Genetic Analysis Workshop 14 simulated data have been analyzed with MASC(marker association segregation chi-squares) in which we implemented a bootstrap procedure to provide the variation intervals of parameter estimates. We model here the effect of a genetic factor, S, for Kofendrerd Personality Disorder in the region of the marker C03R0281 for the Aipotu population. The goodness of fit of several genetic models with two alleles for one locus has been tested. The data are not compatible with a direct effect of a single-nucleotide polymorphism (SNP)  in the region. Therefore, we can conclude that the functional polymorphism has not been typed and is in linkage disequilibrium with the four studied SNPs. We obtained very large variation intervals both of the disease allele frequency and the degree of dominance. The uncertainty of the model parameters can be explained first, by the method used, which models marginal effects when the disease is due to complex interactions, second, by the presence of different sub-criteria used for the diagnosis that are not determined by S in the same way, and third, by the fact that the segregation of the disease in the families was not taken into account. However, we could not find any model that could explain the familial segregation of the trait, namely the higher proportion of affected parents than affected sibs. The aim of this work is to study and model the marginal effect of one susceptibility locus involved in the determinism of Kofendrerd Personality Disorder (KPD) in the Aipotu population. The presence of a susceptibility locus closely linked to the marker C03R0281 was shown by the existence of both strong association and genetic linkage between this marker and the trait [We want to select the replicates that best represent the distributions in the region of marker M (C03R0281) in terms of both association and linkage. Estimating the parameters and their variation intervals in this sample is then equivalent to evaluating them in the whole set of replicates. In the pooled sample set , we consider one index (an affected case) by family and his genotype for the marker M. For each index, we also consider his identity-by-descent (IBD) sharing for M with one random affected sib (families of the Aipotu population have been selected as having at least two sibs affected with KPD). In order to have a reliable IBD sharing for each sib pair, we ordered one SNP packet (153) surrounding marker M. The IBD sharing is obtained by maximum likelihood estimation using the information provided by the whole set of markers in the M region on chromosome 3.Figure Because we are looking for the model that best explains these four independent distributions (one genotype distribution and three stratified IBD distributions), we first determined the replicates that best reflect these distributions. We computed the distance of each of the 100 replicates to the pooled sample by a chi-square statistics, equal to the sum of the four independent chi-squares obtained by comparing the distributions observed in the replicate and in the pooled sample.We modeled the effect of the susceptibility factor by the MASC method. For a given genetic model, the MASC method computes the four expected distributions described in Figure We implemented a bootstrap procedure for calculating the variation intervals of the parameter estimates in the MASC program. The uncertainty on the parameters is due to the sampling of families and of controls, when the marker allele frequencies are inferred from a control sample. For each bootstrapped family set , we estimated the parameters considering the three possibilities described above for the marker allele frequencies. In situation 1, there is no uncertainty induced by the marker allele frequencies. In situation 2, the bootstrap procedure is applied to both the family sample and a sample of 50 controls randomly drawn among the 100 control samples. In situation 3, the bootstrap is only applied to the family sample. For the three situations, we obtain the distribution of the parameter estimates and provide the 95% intervals.Table 11and c12, whereWe tested a biallelic susceptibility locus model  and estimate four parameters: 2 relative penetrances, λ1 and λ2, and 2 coupling probabilities, cλ1 = P(affected | S1S2) / P(affected | S1S1),λ2 = P(affected | S2S2) / P(affected | S1S1),11 = P(S1 | M1),c12 = P(S1 | M2).cThe frequency (q) of allele S1 at the susceptibility locus can be written as11 P(M1) + c12 P(M2).q = P(S1) = c2 = 14.14; 6 df).The direct effect of marker M, which means S1 is confounded with M1, and S2 with M2, was rejected . The larger the sample size, the closer the estimate to the true value.Note that the true value of the dominance parameter cannot be inferred from the provided answers without extensive work. Indeed, the KPD phenotype is a mixture of different phenotypes, each one corresponding to different models of interaction between D2 and another susceptibility locus. Because there is no generation effect in the simulation, we still cannot explain the greater risk for parents than for sibs.GAW14: Genetic Analysis Workshop 14IBD: Identity by descentKPD: Kofendrerd Personality DisorderLD: Linkage disequilibriumMASC: Marker association segregation chi-squaresSNP: Single-nucleotide polymorphismMB performed the MASC and bootstrap analyses and drafted the manuscript. A-LL carried out the selection of the SNPs and draft the manuscript. EC carried out the study of the LD in the region, CB the association studies, M-CB the linkage studies. FC-D conceived, designed, coordinated the study and helped to draft the manuscript.


Further explorations into genetic and environmental population characteristics that benefit from this approach are warranted.We explored the utility of selecting a genetically predisposed subgroup to increase the finding of a genetic signal in the Genetic Analysis Workshop 14 Collaborative Study on the Genetics of Alcoholism dataset. A subgroup of affected probands with low environmental risk exposures was defined using a susceptibility score calculated from an environmental risk model. Thirty-nine probands with highly positive scores were selected, along with their parents, for use in a genotypic transmission disequilibrium test (TDT) test. We compared the results of the genotypic TDT in this subgroup to the TDT results using all probands and their parents. For some markers, the susceptibility scoring approach resulted in smaller  The most challenging and prevalent traits in genetic epidemiology today are complex phenotypes. Studies of such phenotypes are likely limited by genetic heterogeneity and high phenocopy rates, reducing power to detect linkage and association. Techniques to identify homogenous subgroups that harbor a genetic predisposition should involve the consideration of environmental influences. Accounting for non-genetic influences is crucial for the successful modeling of complex traits .Family-based association tests (FBAT) have been used to detect association and linkage, or association in the presence of linkage. The original transmission disequilibrium test (TDT) considered affected children only . A modifEmploying the modified FBAT method, Poisson et al.  developeFrom this susceptibility score modeling, individuals with highly positive residual values represent those who are affected even though this would not have been expected on the basis of their environmental profile, while those with highly negative residuals represent those who are unaffected, even though they would have been predicted to be affected given their environment. Individuals with highly positive residual values, who show affection despite low environmental risk, may therefore harbor a genetic predisposition for the outcome. Use of these individuals for genetic analyses may then yield a stronger magnitude of effect among this subgroup.When using these residuals as phenotypes "T" for FBAT, those with a residual close to zero do not contribute substantially to the test statistic. These individuals represent those with an environmental profile that predicts the outcome accurately, suggesting that genetic information would not provide additional information to an association test. Thus, it is unclear whether environmental information can increase power to detect linkage or association in this setting by reducing noise, or whether environmental information can reduce power by effectively lowering the contribution of persons with high genetic and environmental risk. In fact, Poisson et al. found that the use of the adjusted FBAT method involving the susceptibility residual oftentimes reduces the ability to detect a genetic signal.To further explore the trade-off between an increased magnitude of effect and loss of power, we have used the susceptibility scoring approach to define a highly predisposed genetic subgroup, and compared results of a genotypic TDT on the subgroup to the entire group. Our strategy differs from the approach taken by Poisson et al. in that we used susceptibility scoring to reduce our analyses to a stratified subgroup, whereas the Poisson et al. paper utilized the entire sample, and compared adjustment of susceptibility score to non-adjustment.p-values below 0.05 were then selected for the genotypic TDT test. Thus, the FBAT procedure was used as an initial screen to increase the opportunity of observing a SNP in linkage disequilibrium with a risk variant in subsequent genotypic TDT analyses.We used cleaned Illumina data from the Genetic Analysis Workshop 14 (GAW14) Collaborative Study on the Genetics of Alcoholism dataset. This dataset provided 137 families with a proband who was diagnosed with alcoholism according to DSM-IIIR + Feighner criteria and had genetic data available. Of these 137 families, 73 probands also had two parents with genetic data. The program FBAT was run for all family members relating to these 73 probands for all single-nucleotide polymorphisms (SNPs) provided for chromosomes 4 (275 SNPs), 15 166 SNPs), and 16 (162 SNPs). These chromosomes were chosen due to previous suggestive linkage findings 6 SNPs, a. The 73 The genotypic TDT focuses on the case, or child, genotype. Three pseudo-controls that represent the other possible genotypes that could have occurred conditional on the parental genotypes are matched to each case genotype. Conditional logistic regression was then performed on the 3:1 matched dataset in SAS v8.2 using the PHREG procedure. Beta coefficients can be interpreted as the change in log odds for the outcome associated with the possession of one copy of the 2 allele compared to the 1 allele and reflect an additive model. We conducted a genotypic TDT test for the 73 probands with parental genetic information, and on a subset of 39 probands that yielded a susceptibility score above 0.30.To determine the susceptibility score for each proband, we selected all full siblings and modeled alcoholism as a function of age at interview, sex, ethnicity, and EEG phenotypes. To account for the correlation among siblings within families for alcoholism, we used generalized estimating equations specifying an exchangeable correlation structure. The difference between observed and expected values for each proband from this model is then the susceptibility residual. Susceptibility modeling was conducted in SAS v8.2 using the GENMOD procedure.p-values between genotypic TDT analysis run with all 73 potentially informative trios versus the same test using a maximum eligible dataset of 39 trios based on a cutoff of 0.3 for genetic susceptibility. The actual number of trios utilized for each SNP is shown in Table p-values attributable to this restriction, and blue numbers specify those cases in which susceptibility scoring resulted in smaller p-values. We colored in only those rows where at least one p-value was less than 0.05 and the difference in p-values between the two methods was more than 0.01. Color was determined by the relative magnitude of the attained significance level. We also created susceptibility scores using the DSM-IV criteria for diagnosing alcoholism. Only two additional trios were gained, and results did not change meaningfully. Failure to include a significant predictor in the susceptibility model also did not alter the patterns observed in Table Table Our results are consistent with previous findings of Poisson et al., in which a covariate-adjusted phenotype, the susceptibility residual, was used in the FBAT procedure. These results were compared with results using an unadjusted phenotype for several simulated combinations of environmental and genetic main effects. It was found that power was consistently reduced when using the susceptibility score, while type I error was unaffected. However, when compared in real data, the disadvantage of the susceptibility score adjusted phenotype versus the unadjusted phenotype was unclear, possibly due to the unknown strength of the genetic effect, or interaction between genes and environment.From our analysis of these GAW14 data, the overall picture again appears mixed, replicating the findings from Poisson et al. in a new context. This procedure is slightly different from Poisson's method but reflects the same underlying conceptual approach. Unlike Poisson, we are restricted in our ability to make direct inferences about power and type I error. The ideal design in which to apply our method would be a set of simulations in which the true variant is known. Our approach in the applied GAW14 dataset, however, provides some intuition for the main conclusions of a simulation approach.p-values than expected as shown in Table The reduction of our susceptibility scoring approach to 39 high-risk trios led to a very small sample size. It is clear that the FBAT screening procedure led to a subset of markers with a greater number of significant Failure to measure informative covariates might have weakened the strength of susceptibility scoring. In complex diseases including alcoholism, genes and environment may both be active in transmission of a trait from parents to offspring. It is therefore important to distinguish correlation of genetic and environmental precursors of a complex trait within individuals versus gene × environment interaction effects on risk. For example, the risk factors might act independently on the trait but occur in non-random combinations. This problem could affect the relevance and impact of susceptibility scoring as well as other covariate adjustment procedures.The nature of any gene × environment interactions would be important determinants of the relevance of the susceptibility scoring method. To understand more fully the utility of our susceptibility score approach, simulations of varying underlying gene × environment scenarios seem most informative. Gene × environment interactions of a synergistic type may tend to reduce the value of susceptibility scoring because of the probable discounting of highly informative transmissions. If those with high genetic susceptibility are rarely exposed to environmental risk, however, this method is likely to give favorable results by down-weighting potential phenocopies.Susceptibility scoring is a mixed blessing in that it confers benefit in cases in which the amount of information contributed by highly susceptible persons outweighs the sample loss from use of a cut-off point. Further study is needed to identify scenarios in which susceptibility scoring is most useful.GAW14: Genetic Analysis Workshop 14FBAT: Family-based association testsSNP: Single-nucleotide polymorphismTDT: Transmission disequilibrium testKSB carried out analyses and drafted the manuscript. GAC and MDF aided in the design of the analysis and provided intellectual input for the discussion. All authors read and approved the final manuscript.


