center, Brockton, Massachusetts (W.J.C., S. V.F., M. T.T.) In planning for a linkage study, it is important to determine the number of pedigrees needed to show linkage. Our study overcomes some of the limitations of previous power studies by simulating multigeneration pedigrees to be compatible with the demographic and genetic epidemiological features of schizophrenia; these are variable age at onset, reduced fertility, and increased mortality after onset. We evaluate the power of these pedigrees by first simulating an ascertainmentrule requiring at least three ill family members per pedigree and then simulating the trait and marker genotypes according to a single gene model known to fit epidemiological family study data. Our analysis allows for incomplete and age-dependent penetrance, phenocopies, and interpedigree heterogeneity. We present the power to detect linkage at several lod score thresholds since the multiple tests and phenotypic models required for complex diseases may necessitate using a lod score significance level greater than three. The sample size needed to achieve sufficient power is feasible if 50% of the pedigrees are linked to the marker under test. It may not be feasible to detect linkage if only 25% of the pedigrees are linked, even if a very closely linked marker is used. Our results indicate that to be certain of adequate statistical power, linkage analyses of schizophrenia will require very large samples that do not have a marked degree of genetic heterogeneity. o 1992 Wiley-Liss, Inc. Key words: single gene model, lod score, pedigree, genetic heterogeneity
INTRODUCTION In planning for a linkage study, we must determine the number of pedigrees needed to show linkage. Although power calculations are relatively straightforward for simple Received for publication September 3, 1991;revision accepted January 28, 1992. Address reprint requests to Ming T. Tsuang, M.D., Ph.D., D.Sc., Psychiatry Service (116A), Veterans Affairs Medical Center, 940 Belmont Street, Brockton, MA 02401.
0 1992 Wiley-Liss, Inc.
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Mendelian diseases, less is known about power for complex diseases such as schizophrenia and other psychiatric illnesses. Many investigators have addressed the statistical power of linkage methods that do not require knowledge of the genetic parameters of a disease, such as affected sib pair or relative pair methods [Blackwelder and Elston, 1985; Chakravarti et al., 1987; Gershon and Goldin, 1987; Goldin and Gershon, 1988; Bishop and Williamson, 1990; Risch, 19901. However, since these robust methods may have less power than the lod score method, a careful examination of the latter method is warranted. Early power studies of the lod score method assumed Mendelian modes of transmission with complete penetrance and used simple pedigree structures such as nuclear families of a fixed size. These investigations then analytically calculated the expected lod score [Thompson et al., 1978; Elston and Bonney, 1984; Skolnick et al., 1984; Ott, 1985; Cavalli-Sforza and King, 1986; Wong et al., 19861. Kidd [1987] presented power estimates based on the number of fully informative gametes in a sample rather than the pedigree itself. Subsequent studies allowed for more flexible models of disease transmission but pedigree structures were still assumed to be simple and fixed [Martinez and Goldin, 1989; Leboyer et al., 19901. Although the available power studies of the lod score method provide instructive guidelines for the planning of experiments, the pedigrees used for linkage analysis are usually more complex than those assumed in these studies. Since extended pedigrees can provide more linkage information than a combination of their constituent nuclear families [Thompson et al., 19781, considering power for nuclear families only limits the utility of a power study. A simulation approach to evaluate the power of linkage analysis for pedigrees of extended structure was initially proposed by Boehnke [ 19861 and was subsequently generalized to handle reduced penetrance, phenocopies, and heterogeneity [Ott, 1989; Ploughman and Boehnke, 19891. Using simulation methods, several investigators have evaluated the expected lod scores or power to detect linkage for a few ascertained pedigrees [Gershon and Goldin, 1987; Kaufmann et al., 19891 or for pedigrees with fixed structures [Martinez and Goldin, 19901. Previous power studies usually assumed a significance level of 0.05 (for affected relative pairs) or a lod score threshold of 3.0. For simple genetic diseases, the increase in type I error due to multiple markers will be counteracted by the increased prior probability of linkage [Ott, 19851. Thus the threshold of 3.0 is appropriate if the number of tested markers is less than 100 and the threshold for more than 100 markers is easily computed [Ott, 19851. However, as argued by Clerget-Darpoux et al. [1990] and Green [ 19901, since the prior probability of linkage is undefined for complex diseases, a correction for multiple comparisons is still necessary. As shown by ClergetDarpoux et al. [ 19901, the type I error rate at a lod score of 3 could increase 300-fold by a combination of 4 penetrances, 4 phenotype definitions, and 40 marker tests, a practice which is not uncommon in linkage analysis of complex diseases. As a result, the lod score threshold of significance needs to be adjusted upward. The magnitude of this adjustment is difficult to calculate, although we may approximate it by adding log{m} to the lod score criterion if m different tests were made, as suggested by Kidd and Ott [1984]. An alternative is to assess the inflation of the lod score by simulation after analyses are completed [Weeks et al., 19901. For many psychiatric illnesses, the demographic and genetic epidemiological features of illness influence the availability and structure of pedigrees. For example, schiz-
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ophrenic patients have increased mortality and lower marriage rates compared with the general population. Thus, to plan for adequate power prior to data collection, it may be useful to use simulated pedigrees. In this paper we use this approach by simulating multigeneration pedigrees to be compatible with the demographic and genetic epidemiological features of schizophrenia. We then evaluate the power of these pedigrees by simulating the trait and marker genotypes according to the same genetic model used in the simulation of pedigrees. In addition, we present the power to detect linkage at several lod score thresholds since the multiple tests and phenotypic models required for complex diseases may necessitate using a lod score significance level greater than three. METHODS Simulation of Pedigrees
An overview of our procedure for simulatingpedigrees is as follows: (1) we assumed that schizophrenia is caused by a dominant gene with a known age-dependent penetrance; (2) starting with schizophrenic probands, we generated all their first and second degree relatives using sibship sizes and ages that are compatible with population data; (3) we assigned genotypes to each individual according to our genetic model of schizophrenia; (4) we assigned phenotypes to all pedigree members based on their ages and genotypes; ( 5 ) after determining each individual’s mortality status, we further assumed that some living pedigree members refuse to participate; (6) pedigrees were extended to second degree relatives if they or their siblings were affected; (7) we also assumed that some of the cell lines are lost to experimental error. The algorithms that simulated the pedigrees were written in FORTRAN and were executed on an IBM 3090 computer. Our simulation used single gene parameters that provide an adequate fit to systematically collected family data [Matthysse et al., 1986; Holzman et al., 19881. This model specifies that the transmission of schizophrenia is due to a dominant ill allele a with a gene frequency of 0.0223. The genotype specific penetrances of AA, Aa, and aa are 0.1896, 0.1896, 0.0017, respectively. The population prevalence of schizophrenia predicted by this model is 0.01; 16% of cases are expected to be phenocopies. This single