The loss of statistical power to distinguish populations when certain samples are ambiguous

Case-control studies are used to map loci associated with a genetic disease. The usual case-control study tests for significant differences in frequencies of alleles at marker loci. In this paper, we consider the problem of comparing two or more marker loci simultaneously and testing for significant differences in haplotype rather than allele frequencies. We consider two situations. In the first, genotypes at marker loci are resolved into haplotypes by making use of biochemical methods or by genotyping family members. In the second, genotypes at marker loci are not resolved into haplotypes, but, by assuming random mating, haplotypes can be inferred using a likelihood method such as the expectation-maximization (EM) algorithm. We assume that a causative locus has two alleles with a multiplicative effect on the penetrance of a disease, with one allele increasing the penetrance by a factor pi. We find, for small values of pi-1 and large sample sizes, asymptotic results that predict the statistical power of a test for significant differences in haplotype frequencies between cases and a random sample of the population, both when haplotypes can be resolved and when haplotypes have to be inferred. The increase in power when haplotypes can be resolved can be expressed as a ratio R, which is the increase in sample size needed to achieve the same power when haplotypes are resolved over when they are not resolved. In general, R depends on the pattern of linkage disequilibrium between the causative allele and the marker haplotypes but is independent of the frequency of the causative allele and, to a first approximation, is independent of pi. For the special situation of two di-allelic marker loci, we obtain a simple expression for R and its upper bound.     

Role of the apolipoprotein E polymorphism in determining normal plasma lipid and lipoprotein variation.

The structural gene locus for apolipoprotein E (apo E) is polymorphic. Three common alleles (epsilon 2, epsilon 3, epsilon 4) code for three major isoforms in plasma and determine six apo E phenotypes that may be identified by isoelectric focusing on polyacrylamide. To establish what fraction of the inherited variation in a normal plasma lipid and lipoprotein profile is attributable to the segregation of the common alleles at the apo E gene locus, we have estimated the average apo E allelic effects on plasma cholesterol (C), triglycerides, very low-density lipoprotein (VLDL)-C, VLDL-apo B, low-density lipoprotein (LDL)-C, LDL-apo B, and high-density lipoprotein (HDL)-C in a representative sample of normolipidemic individuals from Ottawa, Canada. Data from published studies were also analyzed by the same statistical procedures. As much as 16% of the genetic variance (8.3% of the total variance) for LDL-C could be accounted for by the apo E gene locus. After correction for differences in age, sex, height, and weight, it was found that the epsilon 2 allele lowered and the epsilon 4 allele raised total cholesterol, LDL-C, and LDL-apo B. No other gene has been identified that contributes as much to normal cholesterol variability. Analysis of these data and those of others also indicates that the apo E locus imparts a differential susceptibility to a variety of factors that promote hyperlipidemia. The hypothesis is proposed that the epsilon 2 allele protects against coronary heart disease (CHD) and, hence, gives a reproductive advantage that is balanced by a predisposition to CHD when the epsilon 2 is combined with a second, independent causative factor to give a reproductive disadvantage. A similar mechanism is proposed for the maintenance of the epsilon 4 allele in the population.     

Assessing the role of HLA-linked and unlinked determinants of disease.