Imprinted genes are expressed either from the maternally or paternally inherited copy only, and they play a key role in regulating complex biological processes, including offspring development and mother–offspring interactions. There are several competing theories attempting to explain the evolutionary origin of this monoallelic pattern of gene expression, but a prevailing view has emerged that holds that genomic imprinting is a consequence of conflict between maternal and paternal gene copies over maternal investment. However, many imprinting patterns and the apparent overabundance of maternally expressed genes remain unexplained and may be incompatible with current theory. Here we demonstrate that sole expression of maternal gene copies is favored by natural selection because it increases the adaptive integration of offspring and maternal genomes, leading to higher offspring fitness. This novel coadaptation theory for the evolution of genomic imprinting is consistent with results of recent studies on epigenetic effects, and it provides a testable hypothesis for the origin of previously unexplained major imprinting patterns across different taxa. In conjunction with existing hypotheses, our results suggest that imprinting may have evolved due to different selective pressures at different loci. This theoretical paper demonstrates that maternal-offspring coadaptation (the selection for gene combinations of parents and offspring) could select for genomic imprinting of the offspring trait. The term “genomic imprinting” refers to the phenomenon of parent-of-origin–dependent gene expression, whereby at a given locus, either the maternally or the paternally inherited copy is expressed while the other ‘imprinted' copy remains silent . Since tIgf2  to quantify the adaptive genetic integration of traits expressed in mothers and their offspring I = 1) is expected to double the maternal–offspring genetic covariance. In the absence of imprinting, the covariance has the value (s) to calculate the effect of imprinting on population mean fitness. This confirms that the effect of imprinting on population mean fitness is due to its effect on increasing the maternal–offspring genetic covariance.Genomic imprinting increases population mean fitness because it increases the adaptive genetic integration of maternal and offspring traits. This can be seen by examining the covariance between the maternal and offspring traits [cov [The key result of our model is that natural selection favors the evolution of genomic imprinting, because it increases offspring fitness by enhancing the genetic integration of coadapted offspring and maternal traits. Our model provides a number of testable predictions, which, in conjunction with existing hypotheses, may yield a better understanding of the functional basis for the evolutionary origin of imprinting across the genome. First, central to our model is the prediction that in systems where selection favors coadaptation, the loci involved in the intimate maternal–offspring interaction are more likely to show patterns of imprinting with maternal expression. The predominance of maternally expressed genes in both placental  and earld Igf2r) , where aSecond, we expect the incidence of imprinting at loci affecting traits involved in maternal–offspring interactions to be higher in taxa in which such interactions have a greater impact on offspring fitness. Like the intralocus sexual conflict model, our results also suggest that imprinting may occur in systems that previously have not been the focus of research on genomic imprinting because they do not offer the opportunity for conflict over maternal investment, but in which coadapted maternal–offspring traits have potentially large fitness effects . Our moFinally, although we offer an alternate theory for the evolution of genomic imprinting, we stress that the diversity of imprinting patterns and associated phenotypic effects found across different genes suggests the possibility that imprinting may have evolved for different reasons in different taxa and loci ,49. ThisAt present, most evidence in support of or against theories of imprinting comes from studies on phenotypic effects identified by gene targeting . HoweverA locus were derived under the assumption of a standard random mating population [For the single-locus model, the frequencies of the 16 possible maternal–offspring genotype combinations at the pulation . As a reM-locus–offspring O-locus genotype combinations. These frequencies are important because they are the unit to which fitness is assigned. See also [We start by deriving the frequencies of maternal–offspring two-locus genotype combinations with the ultimate goal of deriving the frequencies of maternal See also  for a di,,, where d is a measure of linkage disequilibrium [d in the offspring generation is equal to the value in the parental generation minus the amount lost due to recombination, such that doffspring = dparents – rdparents , where r is the recombination rate between these two loci. Therefore, we use only one parameter, d, to measure linkage disequilibrium [The frequencies of the four possible two-locus haplotypes in the parental generation measured after selection but before gamete production  in that ilibrium . The valilibrium .M1O1, M1O2 , M2O1, and M2O2. Although we assume that the maternal trait is not imprinted, we track parent-of-origin of alleles in mothers for simplicity, because we need to differentiate the parent-of-origin of alleles in offspring. The frequencies of the 16 maternal genotypes are denoted ϕij , where i and j correspond to the two haplotypes  from which the genotypes are derived. The values ϕij are calculated as the products of the frequencies of the two haplotypes that make up each particular genotype . The frequencies of the paternal gamete types, denoted iγ with subscripts following ih above, have the values of ih minus the proportion of linkage disequilibrium lost to recombination during gametogenesis in males  the frequencies of the 16 maternal genotypes, ϕij , (b) the frequencies of the paternal gametes, i ,γ  and (c) the rate of recombination, r. These frequencies are given in The frequencies of the 256 possible maternal–offspring genotype combinations are denoted M-locus–offspring O-locus combinations to which we assign fitness. These maternal–offspring genotype frequencies are denoted ijF, with subscripts following ijw (given in the Description of the Model section above) and are presented in O locus genotypes in offspring are associated with the various M locus genotypes in their mothers.These two-locus genotype frequencies can be used to derive the frequencies of the maternal Table S1Maternal and offspring genotypes are given by their maternal and paternal haplotypes . Empty cells have zero frequencies under Mendelian inheritance.(112 KB DOC)Click here for additional data file.