major gene model satisfactorily accounts for the observed morbidity risk among first, second, and third degree relatives. The simulation of each pedigree started with a schizophrenic proband. We then extended that pedigree by generating the proband’s siblings, parents, offspring, nephews, nieces, uncles, aunts, and grandparents. To simulate sibship size we used the truncated negative binomial distribution that had been fitted to the population data of the United States in 1950 [Brass, 1958; Cavalli-Sforza and Bodmer, 19711. The parameter values are n = 2.84, P = 0.93, and mean = 2.6. We used the 1980 vital statistics on marriage in the United States to estimate the probability of marriage for pedigree members 30 years of age and older. For pedigree members without schizophrenia we assumed that 93% of females and 88% of males are married if their ages are equal to or older than 30 years. Since schizophrenic patients have a decreased probability of marriage, which is the main cause of their reduced fertility [Haverkamp et al., 19821, we assumed that the probability of marriage for schizophrenic pedigree members was 46.5 and 44% for females and males, respectively. We did not simulate marriage for individuals younger than 30 be-
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cause their children would be too young to be informative for the diagnosis of schizophrenia. Since there is increased mortality after the onset of schizophrenia, the age distribution of schizophrenic patients differs from that of the general population. We applied the method of Heimbuch et al. [ 19801 to model this altered age distribution. Assuming that the general population is in a stable age equilibrium with constant mortality and fertility rates and that the age distribution, c b ) , is proportional to e-KYly,the density function for age is c(y) = N1,,e.-RY
The normalizing constant, N , makes the integral of c(y) equal to one; Zy is the survival function from the life table for the population, and R is the natural intrinsic growth rate. For schizophrenics, we replaced l,, by a conditional survival function, rnblx), where x is the age at onset. According to follow-up studies of the mortality in schizophrenic patients [Tsuang and Woolson, 1978; Brook, 1985; Simpson, 19881, the increased mortality of schizophrenic patients is most prominent in the first decade after onset and only slightly higher in subsequent years. Thus, we approximated the increased mortality of schizophrenic patients by a standardized mortality ratio of 2.5 in the first decade and 1.25 afterward. We obtained rnO,Ix) as follows. For an age at onset belonging to the 5-year interval starting at x , we multiply the number of deaths in the life table by 2.5 for the age intervals x to x 5 and x + 5 to x 10, and by 1.25 for subsequent age intervals. We then derived rn(ylx) by subtracting the newly calculated number of deaths from the number alive at birth incrementally by 5-year interval until the interval that contains y. We generated the proband’s age at onset from a lognormal distribution with a p parameter of 3.25 and a CJ parameter of 0.50. This corresponds to a median age at onset of 25.8. Using the conditional survival distribution for the generated age at onset, we generated the proband’s current age. If the proband’s current age was greater than or equal to his age at onset, we continued to generate the pedigree. We simulated the ages of a proband’s siblings by randomly assigning a birth order to the proband using a uniform distribution to generate random numbers and assuming that the ages of successive siblings in a family are 1 or 2 years apart with equal probability. The difference between the father’s age and the oldest child’s age is a random integer between 20 and 35. The age of the proband’s mother was set to be within 3 years of the age of the proband’s father. These rules were also applied to each nuclear family in the pedigree to determine the ages of offspring, nephews, nieces, uncles, aunts, and grandparents. We simulated the proband’s genotype based on the genetic model and the assumption of Hardy-Weinberg equilibrium. Given the proband’s genotype, there are several possible pairs of parental genotypes; however, rare genotype combinations were omitted from consideration because they do not substantially affect the results and markedly increase the computer time needed for the simulation. For example, according to our dominant gene model for schizophrenia, the genotype of a proband could be aa, aA, or AA with probabilities of 0.009,0.828, and 0.163, respectively. If the genotype d)or (d, AA); the of a proband was aA, the genotypes of the parents could be (d, rare combinations (aa, AA) and (aa, aA) were not considered. Since the a allele is the
+
+
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disease gene, bilineal pedigrees such as (uA, uA) are allowed in our simulations. After the parents’ genotypes were assigned, the genotypes of the siblings were determined by Mendelian laws of transmission. We simulated the genotypes of other relatives by following the same principles. The genotypes of people marrying into the pedigree were determined by the population gene frequency and the assumption of Hardy-Weinberg equilibrium. The phenotypes of the relatives were determined from their ages, ages at onset, and genotypes. For each relative, we generated an age at onset according to the lognormal distribution used for the probands. A relative was designated as ill if his or her age at onset was less than or equal to his or her current age and a random number generated from a uniform distribution was less than or equal to the penetrance of his or her genotype. The genotype specific penetrances of parents and grandparents of probands decrease as a result of the reduced fertility associated with schizophrenia. The risk to siblings, uncles, aunts, and offspring are minimally affected [Risch, 1983;Kendler and MacLean, 19891. Following Kendler and MacLean [ 19891, the genotype specific proportion affected (GSPA) for parents and grandparents was approximated by
GSPA
=
FQ FQ+l-Q
where F is the relative fitness of affected individuals and Q is the genotype-specific penetrance. Since we assumed F = 0.5, the modified penetrance vector for parents and grandparents of probands was (0.1047,O.1047,0.0009). After we generated the structure of the pedigree and the phenotypes of its members, we simulated three additional factors: the mortality status of each pedigree member, the availability of living members, and the branches of the family selected for linkage analysis. The mortality status of relatives was a function of their current ages, phenotypes, and corresponding survival functions. The mortality status of unaffected relatives was determined by their current age and survival function of the United States in 1980. For affected relatives, this survival function was conditioned on their ages at onset as discussed above. After hving simulated them as being alive, first and second degree relatives were simulated to be available with probabilities of 0.95 and 0.70, respectively. We selected branches of the simulated pedigree for inclusion in the linkage analysis as follows. All first degree relatives of a proband were included. Cox et al. [1988] have shown that including information on unaffected second degree relatives adds little to linkage studies of affected individuals and their first degree relatives. Thus, second degree relatives were included only if at least one member of their sibship was affected or one of the parents of the sibship was affected. For example, if an unaffected sibling of the proband did not produce any affected offspring, his or her spouse and offspring were not included in the pedigree. In contrast, the nuclear family of affected siblings were included even if none of their offspring was affected. However, in a large scale pedigree study, occasional failures of cell lines occur. Thus the power that a pedigree can provide for linkage analysis will be less than that expected by the initial number of blood samples collected. In this study we simulated a 5% rate of failure for each cell line regardless of each individual’s phenotype.
