Estimation of Trait-Model Parameters in a MOD Score Linkage Analysis

Markus Brugger; Susanne Rospleszcz; Konstantin Strauch

doi:10.1159/000479738

. 2017 Nov 1;82(3-4):103–139. doi: 10.1159/000479738

Estimation of Trait-Model Parameters in a MOD Score Linkage Analysis

Markus Brugger ^1,^*, Susanne Rospleszcz ¹, Konstantin Strauch ¹

PMCID: PMC6187844 PMID: 29131067

Abstract

Background/Aims

Theoretically, the trait-model parameters (disease allele frequency and penetrance function) can be estimated without bias in a MOD score linkage analysis. We aimed to practically evaluate the MOD score approach regarding its ability to provide unbiased trait-model parameters for various pedigree-type and trait-model scenarios. We further investigated the ability of the MOD score approach to detect imprinting using affected sib pairs (ASPs) and affected half-sib pairs (AHSPs) when all parental genotypes are missing.

Methods

Simulated pedigree data were analyzed using the GENEHUNTER-MODSCORE software package. Parameter estimation performance in terms of bias and variability was evaluated with regard to trait-model type and pedigree complexity.

Results

Generally, parameters were estimated with lower bias and variability with increasing pedigree complexity, especially for recessive and overdominant models. However, dominant and additive models could hardly be distinguished even when using 3-generation pedigrees. Imprinting could clearly be detected for mixtures of mainly ASPs and only few AHSPs with the common parent of the imprinted sex, even though no parental genotypes were available.

Conclusion

Our results provide guidance to researchers regarding the possibility to estimate trait-model parameters by a MOD score analysis, including the degree of imprinting, with certain types of pedigrees.

Keywords: Parametric linkage analysis, MOD scores, Trait-model parameters, Identifiability, Estimation bias

Introduction

Trait Inheritance and Pedigree Analysis

The inheritance of a trait is defined as the mechanism by which the joint phenotypic distribution of the particular trait in pedigree members can explicitly be described [1]. A pedigree can be considered as a discrete unit of a population for which the relationship connecting any pair of pedigree members is unambiguously known. There is hence no other individual for which a relationship to any of these pedigree members can be established. Under the assumption that pedigrees implicitly contain information about details of the mode of inheritance of a trait through the covariation and cosegregation of the trait characteristics among its members, collecting and analyzing samples of pedigrees can be used to study the trait inheritance. In genetics, inference about trait inheritance by pedigree analysis is made assuming that the main factors underlying the inheritance are genes. Mathematical-genetic models can then be used to describe the trait inheritance, and these models are tested using pedigree samples drawn from the population.

If the genetic model of trait inheritance is inferred on the basis of the pedigree sample, which contains the necessary information through the joint phenotypic cosegregation in the pedigree members, such an analysis is called “segregation analysis” [1]. If the purpose of the analysis is to map the putative disease gene(s), whose existence may have been previously established by segregation analysis to specific chromosomal segments by investigation of cosegregation of DNA marker alleles and the trait phenotype, such an analysis is called “linkage analysis” [1]. Nowadays, pure segregation analysis is of less practical importance than it has been a few decades ago. With increasing availability of DNA marker maps and rapid and cost-effective DNA genotyping techniques, linkage analysis has become the state-of-the-art technique of pedigree analysis. In addition, association analysis can be performed with pedigrees as well as samples of unrelated individuals. However, software packages for segregation analysis like PAP [2], S.A.G.E. [3], and MORGAN [4, 5] continue to be available and provide great flexibility with respect to fitting the model for the mode of inheritance (see also e.g. Kriszt et al. [6] for a recent publication using complex segregation analysis with keratoconus pedigrees).

Linkage Analysis

In earlier times, linkage analysis has been used to map genes that were already known to exist. In the meantime, linkage analysis serves 2 purposes: (1) to prove the existence of a disease gene and (2) to map it [7]. Linkage analysis methods can be distinguished as model-based or model-free [8]. The former is also known as parametric or LOD score linkage analysis for which a certain set of trait-model parameters regarding the segregation of the disease is explicitly assumed in the genetic likelihood. The latter, which is also known as nonparametric linkage analysis, proceeds without such explicit models. These 2 types of linkage analysis are, however, closely related to each other. It can be shown that certain nonparametric and parametric linkage tests are equivalent for any type of pedigree [9, 10] and can be considered as different ways to parametrize the allele-sharing probabilities, i.e., the probabilities of allele(s) shared identical-by-descent (IBD) by affected pedigree members, in the genetic likelihood.

Mode of Inheritance and Trait-Model Parameters

A crucial factor in linkage analysis is the true mode of inheritance. Under the term “mode of inheritance,” 2 concepts are often subsumed that need, however, to be distinguished. The first concept is the genetic mechanism of the disease involving the number of loci, the number of alleles at each locus, and the segregation parameters including the recombination fraction among the trait loci as well as between them and any marker(s) [11]. The second concept is the genotype-phenotype relation, which is defined by the penetrance function, i.e., the probability that an individual with a certain number of copies of the disease allele is affected by the disease. The genetic mechanism of the disease, apart from the recombination fraction, is assumed to be known for linkage analysis. In the case of a binary trait governed by a single diallelic autosomal locus, which is assumed throughout this paper, the disease allele frequency p and the 3 penetrances f0, f1, and f₂, with f_i denoting the probability that an individual with i copies of the disease allele is affected by the disease, can be subsumed under the term “trait-model parameters.” In the case of parametric linkage analysis, trait-model parameters can either be prespecified according to results from previous segregation analyses or maximized along with the recombination fraction in a joint segregation and linkage (JSL) analysis. A specific type of this approach is the MOD score analysis, which was first proposed by Risch [12]. If the genetic mechanism of the trait is not modelled correctly, however, which is expected in practice due to the large number of possible inheritance modes, parameter estimates obtained from a MOD score analysis will be asymptotically biased [11, 13].