We explore a philosophically different criterion, recently proposed in the literature, which controls the false discovery rate. The test statistics are assumed to arise from a mixture of distributions under the null and non-null hypotheses. We fit the mixture distribution using both a nonparametric approach and commingling analysis, and then apply the local false discovery rate to select cut-off points for regions to be declared interesting. Another criterion, the minimum total error, is also explored. Both criteria seem to be sensible alternatives to controlling the classical type I and type II error rates.The simultaneous testing of a large number of hypotheses in a genome scan, using individual thresholds for significance, inherently leads to inflated genome-wide false positive rates. There exist various approaches to approximating the correct genomewide  The increase in genome-wide experiments and sequencing of multiple genomes has resulted in the analysis of large data sets that involve the simultaneous testing of statistical hypotheses on a large number of features in a genome. Traditionally, researchers have tackled the problem of multiple testing in linkage analysis by proposing a common threshold to control the family-wise  error rate. The simplest and frequently used Bonferroni correction is often conservative and is mainly useful when the number of tests involved is not very large. Recently, new criteria have been proposed, the false discovery rate  being onz-values , converted from the p-values of the test statistics has been introduced [po, with corresponding density for the z-values being fo(z), and let the "interesting" hypotheses  have a prior proportion p1, with corresponding density for the z-values being f1(z). Thus, f(z) = po fo(z) + p1 f1(z) gives the distribution of the z-values under the mixture of the populations of interesting and uninteresting hypotheses. The posterior probability for a particular z-value,z, to be from an uninteresting hypothesis, is p0f0(z)/f(z), and that to be from an interesting hypothesis is 1 - po fo(z)/f(z) The LFDR is defined to be fdr(z) = f0(z)/f(z), which should be very close to the posterior probability P(hypothesis is uninteresting | z) = p0 f0(z)/f(z) under the often valid assumption that p0 is close to 1. In Efron [z-values for which fdr(z) ≤ 0.10 are declared to be from interesting hypotheses. A threshold of 0.10 or smaller is desirable in that the proportion of false discoveries will in this way be controlled. We have devised another criterion, the minimum total error (MTE), for identifying "interesting" regions in the genome. This method is explained in the methods section.A nonparametric empirical Bayes approach that makes simultaneous inferences based on troduced ,3. The aIn Efron , all thef(z) nonparametrically by fitting a Poisson regression to the z-values. In addition to implementing this approach, we also estimate the mixture distribution parametrically. For the nonparametric approach, a natural spline, as proposed by Efron [z-values resulting from the genome scan. The normal empirical null is then estimated from the central peak. The results of the analysis can change drastically depending on whether the null distribution is set to be the usual N or is estimated from the data. We use the notation N-LFDR for the procedure of estimating the mixture nonparametrically via a spline and then applying the LFDR to determine the cut-off point.Efron  estimateby Efron , is fittf0(z) and f1(z) are normal distributions with common variances but a larger mean for f1(z). This is a common practice in statistics as a first step in comparing two treatments, with the difference in the means of the normals being the treatment effect. After fitting the mixture distribution via commingling analysis as implemented in S.A.G.E. [We also propose, as an alternative, to assume that both S.A.G.E. , both thLinkage analysis was performed on the Kofendrerd Personality Disorder (KPD) data with disease being expressed as a binary trait. The true regions causing the disease were known. This study is based on the nuclear family data available from the Aipotu (AI), Karangar (KA), and Danacaa (DA) populations. The analysis was performed on both the microsatellite (MS) and the single-nucleotide polymorphism (SNP) data sets. For the parametric approaches, ten replicates , were randomly chosen and analyzed. Another ten replicates of the AI population  were used in calculations studying the effects of increasing sample size. For the N-LFDR, we used all twenty of the above replicates. All methods of analysis were performed on each population, for the SNPs and the MS markers separately.p-values for a whole multipoint genome scan at 2-cM intervals were converted to standard normal z-values, that is, z = Φ-1(1-p), where Φ(●) is the standard normal cumulative distribution function, so that interesting hypotheses are more likely to produce larger z-values.In each case, the Haseman-Elston  regressiz-values were plotted in a histogram of 60 equi-length intervals, each of 0.1 unit. Then a natural spline of degree seven was fitted to the histogram  is assumed to be normal with mean δ0 and variance 02σ. These parameters were estimated from the central peak of the observed mixture distribution. Specifically, the mean is estimated by the mode of the spline,  = argmax . A potential estimate for the standard deviation isThe empirical null distribution,   was fitted by least squares, and the estimated standard deviation is . Table  and those chromosomal locations for which the N-LFDR is less than or equal to 0.1 were deemed interesting.However, this estimate is numerically unstable and a smoothing technique, as suggested by Efron , was usez-values using the SEGREG program in S.A.G.E. 4.5 [ with a smaller mean than that of . Thus, z-values less than – 2, which are classified as uninteresting, were trimmed off to allow the fitting. The percentage of z-values below -2 was on average 1%. Promising results were then found after fitting a mixture of two normal distributions without any transformation and with both means estimated from the data. The variances were constrained to be equal to avoid numerical instability.Commingling analysis was performed on the G.E. 4.5  was atte0 is the intersection point of  and , then locations with z-values > x0 are declared interesting. In Figure 0 = 1.45. Note that the area under the null distribution curve to the right of this point is the probability of type I error, that is, we make an error by declaring an underlying uninteresting region as interesting. Similarly, the area under the alternative distribution curve to the left of this point is the probability of type II error, that is, genuine interesting regions are declared uninteresting. The P-MTE method minimizes the total probability of these two types of errors. Note that the P-MTE criterion implicitly assigns equal costs to committing either type of error, which may not be desirable. The P-LFDR criterion, on the other hand, assumes most of the large number of hypotheses are uninteresting and controls the false discovery rate. As for N-LFDR, we declared a region as interesting when fdr(z) < 0.1.After fitting the mixture distribution, a criterion is called for to identify z-values that are likely to be classified as interesting, i.e., locations that are potentially genetically linked to disease causing regions. Two criteria, the P-LFDR and the P-MTE, were explored for this purpose. The P-MTE method sets the cut-off at the intersection of the fitted null and alternative distributions. Figure For each population in each replicate, analysis yields p-values at 1,921 locations on the ten chromosomes with the MS data, and at 2,380 locations with the SNP data. The p-values were converted to z-values by inverting the distribution function of a standard normal. Mixture distributions were fitted to the z-values from each population, each replicate, separately for each type of marker data. Both the nonparametric and the parametric approaches were used in the fitting. The interesting regions obtained were compared with the true regions. The disease causing loci are D1, D2, D3, and D4 for the AI and KA populations and D1 and D2 for the DA population. The modifying loci, D5 and D6, were not considered as disease loci, as our focus is to compare the performance of the new criteria rather than to seek the best linkage analysis model. We defined a location as being "linked" to the disease if it is within 10 cM of a disease locus. We defined loci more than 20 cM from any disease locus as "unlinked". Chromosomal regions between 10 cM and 20 cM away from any disease locus were intentionally ignored to allow for a sharp distinction between "linked" and "unlinked" cases.For N-LFDR, where the mixture distribution is fitted by a spline, in all the replicates of the DA population the true regions were identified 39.52% of the time using the MSs and 46.67% of the times using the SNPs. Given that the SNPs are more than twice as dense as the MSs, the power of using either is comparable in our data. Figure The parametric approach fitted the mixture distributions to the z-values via commingling analysis. Both the P-MTE and the P-LFDR criteria were applied to the fitted mixture distributions to determine the threshold on the z-values for declaring a genome location interesting. The results for the DA population with MS data are shown in Figure For each of the three populations and each of the two types of marker data, the probability of committing each of the two types of errors, pooled over all replicates considered, has been calculated and is reported in Table We also investigated whether increasing the sample size will provide better results. For this, we merged replicates to form larger samples and repeated the analysis on these samples. The results on the AI population are reported in Tables The results for N-LFDR are provided in Table Results for the parametric approach are presented in Table Appropriate control of various types of error rates in multiple testing scenarios has long been an intriguing research problem. However, despite the immense efforts spent on this subject, satisfactory solutions are not available. For example, the application of the genome-wide significance criterion suggested by Lander and Kruglyak  to the sIn this article, we explored the control of false discovery rates, a philosophically different approach, instead of the classical type I and type II error rates in multiple testing problems. Individual test statistics, after appropriate scaling, are viewed as having arisen from a mixture of null and non-null hypotheses. The ratio of the null density versus the mixture density at a given test statistic provides a measure of the LFDR. A procedure that places a cut-off on this ratio controls the FDR.The mixture distributions were fitted using two different procedures: by fitting 1) a spline (N-LFDR), and 2) a mixture of normal distributions with differing means (P-MTE and P-LFDR). A direct comparison of these approaches with methods in the literature that control the classical types of error rates is not necessarily meaningful because of the philosophical difference between them. Both the local FDR and the MTE criteria have intuitively simple and appealing interpretations.P-MTE and P-LFDR were compared in terms of finding genuine regions containing disease genes. We observed that the P-LFDR method is more conservative than the P-MTE method in declaring a location "interesting". Therefore, while fewer type I errors are incurred, more truly interesting locations are missed. Table The DA population had a simpler genetic model and hence the associated genes were identified with lower type I and type II error rates than in the other populations, as expected. As the sample size increases Tables  and 4 thFinally, although both methods for fitting the mixture distributions implicitly assume that the test statistics are independent, which is surely violated in a true multipoint genome scan, the estimate of the mixture distribution is nevertheless consistent and requires a large sample size to be accurate.KPD: Kofendrerd Personality DisorderLFDR: Local false discovery rateMS: MicrosatelliteMTE: Minimum total errorP-LFDR: Parametric-local false discover rateP-MTE: Parametric-minimum total errorSNP: Single-nucleotide polymorphismRS, MS, and GM are equally involved in the statistical analysis, interpretation, preparation of the draft and in the final revision of the manuscript. RCE and YL contributed to the initial conception and design, and critical revision of the manuscript. All authors approved of the manuscript.