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Simulation of Power
After simulating a pedigree, we selected it for the power analysis if at least three members of the pedigree members were ill and willing to participate. We continued simulating pedigrees until 200 pedigrees meeting this criterion had been created. We then assessed the power of these pedigrees to detect linkage by using version 4.1 of the program SIMLINK [Ploughman and Boehnke, 19891. Although we know the trait genotypes of our pedigree members, these cannot be used in the power simulation because they would not be known in a real linkage analysis. Therefore, genotypes must be simulated in the power analysis. As discussed by Ploughman and Boehnke [ 19891, an unbiased simulation of trait genotypes requires calculations of the conditional probabilities of genotypes given phenotypes. If we randomly assign a genotype vector to a pedigree based on the conditional probabilities of all possible genotypes given the phenotypes of that pedigree, we can perform an unbiased simulation of the genotypes that is consistent with the observed phenotypes. Thus SIMLINK needs only information about phenotype, age, and pedigree structure from generated pedigrees. To simulate genotypes, SIMLINK requires the genetic parameters of the disease locus, including the gene frequency, penetrance vector, and age at onset distribution. For the power analysis we used the same genetic parameters that had been used to generate the pedigrees. After generating trait genotypes, SIMLINK generates marker genotypes according to a given marker allele frequency, Mendelian transmission, and the recombination fraction (0) between trait and marker. Since highly polymorphic markers such as variable number of tandem repeats (VNTR) [Nakamura et al., 1987, 19881 and GT repeats [Litt and Luty, 1989; Weber and May, 19891 are now available, it is realistic to assume a codominant marker of 4 alleles with equal frequency. The polymorphism information content (PIC) value of this marker is equal to 0.70 according to the formula of Botstein et al. [1980]. In our simulations, male and female recombination fractions are assumed to be equal. To compute the statistical power under genetic heterogeneity, SIMLINK allows one to specify that only a proportion of (Y of pedigrees is linked to the marker. After simulating trait and marker genotypes, the likelihood of the pedigrees is computed and maximized over either 0 only (the single-parameter test) or both 0 and cy (the two-parameter test) to get the maximum lod score. For a user specified number of replications, SIMLINK simulates the cosegregation of the disease locus with a marker locus under the assumption of a true recombination fraction r . For each replication, SIMLINK computes the lod scores Z(Oi;r>,i = 1 to 7, over a range of test recombination fractions, 8 (e.g., 0.01, 0.05, 0.10, 0.20, 0.30, 0.40,0.50). We modified SIMLINK to output the Z(0,;r)for each replication for every pedigree and store them for later summation. In this manner we can assess the power of different numbers of pedigrees without repeatedly running SIMLINK on the same pedigrees. The joint linkage information in a set of N pedigrees was calculated as follows. For each replication, we summed the lod scores Z(Bi; r ) over N pedigrees and then determine the approximate maximum summed lod score, Z*(r), for that replication. After 200 replications, each set of pedigrees was characterized by a distribution of 200 maximum summed lod scores. This distribution provides the expected maximum summed lod score for the pedigrees and, for a specified lod score threshold of declaring linkage, the relative frequency that the maximum summed lod score exceeds the threshold (i.e., the power of the linkage test).
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In total, we ran SIMLINK over the 200 generated pedigrees and stored all lod scores calculated for 200 replications of each pedigree. Thus there was a total of 40,000 simulation trials. Given that each pedigree has been replicated 200 times, the power estimate obtained from small subsamples of the 200 generated pedigrees was very stable. For example, to estimate the power of 10 pedigrees, we randomly sampled 10 pedigrees from these 200 pedigrees and repeated this without replacement 20 times. The final estimate is obtained by averaging the powers of the 20 samples of 10 pedigrees. The standard errors of the power estimates using various lod score thresholds ranged from 0 to 0.005. The powers of other number of pedigrees (Ns ranging from 20 to 200 in increments of 10) were calculated similarly although the number of repetitions of sampling was decreased as the number of pedigrees increased. The standard errors of these power estimates were consistently small. All of the standard errors were less than 0.05; 60% were less than 0.02. When only a proportion, a , of pedigrees is linked, there are two ways to test for linkage. Following the modeling of Smith [1976], the likelihood for a set of families can be expressed as follows:
where Li(8) is the likelihood of observing the ith family and the product is taken over all families. The two ways to test for linkage are the single-parametertest, which assumes homogeneity( a= l), andthe two-parametertest,whichassumes heterogeneity (O3). Of course, excellent power to exclude linkage could be a liability if the probability of exclusion remained high when a fraction of the pedigrees were truely linked. Unfortunately, our simulations indicate that the probability of erroneously excluding linkage is nonnegligible, even when a =0.5. These results suggest that when genetic heterogeneity is possible, we should be cautious in interpreting lod scores less than - 2. Our results are consistent with the work of Martinez and Goldin [ 19891who determined power analytically for nuclear families with fixed structures. Compared with that study, we assumed a much lower penetrance (0.189 vs. 0.5), a higher phenocopy rate (16 vs. 5 % ) , and a less informative marker (PIC=O.7 vs. 1.0). Nevertheless, for a given level of power, the sample size needed under our assumptions is not much greater than that reported by Martinez and Goldin. For example, consider the sample size needed to attain 50% power using a tightly linked marker ( r = 0 for our study and 0.01 for Martinez and Goldin) when a is 50%. Martinez and Goldin [1989] showed that any one of the following three types of families meet this requirement: (1) 101 nuclear family of sibship size 3 with at least 1 affected child (202 of 505 individuals affected), (2) 53 nuclear family of sibship size 3 but with at least 2 affected children
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(146 of 265 individuals affected), or ( 3 ) 41 nuclear families of sibship size 4 with at least 2 affected children (117 of 246 individuals affected). In our simulation, 50 multigeneration pedigrees with at least 3 affected individuals were needed to achieve the same level of power. These pedigrees contain approximately 150 affected individuals and 500 subjects in total. Despite the differences in total number of subjects, the number of affected is similar between these two studies. It may be that our simulated ascertainment procedure is fairly robust to conditions of low penetrance and high phenocopies. In part this is due to not limiting ascertainment to the nuclear family. It is well known that extended multiplex pedigrees can help to resolve the linkage phase and thus provide more linkage information than a combination of nuclear families with the same total number of individuals [Thompson et al., 19781. Although it is obvious that requiring three ill relatives per family handles the problem of low penetrance, it also appears to have reduced the proportion of phenocopies in the sample. What is perhaps most notable in our comparison with Martinez and Goldin [1989] is that the number of affected individuals required to achieve the same power is very similar. This suggests that the contribution of unaffected individuals to the linkage analysis is limited. Indeed, Goldin and Martinez [1989] have demonstrated that ignoring marker and disease loci information of unaffected individuals in phase unknown nuclear families results in only a small increase in the average sample size needed to detect linkage. Whether their result can be generalized to our simulated pedigrees remains to be tested. Because our results of power to detect linkage are based on simulated pedigrees, we can generalize them to data that would be systematically collected in an epidemiological study following our ascertainment scheme. Our simulation also provides some information about the probability of ascertaining informative pedigrees of schizophrenic patients. This would not be possible if we had simulated power using published pedigrees. Only 9.8% of simulated patients had available families with at least 3 affected individuals. This ascertainment probability is comparable to that reported by empirical studies using family study method [Kendler