Likelihood and Sample Space

In pedigree analysis, the likelihood given a particular sample of pedigrees can be defined as the probability to observe the data available for the individuals in the pedigree, constructed under a certain genetic model. In fact, any formulation that is proportional to this probability can be used as the likelihood. The pedigree samples used for pedigree analysis are collected from what is called the “real” population that is defined on the basis of usually unknown factors like the population's origin and history. This real population is mapped into a set of disjoint pedigrees by the use of those relationships between members of the real population that can unambiguously be established [1]. These disjoint pedigrees are then further determined by the predefined sampling design, which partitions the pedigrees into substructures of certain inheritance relations, e.g., sibships with all other relationships outside sibships being ignored. The resulting structures are called “true pedigrees.” As described in Ginsburg et al. [1], pedigree analysis is performed on sampled pedigrees collected from the set of true pedigrees. The subset of pedigrees that in principle can be sampled according to the sampling design is called the “sample space.” The sampling procedure involves the pedigree ascertainment (primary selection), the intrafamilial extension (inclusion of additional relatives), and the selective inclusion in the analysis (censoring).

In the following, we will assume that ascertainment takes place through probands. For each true pedigree, there are members who could “potentially” become probands due to prespecified proband characteristics, e.g., geographic area, age, sex, but independently of their phenotypes. This subset of potential probands in the true pedigree, including both their relationships and phenotypes, is called the “proband sampling frame” (PSF, [14]). It can be shown that assuming the wrong mode of inheritance and/or the wrong model for the sampling procedure leads to asymptotically biased trait-model parameters and nuisance parameters of the sampling model when performing maximum likelihood estimation [15]. In order to obtain unbiased parameter estimates, the pedigree likelihood is defined as the probability of the particular pedigree data having been sampled (ascertained, extended, and included in the analysis) on the sample space generated by the sampling procedure under the given mode of inheritance [1]. The sample space for the given sampling procedure is the probability that at least 1 pedigree is sampled from the set of true pedigrees [1]. In this general form, however, the pedigree likelihood cannot be calculated using only the sampled data [1]. This would demand knowledge about the distribution of possible PSFs to calculate the sample space on which the likelihood is defined. Therefore, pedigree likelihoods are conditioned on specific parts of the sampled data to circumvent this problem and – by the same token – to retain unbiasedness of parameter estimates. In the following sections, pedigree likelihoods, which are conditioned on specific parts of the sampled data, are briefly introduced in the context of JSL analysis.

Sampling Model-Based Likelihood

As was explained in the previous section, the pedigree likelihood provides consistent estimates of the trait-model parameters if it is conditioned on the pedigree having been sampled, i.e., ascertained, extended, and included in the sample under analysis [16]. This also holds true for JSL analysis. In parametric JSL analysis, which is the main focus of this paper, the likelihood is formulated using the trait-model parameters, i.e., the disease allele frequency p and the penetrances f₀, f₁, and f₂, as well as the marker allele frequencies and the recombination fractions – and, if applicable, linkage disequilibria (LD) between loci. These parameters can be subsumed under the term “joint trait-marker inheritance parameters” [16]. In addition, information about the following aspects must also be included in the likelihood: (1) the whole PSF structure and its population distribution, which is relevant for ascertainment, (2) the pedigree extension procedure, and (3) the conditions relevant to inclusion, which could be specific marker genotypes of certain pedigree members. Since the population distribution of the PSF structure is unknown, the pedigree likelihood can be conditioned on the substructure of the pedigree that is “relevant to sampling” (RS), in order to make the likelihood calculable and to properly take the sampling procedure into account. The structure RS corresponds to all PSF members of the true pedigree under study – i.e., the part of the pedigree “relevant to ascertainment” (RA) – and those pedigree members responsible for the inclusion of the pedigree in the sample. Importantly, the likelihood is only conditioned on the structure RS but not on the phenotypes of the corresponding pedigree members. Since the likelihood includes explicit details of the sampling procedure, it is termed “sampling model-based (SMB) likelihood” [16]. The SMB likelihood provides asymptotically unbiased estimates of all joint trait-marker inheritance parameters, including the mode of inheritance, as well as of the parameters determining the ascertainment, extension, and inclusion procedure [1].

Sampling Model-Free Likelihood

A sampling model-free (SMF) likelihood can be formulated using a more robust procedure initially proposed by Ewens and Shute [17] in the context of segregation analysis, in which uncertainties about the ascertainment procedure are controlled by conditioning the likelihood on that part of the pedigree data RA. The latter approach is called “ascertainment assumption-free” (AAF) and can readily be extended to be SMF, if the likelihood is also conditioned on that part of the data RS [16]. The part of the data RS is the data RA and that part of the data relevant to inclusion, which could be, e.g., certain parental marker genotypes. In contrast to the SMB likelihood, which is conditioned only on the structure RS, the SMF likelihood is conditioned on the data RS, i.e., structure as well as marker and trait values RS. This SMF likelihood provides asymptotically unbiased estimates of all joint trait-marker inheritance parameters, including the mode of inheritance, as well as of the extension parameter [1].