In this paper we propose the use of the within-subject coefficient of variation as an index of a measurement's reliability. For continuous variables and based on its maximum likelihood estimation we derive a variance-stabilizing transformation and discuss confidence interval construction within the framework of a one-way random effects model. We investigate sample size requirements for the within-subject coefficient of variation for continuous and binary variables.We investigate the validity of the approximate normal confidence interval by Monte Carlo simulations. In designing a reliability study, a crucial issue is the balance between the number of subjects to be recruited and the number of repeated measurements per subject. We discuss efficiency of estimation and cost considerations for the optimal allocation of the sample resources. The approach is illustrated by an example on Magnetic Resonance Imaging (MRI). We also discuss the issue of sample size estimation for dichotomous responses with two examples.For the continuous variable we found that the variance stabilizing transformation improves the asymptotic coverage probabilities on the within-subject coefficient of variation for the continuous variable. The maximum like estimation and sample size estimation based on pre-specified width of confidence interval are novel contribution to the literature for the binary variable.Using the sample size formulas, we hope to help clinical epidemiologists and practicing statisticians to efficiently design reliability studies using the within-subject coefficient of variation, whether the variable of interest is continuous or binary. Measurement errors can seriously affect statistical analysis and interpretation; it therefore becomes important to assess the magnitude of such errors by calculating a reliability coefficient and assessing its precision. For instance medical diagnosis, clinicians have now become cognizant to the paramount importance of obtaining accurate measurements to ensure safe and efficient delivery of care to their patients. Experiments designed to measure validity and precision of instruments used in biomedical and epidemiological research are ubiquitous. For example, Ashton  demonstrScientists have developed several indices to assess the reliability and reproducibility of quantitative measurements. The intra-class correlation (ICC), the proportion of the between-subject variance to the total variance, has been widely used as an index of measurement reliability. For a comprehensive review on the ICC and its applications, we refer the reader to Fleiss , Dunn 55 and Shok) and numbers of replicates (n). These power results were then used to identify optimal designs that minimize the study costs. Assuming a constant number of replicates per subject, Walter et al. [k and n that allocate the sample resources optimally and minimize the variance of the estimated ICC under cost constraints. The cost structure that was considered was general and followed the general guidelines identified by Flynn et al. [When the ICC is used as the index of reliability, Donner and Eliasziw  providedr et al.  considerr et al.  calculatr et al.  derived n et al. .n, k) that minimize the variance of the MLE of WSCV for normally distributed variables. The use of the WSCV for dichotomous data has never been investigated before, and a novel contribution in this paper is the estimation of WSCV for binary outcome measurements, and sample size requirements, with emphasis on the case of two ratings per subject (i.e. n = 2). We devote Section 4 to the binary data, and general discussion is presented in Section 5.In this paper, we derive the optimal allocation for the number of subjects and the number of repeated measurements needed to minimize the variance of the maximum likelihood estimator (MLE) of the WSCV. In Section 2 we present the random effects model, the definition of the WSCV, and the asymptotic distribution of its MLE for continuous data. In Section 3, we use the calculus of optimization to find the optimal combinations . In a test-retest scenario, and under the assumption of no reader's effect (i.e. the readings within a specific subject are exchangeable), Yij denotes the reading of the jth trial made on the ith subject. A useful model for analyzing such data is given by:Consider a random sample of Yij = μ + si + eij     i = 1,2,...k; j = 1,2,...n     (1)μ is the mean of Yij, the random subject effects si are normally distributed with mean 0 and variance NWith model (1), it is assumed that the within subject variance is the same for all subjects.Under the above set-up, the log-likelihood function has the formσ2 = ρ = ρσ2 and σ2(1-ρ). Because the design is balanced the maximum likelihood estimators (MLE) for μ, σ2, and ρ are given in closed forms by: MSB does not exist for k = 1, which means that to obtain a sensible estimate of ρ as an index of reliability, the study should include more than one subject.Define The asymptotic variance-covariance matrix of the MLE's is obtained by inverting Fisher's information matrix. The large sample variance of l vol. 1 ) and wask, θ) follows a normal distribution with mean 0 and variance A. An approximate 100(1 - α)% confidence interval on θ can be given as α/2 is the 100(1 - α/2)% cut off point of the standard normal distribution.To construct an approximate confidence interval on θ itself, we found that the asymptotic coverage deviates from its nominal levels for some values of θ. To improve the coverage probability we suggest a variance stabilizing transformation to remove the dependence of var(θ.Due to the dependence of the variance of -1/2 dθ. With θ defined as in equation (2), it can be shown that ρ* = ρ/(1 - ρ) Letting g is bounded and differentiable, that, f is asymptotically normally distributed with mean f and variance 1/k. Therefore, we can construct 100(1- α)% confidence limits on θ based on the above transformation. The upper and lower (1- α/2)100% confidence bound on θ are respectively given by:To improve the estimated coverage proportion, we propose a variance stabilizing transformation g  where, ξ1 = f + zα/2/ξ2 = f - zα/2/where, Note that the limits of the interval depend on the unknown value of the intra-class correlation, which can be replaced by its MLE as defined in section 2.1.ρ were, 0.3, 0.4, 0.6, 0.7, and 0.8; μ = 10, and θ = 4%, 10% or 20%. Sample size (k) = 12, 25, 50, 75, and number of replicates (n) = 2, 3, and 5. The number of repetitions for each simulation was 1000. Tables To examine the finite sample behavior of the VST based confidence interval estimator, a Monte-Carlo study was conducted under model 1) using the S-Plus program. The values of  using thet al. [Accurate and reproducible quantification of brain lesion count and volume in multiple sclerosis (MS) patients using magnetic resonance imaging (MRI) is a vital tool for evaluation of disease progression and patient response to therapy. Current standard methods for obtaining these data are largely manual and subjective and are therefore error-prone and subject to inter-and intra-operator variability. Therefore, there is a need for a rapid automated lesion quantification method. Ashton et al.  comparedBased on the guidelines for the levels of reliability provided by Fleiss , a valuen, per subject, so that the variance of the estimate of θ is minimized, given that the total number of measurements is fixed a priori at N = nk.In the following development we discuss the second objective of this paper. We assume that the investigator is interested in the number of replicates, N = nk, equation (3) gives:For fixed total number of measurements n = 0. This, and the additional condition that ∂2 var(n2 > 0 are both satisfied so long as 0 <ρ < 1. Differentiating (4) with respect to n, equating to zero and solving for n we obtainThe necessary condition for var for ρ = 0.6, 0.7, and 0.8, θ = 0.1, 0.2, 0.3 and 0.4, when N = 24.Table n, k) are used, and because N = nk is fixed a priori, we first round the optimum values of n to the nearest integer; then k = N/n was rounded to the nearest integer. The values of var100%CI on θ is, 2zα/2 var100% CI for θ with a desired width w obtains by setting w = 2zα/2 {var(θ)}1/2 and then solving for k:Bonett  discussen and θ, larger values of ρ require larger number of subjects to satisfy the criterion. As an example, suppose that it is of interest to construct 95% CI on θ with expected width w = 0.05, ρ = 0.3, and an afforded number of replicates n = 2. If the hypothesized value of θ is 0.10, then k = 31, and if θ is 0.3 (i.e. lower reliability), then k = 323.We observe that, for fixed Funding constraints will often determine the cost of recruiting subjects for a reliability study. Although too small a sample may lead to a study that produces an imprecise estimate of the reproducibility coefficient, too large a sample may result in a waste of resources. Thus, an important decision in a typical reliability study is to balance the cost of recruiting subjects with the need for a precise estimate of the parameter summarizing reliability.n, k) that minimize the variance of In this section, we determine the combinations c0 is the fixed cost, c1 the cost of recruiting a single subject, and c2is the cost of making one observation. Using the method of Lagrange multipliers and following Shoukri et al. [Ψ in this formwhere, i et al. , we writΨ = var(λ(C - c0 - kc1 - nkc2)     (7)λ is the Lagrange multiplier. Differentiating Ψ with respect to n, k and λ and equating to zero, we obtainwhere, var(θ2 ρ* n4 - 4θ2 ρ* n3 -(2θ2r + r - 2θ2 ρ* + 1)n2 + 4θ2rn - 2θ2r = 0     (8)2r = c1/c2, and ρ* = ρ/(1 - ρ)where th degree polynomial in the left side of (8) has two imaginary roots, one negative and one admissible (positive) root for n. Table n for various values of θ, ρ, and r, noting that:Although an explicit solution to (8) is available, the resulting expression is complicated and does not provide any useful insight. The 4r = c1/c2 increases, the required number of replicates per subject (n) increases, because the cost of making a single observation (c2) decreases and the cost of recruiting a subject (c1) increases. When r is fixed, an increase in ρ results in a decline in the required value of n and accordingly an increase in k. An increase in θ also results in a decrease in n. The general conclusion is that it is sensible to decrease the number of items associated with a higher cost, while increasing those with lower cost.From Table c1 = 0 in Equation (8), we obtain c1 = 0 is quite plausible, at least approximately if the major cost is in actually making the observations . This means that a special cost structure is implied by the optimal allocation procedure discussed earlier.We note that by setting (ρ = 0.80) in estimating AVA using the "velocity integral method" with a planned value for the WSCV = 0.10. Suppose that the total cost of making the study is fixed at $1600.0. It is assumed that the overhead fixed cost c0 is absorbed by the hospital. Moreover, we assume that travel cost is $200.0, and the administrative cost using the DE is $200.0 per visit. From Table noptfor r = 1, ρ = 0.8, and θ = 0.10 is 6. From (9), kopt = (1600/15)/(1 + 6) = 15. That is we need 15 patients, with 6 measurements each to minimize var(To assess the accuracy of Doppler Echocardiography (DE) in determining aortic valve area (AVA) prospective evaluation on patients with aortic stenosis, an investigator wishes to demonstrate a high degree of reliability k subjects, each is blindly evaluated n times by the same rater. We assume that all subject responses yij  are dichotomous and are conditionally independent with probabilities P(yij = 1) = pi  and p(yij = 0) = 1 - pi. Thus, for fixed pi, the conditional distribution of the random variable n and pi. To account for the variation of response probabilities between subjects, as considered by Mak [pi are independently and identically distributed as a beta distribution, Beta , with mean π = α/(α + β) and variance π (1 - π)ρ. Given these assumptions, one can show that the correlation between yij and yil is in fact ρ. Define i• = yi•/n and n - 1), Consider a random sample of d by Mak , we assun = 2, or a test re-test reliability study involving two readings per subject. For this case we investigate the sample size issue in the following section.A case of special interest to clinical epidemiologists is when Yij, Yil):Under the above set-up, the common correlation model (CCM)  = π2 + ρπ(1 - π).P10 = P01 = P = P = (1 - ρ)π(1 - π).     (10)P00 = P = (1 - π)2 + ρπ(1 - π).The data layout can be summarized as in Table k = k11 + k01 + k10 + k11.where, ith subject, the mean of two measurements is μi = Yi1 + Yi2) when summed up over all subjects we have:Since for the π is:Therefore, an unbiased estimator of the population mean S2 = k10 + k01). Hence the sample coefficient of variation for binary responses is:and the MSW, is E(π and E(SD)= π(1 - π)(1 - ρ) we define ρ. Applying the delta method, the first order approximation to the variance of Since ∂) = k-1 (a1 + a2 + a3),var(a1 = υ2(1 - υ2π)(1 - π + υ2π2)/π,where a2 = (1 - 2πυ2)2(1 - 2π2υ2)/8π2, anda3 = υ2(1 - 2υ2π)(1 - υ2π).     (12)α)100% confidence interval as We suggest an approximate . To construct an estimate of the confidence interval on υ, the MLE of ρ, Note that if the observed frequencies in the sample of Eliasziw π^=n1+2n2There has been increasing attention given recently to estimation of sample size using a confidence interval rather than a significance testing approach (e.g. Gardner and Altman ). This iH0 :υ = υ0, where υ0 is some hypothesized value of the WSCV υ.For comparative interest we also present sample size requirements needed to test α)100% CI on υ is, 2zα/2{var(1/2. An approximate sample size that yields an (1 - α)100% CI for υ with a desired width w obtains by setting w = 2zα/2{var/w2.     (13)H0 : υ = υ0 then:Donner and Eliasziw (DE)  developeH1 : υ = υ1 with non-centrality parameterhas a non-central chi-square distribution with one degree of freedom under the alternative hypothesis α and power 1 - β is:Following DE it can be shown that the sample size needed to conduct a two-sided test with significance level α/2 1-and zβ 1-are the critical values of the standard normal distribution corresponding to α and β.where zH0: v = 0.04 versus H1: v = 0.1, where v0 corresponds to high reliability. To ensure with 80 per sent probability a significant result at α = 5% and π = 0.30 when v1 = 0.10, we compute the required number of subjects from the above equation as k = 986 and when π = 0.50, k = 355. For the sake of comparison to the fixed width CI procedure, suppose it is of interest to construct 95% CI on v with expected width w = 0.10. If the hypothesized values of v is 0.10 and π = 0.30, then from (13) k = 1100, and if π = 0.5, then k = 400.As an example suppose it is of interest to test The ICC has been traditionally used to assess the reliability of a measurement. QS considered the WSCV as an alternative measure of reproducibility for continuous scale measurements. It should be emphasized that our investigation has not allowed for forms of systematic error . A reviewer of this paper indicated that this is beyond our scope. In this paper we have dealt with the issue of sample size estimation of the WSCV from continuous and binary scale measurements focusing on random measurement error, in the conventional way that reliability is usually discussed.n, k) that maximizes the precision of the estimated coefficient under cost constrains. We note that cost implications, for dichotomous assessments, are quite important particularly when n is larger than two, which we intend to report on in a future paper.As in any reliability study, a crucial decision that a researcher faces in the design stage is the determination of the number of subjects, k and the number of measurements per subject, n. We have discussed two alternative statistical techniques to determine an optimal allocation. When we have prior knowledge of what constitutes an acceptable level of reproducibility, a hypothesis testing approach may be used. We used this approach in the case of binary outcome variable, following the GOF approach proposed by DE. The application of the GOF was straightforward because the number of replicates n = 2 was fixed. However, there are situations, when appropriate values of the reliability coefficient under the null and alternative hypotheses may be difficult to specify. An alternative to hypotheses testing is the efficient allocation of the sample, and the guidelines provided in this article for the continuous scale measurements allow selection of the pair  was negligible. Ideally one should adopt one of the available optimization algorithms, often referred to as integer programming models. These models are suited for the optimal allocations problems since the main concern was to find the best solution(s) in a well-defined discrete space.The WSCV is a useful index measure of measurements reliability. Investigators may design reliability studies using either efficiency or cost considerations. For continuous measurements, optimal allocation of the sample may be achieved with as few as two replications per subject. For dichotomous data, when each subject is measured twice, investigators may use, either fixed length confidence interval, or power considerations is estimating the sample size. Both methods produce comparable results.The author(s) declare that they have no competing interests.The authors contributed equally to this workThe pre-publication history for this paper can be accessed here:


ENPP1 locus is involved in the etiology of osteoarthritis of the hand. The study population comprised 126 nuclear families with 574 adult individuals living in small villages in the Chuvasha and Bashkirostan autonomies of the Russian Federation. The extent of osteoarthritis was determined by analyzing plain hand radiographs. The outcome of a principal component analysis of osteoarthritis scores of a total of 28 joints of both hands was used as a primary phenotype in this study. Maximum likelihood estimates of the variance component analysis revealed a substantial contribution of genetic factors to the overall trait variance of about 25% in this homogeneous population. Three short tandem repeat (STR) polymorphisms – one intragenic and two flanking markers – and four single-nucleotide polymorphisms were tested. The markers tagged the ENPP1 locus at nearly equal intervals. We used three different transmission disequilibrium tests and obtained highly significant association signals. Alleles of the upstream microsatellite marker as well as several single-nucleotide polymorphism haplotypes consistently revealed the association. Thus, our data highlights variability of ENPP1 as an important genetic factor in the pathogenesis of idiopathic osteoarthritis.Periarticular calcification is a common attendant symptom of generalized arterial calcification of infancy, a rare Mendelian disorder caused by mutations of the gene coding for ectonucleotide pyrophosphatase/phosphodiesterase 1 (ENPP1). This prompted us to perform a family-based association study to test the hypothesis that genetic variation at the  Osteoarthritis (OA) is the most common form of arthritis and is among the leading causes of disability throughout the world. It is a multifactorial disorder with multiple risk factors contributing to its onset and progression, such as age, genes, hormones, and lifestyle . The mosEvidence of a genetic influence on OA originates from various studies, including those on family history and familial clustering, twin studies, and examination of rare monogenic disorders. Estimates of the heritability of OA have ranged from 27% to 65% -5. A numi), a solute that triggers cell differentiation and serves as an essential physiological inhibitor of hydroxyapatite deposition [ttw/ttw) mouse, ectopic ossification of the joints and the ligament of the spine are the striking features, while spontaneous arterial calcification seems to be of minor relevance for the health of the mice [ENPP1 mutations in men and mice suggest that genetic variability of ENPP1 activity may contribute to common forms of articular disorders [Recently, we identified the genetic defect in patients with generalized arterial calcification of infancy (MIM#208000) . In addiposition . In the the mice ,15. The isorders .ENPP1 gene to analyze whether they have effects on the development of hand OA in a large sample of Caucasian nuclear families. We focused our attention on a relatively isolated population in southern Russia, the Chuvashians. They live in small villages in the Chuvasha and Bashkirostan autonomies of the Russian Federation. They have lived there for generations, their family structures are stable, and their environmental conditions have been relatively constant. Our data support the hypothesis that ENPP1 is a major candidate gene for OA susceptibility.In this study, we investigated variants of the We studied 574 adults, 294 men and 280 women, ranging between 18 and 90 years of age. They belonged to 126 two- to four-generation pedigrees. The subjects were uniformly distributed with respect to age between 20 and 70 years. The pedigrees comprised 3 to 14 persons. The studied individuals were all Chuvashians (Caucasians) from small villages in the Chuvasha and Bashkirostan autonomies of the Russian Federation. Their population is demographically stable and they have lived there for centuries. The environmental conditions, particularly dietary influences, have been relatively constant and genetic flow has been minimal . FurtherP < 0.01) and was in good agreement with that found in other studies on a similar subject [Our genetic field work in Chuvasha consisted of a complete history, physical examination, and health questionnaire conducted by a native speaker. All diagnostic measures were in compliance with the Helsinki Declaration. Written, informed consent was obtained after due approval by the local ethics committee. Plain posteroanterior radiographs of both hands were taken from each study participant, with the x-ray source located 60 cm above and using a standard radiographic technique . Digital subject .ENPP1 gene in 23 unrelated patients with generalized arterial calcification of infancy, together with their parents and a number of control individuals [DNA was prepared from peripheral blood lymphocytes by standard techniques. Single-nucleotide polymorphisms (SNPs) were previously identified when sequencing the ividuals ,12. All ividuals  and may ENPP1 locus at 6q, one of the (tcct)n tetranucleotide repeat type (M06NR1A) and two of the common (ca)n dinucleotide repeat type (M06NR2A and M06NR3A), and genotyped them in all samples in addition to the SNPs. The exact position in base pairs of the three STR markers in contig NM_006208.1 is at 8776828ff base pairs for M06NR1A, 8862795ff base pairs for M06NR2A, and 8911166ff for M06NR3A. For each marker, 6 ng of genomic DNA was amplified in a 10 μl reaction volume on an MJ PTC 225 Tetrad Cycler. The PCR mix contained 0.53 μM each primer (sequences are available upon request), 0.1 μM each dNTP, 0.5 U Taq polymerase, 1 × reaction buffer with 1.5 mM MgCl2. The forward primers were labelled at their 5' ends with FAM. Genotypes were determined on a MegaBACE 1000 automated sequencer . For allele calling, the proprietary Genetic Profiler software version 1.5 from Amersham Biosciences was used.We identified three new short tandem repeat (STR) markers at the Genotyping data were checked for Mendelian errors with the PedCheck program . In two Haplotypes for all individuals were determined by using the program Genehunter version 2.1 . SNPs weAD) and common environment shared by parents (VSP), by siblings (VSB), and by all members of nuclear pedigrees/household (VHS). First factor scores (FS1-OAs) were adjusted for all significant covariates, that is, age and sex, before proceeding to the second step of the analysis.Data analysis was carried out in two steps. We first used variance component analysis as implemented in the FISHER program  to assest-test (EOT) proposed by Malkin and colleagues [As a second step, we then employed transmission disequilibrium tests (TDTs) to detect an association between hand OA as a quantitative trait (FS1-OA) and selected DNA marker alleles. Three TDT-like tests were carried out for each pair of dependent variable and specific genetic marker or haplotype. They included the orthogonal test (OT) proposed by Abecasis and colleagues  and impllleagues  and impllleagues  by optimP values obtained in testing the same pedigree sample. We computed the combined P value, the probability of erroneous rejection of the general null hypothesis of no linkage disequilibrium, which unites certain null hypotheses for separate tests. The combined test can be constructed in two ways: one (A) using the asymptotic χ2 distribution, and the other (B) using the simulated joint distribution of three separate test values.The simultaneous application of different tests requires interpretation and pooling together of the various 2 distribution. The FBAT and EOT use normal and Student's distributions, respectively. Theoretically, the square of FBAT and asymptotically the square of EOT are also distributed as χ2. If the above TDTs are considered as independent investigations, the corresponding P values will also be independent and the sum of three tests is distributed as χ2. This distribution then can be used to obtain an overall P value for all three tests as a combined probability of all three null hypotheses together [2 to an extent depending on the overlap of the areas of false-positive results in the separate tests.The orthogonal test is the likelihood ratio test with χtogether . HoweverP value triad α1<α2<α3, the probability can be found that the following condition is true: all three separate tests have P values not greater than α3, at least two of them have P values not greater than α2, and at least one of them has a P value less than α1. If we now use the defined condition as a rejection of the general null hypothesis of no linkage disequilibrium, then the described probability is the probability of erroneous rejection of the general null hypothesis based on the simultaneous combination of separate test results. The obtained value can be treated as a combined P value, accounting for the extent of overlap of the areas of false-positive results in different TDTs on a sample of given structure.We generated 20,000 simulation replicates to generate the joint three-dimensional null distribution for three tests. We used the pedigree structure of our sample and the trait inheritance model