et al., 19851 or family history method [Pulver and Bale, 19891. It is also consistent with other simulation studies [McGue and Gottesman, 19891. Our results are limited by the genetic and demographic parameters used to generate the pedigrees and to perform the linkage analyses. We believe that our choices for demographic parameters (sibship size, age at onset, age, marriage, increased mortality, and reduced fertility after onset) are reasonable. They are based on epidemiological data and have produced pedigree structures that are similar to those reported in the literature. Nevertheless, further work would be useful to determine the degree of power change across relevant portions of the genetic and demographic parameter space. It would also be useful to examine how different ascertainment strategies affect power. Our simulations of power under conditions of genetic heterogeneity assumed that except for the recombination fraction, the parameters of genetic transmission were identical for the linked and unlinked families. However, as shown by Martinez and Goldin [ 19901, heterogeneity due to two independent loci complicates the assessment of the power to detect linkage. Our simulations also did not examine how multipoint analyses, model misspecification, and laboratory error would affect statistical power. These questions have been addressed by other investigators [Lathrop et al., 1984; Lander and Botstein, 1986; Martinez and Goldin, 1989; Risch and Giuffra, 19901 (multipoint
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analyses), [Clerget-Darpoux et al., 1986; Greenberg and Hodge, 1989; Martinez et al., 1989; Weeks et al., 1990; Williamson and Amos, 19901 (model misspecification), and [Tenvilliger et al., 19901 (laboratory error). In summary, we have investigated the number of pedigrees needed to detect linkage between a dominant schizophrenia gene and a highly polymorphic DNA marker. By incorporating demographic parameters into our simulation of pedigrees, we have presented a useful approach to the investigation of power for linkage studies of complex diseases. By modeling the structure of pedigrees according to both the demographic and genetic parameters of a disease, one can derive a reasonable estimate of the power to be expected before a study is performed. When pedigrees with at least three affected members are ascertained, the sample size needed to achieve sufficient power is feasible if 50% of the pedigrees are linked to the marker under test. It may not be feasible to detect linkage if only 25% of the pedigrees are linked, even if a very closely linked marker is used. Our results indicate that to be certain of adequate statistical power, linkage analyses of schizophrenia will require very large samples that do not have a marked degree of genetic heterogeneity. ACKNOWLEDGMENTS The authors wish to acknowledge Dr. Endel John Orav for his insightful help in design of simulation programming, and Dr. Chung-Cheng Hsieh for his comments on our earlier manuscript. We also thank four anonymous reviewers for their constructive comments. Preparation of this article was supported in part by the Veterans Administration’s Medical Research and Health Services Research and Development Programs and National Institute of Mental Health Grants 1 R01MH4 1879-01 ,5 UO 1 MH46318-02, and 1 R37MH43518-01 to Dr. Ming Tsuang. This manuscript was prepared while Dr. Tsuang was a Fellow at the Center for Advanced Study in the Behavioral Sciences. We are grateful for financial support provided to the Center by the John D. and Catherine T. MacArthur Foundation and the Foundations Fund for Research in Psychiatry Endowment. REFERENCES Aschauer HN, Aschauer-Treiber G, Isenberg KE, Todd RD, Knesevich MA, Garver DL, Reich T, Cloninger CR (1990): No evidence for linkage between chromosome 5 markers and schizophrenia. Hum Hered 40:109-115. Bishop DT, Williamson JA (1990); The power of identity-by-state methods for linkage analysis. Am J Hum Genet 46:254-265. Blackwelder WC, Elston RC (1985): A comparison of sib-pair linkage tests for disease susceptibility loci. Genet Epidemiol2:85-97. Boehnke M (1986): Estimating the power of a proposed linkage study: A practical computer simulation approach. Am J Hum Genet 3 9 5 13-527. Botstein D, White RL, Skolnick M, Davis RW (1980): Construction of a genetic linkage map in man using restriction fragment length polymorphisms. Am J Hum Genet 32:314-331. Brass W (1958): The distribution of births in human populations. Popul Stud 1251-72. Brook OH (1985): Mortality in the long-stay population of Dutch mental hospitals. Acta Psychiatr Scand 7 1:626-635. Cavalli-Sforza LL, Bodmer WF (197 1): “The Genetics of Human Populations.” San Francisco: Freeman. Cavalli-Sforza LL, King MC (1986): Detecting linkage for genetically heterogeneous diseases and detecting heterogeneity with linkage data. Am J Hum Genet 38399-616.
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Martinez M, Khlat M, Leboyer M, Clerget-Darpoux F (1989): Performance of linkage analysis under misclassification error when the genetic model is unknown. Genet Epidemiol6:253-258. Martinez MM, Goldin LR (1989): The detection of linkage and heterogeneity in nuclear families for complex disorders: one versus two loci. Am J Hum Genet 44:552-559. Matthysse S , Holzman PS, Lange K (1986): The genetic transmission of schizophrenia: Application of Mendelian latent structure analysis to eye tracking dysfunctions in schizophrenia and affective disorder. J Psychiatr Res 2057-76. McGue M, Gottesman I1 (1989): Genetic linkage in schizophrenia: Perspectives from genetic epidemiology. Schizophr Bull 15:453-464. McGuffin P, Sargeant M, Hetti G, Tidmarsh S, Whatley S, Marchbanks RM (1990): Exclusion of a schizophrenia susceptibility gene from the chromosome 5qll-q13 region: New data and a reanalysis of previous reports. Am J Hum Genet 47524-535. Nakamura Y , Carlson M, Krapcho D, Kanamori M, White R (1988): New approach for isolation of VNTR markers. Am J Hum Genet 435354-859. Nakamura Y, Eppert M, O’Connell P, Wolff R, Holm T, Culver M, Kumlin E, White R (1987): Variable number of tandem repeat (VNTR) markers for human gene mapping. Science 235:1616-1622. Ott J (1985): “Analysis of Human Genetic Linkage.” Baltimore: The Johns Hopkins University Press. Ott J (1989): Computer-simulation methods in human linkage analysis. Proc Natl Acad Sci USA 86: 4175-4178. Ploughman LM, Boehnke M (1989): Estimating the power of a proposed linkage study for a complex genetic trait. Am J Hum Genet 44543-55 1. Pulver AE, Bale SJ (1989): Availability of schizophrenic patients and their families for genetic linkage studies: Findings from the Maryland epidemiology sample. Genet Epidemiol6:671-680. Risch N (1983): Estimating morbidity risks in relatives: The effect of reduced fertility. Behav Genet 13~441-451. Risch N (1989): Linkage detection tests under heterogeneity. Genet Epidemiol6:473-480. Risch N (1990): Linkage strategies for genetically complex traits. 11. The power of affected relative pairs. Am J Hum Genet 46:229-241. Risch N, Giuffra L (1990): Multipoint linkage analysis of genetically complex traits. (Abstract). Am J Hum Genet 47:A197. Shemngton R, Brynjolfsson J, Petursson H, Potter M, Dudleston K, Barraclough B, Wasmuth J, Dobbs M, Curling H (1988): Localization of a susceptibility locus for schizophrenia on chromosome 5 . Nature (London) 336: 164-167. Simpson JC (1988): Mortality studies in schizophrenia. In Tsuang MT, Simpson JC (eds.): “Nosology, Epidemiology and Genetics of Schizophrenia. ” Amsterdam: Elsevier, pp. 245-273. Skolnick MH, Bishop DT, Cannings C, Hasstedt SJ (1984): The impact of RFLPs on human gene mapping. In Rao DC, Elston RC, Kuller LH, Feinleib M, Carter C, Havlik R (eds.): “Genetic Epidemiology of Coronary Heart Disease: Past, Present, and Future.” New York: Alan R. Liss, pp. 271-292. Smith CAB (1976): Statistical resolution of genetic heterogeneity in familial disease. Ann Hum Genet 39:281-291. St. Clair D, Blackwood D, Muir W, Baillie D, Hubbard A, Wright A, Evans HJ (1989): No linkage of chromosome 5qll-ql3 markers to schizophrenia in Scottish families. Nature (London) 339:305-309. Tenvilliger JD, Weeks DE, Ott J (1990): Laboratory errors in the reading of marker alleles cause massive reductions in lod score and lead to gross overestimates of the recombination fraction. (Abstract). Am J Hum Genet 47:A201. Thompson EA, Kravitz K, Hill J , Skolnick MH (1978): Linkage and the power of a pedigree structure. In Morton NE, Chung CS (eds.): “Genetic Epidemiology.” New York Academic Press, pp. 247-253. Tsuang MT, Woolson RF (1978): Excess mortality in schizophrenia and affective disorders. Arch Gen Psychiat 35: 1181-1 185. Weber JL, May PE (1989): Abundant class of human DNA polymorphisms which can be typed using the polymerase chain reaction. Am J Hum Genet 44:388-396. Weeks DE, Lehner T, Squires-Wheeler E, Kaufmann C, Ott J (1990): Measuring the inflation of the lod score due to its maximization over model parameter values in human linkage analysis. Genet Epidemiol 71237-243.