Likelihood in a MOD Score Analysis

The question arises which kind of likelihood underlies a JSL analysis using the MOD score, and if it is in principle possible to obtain unbiased parameters from this procedure. As shown by Clerget-Darpoux et al. [18] and later also by Elston [11], maximizing the LOD score in the context of a MOD score analysis is equivalent to maximizing the likelihood of the marker data, conditional on the pedigree structure and conditional on all the trait data, i.e., not only on that part RS. This conditional likelihood – from now on referred to as “MOD score likelihood” – does not depend on the ascertainment scheme, provided that the sampling of pedigrees is independent of marker data. Hence, this means that selective inclusion of pedigrees based on marker genotypes (i.e., marker-dependent sampling) is not controlled in the MOD score likelihood, because it does not contain the inclusion parameter. As a consequence, the MOD score will yield biased estimates of the joint trait-marker inheritance parameters if there is association between disease and marker alleles (LD > 0), because ascertainment is no longer marker-independent in that case [19].

The following conditions must be satisfied to obtain unbiased estimates of the joint trait-marker inheritance parameters from a MOD score analysis [1, 19]: (i) the marker locus must be truly linked to the trait locus, (ii) the genetic mechanism of the trait (number of loci and number of alleles at each locus) is known, (iii) sampling is marker-independent, (iv) the model for the pedigree extension procedure is known, and either (v) trait values are available for all members of the PSF, which has to be completely known, or (vi) the ascertainment is proband-independent (PI) or single in the sense described by Hodge and Vieland [20], i.e., all pedigrees have equal probabilities of being ascertained, independent of pedigree size or structure, or (vii) the joint probability of the unobserved trait phenotypes of the members of the PSF, conditional on the trait and marker phenotypes of all the observed pedigree members, does not depend on the marker phenotypes. Condition (v) reflects that the MOD score likelihood can be derived from the SMB likelihood by conditioning the latter on the trait values of all individuals, including all PSF members, in addition to the structure RS. Condition (vi) is due to the fact that the MOD score likelihood does not include an ascertainment parameter as opposed to the SMB likelihood, which contains such a parameter. The probability of ascertainment, however, actually depends on the joint trait-marker inheritance parameters, if sampling is not PI or single [21]. Only with PI or single ascertainment, the probability of ascertainment no longer depends on these parameters and can, therefore, be omitted in the likelihood without influencing the estimates of the parameters [1]. Without specifying details of the sampling procedure, parameter estimates are also consistent when missing trait values of the PSF members do not depend on marker phenotypes (condition [vii]). However, this only holds in the case of no LD and no linkage between trait and marker locus, or if the trait phenotype unambiguously defines the trait genotype [19].

The MOD score likelihood differs from the SMF likelihood by the fact that it is conditioned on all trait values (i.e., not only of the PSF members) in addition to the data RS, and that it assumes PI or single ascertainment as well as marker-independent sampling, rather than specifying some value for the ascertainment probability in the likelihood. This is why the MOD score likelihood can be considered to be somewhere between SMB and SMF. If sampling is marker-independent, but conditions (i), (ii), and (iv) are not simultaneously satisfied, parameter estimates obtained from MOD score analyses will be biased. If conditions (i)–(iv) hold, but neither condition (v), (vi), nor (vii) is met, the estimate of the recombination frequency will only slightly be biased [1]. In this case, it is of note that estimates of the recombination fraction are biased even when trait-model parameters are fixed at their true values [22].

Summary of Conditions to Obtain Unbiased

Parameter Estimates from a MOD Score Analysis

The pedigree likelihood of the MOD score approach delivers asymptotically unbiased estimates of the joint trait-marker inheritance parameters (recombination fraction, allele frequency, and penetrances, but not the LD parameter), if the following conditions are satisfied (see also Malkin and Elston [19]):

i The marker is truly linked.

AND
ii The genetic mechanism of the trait (number of loci and number of alleles at each locus) is known.

AND
iii Sampling (ascertainment, extension, inclusion) of pedigrees is independent from marker data.

AND
iv The model of extension is known.

AND

At least 1 of the following 3 conditions is satisfied:
v All members of the pedigree PSF must have measured trait values (if not sampled, information on trait values can be gathered using a questionnaire as proposed by Ginsburg et al. [16]).

OR
vi The ascertainment procedure is PI or single in the sense of Hodge and Vieland [20].

OR
vii The joint probability of the unobserved trait phenotypes of the members of the PSF, conditional on the trait and marker phenotypes of all the observed pedigree members, does not depend on the marker phenotypes.

Hence, unbiased estimates of the joint trait-marker inheritance parameters can in principle be obtained without explicitly formulating the ascertainment and inclusion procedures. It should further be noted that the likelihood correction in a MOD score analysis directly follows from the AAF method proposed in Ewens and Shute [17]. Whereas conditions (i)–(iii) are crucial, conditions (v)–(vii) may be of minor impact on the bias of parameter estimates in practice [20, 23]. With respect to condition (v), if members of the pedigree PSF are not sampled and trait values cannot be gathered using a questionnaire, an approximate likelihood using the sample mean of the trait value can be constructed [1]. Condition (iv) could be satisfied as follows. PI sampling implies that fixed pedigree structures are sampled, which renders a specification of the extension parameter pointless. With single ascertainment, the pedigree extension model could be chosen to be trait-independent, such that any initially sampled subpedigree is further extended using all available relatives, regardless of their phenotypes and with a random, trait-independent stopping rule. If this holds true, an extension parameter does not have to be formulated in the likelihood. Despite being hard to achieve in practice, conditions (iv)–(vii) can in theory be fulfilled. If not, the resulting bias in parameter estimates is argued to be small [20], but numerical quantification of the bias of the joint trait-marker inheritance parameters obtained from a MOD score analysis under many different sampling schemes is not available so far. This would demand an extensive simulation study to prove that the MOD score approach is robust with regard to its ability to estimate parameters, even if some necessary assumptions do not hold. Even if all necessary conditions are satisfied, a bias of maximum likelihood estimates can nevertheless occur for finite sample sizes. In addition, variances of the obtained estimates are expected to be rather large using the MOD score likelihood due to a loss of pedigree information by conditioning not only on the pedigree structure but also on the trait data of all individuals [24].