The relationship between increased risk in relatives over population prevalence (lambda R = KR/K) and probability of sharing zero marker alleles identical by descent (ibd) at a linked locus (such as HLA) by an affected relative pair is examined. For a model assuming a single disease-susceptibility locus or group of loci tightly linked to a marker locus, the relationship is remarkably simple and general. Namely, if phi R is the prior probability for the relative pair to share zero marker alleles identical by descent, then P (sharing 0 markers/both relatives are affected) is just phi R/lambda R. Alternatively, lambda AR, the increased risk over population prevalence to a relative R due to a disease locus tightly linked to marker locus A, equals the prior probability that the relative pair share zero A alleles ibd divided by the posterior probability that they share zero alleles ibd, given that they are both affected. For example, for affected sib pairs, P (sharing 0 markers/both sibs are affected) = .25/lambda S. This formula holds true for any number of alleles at the disease locus and for their frequencies, penetrances, and population prevalence. Similar formulas are derived for sharing one and two markers. Application of these formulas to several well-studied HLA-associated diseases yields the following results: For multiple sclerosis, insulin-dependent diabetes mellitus, and coeliac disease, a single-locus model of disease susceptibility is rejected, implying the existence of additional unlinked familial determinants. For all three diseases, the effect of the HLA-linked locus on familiality is minor: for multiple sclerosis, it accounts for only a 2.5-fold increased risk to sibs over the population prevalence, compared to an observed value of 20; for coeliac disease, it accounts for approximately a 5.25-fold increased risk to sibs, while the observed value is on the order of 60; for insulin-dependent diabetes mellitus, it accounts for a 3.42-fold increased risk in sibs, while the observed value is 15. In all cases, the secondary determinants must be outside the HLA region. For tuberculoid leprosy, an unlinked familial determinant is also implicated (increased risk to sibs due to HLA = 1.49; observed value = 2.38). For hemochromatosis and Hodgkin's disease, there is little evidence for HLA-unlinked familial determinants. With this formula, it is also possible to examine the hypothesis of pleiotropy versus linkage dis-equilibrium by comparing lambda AS with the increased risk to sibs due to the associated allele(s).(ABSTRACT TRUNCATED AT 400 WORDS)     

Short tandem repeat polymorphism of the D2S1242 locus in a Japanese population

Allele frequencies and sequence characteristics of the D2S1242 short tandem repeat (STR) locus were studied in a Japanese population sample. A total of 10 D2S1242 alleles and 34 genotypes were identified in 273 unrelated Japanese individuals. The five most common alleles detected had frequencies of over 10%. No deviations from Hardy-Weinberg equilibrium were found when the expected allele values were compared with the observed values. Sequence analysis of each allele showed a tetranucleotide polymorphism. Alleles 9 to 14 had different sequence structures than alleles 15 to 19. Allele 18 had a different sequence in the Japanese sample compared to an Austrian sample. The power of discrimination was 0.95. The present results demonstrate that the D2S1242 STR locus is a useful genetic marker in the Japanese population.     

Identification of repeat sequence heterogeneity at the polymorphic short tandem repeat locus HUMTH01[AATG]n and reassignment of alleles in population analysis by using a locus-specific allelic ladder.

An allelic ladder containing amplified sequences of seven alleles of the polymorphic human tyrosine hydroxylase locus, HUMTH01, was constructed and employed as a standard marker. Sequence analysis of each ladder component indicates that fragments differ by integral multiples of the AATG core repeat sequence characteristic of this locus. Individual alleles are designated "5" through "11," according to the number of complete reiterations of the core repeat contained within them. Comparison of the HUMTH01 allelic ladder with DNA samples amplified at this locus revealed core repeat length heterogeneity (i.e., deletions or insertions shorter than one core repeat) within the human population. In particular, a common allele was identified which migrates more quickly than allele 10, but more slowly than allele 9, on electrophoresis through a denaturing polyacrylamide gel. Sequence analysis of this allele, designated "10-1," reveals lack of a single adenine normally present in the seventh copy of the AATG. The allelic ladder was used to reevaluate previously published population data. Results of testing for Hardy-Weinberg equilibrium and population substructure were not altered significantly by these modifications.     

Population aspects of cancer genetics

A number of relatively rare, high-risk genes have been identified which predispose to common cancers such as breast, colon, and melanoma. Although these are clearly important in the clinical setting, it is also relevant to discuss the impact of these genes at the population level and to contrast these with that which could be ascribed to more common genetic variants which only confer a modest increased risk of cancer. In this review, we examine inferences about the role of genetics in cancer from ecological studies of incidence patterns from a number of population-based studies of familial and attributable risk. The relationship between the genetic model (genotypic risk, allele frequency, mode of inheritance) and the expected impact in the population in terms of both attributable risk and familial risk is presented. The advantages and limitations of using cancer occurrence in twins to measure the genetic contribution to specific cancer sites is discussed. The potential role of lower-penetrance genes in the overall cancer burden may be significant but may pose significant problems in the public health arena.     

