Abstract
While a variety of methods have been developed to deal with incomplete parental genotype information in familybased association tests, sampling design issues with incomplete parental genotype data still have not received much attention. In this article, we present simulation studies with four genetic models and various sampling designs and evaluate power in familybased association studies. Efficiency depends heavily on disease prevalence. With rare diseases, sampling affecteds and their parents is preferred, and three sibs will be required to have close power if parents are unavailable. With more common diseases, sampling affecteds and two sibs will generally be more efficient than trios. When parents are unavailable, siblings need not be phenotyped if the disease is rare, but a loss of power will result with common diseases. Finally, for a class of complex traits where other genetic and environmental factors also cause phenotypic correlation among siblings, little loss of efficiency occurs to rare disease, but substantial loss of efficiency occurs to common disease.
FAMILYbased tests of association have become popular methods for use in genemapping studies. They narrow down chromosome regions containing trait loci by searching for association between marker alleles and traits. Unlike the usual casecontrol study, genotype information of related individuals is used to control for potential confounding due to population admixture and stratification.
Historically, parental genotypes have been considered the best data to adjust for population admixture. See, for example, the transmission/disequilibrium test (TDT) of Spielman et al. (1992). However, in many cases, especially where the disease is a lateonset one, some of the parents may not be available. Numerous methods that use genotype information from other family members to adjust for population admixture have been introduced. Among them, Curtis (1997), Boehnke and Langefeld (1998), Spielman and Ewens (1998), and Horvath and Laird (1998) use genotypes and traits of siblings to adjust for population admixture, if at least one parent is missing; Knapp (1999a) includes only those families where complete parental genotypes can be inferred from observed data into a TDTtype statistic with correct variance. Rabinowitz and Laird (2000) present a general approach that efficiently uses data from all the available family members.
While it is likely that a nontrivial portion of parental genotype data may be incomplete in many genetic studies, the sampling design issues with missing parental genotype data in association studies have not received much attention. Knapp (1999b) did intensive simulation studies to estimate sample sizes needed for 80% power for TDT with complete genotype data from both parents for both one and two probands, assuming various genetic models. Risch (2000) compared the use of both parents as controls to designs with two unrelated controls and designs with two unaffected siblings as controls. When the parental genotype data are incomplete, we study how many siblings in addition to the proband need to be genotyped and phenotyped to achieve power equivalent to having both parents genotyped. A related question is: Which strategy is more efficient after obtaining a proband child, sampling siblings or sampling parents?
In many settings, for example, with psychiatric disorders, phenotyping additional siblings may be very expensive and time consuming. Thus we consider: What is the effect on power to phenotype siblings of the proband child? Many simulations for familybased association tests assume simplex traits, by which we mean that given the genotypes at the single locus tested, the phenotypes of siblings are independent; this is unlikely to be true if multiple genes and/or environmental factors are involved in the regulation of traits. We investigate the effect on power when we relax that assumption.
On the basis of our simulations, we provide evaluations of the relative efficiency of several sampling designs in familybased association studies. In what follows, we present the genetic models, sampling designs and test statistic used in the simulation studies, and conclusions drawn from the simulation results to answer the above concerns.
DESIGNS AND METHODS
Genetic models: We assume the marker under examination for association is the disease gene itself. We use p to denote the allele frequency of the disease gene and K to denote the population prevalence of the disease. We assume the disease locus is biallelic and use f_{2}, f_{1}, and f_{0} to denote the penetrance associated with carrying two, one, and zero copies of the disease gene, respectively. The attributable fraction (AF) of the disease due to carrying at least one copy of disease gene thus is (K – f_{0})/K. For all the simulations reported here, we assume an additive model of inheritance (MOI), i.e., f_{1} = (f_{2} + f_{0})/2, but we discuss effects of MOI choice for a subset of the models.