The focus of the present paper is the proof-of-principle of the ability of a MOD score analysis to obtain asymptotically unbiased joint trait-marker inheritance parameters in practice, given that conditions (i)–(iv) and at least one of (v)–(vii) are satisfied. In particular, the identifiability (see also next section) of these parameters using various pedigree types and realistic sample sizes will be investigated.

Identifiability of Inheritance Parameters

Even if the conditions under which the MOD score provides unbiased estimates of the joint trait-marker inheritance parameters are fulfilled, the identifiability of these parameters is restricted by the type(s) of pedigrees in a given sample. In a model-based linkage analysis, such as a MOD score analysis, the penetrances, disease allele frequency, and the recombination fraction represent a reparametrization of the truly underlying allele-sharing classes [9, 10, 25, 26]. In other words, allele-sharing probabilities (classes) of a given pedigree type can be expressed in terms of the joint trait-marker inheritance parameters. In the case of an affected sib pair (ASP), these allele-sharing classes are z₀, z₁, and z₂ that an ASP shares 0, 1, or 2 allele(s) IBD with restrictions to genetically possible models [27]. With z₂ = 1 − z₀ - z₁ and restrictions z₁ ≤ 0.5 and 2×z₀ ≤ z₁, the allele-sharing classes of ASPs form a 2-dimensional parameter space - the so-called “possible triangle” [27]. Hence, as there are only 3 − 1 = 2 free parameters that can be estimated from ASP data, there will be many sets of f₀, f₁, f₂, p, and the recombination fraction θ that correspond to the estimated ẑ₀, ẑ₁, and ẑ₂. With larger pedigrees, and hence more allele-sharing classes, the degree to which the trait-model parameters can be correctly determined should be higher. However, the corresponding allele-sharing configurations have hitherto only been formulated for unilineal, affected relative pairs (e.g., affected half-sib pairs [AHSPs] [10]), ASPs [27], and affected sib triplets (ASTs) [28]. The parameter space for AHSPs is degenerated to a single line [10]. Hence, many different sets of trait-model parameters correspond to the same point on this so-called “possible line.”

Using the formulas in Knapp [28], it is possible to draw the 3-dimensional parameter space for ASTs with empirically assessed restrictions for genetically possible models (Fig. 1). However, the parameter restrictions have not been derived in closed form so far. The parameter spaces for larger pedigrees involve a larger number of dimensions, and the corresponding restrictions for genetically possible models are expected to have an even more complicated form [10, 28]. It is of note that for any type of affecteds-only analysis, the absolute values of penetrances cannot be determined, because multiplication of all penetrances by the same factor does not change the result. However, their ratios are not defined if the penetrance in the denominator of the ratio is estimated to be 0. Additionally, the ratio is subject to the estimation variance of both the penetrance in the numerator and in the denominator.

Fig. 1. — Graphical depiction of the allele-sharing parameter space for affected sib triplets (ASTs). The axes notations are defined as follows (see also Knapp [28]). Axis z1: allele-sharing class z₁ with range {0; 3/14}. Axis z2: z₂ with range {0; 0.75}. Axis z3: z₃ with range {0; 1}. The panels top and at the left correspond to “top view.” The boundary of the parameter space, which is defined by the genetically possible models, was empirically determined by varying the trait-model parameters {f₀, f₁, f₂, p} in the formulas given in Knapp [28]. p, disease allele frequency; *f_i*, penetrances, with *f_i* denoting the probability that an individual with i copies of the disease allele is affected by the disease. Light green, dark green, and black lines were drawn by varying p between 0 and 1. For more details, see table below. Figures were drawn using rgl: 3D Visualization Using OpenGL, R package version 0.95.1441 (2016) by Adler, Murdoch, and others.

Imprinting

Genomic imprinting implies dependence of an individual's liability to develop a disease on the parental origin of the mutated allele(s), leads to a deviation from the classic Mendelian assumption of equal contribution of parental genomes to the progeny and is, therefore, called a “parent-of-origin effect” [29]. In the context of a parametric linkage analysis, imprinting can be modelled using a 4-penetrance formulation distinguishing the heterozygotes according to the parental origin of the disease allele: f = (f₀, f_1,pat, f_1,mat, f₂), as implemented in the program GENEHUNTER-MODSCORE (GHM) [25, 30, 31, 32], which is a further development of GENEHUNTER-IMPRINTING [33]. In the nonparametric context, the allele-sharing class z₁ of an ASP is split up into z_1,pat and z_1,mat according to the parental origin of the shared allele. The corresponding parameter space of ASPs, hence, extends to a 3-dimensional tetrahedron which accounts for disease models with z_1,pat ≠ z_1,mat, i.e., for imprinting [34]. In the case of AHSPs, the allele-sharing class z₁ is distinguished as either being z_1,pat or z_1,mat, depending on the sex of the common parent, i.e., male or female, respectively. Although the information contained in AHSPs on all trait-model parameters is limited, the information for imprinting may be high, such that parameter estimates for f_1,pat and f_1,mat using a sample of AHSPs having a common father and of AHSPs having a common mother should indicate imprinting if it was really present. In the case of an informative marker, this even holds if parental genotypes are missing.