exhibiting the observed familial correlations, but assuming no effect of the tested marker on the trait variation. Then for each P value obtained as described in the previous paragraph under (A) and (B). The STR markers presented with 6 to 13 alleles. To facilitate analysis, low-frequency alleles were combined and the three most frequent alleles of each marker were used as they presented.For the multiallelic STR markers, we examined only those alleles for which the dichotomy factorization produced more than 40 informative nuclear families in our sample. This minimum-number-of-nuclear-families criterion was introduced after the simulation study, investigating the dependence of the ratio of type I error to the test power on the number of informative families. To account for the number of tested alleles for each STR marker, the Bonferroni correction was made. It was performed for each STR marker, using for the separately tested allele the combined n-dimensional normal with familiar partial correlation coefficients estimated as parameters. The LRT is used to reject the null hypothesis that all marker genotypes exhibit the same mean trait value. Here the complete pedigree data are analyzed instead of only members of informative nuclear families as in TDT. This test is very sensitive to disequilibrium. Our comparison of power of the PDT to detect the simulated linkage disequilibrium against TDTs  in the present sample for markers influencing only a small portion (<0.10) of the total trait variance consistently showed substantial superiority of the PDT.We also applied the pedigree disequilibrium test (PDT). The PDT examines the trait inheritance under the assumption that the marker locus itself is the gene controlling a part of the trait variation. The distribution of residuals is modeled as an Table P < 0.001). Nearly 25% of the age-adjusted FS1-OA variation was attributable to genetic factors. Common environment shared by spouses also made a significant contribution (approximately 13%) to FS1-OA variation. Constraining this effect to zero was rejected by the LRT (P < 0.05).Variance decomposition analysis was performed to estimate the contribution of genetic factors to the interindividual FS1-OA variation in comparison with the effect of the potential covariates Table . The ageENPP1 locus to obtain a fairly complete coverage of the gene region . The markers were genotyped in all individuals of the population sample. To test whether particular alleles of any of the markers were significantly associated with age-adjusted FS1-OA, we used three different TDT-like tests and PDT (Table P < 0.05) or even highly significant (P < 0.001) associations between the rare-allele pool of M06NR1A , the C-allele of K173Q, and various haplotypes of two or three adjacent SNPs . This enzyme directly antagonises the PPi-generating function of ENPP1 by cleaving PPi into phosphate [ANKH have been identified in patients with familial articular chondrocalcinosis type 2 (MIM#118600) [i. Nevertheless, both diseases underline that a concerted regulation of PPi by the three genes mentioned is critical to avoid mineralization disorders [Two other genes are also involved in the regulation of extracellular PPthe cell . The othhosphate . Interes#118600) -36. The isorders . Therebyisorders .ENPP1 locus explains up to 3.2% of the population variation of FS1-OA. The contribution of the whole PPi pathway to the genetic component of the disease development may be considerably larger. To better understand the role of PPi in OA, studies are in progress to analyze the influence of the genetic variation of all three PPi regulator genes on the variation of the disease phenotype.The association that we have found with marker alleles at the t-test; FBAT = family-based association test; FS1-OA = first factor score obtained from principal component analysis of OA; kb = kilobases; K–L = Kellgren and Lawrence; LRT = likelihood ratio test; OA = osteoarthritis; OT = orthogonal test; PDT = pedigree disequilibrium test; PPi = inorganic pyrophosphate; QTDT = quantitative transmission disequilibrium test; SNP = single-nucleotide polymorphism; STR = short tandem repeat; TDT = transmission disequilibrium test; TNSALP = tissue nonspecific alkaline phosphatase.BMI = body mass index; ENPP1 = ectonucleotide pyrophosphatase/phosphodiesterase 1; EOT = extreme offspring design The authors declare that they have no competing interests.ES performed most of the genotyping and contributed to the interpretation of the data. IM developed software for the statistical analysis of the data, performed the association tests, and contributed substantially to the first draft of the manuscript. SD independently analyzed the data to confirm the results. Assessment of osteoarthritis from hand radiographs was performed by LK. NR established the new microsatellite markers and supervised the allele calling. Field work in Russia was organized by EK. MT designed the pyrosequencing assays for SNP typing. FR suggested testing the candidate gene and contributed to the manuscript. PN and GL together designed the study, supervised and directed the whole project, and wrote the manuscript. All authors read and approved the final manuscript.


The correlations between systolic blood pressure (SBP) and total cholesterol levels (CHOL) might result from genetic or environmental factors that determine variation in the phenotypes and are shared by family members. Based on 330 nuclear families in the Framingham Heart Study, we used a multivariate normal model, implemented in the software FISHER, to estimate genetic and shared environmental components of variation and genetic and shared environmental correlation between the phenotypes. The natural logarithm of the phenotypes measured at the last visit in both Cohort 1 and 2 was used in the analyses. The antihypertensive treatment effect was corrected before adjustment of the systolic blood pressure for age, sex, and cohort.The univariate correlation coefficient was statistically significant for sibling pairs and parent-offspring pairs, but not significant for spouse pairs. In the bivariate analysis, the cross-trait correlation coefficients were not statistically significant for all relative pairs. The shared environmental correlation was statistically significant, but the genetic correlation was not significant.There is no significant evidence for a close genetic correlation between systolic blood pressure and total cholesterol levels. However, some shared environmental factors may determine the variation of both phenotypes. In an early analysis of the Framingham Heart Study data, it was observed that the modest correlation in blood pressure between spouses was reduced when blood pressure was adjusted for weight . This fiIn this article, we applied similar analysis techniques to investigate the genetic and shared environmental components of variance between SBP and total cholesterol levels (CHOL) based on data from the Framingham Heart Study.A total of 330 nuclear families, comprising 1776 individuals , were used in this analysis. When there are multiple nuclear families in a pedigree, a nuclear family that has at least one offspring in Cohort 2 was randomly selected for the analysis. We used the phenotypes of SBP and CHOL, which were measured at the last visit, and transformed them into the logarithms. This ensured that their distributions were close to the normal distribution. We also used the corresponding ages at the last visit in the analysis. A total of 72 parents (31 fathers and 42 mothers) did not have any information on age and the phenotypes. This accounted for 4% (72/1776) of all individuals. These parents did not receive any antihypertensive treatment. There are no missing values in age and the phenotypes in the offspring generation. We analyzed the data with and without imputation. To correct for the antihypertensive treatment effect, we added 10 mm Hg to the measured SBP for the treated subjects and used the original measurements for the untreated subjects  and S.A.Y was given by σa2 + σse2 + σe2, where σa2, σse2, and σe2 are the genetic, shared environmental, and individual-specific environmental variances, respectively.Statistical analyses were carried out under a multivariate normal model for pedigree analysis using the software FISHER ,8. The ma2 + γσse2, where φ is the kinship coefficient between the two individuals and γ (-1 ≤ γ ≤ 1) is the coefficient for shared environmental effect. We estimated the coefficients for sibling pairs, γSib, and for parent-offspring pairs, γPO, from the data. The covariance between spouses was modeled separately by ρSP(σa2 + σse2 + σe2), where ρSP is the correlation coefficient between spouses for that trait.Univariate variance component analyses were conducted for each phenotype separately. The covariance between a pair of individuals depends on the type of relationship. For non-spouse relationships the covariance is given by 2φσaσa1σa2 + ρseσse1σse2 + ρeσe1σe2, where ρa, ρse, and Pe are the correlation coefficients for genetic, shared environmental, and individual environmental effect between the phenotypes, respectively. Correspondingly, σai, σsei, and σei are the genetic, shared environmental, and individual-specific environmental standard deviations of phenotype Yi .As an extension to the univariate analysis, bivariate variance component analysis considers correlation of two phenotypes simultaneously. The covariance of two phenotypes for an individual was given by ρaσa1σa2 + γ1γ2ρseσse1σse2, where γi is the coefficient for shared environment of phenotype Yi. As for univariate analyses, we estimated γSib and γPO separately from the data for sibling and parent-offspring relationships, respectively. For spouse-spouse relationship, the covariance was given by RSPσ1σ2, where σi is the total standard deviation of phenotype Yi  and RSP is the cross-trait correlation coefficient between spouses.The covariance of the two phenotypes between a pair of individuals was parameterized in terms of the different types of relationships. For non-spouse relationship, the covariance was 2φρP < 0.001). Within each cohort, there was no significant difference in SBP between males and females. Total cholesterol levels (CHOL) were significantly higher (P < 0.001) in parents who have a mean of 219 mg/dl (SD 41.0) than in offspring who have a mean of 119 mg/dl (SD 37.2). Among the parents, total cholesterol levels were significantly higher (P = 0.001) in mothers who have a mean of 229 mg/dl (SD 40.6) than in fathers who have a mean of 209 mg/dl (SD 39.1). There was no statistically significant difference in CHOL between males and females in the offspring cohort.Based on data before imputing the missing values, the mean age was 72.5 years (SD 9.3) for the parent cohort and 48.9 years (SD 10.8) for the offspring cohort. Within each cohort, there was no significant difference in age between males and females. Systolic blood pressure (SBP) was significantly higher in the parental cohort, who have a mean of 148 mm Hg (SD23.9), than in the offspring cohort, who have a mean of 124 mm Hg (SD 19.9), based on two-sample t-test (Sib) than among parent and offsprings (γPO) in both phenotypes. This suggests that individuals within the same generation share more environmental factors than those between generations.Table Table There are 72 parents with missing values (the two phenotypes and the corresponding age). We imputed these missing values, re-analyzed the data, and compared the results with the above analyses when the missing values were not imputed. The missing value of the parent's phenotypes were generated according to cohort- and sex-specific normal distribution of relevant non-missing phenotypes. The missing age was calculated according to the spouse's non-missing age, which is available for at least one spouse. The husband's age was imputed by adding 2.8 years onto the wife's age, and vice versa. No significant difference in the mean SBP and CHOL in each cohort and gender category was found before and after the imputation. For example, the mean SBP of the parents was 147.3 mm Hg (SD 23.85) after the imputation compared with 147.7 mm Hg (SD 23.87) before the imputation. Correlation and variance component estimates from both univariate and bivariate analyses were very similar to those given in Table Different statistical analysis methods were used in this paper to investigate the genetic and shared environmental correlations between ln(SBP) and ln(CHOL) based on nuclear families from the Framingham Heart Study. We found significant correlation in sibling-sibling pairs and parent-offspring pairs for ln(SBP) and also for ln(CHOL). No significant correlation was found in spouse-spouse pairs Table . Both unHowever, the two measurements of blood pressure (SBP and DBP) shared both genetic and environmental factors . The FraCui et al.  have disFrom this investigation, we did not find any significant evidence to suggest that variation in systolic blood pressure and total cholesterol levels is determined by the same genes. However, there seems to be some common family living environment factors determining the variation in both phenotypes. This information is important in considering analysis methods and components of variance for linkage analysis to discover new genes for SBP using the Framingham Heart Study data.