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Williamson JA, Amos CI (1990): On the asymptotic behavior of the estimate of the recombination fraction under the null hypothesis of no linkage when the model is misspecified. Genet Epidemiol7:309-3 18. Wong FL, Cantor RM, Rotter JI (1986): Sample-size considerations and strategies for linkage analysis in autosomal recessive disorders. Am J Hum Genet 39:25-37.
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Linkage Studies of Schizophrenia: A Simulation Study of Statistical Power Wei J. Chen, Stephen V. Faraone, and Ming T. Tsuang Department of Epidemiology, Harvard School of Public Health, Boston, Massachusetts (W.J.C., M.T.TJ, Section of Psychiatric Epidemjology and Genetics, Department of Psychiatry, Harvard Medical School, Boston, Massachusetts (S.V.F., M. T.T.), and Psychiatry Service, Brockton-West Roxbury Veterans Affairs Medical Center, Brockton, Massachusetts (W.J.C., S. V.F., M. T.T.) In planning for a linkage study, it is important to determine the number of pedigrees needed to show linkage. Our study overcomes some of the limitations of previous power studies by simulating multigeneration pedigrees to be compatible with the demographic and genetic epidemiological features of schizophrenia; these are variable age at onset, reduced fertility, and increased mortality after onset. We evaluate the power of these pedigrees by first simulating an ascertainmentrule requiring at least three ill family members per pedigree and then simulating the trait and marker genotypes according to a single gene model known to fit epidemiological family study data. Our analysis allows for incomplete and age-dependent penetrance, phenocopies, and interpedigree heterogeneity. We present the power to detect linkage at several lod score thresholds since the multiple tests and phenotypic models required for complex diseases may necessitate using a lod score significance level greater than three. The sample size needed to achieve sufficient power is feasible if 50% of the pedigrees are linked to the marker under test. It may not be feasible to detect linkage if only 25% of the pedigrees are linked, even if a very closely linked marker is used. Our results indicate that to be certain of adequate statistical power, linkage analyses of schizophrenia will require very large samples that do not have a marked degree of genetic heterogeneity. o 1992 Wiley-Liss, Inc. Key words: single gene model, lod score, pedigree, genetic heterogeneity
INTRODUCTION In planning for a linkage study, we must determine the number of pedigrees needed to show linkage. Although power calculations are relatively straightforward for simple Received for publication September 3, 1991;revision accepted January 28, 1992. Address reprint requests to Ming T. Tsuang, M.D., Ph.D., D.Sc., Psychiatry Service (116A), Veterans Affairs Medical Center, 940 Belmont Street, Brockton, MA 02401.
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Mendelian diseases, less is known about power for complex diseases such as schizophrenia and other psychiatric illnesses. Many investigators have addressed the statistical power of linkage methods that do not require knowledge of the genetic parameters of a disease, such as affected sib pair or relative pair methods [Blackwelder and Elston, 1985; Chakravarti et al., 1987; Gershon and Goldin, 1987; Goldin and Gershon, 1988; Bishop and Williamson, 1990; Risch, 19901. However, since these robust methods may have less power than the lod score method, a careful examination of the latter method is warranted. Early power studies of the lod score method assumed Mendelian modes of transmission with complete penetrance and used simple pedigree structures such as nuclear families of a fixed size. These investigations then analytically calculated the expected lod score [Thompson et al., 1978; Elston and Bonney, 1984; Skolnick et al., 1984; Ott, 1985; Cavalli-Sforza and King, 1986; Wong et al., 19861. Kidd [1987] presented power estimates based on the number of fully informative gametes in a sample rather than the pedigree itself. Subsequent studies allowed for more flexible models of disease transmission but pedigree structures were still assumed to be simple and fixed [Martinez and Goldin, 1989; Leboyer et al., 19901. Although the available power studies of the lod score method provide instructive guidelines for the planning of experiments, the pedigrees used for linkage analysis are usually more complex than those assumed in these studies. Since extended pedigrees can provide more linkage information than a combination of their constituent nuclear families [Thompson et al., 19781, considering power for nuclear families only limits the utility of a power study. A simulation approach to evaluate the power of linkage analysis for pedigrees of extended structure was initially proposed by Boehnke [ 19861 and was subsequently generalized to handle reduced penetrance, phenocopies, and heterogeneity [Ott, 1989; Ploughman and Boehnke, 19891. Using simulation methods, several investigators have evaluated the expected lod scores or power to detect linkage for a few ascertained pedigrees [Gershon and Goldin, 1987; Kaufmann et al., 19891 or for pedigrees with fixed structures [Martinez and Goldin, 19901. Previous power studies usually assumed a significance level of 0.05 (for affected relative pairs) or a lod score threshold of 3.0. For simple genetic diseases, the increase in type I error due to multiple markers will be counteracted by the increased prior probability of linkage [Ott, 19851. Thus the threshold of 3.0 is appropriate if the number of tested markers is less than 100 and the threshold for more than 100 markers is easily computed [Ott, 19851. However, as argued by Clerget-Darpoux et al. [1990] and Green [ 19901, since the prior probability of linkage is undefined for complex diseases, a correction for multiple comparisons is still necessary. As shown by ClergetDarpoux et al. [ 19901, the type I error rate at a lod score of 3 could increase 300-fold by a combination of 4 penetrances, 4 phenotype definitions, and 40 marker tests, a practice which is not uncommon in linkage analysis of complex diseases. As a result, the lod score threshold of significance needs to be adjusted upward. The magnitude of this adjustment is difficult to calculate, although we may approximate it by adding log{m} to the lod score criterion if m different tests were made, as suggested by Kidd and Ott [1984]. An alternative is to assess the inflation of the lod score by simulation after analyses are completed [Weeks et al., 19901. For many psychiatric illnesses, the demographic and genetic epidemiological features of illness influence the availability and structure of pedigrees. For example, schiz-