Inheritance of low-density lipoprotein subclass patterns: results of complex segregation analysis

Heterogeneity in the size of low-density lipoprotein (LDL) particles was used to identify two distinct patterns based on gradient gel electrophoresis analysis. These two phenotypes, LDL subclass pattern A and pattern B, were characterized by a predominance of large, buoyant LDL particles and small, dense LDL particles, respectively. The inheritance of these LDL subclass patterns was investigated in a sample of 61 healthy families including 301 individuals. LDL subclass pattern B was present in 31% of the subjects, with the prevalence varying by gender, age, and (in women) menopausal status. Complex segregation analysis suggested a major locus controlling LDL subclass patterns. The model providing the best fit to the data included a dominant mode of inheritance with a frequency of .25 for the allele determining LDL subclass pattern B and reduced penetrance for men under age 20 and for premenopausal women. Thus, the allele for the LDL subclass pattern characterized by a predominance of small, dense LDL particles appears to be very common in the population, although not usually expressed until adulthood in men and until after menopause in women. The presence of a major gene controlling LDL subclass could explain much of the familial aggregation of lipid and apolipoprotein levels and may be involved in increased risk of coronary heart disease.     

Mapping genes that underlie ethnic differences in disease risk: methods for detecting linkage in admixed populations, by conditioning on parental admixture.

Genes that underlie ethnic differences in disease risk can be mapped in affected individuals of mixed descent if the ancestry of the alleles at each marker locus can be assigned to one of the two founding populations. Linkage can be detected by testing for association of the disease with the ancestry of alleles at the marker locus, by conditioning on the admixture (defined as the proportion of genes that have ancestry from the high-risk population) of both parents. With regard to exploiting the effects of admixture, this test is more flexible and powerful than the transmission-disequilibrium test. Under the assumption of a multiplicative model, the statistical power for a given sample size depends only on parental admixture and the risk ratio r between populations that is generated by the locus. The most informative families are those in which mean parental admixture is .2-.7 and in which admixture is similar in both parents. The number of markers required for a genome search depends on the number of generations since admixture and on the information content for ancestry (f) of the markers, defined as a function of allele frequencies in the two founding populations. Simulations using a hidden Markov model suggest that, when admixture has occurred 2-10 generations earlier, a multipoint analysis using 2,000 biallelic markers, with f values of 30%, can extract 70%-90% of the ancestry information for each locus. Sets of such markers could be selected from libraries of single-nucleotide polymorphisms, when these become available.     

Mutation and polymorphism of the prion protein gene in Libyan Jews with Creutzfeldt-Jakob disease (CJD).

The inherited prion diseases are neurodegenerative disorders which are not only genetic but also transmissible. More than a dozen mutations in the prion protein gene that result in nonconservative amino acid substitutions segregate with the inherited prion diseases including familial Creutzfeldt-Jakob disease (CJD). In Israel, the incidence of CJD is about 1 case/10(4) Libyan Jews. A Lys200 substitution segregates with CJD and is reported here to be genetically linked to CJD with a lod score of > 4.8. Some healthy elderly Lys200 carriers > age 65 years were identified, suggesting the possibility of incomplete penetrance. In contrast, no linkage was found between the development of familial CJD and a polymorphism encoding either Met129 or Val129. All Libyan Jewish CJD patients with the Lys200 mutation encode a Met129 on the mutant allele. Homozygosity for Met129 did not correlate with age at disease onset or the duration of illness. The frequency of the Met129 allele was higher in the affected pedigrees than in a control population of Libyan Jews. The frequency of the Met129 and Val129 alleles in the control Libyan population was similar to that found in the general Caucasian population. The identification of three Libyan Jews homozygous for the Lys200 mutation suggests frequent intrafamilial marriages, a custom documented by genealogical investigations.     

Detecting linkage for genetically heterogeneous diseases and detecting heterogeneity with linkage data.