In contrast, imprinting information contained in ASPs with untyped parents is 0, even in the case of a fully informative marker, because alleles shared IBD through the father cannot be distinguished from those shared IBD through the mother. However, we hypothesize that the information on linkage and imprinting gained from AHSPs can be combined with the pure linkage information contained in ASPs in the analysis to compensate for missing parental marker genotypes. If there is sufficient evidence for linkage, this pedigree scenario should lead to trait-model parameter estimates reflecting at least some degree of imprinting. Using GHM, imprinting can be quantified by looking at the imprinting index I [35], calculated from the estimated penetrances. The imprinting index equals the difference between the 2 heterozygote penetrances, normalized by the difference of the homozygote penetrances in order to properly take the case of a non-0 phenocopy rate or reduced penetrance into account:

I = \frac{f_{1, p a t} - f_{1, m a t}}{f_{2} - f_{0}} .

An imprinting index of I = 1, therefore, indicates complete maternal imprinting (cmi), whereas I = −1 indicates complete paternal imprinting (cpi). If penetrances are not restricted to f₀ < f₁ < f₂ in the analysis, the penetrances f_1,pat and f_1,mat can, therefore, be estimated to be <f₀ and >f₂. Thus, the imprinting index may exceed 1 or fall below −1. In the case of f₀ = f₂, the imprinting index is defined to be 0. In a work by Haghighi and Hodge [36], it was shown that asymptotically unbiased estimates of parent-of-origin effects can be obtained using a likelihood formulation for segregation analysis without including an ascertainment parameter when ascertainment is single. The same should hold true for the method by Strauch et al. [33] applied in this paper in the context of parametric linkage analysis according to the arguments given by Ginsburg et al. [1] and Malkin and Elston [19], provided that the formulation with 4 penetrances correctly reflects the genetic mechanism of genomic imprinting.

Aims of the Present Study

The aim of the present study was to evaluate how accurately penetrances, or penetrance ratios in the case of affecteds-only analyses, and the disease allele frequency of a monogenic, dichotomous trait can be estimated in a MOD score analysis. To this end, we performed a simulation study to determine the bias and variability of trait-model parameter estimation for 6 pedigree types (AHSP, ASP, AST, discordant sib triplets [DST], discordant sib quadruplets [DSQ], and 3-generation [3-G] pedigrees) and 4 types of generic models (recessive, dominant, additive, and overdominant) as well as an imprinting model. A single marker locus linked with θ = 0 to the disease locus was considered. It is of note that we did not consider the estimation of the recombination fraction θ or any LD parameter in our analysis. That is because the primary focus of this paper is on the estimation of trait-model parameters, which do not include the recombination fraction. However, the recombination fraction is confounded with the trait-model parameters, especially for smaller pedigree types, like the ones considered in our study, having only a limited number of allele-sharing classes (see also “Identifiability of Inheritance Parameters” above). In addition, LD parameters cannot be estimated using GHM so far.

We avoided the problem of an additional bias due to a possible misspecification of the sampling model for the likelihood correction. This was done by designing the simulation study in a way that conditions (i)–(iv) and (vi) mentioned above to obtain asymptotically unbiased parameter estimates from a MOD score analysis were satisfied as follows (note that only one of conditions [v]–[vii] needs to be fulfilled):

i The marker was truly linked (θ = 0).
ii A diallelic autosomal binary trait locus, which is usually assumed as the mode of inheritance in a MOD score analysis, was used for the simulation of pedigree data.
iii Sampling of pedigrees was marker-independent.
iv Extension of pedigrees was trait-independent.
v -
vi Ascertainment was single in the sense of Hodge and Vieland [20].
vii -

Hence, the questions we aimed to answer in our study were:

1. For each pedigree type, can the MOD score approach differentiate between the trait-model types? That is, are, for example, recessive models recognized as being recessive, irrespective of the accuracy of the individual parameter estimates?
2. How does the estimation accuracy change from ASP to AST, i.e., when adding an affected sibling?
3. How does the estimation differ between an analysis using only affecteds vs. both affecteds and unaffecteds?
4. How does the estimation accuracy change from DST to DSQ, i.e., when adding a second unaffected sibling?
5. How does the estimation accuracy change when more complex pedigrees are considered?
6. How well can imprinting be detected and estimated in a sample of AHSPs and in a mixture sample of AHSPs and ASPs when parental genotypes are missing?

The answers to these questions are summarized in the Results section.

Methods

Nomenclature

Parameters written in capital letters (P, D, F₀, F₁,p_at, F_1,mat, F₁, F₂, I) denote theoretical parameters and the parameters that were used for simulation (“true” parameters). Lowercase letters (p, d, f₀, f_1,pat, f_1,mat, f₁, f₂, i) denote the parameters that were estimated from simulated data.

Data Generation

The 5 pedigree types shown in Figure 2 (top and middle row) were chosen for the simulations. We used a sample size of 500 families for each pedigree type to ensure sufficient power to detect linkage while maintaining reasonable computation times. For certain trait-model scenarios, we performed additional analyses with a sample size of 1,000 families to assess the degree by which parameter estimates are biased due to finite sample sizes. Disease and marker locus genotypes were simulated using FastSLINK [37, 38, 39]. For each pedigree-type-trait-model scenario, we simulated 1,000 replicates. Affection statuses were assumed to be unknown for all founders. Nonfounders were either affected or unaffected (Fig. 2).