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ophrenic patients have increased mortality and lower marriage rates compared with the general population. Thus, to plan for adequate power prior to data collection, it may be useful to use simulated pedigrees. In this paper we use this approach by simulating multigeneration pedigrees to be compatible with the demographic and genetic epidemiological features of schizophrenia. We then evaluate the power of these pedigrees by simulating the trait and marker genotypes according to the same genetic model used in the simulation of pedigrees. In addition, we present the power to detect linkage at several lod score thresholds since the multiple tests and phenotypic models required for complex diseases may necessitate using a lod score significance level greater than three. METHODS Simulation of Pedigrees
An overview of our procedure for simulatingpedigrees is as follows: (1) we assumed that schizophrenia is caused by a dominant gene with a known age-dependent penetrance; (2) starting with schizophrenic probands, we generated all their first and second degree relatives using sibship sizes and ages that are compatible with population data; (3) we assigned genotypes to each individual according to our genetic model of schizophrenia; (4) we assigned phenotypes to all pedigree members based on their ages and genotypes; ( 5 ) after determining each individual’s mortality status, we further assumed that some living pedigree members refuse to participate; (6) pedigrees were extended to second degree relatives if they or their siblings were affected; (7) we also assumed that some of the cell lines are lost to experimental error. The algorithms that simulated the pedigrees were written in FORTRAN and were executed on an IBM 3090 computer. Our simulation used single gene parameters that provide an adequate fit to systematically collected family data [Matthysse et al., 1986; Holzman et al., 19881. This model specifies that the transmission of schizophrenia is due to a dominant ill allele a with a gene frequency of 0.0223. The genotype specific penetrances of AA, Aa, and aa are 0.1896, 0.1896, 0.0017, respectively. The population prevalence of schizophrenia predicted by this model is 0.01; 16% of cases are expected to be phenocopies. This single major gene model satisfactorily accounts for the observed morbidity risk among first, second, and third degree relatives. The simulation of each pedigree started with a schizophrenic proband. We then extended that pedigree by generating the proband’s siblings, parents, offspring, nephews, nieces, uncles, aunts, and grandparents. To simulate sibship size we used the truncated negative binomial distribution that had been fitted to the population data of the United States in 1950 [Brass, 1958; Cavalli-Sforza and Bodmer, 19711. The parameter values are n = 2.84, P = 0.93, and mean = 2.6. We used the 1980 vital statistics on marriage in the United States to estimate the probability of marriage for pedigree members 30 years of age and older. For pedigree members without schizophrenia we assumed that 93% of females and 88% of males are married if their ages are equal to or older than 30 years. Since schizophrenic patients have a decreased probability of marriage, which is the main cause of their reduced fertility [Haverkamp et al., 19821, we assumed that the probability of marriage for schizophrenic pedigree members was 46.5 and 44% for females and males, respectively. We did not simulate marriage for individuals younger than 30 be-
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cause their children would be too young to be informative for the diagnosis of schizophrenia. Since there is increased mortality after the onset of schizophrenia, the age distribution of schizophrenic patients differs from that of the general population. We applied the method of Heimbuch et al. [ 19801 to model this altered age distribution. Assuming that the general population is in a stable age equilibrium with constant mortality and fertility rates and that the age distribution, c b ) , is proportional to e-KYly,the density function for age is c(y) = N1,,e.-RY
The normalizing constant, N , makes the integral of c(y) equal to one; Zy is the survival function from the life table for the population, and R is the natural intrinsic growth rate. For schizophrenics, we replaced l,, by a conditional survival function, rnblx), where x is the age at onset. According to follow-up studies of the mortality in schizophrenic patients [Tsuang and Woolson, 1978; Brook, 1985; Simpson, 19881, the increased mortality of schizophrenic patients is most prominent in the first decade after onset and only slightly higher in subsequent years. Thus, we approximated the increased mortality of schizophrenic patients by a standardized mortality ratio of 2.5 in the first decade and 1.25 afterward. We obtained rnO,Ix) as follows. For an age at onset belonging to the 5-year interval starting at x , we multiply the number of deaths in the life table by 2.5 for the age intervals x to x 5 and x + 5 to x 10, and by 1.25 for subsequent age intervals. We then derived rn(ylx) by subtracting the newly calculated number of deaths from the number alive at birth incrementally by 5-year interval until the interval that contains y. We generated the proband’s age at onset from a lognormal distribution with a p parameter of 3.25 and a CJ parameter of 0.50. This corresponds to a median age at onset of 25.8. Using the conditional survival distribution for the generated age at onset, we generated the proband’s current age. If the proband’s current age was greater than or equal to his age at onset, we continued to generate the pedigree. We simulated the ages of a proband’s siblings by randomly assigning a birth order to the proband using a uniform distribution to generate random numbers and assuming that the ages of successive siblings in a family are 1 or 2 years apart with equal probability. The difference between the father’s age and the oldest child’s age is a random integer between 20 and 35. The age of the proband’s mother was set to be within 3 years of the age of the proband’s father. These rules were also applied to each nuclear family in the pedigree to determine the ages of offspring, nephews, nieces, uncles, aunts, and grandparents. We simulated the proband’s genotype based on the genetic model and the assumption of Hardy-Weinberg equilibrium. Given the proband’s genotype, there are several possible pairs of parental genotypes; however, rare genotype combinations were omitted from consideration because they do not substantially affect the results and markedly increase the computer time needed for the simulation. For example, according to our dominant gene model for schizophrenia, the genotype of a proband could be aa, aA, or AA with probabilities of 0.009,0.828, and 0.163, respectively. If the genotype d)or (d, AA); the of a proband was aA, the genotypes of the parents could be (d, rare combinations (aa, AA) and (aa, aA) were not considered. Since the a allele is the
+
+