Interest in searching for genetic linkage between diseases and marker loci has been greatly increased by the recent introduction of DNA polymorphisms. However, even for the most well-behaved Mendelian disorders, those with clear-cut mode of inheritance, complete penetrance, and no phenocopies, genetic heterogeneity may exist; that is, in the population there may be more than one locus that can determine the disease, and these loci may not be linked. In such cases, two questions arise: (1) What sample size is necessary to detect linkage for a genetically heterogeneous disease? (2) What sample size is necessary to detect heterogeneity given linkage between a disease and a marker locus? We have answered these questions for the most important types of matings under specified conditions: linkage phase known or unknown, number of alleles involved in the cross at the marker locus, and different numbers of affected and unaffected children. In general, the presence of heterogeneity increases the recombination value at which lod scores peak, by an amount that increases with the degree of heterogeneity. There is a corresponding increase in the number of families necessary to establish linkage. For the specific case of backcrosses between disease and marker loci with two alleles, linkage can be detected at recombination fractions up to 20% with reasonable numbers of families, even if only half the families carry the disease locus linked to the marker. The task is easier if more than two informative children are available or if phase is known. For recessive diseases, highly polymorphic markers with four different alleles in the parents greatly reduce the number of families required.     

Polymorphism of trinucleotide repeats at loci FRAXA and FRAXE in the population of Tomsk

Polymorphism of CGG and GCC trinucleotide repeats, whose expansions at the FRAXA and FRAXE loci have been identified as causative mutations in two forms of mental retardation, was studied in Slavic population of Tomsk. At the FRAXA locus a total of 31 allelic variants ranging from 8 to 56 copies of CGG repeat with two modal classes of 28-29 and 18-20 repeat units (with the frequencies of 24.6 and 11.5% respectively) were revealed. Compared to other populations, this locus was characterized by unusually high frequency of intermediate alleles with the sizes of more than 40 CGG repeat units (12.4%). Since intermediate repeats of the FRAXA locus were more prone to instability than normal alleles, it was suggested that Slavic population of Siberia had higher risk of the development of FMR1 dynamic mutations, giving rise to the Martin-Bell syndrome. The FRAXE allele frequency distribution was demonstrated to be normal with 18 allelic variants ranging from 9 to 27 GCC repeat units. In the population of Tomsk this locus had higher than in other populations frequency (26.7%) of short (less than 15 repeat units in size) alleles. In addition, in the Tomsk population both loci were characterized by high level of heterozygosity and low frequencies of modal allele classes. These results can be explained by the high level of outbreeding typical of the population of Siberia.     

Marker-assisted introgression using non-unique marker alleles I: selection on the presence of linked marker alleles

This paper investigates marker-assisted introgression of a major gene into an outbred line, where identification of the introgressed gene is incomplete because marker alleles are not unique to the base populations (the same marker allele can occur in both donor and recipient population). Those markers are used to identify the introgressed allele as well as the background genotype. The effect of using those markers, as if they were completely informative on the retention of the introgressed allele, was examined over five generations of backcrossing by using a single marker or a marker bracket for different starting frequencies of the marker alleles. Results were calculated by using both a deterministic approach, where selection is only for the desired allele, and by a stochastic approach, where selection is also on background genotype. When marker allele frequencies in donor and recipient population diverged from 1 and 0 (using a diallelic marker), the ability to retain the desired allele rapidly declined. Marker brackets performed notably better than single markers. If selection on background marker genotype was applied, the desired allele could be lost even more quickly than expected at random because the chance that the allele, which is common in the donor line, is present on the locus identifying the introgressed allele and is surrounded by alleles common in the recipient line on the background marker loci, will descend from the donor line (double recombination has taken place), is a lot smaller than the chance that this allele will stem from the recipient line (in which the allele occurs in low frequency). Marker brackets again performed better. Preselection against marker homozygotes (producing uninformative gametes) gave a slightly better retention of the introgressed allele.     

MC1R variation and melanoma risk in the Swedish population in relation to clinical and pathological parameters

The genetic background of cutaneous malignant melanoma (CMM) includes both germ line aberrations in high‐penetrance genes, like CDKN2A, and allelic variation in low‐penetrance genes like the melanocortin‐1 receptor gene, MC1R. Red‐hair colour associated MC1R alleles (RHC) have been associated with red hair, fair skin and risk of CMM. We investigated MC1R and CDKN2A variation in relation to phenotype, clinical factors and CMM risk in the Swedish population. The study cohort consisted of sporadic primary melanoma patients, familial melanoma patients and a control group. An allele‐dose dependent increase in melanoma risk for carriers of variant MC1R alleles (after adjusting for phenotype), with an elevated risk among familial CMM patients, was observed. This elevated risk was found to be significantly associated with an increased frequency of dysplastic nevi (DN) among familial patients compared to sporadic patients. MC1R variation was found to be less frequent among acral lentiginous melanomas (ALM) and dependent on tumour localisation. No association was found between CDKN2A gene variants and general melanoma risk. Two new variants in the POMC gene were identified in red haired individuals without RHC alleles.     