To investigate the power of the familybased association test with incomplete parental genotype data, we assume two rare disease models (K ≈ 0.01) and three common disease models (K = 0.05, 0.2, and 0.3). See Table 1. Since the test statistic relies on asymptotic normality, we first use simulation to examine test validity under H_{0}.
A complex trait model: So far, most of the simulation studies evaluating power of familybased association tests are based on singlediseasegene models. Since many diseases are believed to be caused by multiple genes, environmental factors, and interactions among them, it is desirable to evaluate the power with considerations for multiple factors. A universal model, however, that can be applied to all kinds of complex traits may not exist. Here we propose a generalizedlinear model that grasps some general aspects of some complex traits to simulate phenotypic data so that the power of familybased association tests with those complex traits could be more precisely studied.
Our model targets a class of dichotomous complex traits where multiple genes besides the candidate gene and environmental factors are involved in the regulation of the traits. Values of environmental factors are positively correlated among family members so that the observed sibrecurrent risk ratio is higher than that implied by the singlediseasegene model.
We propose the following model to simulate dichotomous traits for siblings,
Sampling design: We estimate the power under each of the following sampling designs: (a) each family consists of a parentsproband trio; (b) both parents are missing and one, two, and three extra sibs other than the proband child are genotyped, with or without phenotypes; and (c) one parent is missing and one, two, and three extra sibs other than the proband child are genotyped and with or without phenotypes. We focus on singleproband designs, but also perform limited comparisons with twoproband designs.
For simplex traits, we conduct simulations to estimate powers with sampling designs a, b, and c. For complex traits, we estimate only the power with sample design b, and the observed sibrecurrence risk ratios are assumed to be 10 and 5 for the rare disease models and 8 and 4 for the common disease models.
Simulation method: To simulate genotype and phenotype data, randommating and HardyWeinberg equilibria are assumed in the parental generation and families in the data are assumed to be ascertained through one or twoproband children.
For simplex traits, we first generate the genotypes of parents on the basis of assumed allele frequencies. Then genotypes of one, two, three, or four children are simulated according to Mendelian law of transmission. Then children's phenotypes are generated on the basis of penetrance associated with their genotypes. And last, only families with at least one affected child are retained in the data. According to the sampling design, if both parents are missing, we set their genotypes as “unknown”; if only one parent is missing, we randomly select one parent and set his/her genotype as unknown. If only one affected child is phenotyped, we randomly select one affected child as the proband and set the affection status of the other children as unknown.
For complex traits, we first generate parental and children's genotypes for each family in the same way as generating such data for a simplex trait. Then n_{i} (equal to the number of sibs in that family) α's are generated from the multivariate normal distribution MN(μ, Σ). Affection status of each sib in the family is generated according to the penetrance function given in Equation 1. Only families with at least one affected sib are kept in the sample.
Test statistic: We use the general approach in Rabinowitz and Laird (2000) and Laird et al. (2000) to construct the test statistic. Suppose we sample N nuclear families. In the ith family, the number of sibs is n_{i}. Let T_{ij} be the affection status of the jth sibling in the ith family. It is coded as 1 if affected and 0 if unaffected or unknown. For power simulations when extra sibs' phenotypes are unknown, T_{ij} = 0 for all sibs except the proband(s) who has T_{ij} = 1. The genotype of the jth sibling in the ith family is G_{ij}, and X(G_{ij}) is a coding of it. Here, we let X(·) be the count of disease alleles. Let P_{i} denote the parental genotypes of the ith family. The test statistic is given by S:
The standardized test statistic Z, which is S standardized by its conditional mean and variance given parental genotypes and phenotypes of all the family members under the null hypothesis, has a standard normal distribution under the null hypothesis.