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disease gene, bilineal pedigrees such as (uA, uA) are allowed in our simulations. After the parents’ genotypes were assigned, the genotypes of the siblings were determined by Mendelian laws of transmission. We simulated the genotypes of other relatives by following the same principles. The genotypes of people marrying into the pedigree were determined by the population gene frequency and the assumption of Hardy-Weinberg equilibrium. The phenotypes of the relatives were determined from their ages, ages at onset, and genotypes. For each relative, we generated an age at onset according to the lognormal distribution used for the probands. A relative was designated as ill if his or her age at onset was less than or equal to his or her current age and a random number generated from a uniform distribution was less than or equal to the penetrance of his or her genotype. The genotype specific penetrances of parents and grandparents of probands decrease as a result of the reduced fertility associated with schizophrenia. The risk to siblings, uncles, aunts, and offspring are minimally affected [Risch, 1983;Kendler and MacLean, 19891. Following Kendler and MacLean [ 19891, the genotype specific proportion affected (GSPA) for parents and grandparents was approximated by
GSPA
=
FQ FQ+l-Q
where F is the relative fitness of affected individuals and Q is the genotype-specific penetrance. Since we assumed F = 0.5, the modified penetrance vector for parents and grandparents of probands was (0.1047,O.1047,0.0009). After we generated the structure of the pedigree and the phenotypes of its members, we simulated three additional factors: the mortality status of each pedigree member, the availability of living members, and the branches of the family selected for linkage analysis. The mortality status of relatives was a function of their current ages, phenotypes, and corresponding survival functions. The mortality status of unaffected relatives was determined by their current age and survival function of the United States in 1980. For affected relatives, this survival function was conditioned on their ages at onset as discussed above. After hving simulated them as being alive, first and second degree relatives were simulated to be available with probabilities of 0.95 and 0.70, respectively. We selected branches of the simulated pedigree for inclusion in the linkage analysis as follows. All first degree relatives of a proband were included. Cox et al. [1988] have shown that including information on unaffected second degree relatives adds little to linkage studies of affected individuals and their first degree relatives. Thus, second degree relatives were included only if at least one member of their sibship was affected or one of the parents of the sibship was affected. For example, if an unaffected sibling of the proband did not produce any affected offspring, his or her spouse and offspring were not included in the pedigree. In contrast, the nuclear family of affected siblings were included even if none of their offspring was affected. However, in a large scale pedigree study, occasional failures of cell lines occur. Thus the power that a pedigree can provide for linkage analysis will be less than that expected by the initial number of blood samples collected. In this study we simulated a 5% rate of failure for each cell line regardless of each individual’s phenotype.
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Simulation of Power
After simulating a pedigree, we selected it for the power analysis if at least three members of the pedigree members were ill and willing to participate. We continued simulating pedigrees until 200 pedigrees meeting this criterion had been created. We then assessed the power of these pedigrees to detect linkage by using version 4.1 of the program SIMLINK [Ploughman and Boehnke, 19891. Although we know the trait genotypes of our pedigree members, these cannot be used in the power simulation because they would not be known in a real linkage analysis. Therefore, genotypes must be simulated in the power analysis. As discussed by Ploughman and Boehnke [ 19891, an unbiased simulation of trait genotypes requires calculations of the conditional probabilities of genotypes given phenotypes. If we randomly assign a genotype vector to a pedigree based on the conditional probabilities of all possible genotypes given the phenotypes of that pedigree, we can perform an unbiased simulation of the genotypes that is consistent with the observed phenotypes. Thus SIMLINK needs only information about phenotype, age, and pedigree structure from generated pedigrees. To simulate genotypes, SIMLINK requires the genetic parameters of the disease locus, including the gene frequency, penetrance vector, and age at onset distribution. For the power analysis we used the same genetic parameters that had been used to generate the pedigrees. After generating trait genotypes, SIMLINK generates marker genotypes according to a given marker allele frequency, Mendelian transmission, and the recombination fraction (0) between trait and marker. Since highly polymorphic markers such as variable number of tandem repeats (VNTR) [Nakamura et al., 1987, 19881 and GT repeats [Litt and Luty, 1989; Weber and May, 19891 are now available, it is realistic to assume a codominant marker of 4 alleles with equal frequency. The polymorphism information content (PIC) value of this marker is equal to 0.70 according to the formula of Botstein et al. [1980]. In our simulations, male and female recombination fractions are assumed to be equal. To compute the statistical power under genetic heterogeneity, SIMLINK allows one to specify that only a proportion of (Y of pedigrees is linked to the marker. After simulating trait and marker genotypes, the likelihood of the pedigrees is computed and maximized over either 0 only (the single-parameter test) or both 0 and cy (the two-parameter test) to get the maximum lod score. For a user specified number of replications, SIMLINK simulates the cosegregation of the disease locus with a marker locus under the assumption of a true recombination fraction r . For each replication, SIMLINK computes the lod scores Z(Oi;r>,i = 1 to 7, over a range of test recombination fractions, 8 (e.g., 0.01, 0.05, 0.10, 0.20, 0.30, 0.40,0.50). We modified SIMLINK to output the Z(0,;r)for each replication for every pedigree and store them for later summation. In this manner we can assess the power of different numbers of pedigrees without repeatedly running SIMLINK on the same pedigrees. The joint linkage information in a set of N pedigrees was calculated as follows. For each replication, we summed the lod scores Z(Bi; r ) over N pedigrees and then determine the approximate maximum summed lod score, Z*(r), for that replication. After 200 replications, each set of pedigrees was characterized by a distribution of 200 maximum summed lod scores. This distribution provides the expected maximum summed lod score for the pedigrees and, for a specified lod score threshold of declaring linkage, the relative frequency that the maximum summed lod score exceeds the threshold (i.e., the power of the linkage test).