Fig. 2. — Pedigree types used for the simulations. ASP, affected sib pair; AST, affected sib triplet; DST, discordant sib triplet; DSQ, discordant sib quadruplet; 3-G, three-generation pedigree; AHSP 1, affected half-sib pair with common father; AHSP 2, affected half-sib pair with common mother;?, unknown phenotype; filled symbols, affected; empty symbols, unaffected.

Recessive, additive, dominant, and overdominant trait models were considered in the simulations. An overview of the simulated trait models is given in Table 1. Trait models were named according to their generic type, i.e., “R” for a recessive model, “D” for a dominant model, “A” for an additive model, and “U” for an overdominant model. For each of the 4 generic types, 3 trait models with a particular combination of penetrances were simulated (trait-model names 1–3; Table 1). The setup of the trait-model parameters was inspired by Xing and Elston [40]. Each of the 3 trait models was simulated with a disease allele frequency P = 0.1 or 0.01. For the lower disease allele frequency P = 0.01, an additional trait model was simulated with a sample size of 1,000 families per replicate for each of the 4 generic types (Table 1). This amounts to 28 simulated scenarios. Furthermore, an overdominant model with a different combination of penetrances was simulated (model U4). For the recessive, dominant, and additive trait models, 2 further models similar to those in Flaquer and Strauch [41] were considered (models preceded by “AF” in Table 1). One of these models was simulated with sample sizes 500 and 1,000, whereas the other model was simulated with sample size 500 only. The total number of simulated scenarios, therefore, amounts to 38.

Table 1.

Overview of the simulated scenarios using trait models of the generic types “recessive,” “dominant,” “additive,” and “overdominant”

Model type	Name	P	F₀	F₁	F₂	Sample size
Recessive	R1	0.01; 0.1	0.01	0.01	0.2	500 and 1,000; 500
	R2	0.01; 0.1	0.01	0.01	0.5	500
	R3	0.01; 0.1	0.01	0.01	0.8	500
	AFRI	0.2	0.04	0.04	0.2	500 and 1,000
	AFR2	0.25	0.003	0.05	0.5	500

Dominant	D1	0.01; 0.1	0.01	0.2	0.2	500 and 1,000; 500
	D2	0.01; 0.1	0.01	0.5	0.5	500
	D3	0.01; 0.1	0.01	0.8	0.8	500
	AFD1	0.05	0.04	0.2	0.2	500 and 1,000
	AFD2	0.25	0.003	0.5	0.5	500

Additive	A1	0.01; 0.1	0.01	0.1	0.2	500 and 1,000; 500
	A2	0.01; 0.1	0.01	0.2	0.5	500
	A3	0.01; 0.1	0.01	0.5	0.8	500
	AFA1	0.1	0.03	0.13	0.23	500 and 1,000
	AFA2	0.5	0.003	0.25	0.5	500

Overdominant	U1	0.01; 0.1	0.01	0.2	0.01	500
	U2	0.01; 0.1	0.01	0.5	0.01	500
	U3	0.01; 0.1	0.01	0.8	0.01	500 and 1,000; 500
	U4	0.35	0.01	0.9	0.01	500

	Pedigree structure	Model name	P	F₀	F_1,pat	F_1,mat	F₂
Model with imprinting	1. 100 AHSPs with a common father + 100 AHSPs with a common mother	cpi	0.01	0	0	1	1

	2. 100 AHSPs with a common mother + 100 ASPs	cpi	0.01	0	0	1	1

	3. 20 AHSPs with a common mother + 180 ASPs	cmi	0.01	0	1	0	1

Comparison model	All structures	ni	0.01	0	0.5	0.5	1

Model name	P	Estimated median MOD (MAD)
		ASP	AST	DST	DSQ	3-G
R1	0.1	33.07 (5.51)	134.47 (11.19)	32.41 (5.32)	31.25 (5.36)	37.85 (5.45)
R1	0.01	0.07 (0.11)	2.89 (1.69)	0.23 (0.32)	0.27 (0.35)	0.49 (0.49)
R1n1000	0.01	0.09 (0.14)	5.58 (2.31)	0.26 (0.36)	0.31 (0.38)	0.52 (0.5)
R2	0.1	100.76 (8.31)	207.56 (12.31)	108.46 (9.06)	114.38 (9.76)	89.51 (7.01)
R2	0.01	1.08 (0.97)	116.89 (11.4)	1.15 (0.88)	1.0 (0.88)	1.84 (1.1)
R3	0.1	127.37 (9.16)	225.05 (14.13)	162.37 (10.9)	189.64 (11.69)	158.31 (9.82)
R3	0.01	5.37 (2.29)	283.25 (15.7)	4.76 (2.03)	3.86 (1.87)	8.35 (2.72)
AFR1	0.2	2.69 (1.53)	13.02 (3.53)	2.81 (1.49)	2.71 (1.52)	3.78 (1.69)
AFR1n1000	0.2	5.26 (2.19)	25.87 (5.26)	5.13 (2.07)	5.08 (1.95)	7.08 (2.39)
AFR2	0.25	40.43 (5.97)	62.95 (7.65)	50.02 (6.3)	59.07 (6.24)	33.99 (4.87)