Complex segregation analysis of Parkinson's disease in the Finnish population

The risk of Parkinson's disease (PD) is higher among relatives of affected individuals than among other members of the population, and most family studies have suggested autosomal dominant inheritance, although both autosomal dominant and recessive susceptibility genes have recently been identified. We carried out a complex segregation analysis with POINTER to assess the mode of inheritance of PD in the population of northern Finland. Nuclear families (n=265) were identified through a proband with idiopathic PD. The analysis was first carried out for the total data set, and then the heterogeneity between early-onset (proband under 55 years at onset) and late-onset families was examined. Finally, families with more than one affected individual were analyzed separately. The sporadic model was rejected (P<0.0001). Significant heterogeneity was found between the early-onset and late-onset families, suggesting that major genes have a greater role in early-onset PD than in late-onset PD and that the etiology of idiopathic PD is heterogeneous, even in the Finnish population, which has evolved from a small group of founders. The analysis of familial PD supported the hypothesis that a major locus was present in this subset, but it was not possible to distinguish between a recessive model with a high penetrance and a dominant model with lower penetrance.     

Prospects for association-based fine mapping of a susceptibility gene for a complex disease

The potential of association studies for fine-mapping loci with common disease susceptibility alleles for complex genetic diseases in outbred populations is unclear. For a battery of tightly linked anonymous genetic markers spanning a candidate region centered around a disease locus, simulation methods based on a coalescent process with mutation, recombination, and genetic drift were used to study the spatial distribution of markers with large noncentrality parameters in a case-control study design. Simulations with a disease allele at intermediate frequency, presumably representing an old mutation, tend to exhibit the largest noncentrality parameter values at markers near the disease locus. In contrast, simulations with a disease allele at low frequency, presumably representing a young mutation, often exhibit the largest noncentrality parameter values at markers scattered over the candidate region. In the former case, sample sizes or marker densities sufficient to detect association are likely to lead to useful localization, whereas, in the latter case, localization of the disease locus within the candidate region is much less likely, regardless of the sample size or density of the map. The simulations suggest that for a single marker analysis, the simple strategy of choosing the marker with smallest associated P value to begin a laboratory search for the disease locus performs adequately for a common disease allele.     

Polymorphism of Trinucleotide Repeats at Loci FRAXA and FRAXE in the Population of Tomsk

Polymorphism of CGG and GCC trinucleotide repeats, whose expansions at the FRAXA and FRAXE loci have been identified as causative mutations in two forms of mental retardation, was studied in Slavic population of Tomsk. At the FRAXA locus a total of 31 allelic variants ranging from 8 to 56 copies of CGG repeat with two modal classes of 28–29 and 18–20 repeat units (with the frequencies of 24.6 and 11.5% respectively) were revealed. Compared to other populations, this locus was characterized by unusually high frequency of intermediate alleles with the sizes of more than 40 CGG repeat units (12.4%). Since intermediate repeats of the FRAXAlocus were more prone to instability than normal alleles, it was suggested that Slavic population of Siberia had higher risk of the development of FMR1 dynamic mutations, giving rise to the Martin–Bell syndrome. The FRAXE allele frequency distribution was demonstrated to be normal with 18 allelic variants ranging from 9 to 27 GCC repeat units. In the population of Tomsk this locus had higher than in other populations frequency (26.7%) of short (less than 15 repeat units in size) alleles. In addition, in the Tomsk population both loci were characterized by high level of heterozygosity and low frequencies of modal allele classes. These results can be explained by the high level of outbreeding typical of the population of Siberia.     