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In total, we ran SIMLINK over the 200 generated pedigrees and stored all lod scores calculated for 200 replications of each pedigree. Thus there was a total of 40,000 simulation trials. Given that each pedigree has been replicated 200 times, the power estimate obtained from small subsamples of the 200 generated pedigrees was very stable. For example, to estimate the power of 10 pedigrees, we randomly sampled 10 pedigrees from these 200 pedigrees and repeated this without replacement 20 times. The final estimate is obtained by averaging the powers of the 20 samples of 10 pedigrees. The standard errors of the power estimates using various lod score thresholds ranged from 0 to 0.005. The powers of other number of pedigrees (Ns ranging from 20 to 200 in increments of 10) were calculated similarly although the number of repetitions of sampling was decreased as the number of pedigrees increased. The standard errors of these power estimates were consistently small. All of the standard errors were less than 0.05; 60% were less than 0.02. When only a proportion, a , of pedigrees is linked, there are two ways to test for linkage. Following the modeling of Smith [1976], the likelihood for a set of families can be expressed as follows:
where Li(8) is the likelihood of observing the ith family and the product is taken over all families. The two ways to test for linkage are the single-parametertest, which assumes homogeneity( a= l), andthe two-parametertest,whichassumes heterogeneity (O3). Of course, excellent power to exclude linkage could be a liability if the probability of exclusion remained high when a fraction of the pedigrees were truely linked. Unfortunately, our simulations indicate that the probability of erroneously excluding linkage is nonnegligible, even when a =0.5. These results suggest that when genetic heterogeneity is possible, we should be cautious in interpreting lod scores less than - 2. Our results are consistent with the work of Martinez and Goldin [ 19891who determined power analytically for nuclear families with fixed structures. Compared with that study, we assumed a much lower penetrance (0.189 vs. 0.5), a higher phenocopy rate (16 vs. 5 % ) , and a less informative marker (PIC=O.7 vs. 1.0). Nevertheless, for a given level of power, the sample size needed under our assumptions is not much greater than that reported by Martinez and Goldin. For example, consider the sample size needed to attain 50% power using a tightly linked marker ( r = 0 for our study and 0.01 for Martinez and Goldin) when a is 50%. Martinez and Goldin [1989] showed that any one of the following three types of families meet this requirement: (1) 101 nuclear family of sibship size 3 with at least 1 affected child (202 of 505 individuals affected), (2) 53 nuclear family of sibship size 3 but with at least 2 affected children
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(146 of 265 individuals affected), or ( 3 ) 41 nuclear families of sibship size 4 with at least 2 affected children (117 of 246 individuals affected). In our simulation, 50 multigeneration pedigrees with at least 3 affected individuals were needed to achieve the same level of power. These pedigrees contain approximately 150 affected individuals and 500 subjects in total. Despite the differences in total number of subjects, the number of affected is similar between these two studies. It may be that our simulated ascertainment procedure is fairly robust to conditions of low penetrance and high phenocopies. In part this is due to not limiting ascertainment to the nuclear family. It is well known that extended multiplex pedigrees can help to resolve the linkage phase and thus provide more linkage information than a combination of nuclear families with the same total number of individuals [Thompson et al., 19781. Although it is obvious that requiring three ill relatives per family handles the problem of low penetrance, it also appears to have reduced the proportion of phenocopies in the sample. What is perhaps most notable in our comparison with Martinez and Goldin [1989] is that the number of affected individuals required to achieve the same power is very similar. This suggests that the contribution of unaffected individuals to the linkage analysis is limited. Indeed, Goldin and Martinez [1989] have demonstrated that ignoring marker and disease loci information of unaffected individuals in phase unknown nuclear families results in only a small increase in the average sample size needed to detect linkage. Whether their result can be generalized to our simulated pedigrees remains to be tested. Because our results of power to detect linkage are based on simulated pedigrees, we can generalize them to data that would be systematically collected in an epidemiological study following our ascertainment scheme. Our simulation also provides some information about the probability of ascertaining informative pedigrees of schizophrenic patients. This would not be possible if we had simulated power using published pedigrees. Only 9.8% of simulated patients had available families with at least 3 affected individuals. This ascertainment probability is comparable to that reported by empirical studies using family study method [Kendler et al., 19851 or family history method [Pulver and Bale, 19891. It is also consistent with other simulation studies [McGue and Gottesman, 19891. Our results are limited by the genetic and demographic parameters used to generate the pedigrees and to perform the linkage analyses. We believe that our choices for demographic parameters (sibship size, age at onset, age, marriage, increased mortality, and reduced fertility after onset) are reasonable. They are based on epidemiological data and have produced pedigree structures that are similar to those reported in the literature. Nevertheless, further work would be useful to determine the degree of power change across relevant portions of the genetic and demographic parameter space. It would also be useful to examine how different ascertainment strategies affect power. Our simulations of power under conditions of genetic heterogeneity assumed that except for the recombination fraction, the parameters of genetic transmission were identical for the linked and unlinked families. However, as shown by Martinez and Goldin [ 19901, heterogeneity due to two independent loci complicates the assessment of the power to detect linkage. Our simulations also did not examine how multipoint analyses, model misspecification, and laboratory error would affect statistical power. These questions have been addressed by other investigators [Lathrop et al., 1984; Lander and Botstein, 1986; Martinez and Goldin, 1989; Risch and Giuffra, 19901 (multipoint
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analyses), [Clerget-Darpoux et al., 1986; Greenberg and Hodge, 1989; Martinez et al., 1989; Weeks et al., 1990; Williamson and Amos, 19901 (model misspecification), and [Tenvilliger et al., 19901 (laboratory error). In summary, we have investigated the number of pedigrees needed to detect linkage between a dominant schizophrenia gene and a highly polymorphic DNA marker. By incorporating demographic parameters into our simulation of pedigrees, we have presented a useful approach to the investigation of power for linkage studies of complex diseases. By modeling the structure of pedigrees according to both the demographic and genetic parameters of a disease, one can derive a reasonable estimate of the power to be expected before a study is performed. When pedigrees with at least three affected members are ascertained, the sample size needed to achieve sufficient power is feasible if 50% of the pedigrees are linked to the marker under test. It may not be feasible to detect linkage if only 25% of the pedigrees are linked, even if a very closely linked marker is used. Our results indicate that to be certain of adequate statistical power, linkage analyses of schizophrenia will require very large samples that do not have a marked degree of genetic heterogeneity. ACKNOWLEDGMENTS The authors wish to acknowledge Dr. Endel John Orav for his insightful help in design of simulation programming, and Dr. Chung-Cheng Hsieh for his comments on our earlier manuscript. We also thank four anonymous reviewers for their constructive comments. Preparation of this article was supported in part by the Veterans Administration’s Medical Research and Health Services Research and Development Programs and National Institute of Mental Health Grants 1 R01MH4 1879-01 ,5 UO 1 MH46318-02, and 1 R37MH43518-01 to Dr. Ming Tsuang. This manuscript was prepared while Dr. Tsuang was a Fellow at the Center for Advanced Study in the Behavioral Sciences. We are grateful for financial support provided to the Center by the John D. and Catherine T. MacArthur Foundation and the Foundations Fund for Research in Psychiatry Endowment. REFERENCES Aschauer HN, Aschauer-Treiber G, Isenberg KE, Todd RD, Knesevich MA, Garver DL, Reich T, Cloninger CR (1990): No evidence for linkage between chromosome 5 markers and schizophrenia. Hum Hered 40:109-115. Bishop DT, Williamson JA (1990); The power of identity-by-state methods for linkage analysis. Am J Hum Genet 46:254-265. Blackwelder WC, Elston RC (1985): A comparison of sib-pair linkage tests for disease susceptibility loci. Genet Epidemiol2:85-97. Boehnke M (1986): Estimating the power of a proposed linkage study: A practical computer simulation approach. Am J Hum Genet 3 9 5 13-527. Botstein D, White RL, Skolnick M, Davis RW (1980): Construction of a genetic linkage map in man using restriction fragment length polymorphisms. Am J Hum Genet 32:314-331. Brass W (1958): The distribution of births in human populations. Popul Stud 1251-72. Brook OH (1985): Mortality in the long-stay population of Dutch mental hospitals. Acta Psychiatr Scand 7 1:626-635. Cavalli-Sforza LL, Bodmer WF (197 1): “The Genetics of Human Populations.” San Francisco: Freeman. Cavalli-Sforza LL, King MC (1986): Detecting linkage for genetically heterogeneous diseases and detecting heterogeneity with linkage data. Am J Hum Genet 38399-616.
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