When genotype data of both parents are complete, calculation of the conditional mean and variance of S is straightforward. The resulting Z test statistic is identical to the TDT in Spielman et al. (1992). When the parental genotype data are incomplete for some families, the conditional distribution of S given parental genotype data and traits of all the family members is still normal with the same mean and variance as of S with complete parental data, except P_{i} for each family with incomplete parental data is replaced by the minimum sufficient statistic for the missing parental data. The minimum sufficient statistic partitions the sample space of children's genotype patterns into a maximum equivalent class so that within the equivalent class, the ratio of the conditional probabilities of any two children's genotype patterns is invariant to any given parental genotype configurations compatible to observed genotype data. The conditional probability of a children's genotype pattern given the minimum sufficient statistic is thus the conditional probability of the children's genotype pattern given an arbitrary compatible parental genotype configuration, restricted to the equivalent class. With both parents missing, the test for binary traits is nearly equivalent to the RCTDT in Knapp (1999a): The two tests are identical if unique reconstruction of parental genotypes can be made on the basis of children's genotype information and partially overlap otherwise. See Horvath et al. (2001) for details.
We examined the validity of the test under the null hypothesis, by setting f_{0} = f_{1} = f_{2} = 0.2 to estimate the actual type I error rates of the test. Six designs are considered: 100 and 500 nuclear families with parentsproband trios or no parents, one proband with one or two additional phenotyped sibs. To ensure the accuracy of the estimated actual type I error rates, we did 200,000 replicates for each case, which correspond approximately to standard errors 0.0005, 0.0002, 0.00007, and 0.00002 for nominal levels 0.05, 0.01, 0.001, and 0.0001.
RESULTS AND CONCLUSIONS
The results of the simulation under the null hypothesis show that (Table 2) for nominal level 0.05, the actual type I error rate of 100 nuclear families with no parents, one proband, and one additional phenotyped sib is below the nominal α level, while that of the other two designs with 100 nuclear families is very close to nominal level. For nominal levels 0.01, 0.001, and 0.0001, the estimated actual type I error rates of all three designs with 100 nuclear families are below the nominal levels, implying conservative tests. The differences are larger with either smaller nominal levels or fewer members in the family. When the sample size is increased to 500, the estimated actual type I error rates are very close to nominal levels.
The conservativeness of the test with 100 families is due to the small number of informative families with the given genetic model and family structure (See Table 2). The distribution of the test statistic is approximated by a normal distribution based on the central limit theorems, and the P value of the test is calculated using normal distribution. With a small number of families contributing to the test statistic, the tail of the actual distribution of the test statistic may not be well approximated by normal distribution. Although the three designs with 100 families give lower type I error rates at the nominal level 0.01, the actual type I error rates are very close to each other, except that the oneproband, onesib design is more conservative. That indicates that the powers of the three designs are still comparable given the conservative nature of the test for lower nominal levels.
Power simulation results are reported in Tables 3,4,5,6,7,8.
Since we are interested in the relative power of different family structures rather than the absolute levels of power achieved, we presented the estimated mean of Z statistics for 100 families, μ_{Z}, rather than the levels of power. Since we know which allele is the disease allele in the simulation, the higher μ_{Z} thus corresponds to higher power. μ_{Z} is estimated on the basis of 10,000 replicates to ensure accuracy of the first two decimal digits of estimate. The standard error of estimated μ_{Z} is
For the two rare disease models, estimated powers of parentsproband trios are above the corresponding powers of all the other sampling designs (see Tables 3 and 4). When both parents are missing, estimated powers of genotyping and phenotyping three extra siblings are very close to, but still below that of parentsproband trios.
For the common disease models, estimated powers of 100 parentsproband trios are surprisingly below those of some other missing parent designs (see Tables 5,6,7). For the common disease model 1, when three extra sibs are genotyped and phenotyped, the power of the one or noparent design is slightly higher than that of parentsproband trios design. For common disease models 2 and 3, the power of the one or noparent design with two or more siblings available for genotyping and phenotyping is substantially higher than that of the parentsproband trios design. We also note that for all models of common disease, when extra sibs are not phenotyped, the power is generally below that of parentsproband trios.