D1	0.1	19.28 (3.72)	36.43 (5.36)	20.64 (3.87)	21.95 (4.14)	29.17 (4.97)
D1	0.01	25.19 (4.2)	91.2 (7.44)	24.93 (4.28)	24.45 (4.18)	65.87 (6.25)
D1n1000	0.01	49.92 (5.82)	182.62 (9.84)	49.26 (5.95)	48.61 (5.91)	131.43 (8.67)
D2	0.1	24.43 (4.28)	43.74 (6.06)	34.74 (4.87)	45.41 (5.99)	56.48 (6.89)
D2	0.01	47.85 (5.07)	124.71 (7.95)	54.98 (5.39)	60.9 (6.34)	117.79 (7.59)
D3	0.1	25.98 (4.38)	45.93 (6.11)	67.63 (6.82)	115.52 (9.44)	133.19 (10.3)
D3	0.01	54.44 (5.11)	134.19 (7.84)	90.84 (6.9)	125.33 (9.12)	206.27 (10.94)
AFD1	0.05	4.04 (1.9)	15.8 (3.67)	4.19 (1.81)	4.1 (1.76)	9.0 (2.66)
AFD1n1000	0.05	7.94 (2.51)	31.42 (5.15)	7.92 (2.59)	7.85 (2.55)	17.45 (3.51)
AFD2	0.25	8.94 (2.74)	12.37 (3.22)	15.77 (3.77)	24.13 (4.46)	21.30 (4.21)

A1	0.1	13.58 (3.25)	28.6 (4.72)	14.13 (3.4)	14.37 (3.44)	15.05 (3.4)
A1	0.01	7.45 (2.48)	48.12 (6.2)	7.36 (2.44)	7.17 (2.46)	28.56 (4.57)
A1n1000	0.01	14.85 (3.48)	96.12 (8.53)	14.62 (3.25)	13.99 (3.14)	57.27 (6.14)
A2	0.1	21.84 (3.93)	40.65 (5.34)	24.22 (4.08)	26.54 (4.43)	23.93 (4.22)
A2	0.01	25.46 (4.21)	90.54 (7.48)	25.42 (4.22)	25.01 (4.17)	62.34 (6.53)
A3	0.1	24.43 (3.88)	41.74 (5.63)	37.47 (5.02)	50.58 (6.05)	56.32 (6.83)
A3	0.01	47.70 (5.18)	122.61 (8.05)	55.18 (5.67)	61.37 (6.5)	116.46 (7.99)
AFA1	0.1	3.95 (1.79)	11.81 (3.3)	4.22 (1.82)	4.15 (1.72)	5.81 (2.27)
AFA1n1000	0.1	7.69 (2.43)	23.10 (4.54)	8.12 (2.42)	8.06 (2.42)	10.96 (3.12)
AFA2	0.5	2.33 (1.39)	3.28 (1.63)	3.74 (1.73)	5.38 (2.16)	3.7 (1.69)

U1	0.1	21.22 (3.97)	46.18 (5.88)	22.59 (3.98)	23.91 (4.33)	51.61 (5.54)
U1	0.01	25.33 (4.09)	93.75 (7.37)	24.99 (4.09)	24.53 (4.17)	69.53 (6.35)
U2	0.1	27.37 (4.3)	56.77 (6.8)	36.69 (5.19)	46.41 (6.11)	88.48 (6.96)
U2	0.01	48.26 (5.15)	128.75 (7.97)	55.03 (5.79)	60.94 (6.36)	123.23 (7.73)
U3	0.1	29.17 (4.32)	59.87 (6.86)	68.25 (6.92)	109.36 (8.95)	153.82 (9.67)
U3	0.01	54.91 (4.99)	139.16 (7.7)	90.54 (6.97)	124.03 (9.13)	208.56 (10.64)
U3n1000	0.01	109.07 (6.81)	277.17 (11.52)	180.17 (10.03)	247.99 (13.17)	416.58 (15.56)
U4	0.35	12.4 (3.3)	19.99 (4.24)	66.74 (6.53)	136.54 (9.11)	149.85 (8.58)

Model name	P	Estimated median p (MAD)		F₀	Estimated median f₀ (MAD)		F₁	Estimated median f₁ (MAD)		F₂	Estimated median f₂ (MAD)		D	Estimated median d (MAD)
		ASP	AST		ASP	AST		ASP	AST		ASP	AST		ASP	AST
R1	0.1	0.1 (0.01)	0.1 (0.03)	0.01	0.04 (0.007)	0.045 (0.007)	0.01	0.045 (0.022)	0.045 (0.022)	0.2	0.92 (0.074)	0.92 (0.07)	−1.0	−1.0 (0.06)	−1.0 (0.07)
R1	0.01	0.015 (0.02)	0.01 (0.0)	0.01	0.045 (0.052)	0.045 (0.007)	0.01	0.05 (0.059)	0.015 (0.022)	0.2	0.13 (0.178)	0.93 (0.074)	−1.0	0.0 (0.0)	−1.05 (0.12)
R1n1000	0.01	0.05 (0.06)	0.01 (0.0)	0.01	0.05 (0.059)	0.045 (0.007)	0.01	0.08 (0.104)	0.02 (0.03)	0.2	0.14 (0.193)	0.94 (0.044)	−1.0	0.0 (0.06)	−1.03 (0.12)
R2	0.1	0.1 (0.01)	0.1 (0.01)	0.01	0.01 (0.0)	0.01 (0.003)	0.01	0.01 (0.003)	0.01 (0.007)	0.5	0.5 (0.044)	0.51 (0.089)	−1.0	−1.0 (0.01)	−1.0 (0.04)
R2	0.01	0.1 (0.14)	0.01 (0.0)	0.01	0.045 (0.052)	0.01 (0.0)	0.01	0.1 (0.136)	0.01 (0.007)	0.5	0.48 (0.341)	0.5 (0.03)	−1.0	−0.81 (1.2)	−1.0 (0.03)
R3	0.1	0.1 (0.01)	0.1 (0.0)	0.01	0.01 (0.003)	0.008 (0.003)	0.01	0.01 (0.007)	0.01 (0.013)	0.8	0.89 (0.111)	0.74 (0.326)	−1.0	−1.0 (0.01)	−0.99 (0.04)
R3	0.01	0.48 (0.09)	0.01 (0.0)	0.01	0.05 (0.059)	0.01 (0.0)	0.01	0.1 (0.078)	0.01 (0.007)	0.8	0.5 (0.593)	0.85 (0.089)	−1.0	−0.64 (0.95)	−1.0 (0.01)