Reconstructing CNV genotypes using segregation analysis: combining pedigree information with CNV assay

Background

Repeated blocks of genome sequence have been shown to be associated with genetic diversity and disease risk in humans, and with phenotypic diversity in model organisms and domestic animals. Reliable tests are desirable to determine whether individuals are carriers of copy number variants associated with disease risk in humans and livestock, or associated with economically important traits in livestock. In some cases, copy number variants affect the phenotype through a dosage effect but in other cases, allele combinations have non-additive effects. In the latter cases, it has been difficult to develop tests because assays typically return an estimate of the sum of the copy number counts on the maternally and paternally inherited chromosome segments, and this sum does not uniquely determine the allele configuration. In this study, we show that there is an old solution to this new problem: segregation analysis, which has been used for many years to infer alleles in pedigreed populations.

Methods

Segregation analysis was used to estimate copy number alleles from assay data on simulated half-sib sheep populations. Copy number variation at the Agouti locus, known to be responsible for the recessive self-colour black phenotype, was used as a model for the simulation and an appropriate penetrance function was derived. The precision with which carriers and non-carriers of the undesirable single copy allele could be identified, was used to evaluate the method for various family sizes, assay strategies and assay accuracies.

Results

Using relationship data and segregation analysis, the probabilities of carrying the copy number alleles responsible for black or white fleece were estimated with much greater precision than by analyzing assay results for animals individually. The proportion of lambs correctly identified as non-carriers of the undesirable allele increased from 7% when the lambs were analysed alone to 80% when the lambs were analysed in half-sib families.

Conclusions

When a quantitative assay is used to estimate copy number alleles, segregation analysis of related individuals can greatly improve the precision of the estimates. Existing software for segregation analysis would require little if any change to accommodate the penetrance function for copy number assay data.     

Marker selection for the transmission/disequilibrium test, in recently admixed populations.

Recent admixture between genetically differentiated populations can result in high levels of association between alleles at loci that are <=10 cM apart. The transmission/disequilibrium test (TDT) proposed by Spielman et al. (1993) can be a powerful test of linkage between disease and marker loci in the presence of association and therefore could be a useful test of linkage in admixed populations. The degree of association between alleles at two loci depends on the differences in allele frequencies, at the two loci, in the founding populations; therefore, the choice of marker is important. For a multiallelic marker, one strategy that may improve the power of the TDT is to group marker alleles within a locus, on the basis of information about the founding populations and the admixed population, thereby collapsing the marker into one with fewer alleles. We have examined the consequences of collapsing a microsatellite into a two-allele marker, when two founding populations are assumed for the admixed population, and have found that if there is random mating in the admixed population, then typically there is a collapsing for which the power of the TDT is greater than that for the original microsatellite marker. A method is presented for finding the optimal collapsing that has minimal dependence on the disease and that uses estimates either of marker allele frequencies in the two founding populations or of marker allele frequencies in the current, admixed population and in one of the founding populations. Furthermore, this optimal collapsing is not always the collapsing with the largest difference in allele frequencies in the founding populations. To demonstrate this strategy, we considered a recent data set, published previously, that provides frequency estimates for 30 microsatellites in 13 populations.     

Power calculations for the transmission/disequilibrium and affected sib pair tests using elementary probability methods

The transmission/disequilibrium test (TDT) and the affected sib pair test (ASP) both test for the association of a marker allele with some conditions. Here, we present methods for calculating the probability of detecting the association (power) for a study examining a fixed number of families for suitability for the study and for calculating the number of such families to be examined. Both calculations use a genetic model for the association. The model considered posits a bi-allelic marker locus that is linked to a bi-allelic disease locus with a possibly nonzero recombination fraction between the loci. The penetrance of the disease is an increasing function of the number of disease alleles. The TDT tests whether the transmission by a heterozygous parent of a particular allele at a marker locus to an affected offspring occurs with probability greater than 0.5. The ASP tests whether transmission of the same allele to two affected sibs occurs with probability greater than 0.5. In either case, evidence that the probability is greater than 0.5 is evidence for association between the marker and the disease. Study inclusion criteria (IC) can greatly affect the necessary sample size of a TDT or ASP study. IC considered by us include a randomly selected parent at least one parent or both parents required to be heterozygous. It also allows a specified minimum number of affected offspring to be required (TDT only). We use elementary probability calculations rather than complex mathematical manipulations or asymptotic methods (large sample size approximations) to compute power and requisite sample size for a proposed study. The advantages of these methods are simplicity and generality.