For rare disease, the effect on power of not phenotyping extra sibs is very small. That is because, for the rare disease models we studied here, very few extra sibs are actually affected, and therefore very few extra sibs contribute to the test statistic (2). For all the models of common disease (Tables 5,6,7), the differences in power between phenotyping extra sibs or not are larger than those with rare disease models. That is because, for common diseases, more extra sibs could be actually affected when phenotyped and thus contribute to the test statistic and increase the power.
We note that genotyping only one parent offers little advantage over having no parents available for genotyping, whether or not extra sibs are phenotyped and whether the disease is rare or common.
When assuming that the trait is complex, the estimated powers of the tests (except when using trios) are all below that of simplex traits with the same sampling design (Tables 3,4,5,6,7). The impact of complex disease is larger for the common disease models than for the rare disease models. The power with smaller observed sibrecurrence risk ratios is higher than that with larger observed sibrecurrence risk ratios under the same genetic models. There is no effect when sampling trios because the phenotypes of parents are not relevant in the TDT, and α_{i} is absorbed into the baseline prevalence. When we reduce the value of ρ, the power decreases and vice versa (Tables 6 and 8).
Knapp's (1999b) results allow one to study the effect of having two probands relative to one when both parents are present (compare his Tables 4 and 5). Twoproband families are ultraefficient in the sense that one needs fewer than half the number of singleproband families and therefore fewer total affecteds to get power equivalent to trios, unless the disease allele frequency is 0.5 or higher. For disease allele frequency <0.5, the fraction of the twoproband families needed to get the same power as N singleproband families varies depending upon p and the genetic model, but ranges between ∼0.15–0.40 for the models given in Knapp (1999b). Similar results were found in Risch (2000).
To look at the effect of using twoproband families, we study two models similar to Knapp's, both additive with relative risk φ = f_{1}/f_{0} = 4, one with p = 0.1 and K = 0.01 and the other with p = 0.5 and K = 0.2. For the first model (p = 0.1), Knapp's results show that only ∼40% of twoproband families are needed for the same power as oneproband families where both parents are present. For the second model (p = 0.5), the number is 60%. These same results are verified with our program using 100 singleproband trios and 40 and 60 twoproband families with both parents. We then looked at the case where no parents and two extra sibs are available. As with our other models, for K = 0.01, this design loses efficiency compared to having both parents and the proband, but it is more efficient when K = 0.2. However, when we considered two probands with missing parents and two extra sibs, the percentage of twoproband families needed for equivalent power to singleproband families is 55% for p = 0.1 and 80% for p = 0.5. Thus having missing parents seems to lessen the relative efficiency of twoproband vs. singleproband designs, but the effect of missing parents on power does not differ appreciably for one and twoproband designs.
DISCUSSION
The genetic models that we report here all assume additive effects of the disease alleles and ascertainment through one proband child. However, our general conclusions should not depend strongly on those assumptions. To verify this, we repeated the simulations for rare and common disease models 1, using a dominant mode of inheritance, and for two new rare and common disease models (slightly modified from rare disease model 2 and common disease model 3 to ensure valid penetrance values), using a recessive mode of inheritance. Although the absolute power amounts differ, our conclusions regarding the effect of missing parents and adding sibs do not change appreciably.
This article has focused on the power of TDTtype statistics under sampling designs comparing the use of parents vs. sibs as controls. Our main finding is that the most efficient design does not always use both parents as controls. With common diseases, sampling one proband and two extra sibs can give greater power than sampling parentsproband trios, because many of the extra sibs can be expected to be affected. Of course, one can always sample two or more extra sibs even with two parents available for genotyping, for even greater power. But the comparison is not valid, since this would involve genotyping five or more subjects for each family, as opposed to just three for trios or for designs with one proband and two extra sibs. With rare disease, one will need to genotype more than two extra sibs to achieve the same power as parentsproband trios.