Apolipoprotein (Apo) A1 is a protective factor for cardiovascular events. This study aimed to perform complex segregation analyses of Apo A1 levels in families of adolescents systematically ascertained from the junior high school students in a rural community. Both siblings and parents of the adolescent probands were recruited for the study. Apo A1 concentrations were measured by turbidimetric immunoassay methods. After adjustment for gender, age, body mass index, smoking and drinking status, residual values of Apo A1 were subjected to subsequent analyses.Significant mother-father and parent-offspring correlations were found. Commingling analyses indicated that a four-component distribution model was needed to account for the Apo A1 variation. Segregation analysis using regressive models revealed that the best-fit model of Apo A1 was a model of environmental effect plus familial correlation (heritability = 23.9%), in which a significant mother-father correlation existed. Models containing major gene effect could be rejected.These results suggest that variations of Apo A1 levels in the normal range, especially during adolescence, are likely to be influenced by multiple factors without significant contribution from major genes. Apolipoprotein (Apo) A1, one of the structural proteins in high-density lipoprotein particles, is a protective factor against the development of atherosclerotic vascular disease . It promThe mode of inheritance as revealed by complex segregation analysis (CSA) can provide evidence whether there is a major gene effect for Apo A1 concentrations, which is important for subsequent gene localization . Even ifTwo reasons for these inconsistent results are possible. First, since coronary heart diseases have a variety of etiologies and indications for angiography examinations vary, ascertaining probands under these clinical conditions would likely lead to etiological heterogeneity in terms of ApoA1 concentrations. Second, gene-environmental interaction may have an important role in the variation of Apo A1 levels and be more prevalent in the adulthood. One way to overcome these limitations is to conduct CSA in families of younger subjects who do not have clinical symptoms and are systematically ascertained. In this study we employed this approach to recruit adolescent probands and their first-degree relatives from a Taiwanese rural community. The study aimed to assess the possible mechanisms of genetic contribution to Apo A1 concentrations through a series of family-genetic analyses, including familial correlation, commingling analysis, and CSA.There were 368 probands, 333 siblings and 444 parents in this study. Their distributions on demographic and atherosclerotic risk variables were presented in Table 2, BMI, smoking and drinking habits was only 7.3%. The correlation between residual Apo A1 concentrations among family members after adjustment for nonlipid covariates is displayed in Table Proportion of Apo A1 variation in this sample explained by gender, age, age2 for comparing the 4-component with 3-component distribution was 14.21 , while that for comparing the 4-component with 5-component distribution was 4.31 . The finding of multiple distributions is compatible with a major gene hypothesis; however, commingling may also arise through other causes. Thus, segregation analysis was used to determine whether these major effects segregated in families according to Mendelian expectations.Commingling analysis showed that a 4-component rather than a single, two or three-component distribution was the best-fit model for Apo A1 variance. The component means, variances, and proportions for the 4-component distribution model were , , and , respectively. The χP value for each model's χ2 of less than 0.001. The environmental model is the best-fit one. Under this model, factor A is pervasively present in the population (frequency = 0.97) and the mother-father correlation coefficient was 0.30. The mother-offspring correlation was higher than the father-offspring correlation (0.26 vs. 0.15). If the four familial correlations in the general model were reduced to two (ρMF = 0.28 ρMO = ρFO = ρSS = 0.18), the fit was not significantly worse . The estimated heritability was 23.9%.The results of complex segregation analysis of Apo A1 are presented in Table To rule out the possibility that our rejection of the major gene effect was due to numerical failure, sensitivity analysis of likelihood values was conducted for the Mendelian model. The results are plotted in Figure Genetic heterogeneity was tested by plotting -2ln(LE/LM), in which LE refers to the likelihood of the environmental model and LM to that of the Mendelian model . Highly selected samples obtained through a phenotype-based ascertainment scheme may lead to biased estimates. Therefore, the ascertainment scheme is very important on segregation analysis. Although we did not ascertain probands on ApoA1 trait, the likelihoods corrected for ascertainment on probands' ApoA1 levels provided efficient estimation. The results of likelihood without correction of proband ascertainment were similar to corrected likelihood results.Complex segregation analysis was used to ask whether an inheritable trait is controlled by a single major gene plus residual polygenes, if so, gene mapping tasks would be warranted. If not, gene mapping studies would be lack of power because traditional statistical programs and sparse genetic markers would not provide sufficient tools to tackle a trait without major gene effects. However, this situation has changed in the past few years. Now we know a complex trait is less likely to be influenced by a single major gene. Advanced statistical programs, dense genetic markers plus other new technologies has allowed us to investigate a complex trait where genes with small to moderate effects. In the post-genomics era, segregation analysis is considered as an intermediate tool to help investigators to plan further sophisticated genomic studies. Although the method did not give information of exact DNA locus to find the genes, it can provide heritability estimates and parameters for further parametric linkage analysis. So segregation analysis should be useful and not obsolete. Further genotype-based methods such as linkage and association will be planned for elucidating genetic effects and locations.The large proportion of un-response parents was the limitation of the study. Only 60% parents attended the study. The causes of low response rates were as follows. First, the parents were in the productive age (mean 44 years old) and they felt they were relatively healthy and were not interested in the health checkup. Second, the medical history rates of atherosclerotic disease among participant parents were very low. We found there were 6% participant parents who knew their hyperlipidemia disease, and less than 1% had history of coronary heart disease. Although we cannot collect the data on un-response parents, we have checked the questionnaires of lifestyle patterns from the probands, and found the distributions of socioeconomic status and lifestyle patterns were similar between response and non-response parents. It might imply the response parents can be the representatives of all parents.Variations of Apo A1 levels in the normal range, especially during adolescence, are likely to be influenced by multiple factors without significant contribution from major genes.th percentile values among all subjects . The control young probands were ascertained by random sampling of other young students (n = 197). Because the original stratification was not based exclusively on the Apo A1 and the results of CSA for the families of the two strata were similar, only the results of all families together with correction for ascertainment on proband's Apo A1 were reported in this study. After obtaining informed consent from probands' family members, the same measures were performed for each family member. Only first-degree relatives were included for this genetic study, with a total of 1,145 subjects.This family study was part of the Chin-San Community Cardiovascular Study, a prospective cohort study began in 1990 ,28. The The measurement of various lipid profiles have been described in detail elsewhere . BrieflyWe adjusted a person's Apo A1 concentrations for known determinants by regressing for age, gender, body mass index (BMI), smoking and alcohol drinking status. Both age and BMI were centered by subtracting the sample mean, 34.0 for age and 22.3 for BMI, from the corresponding variables. Residual values plus the sample mean of Apo A1 (116.8 mg/dL) were then used for further analyses.The correlations of Apo A1 concentrations between mother-father, parent-offspring, and siblings were estimated by using the FCOR program in SAGE . Next, c2. The frequencies of allele A and B are denoted qA and (1-qA), respectively. The distribution of types in the population is assumed to be in Hardy-Weinberg equilibrium. Individuals of each type are assumed to transmit allele A to their offspring with transmission probabilities τAA, τAB and τBB, respectively. Residual familial resemblance not explained by this major locus is modeled by familial correlations. The correlation between spouses, parent and offspring, mother and offspring, father and offspring, and between siblings, are denoted ρMF, ρPO, ρMO, ρFO, and ρSS, respectively. For this study, we adopted class D regressive models, in which residual sib-sib correlations are equal among all sibs of common parentage and can be due to any cause. If ρPO is held equal to ρSS, these models have been shown to be mathematically and numerically equivalent to the conventional mixed model of inheritance in nuclear families ) ) .We used the adjusted Apo A1 values without logarithm transformation for the segregation analysis because normalizing transformation of a biologically skewed trait would decrease power to detect a major gene effect when one exists . InsteadA sensitivity test of the model fitting, specified by initial values of gene frequency, was performed to differentiate local maxima from global maximum . We calcKLC carried out the data collection, statistical analyses, participated in the study design and processing data. WCC & YTL participated in the design of the study and supervised the ideas developing in hypothesis generation. MFC & HCH performed the laboratory measurement in lipid levels and in charge of quality control. All authors read and approved the final manuscript.