Whittaker and Lewis (1999) concluded that siblingbased STDT could achieve power similar to that of the parentsproband triobased TDT if a certain number of extra siblings were available for genotyping. Our findings confirmed the results of Whittaker and Lewis (1999) but we further examined whether phenotyping extra siblings in families with incomplete parental genotype data brings the power superior to that of the parentsproband triobased design for various disease prevalences. Although siblingbased designs can be more efficient than parentaltriobased designs, as pointed out by the reviewer of this article, one should also consider in practice the unique advantages of having parental data over having just sibling data, such as the possibility of investigating parentoforigin effects. In addition, it is generally easier to check genotyping errors and to infer haplotypes with parental genotype data.
Lange et al. (2002) and Lange and Laird (2002) recently presented analytical methods to compute power for a familybased association study. Their results on relative power of different family structures for simplex traits confirm our simulation findings. We in addition examine the power of the familybased association study for complex traits in this work.
At first sight, our conclusions appear different from those of Risch (2000), who finds that sibcontrolled designs are very inefficient relative to parentcontrolled designs, regardless of population prevalence. However, Risch (2000) considered one, two, and threeproband designs where the controls are always two unaffected siblings, whereas we considered one or twoproband designs with one, two, or three extra siblings without regard to the affection status of these extra siblings. The rationale for our design comes from viewing the function of siblings as reconstructing the missing genotype of the parents. In practice, it is sometimes easier to genotype siblings without regard to their phenotype; hence power calculations for our approach may be more useful in such situations. In fact, our study was motivated in part by determining the power for a proposed study of bipolar patients where it would not be practical to determine the phenotype of siblings.
Risch also showed, as did Knapp (1999b), that for p ≤ 0.5, twoproband designs are highly efficient relative to oneproband designs, which explains why we find that oneproband designs with two siblings of unspecified but known phenotype can have better efficiency than parentsproband trios when disease prevalence is 20% or more.
We have not considered analysis issues in this article. Alternative coding for genotypes [X(G_{ij})] and phenotypes (T_{ij}) may influence the power of these designs. We have always coded affected as one, unaffected or unknown as zero, and the genotype coding is always additive. This is the standard TDTtype approach. Simulation results in Horvath et al. (2001) suggest that the additive coding performs well with other types of models and is thus a good general choice for genotype coding. Whittaker and Lewis (1998) and Lunetta et al. (2000) suggest that using the phenotype coding (1 – K) for affected, –K for unaffected, and 0 for unknown will lead to greater efficiency than using a 0/1 coding scheme. If so, this should increase the power of sib control vs. parent control designs. Although K is not always certain in practice and not generally estimable from samples, the practical implication of this strategy deserves further study.
Though choosing different ρ has resulted in different power for complex traits (Table 8), the conclusion that power for complex traits is lower than that of simplex traits with the same disease allele frequency and expected penetrances still stands. With observed sibrecurrence risk ratio λ_{s} fixed, there is an inverse relationship between ρ and ρ^{2}. See (A11) in the appendix. The inverse relationship means that the higher the correlation among sibs due to polygenic or environmental factors, the lower the variation in the penetrance given a disease genotype is to result in the observed sibrecurrence risk ratio and vice versa. On the other hand, lower variation in the penetrance given a disease genotype will push the power closer to that of a simplex trait with the same disease allele frequency and expected penetrances and vice versa. Finally, caution should be exercised when using our conclusion for complex traits. Since the model we proposed in (1) is not a universal model (e.g., covariates adjustments were not considered) for complex traits, one should evaluate the property of the specific diseases under study and decide if our model is adequate to represent the case.
Acknowledgments
This work is supported in part by National Institutes of Health grant MH59532 and by National Heart, Lung, and Blood Institute contract HC25195.
APPENDIX
Here are the details of obtaining β, μ, and ρ in the complex trait model (1). Note that π_{ij} = P(affectedG_{ij}, α_{ij}); it follows that the penetrances of carrying 2, 1, and 0 copies of the disease gene D are
Footnotes

Communicating editor: C. Haley
 Received March 16, 2002.
 Accepted January 29, 2003.
 Copyright © 2003 by the Genetics Society of America