We explored three approaches to heritability and linkage analyses of longitudinal total cholesterol levels (CHOL) in the Genetic Analysis Workshop 13 simulated data without knowing the answers. The first two were univariate approaches and used 1) baseline measure at exam one or 2) summary measures such as mean and slope from multiple exams. The third method was a multivariate approach that directly models multiple measurements on a subject. A variance components model (SOLAR) was employed in the univariate approaches. A mixed regression model with polynomials was employed in the multivariate approach and implemented in SAS/IML.Using the baseline measure at exam 1, we detected all baseline or slope genes contributing a substantial amount (0.08) of variance (LOD > 3). Compared to the baseline measure, the mean measures yielded slightly higher LOD at the slope genes, and a lower LOD at the baseline genes. The slope measure produced a somewhat lower LOD for the slope gene than did the mean measure. Descriptive information on the pattern of changes in gene effects with age was estimated for three linked loci by the third approach.We found simple univariate methods may be effective to detect genes affecting longitudinal phenotypes but may not fully reveal temporal trends in gene effects. The relative efficiency of the univariate methods to detect genes depends heavily on the underlying model. Compared with the univariate approaches, the multivariate approach provided more information on temporal trends in gene effects at the cost of more complicated modelling and more intense computations. In genetic studies, subjects may be measured repeatedly over a period of time to monitor how the quantitative traits change with age (or other time measure). These types of data offer great opportunity to evaluate whether a gene's influence on traits changes with age. Univariate variance components approaches that use a single measurement or summary statistics such as mean and slopes are easy to implement and the results have a straightforward interpretation. However, the univariate approaches may not be extracting the full information content of the data and may not provide information about differing genetic effects with age. Multivariate variance components approaches that directly model all measurements on one subject by estimating covariance structures within or between subjects may better utilize the information in the data set and provide age-specific estimates of genetic effects at the cost of greater computational burden and more complex interpretation of the linkage information.In this work, we compared three approaches (two univariate and one multivariate) to analyze repeated measures in genetic studies. The first two approaches used univariate phenotypes that were either based on a single exam measurement or summaries from multiple exam measurements. Variance components models for univariate phenotypes were applied . The thiWe applied the three approaches to analyze total cholesterol levels (CHOL) in replicate 8 of the Genetic Analysis Workshop 13 simulated data without prior knowledge of the answers.2) was estimated as the proportion of the total phenotypic variance due to the additive polygenic variance. SOLAR calculates a LOD score by taking log10 of the ratio of the maximum likelihood of a linkage model (containing a quantitative trait loci (QTL) variance and a residual polygenic variance component) to that of a purely polygenic model. The QTL h2 was computed as the proportion of the QTL variance to the total phenotypic variance. In multipoint analyses, linkage to adjacent markers was also considered to evaluate the linkage to the current marker using a regression approach [Baseline measure of CHOL at Exam 1 of both cohorts was used as the dependent variable in variance components model analyses implemented in SOLAR [In calculating summary measures of the repeated CHOL measurements, we looked at three definitions of the mean by imposing restrictions on the selection of the subjects and their measurements. Definition 1 (D1) required that subjects had CHOL measured for at least three exams. This definition resulted in subjects with a wide range of observations used, from 3 to 15. We were concerned that the different number of exams, and hence different standard error associated with the mean measure, would affect the genetic analysis and explored definitions in which each summary measure was based on a similar number of exams. To obtain, approximately, an equal number of exams for both cohorts, definition 2 (D2) included only the first five exams of both cohorts, and all subjects had to have CHOL measured for at least two exams. For D2, Cohort 1 and 2 members had measures taken at approximately the same age (45 years). To obtain measures taken at approximately the same chronological time in the two cohorts, definition 3 (D3) included only exams 10, 14, 15, and 20 for Cohort 1 and exams 1–5 for Cohort 2, and required all subjects have CHOL measured for at least two exams. A slope of CHOL versus age was computed for each individual satisfying D1. Heritability and linkage analyses were conducted in the same way as for the baseline measure.We set up a mixed regression models as followsyij = Xij β + gij + rij + εij,yij is the CHOL at the age j for subject i, Xij and β are vectors of covariates and coefficients of fixed effects, gij and rij are subject-specific additive genetic and environmental effects  respectively, and εij is the residual environmental effect of subject i. To allow age-varying effects, g and r are modelled by Legendre polynomials similar to the approach in Meyer [where in Meyer :im | m = 0, ..., kA - 1} ~ N and {γim | m = 0, ..., kR - 1} ~ N are random regression coefficients of additive genetic and environmental effects for subjects i, φm  is the mth Legendre polynomial [ , kA and kR are the order of the corresponding polynomials. The covariance between two observations of two subjects is then equal to equation (1), assuming g and r independent of each other,where {αlynomial  evaluateCov(αim), αi'l = 2Φii' Cov(αm), αl) and Cov= 2δii' Cov, where Φii, is the kinship coefficient, δii' = 1, if i = i' and 0 otherwise, and , if i = i' and j = j', and 0 otherwise. The total h2 at a standardized age t* is thereforeIt can be further simplified by assuming m, m = 1, ..., kQ, ~ N. The covariance contribution from this QTL to equation (1), assuming its independence of g and r, is , where πii' is the multipoint shared by the two subjects at the QTL. Then the QTL h2 due to this locus isWe extended the model to incorporate the effect of a QTL by adding a Legendre polynomial with random coefficients ηWe utilized kinship coefficients and multipoint identity by descent (IBD) computed in SOLAR and read these values into a matrix using SAS/IML. The other parameterswere estimated via a nonlinear maximization procedure NLPQN in SAS/IML .kA = kR = kQ = 3) for both polygenic and subject-specific environmental effects and 1 for QTL effects. For those individuals who had more than one exam in an age interval, the average phenotype and covariates measured during that age interval were used in the analyses. Since it was time consuming to carry out genome-wide analyses, we only implemented this analysis at the three linked loci  found in the univariate analyses.Since computational load increased quickly with the number of observed ages, we divided the 70 distinct ages (ranging from 20 to 93) into five intervals: below age 30, with 10-year increments from age 30 to 60 and greater than 60. Order of polynomials was set as 2  one of the three slope genes, S7 (QTL h2 = 0.20), and three of the four baseline genes, B30 (QTL h2 = 0.27), B31 (QTL h2 = 0.21), and B32 (QTL h2 = 0.30). Using mean measure D2, we were able to detect the slope gene S7 (QTL h2 = 0.33) and the baseline gene B32 (QTL h2 = 0.26). Using the slope measure, only slope gene S7 (QTL h2 = 0.33) was detected. There were one, two, and one false positives for the Exam 1, mean D2, and slope measures, respectively, and the LOD scores of the false positives were between 3.6 and 4.3  depended heavily on the generating model. Since CHOL were generated using a basic linear model of age (CHOL = Chol_base + Chol_slope * age + random_error), using baseline measure at Exam 1 in which the age of subjects spanned between 20 and 85 enabled us to detect all slope and baseline CHOL genes except three genes with a variance <0.08. The mean measure seemed to contain more noise than Exam 1 data for detecting the baseline genes, but produced a slightly higher LOD than the slope measure to detect slope genes. This observation was confirmed in an experiment: when there is considerable residual random error in the trait, the slope measure could be inferior to the mean measure in power to detect a slope gene [The results of the three definitions of means were not very different for this generating model, though they were designed to avoid possible shortcomings in the other definitions (See Methods section). In practice, one definition may be better than the others depending on the characteristics of the data.2 estimations from the multivariate approach did not vary much with age and were close to those estimated from the univariate approaches using Exam 1 or mean D2. The QTL h2 for B30 and B32 estimated from multivariate analyses were higher than those obtained from univariate analyses, especially at younger ages. The difference at younger ages may be caused by more aged subjects in Exam 1 and mean D2 measures that resulted in lower proportion of total phenotypic variance (increasing with age) explained by the baseline genes for this generating model. The QTL h2 for slope gene S7 estimated using slope measure was close to that estimated using multivariate measure for those aged 30 or less. In theory, QTL variance for S7 from the multivariate measure should be approximately equal to that from slope measures multiplied by age2 for this generating model, which explains the monotonic increase of QTL h2 for S7 observed from the multivariate approach.The total h2 for the two baseline genes were nearly flat and slightly declining with age, but that of the slope gene showed a clear trend of monotonic increase with age, which distinguished the slope gene from the baseline genes.Compared with the univariate approaches, the multivariate approach provided more information regarding the temporal trend of gene effects during aging. We were not able to tell which gene(s) affected the baseline or slope using the univariate approaches, since the univariate measures overlapped with each other in the ability to detect slope and baseline genes. Using the third approach, the QTL hIn conclusion, we found univariate approaches were capable of discovering some of the important trait genes with simple modelling and feasible computational load. The multivariate approaches can provide additional information on age-varying effects of genes but generally involves heavy computation and complex modelling. More work is needed to further develop the multivariate approach in areas such as a sensible test of significance. Nevertheless, the multivariate approach shows promise for genetic analyses of longitudinal measures in linkage studies.