AFR1	0.2	0.44 (0.22)	0.17 (0.18)	0.04	0.05 (0.059)	0.07 (0.067)	0.04	0.1 (0.059)	0.09 (0.044)	0.2	0.49 (0.474)	0.49 (0.104)	−1.0	−0.56 (0.82)	−0.80 (0.56)
AFR1n1000	0.2	0.45 (0.12)	0.21 (0.23)	0.04	0.05 (0.059)	0.08 (0.089)	0.04	0.11 (0.059)	0.09 (0.03)	0.2	0.5 (0.193)	0.48 (0.163)	−1.0	−0.55 (2.02)	−0.77 (0.64)
AFR2	0.25	0.07 (0.04)	0.1 (0.06)	0.003	0.01 (0.003)	0.015 (0.007)	0.05	0.09 (0.044)	0.1 (0.015)	0.5	0.57 (0.489)	0.52 (0.044)	−0.81	−0.79 (0.06)	−0.69 (0.07)

Model name	P	Estimated median p (MAD)		F₀	Estimated median f₀ (MAD)		F₁	Estimated median f₁ (MAD)		F₂	Estimated median f₂ (MAD)		D	Estimated median d (MAD)
		DST	DSQ		DST	DSQ		DST	DSQ		DST	DSQ		DST	DSQ
R1	0.1	0.05 (0.01)	0.05 (0.04)	0.01	0.01 (0.007)	0.008 (0.01)	0.01	0.01 (0.015)	0.003 (0.004)	0.2	0.32 (0.163)	0.26 (0.222)	−1.0	−0.97 (0.06)	–0.97 (0.06)
R1	0.01	0.043 (0.05)	0.045 (0.05)	0.01	0.043 (0.06)	0.09 (0.133)	0.01	0.05 (0.071)	0.1 (0.148)	0.2	0.07 (0.104)	0.195 (0.289)	−1.0	−0.07 (1.12)	–0.83 (1.15)
R1n1000	0.01	0.05 (0.06)	0.05 (0.06)	0.01	0.04 (0.052)	0.085 (0.123)	0.01	0.05 (0.07)	0.09 (0.13)	0.2	0.07 (0.104)	0.19 (0.282)	−1.0	−0.51 (0.76)	–0.97 (1.05)
R2	0.1	0.09 (0.03)	0.09 (0.03)	0.01	0.01 (0.0)	0.01 (0.0)	0.01	0.01 (0.007)	0.01 (0.007)	0.5	0.51 (0.074)	0.5 (0.059)	−1.0	−1.0 (0.03)	–1.0 (0.03)
R2	0.01	0.08 (0.01)	0.05 (0.06)	0.01	0.025 (0.037)	0.03 (0.043)	0.01	0.05 (0.071)	0.045 (0.066)	0.5	0.44 (0.571)	0.46 (0.593)	−1.0	−0.99 (0.44)	–1.0 (0.35)
R3	0.1	0.09 (0.03)	0.09 (0.03)	0.01	0.01 (0.003)	0.01 (0.003)	0.01	0.01 (0.007)	0.01 (0.007)	0.8	0.81 (0.044)	0.8 (0.03)	−1.0	−1.0 (0.03)	–1.0 (0.03)
R3	0.01	0.06 (0.07)	0.05 (0.06)	0.01	0.04 (0.046)	0.035 (0.04)	0.01	0.04 (0.047)	0.035 (0.047)	0.8	0.8 (0.297)	0.82 (0.267)	−1.0	−1.0 (0.14)	–1.0 (0.12)

AFR1	0.2	0.12 (0.16)	0.11 (0.15)	0.04	0.01 (0.015)	0.01 (0.015)	0.04	0.05 (0.062)	0.04 (0.055)	0.2	0.175 (0.256)	0.15 (0.218)	−1.0	−0.87 (0.76)	–0.92 (0.61)
AFR1n1000	0.2	0.15 (0.21)	0.12 (0.16)	0.04	0.01 (0.015)	0.01 (0.015)	0.04	0.05 (0.059)	0.045 (0.052)	0.2	0.19 (0.27)	0.17 (0.23)	−1.0	−0.87 (0.68)	–0.86 (0.6)
AFR2	0.25	0.15 (0.13)	0.22 (0.18)	0.003	0.006 (0.006)	0.005 (0.006)	0.05	0.06 (0.037)	0.05 (0.03)	0.5	0.53 (0.104)	0.52 (0.074)	−0.81	−0.81 (0.09)	–0.84 (0.12)

Model name	P	Estimated median p (MAD)	F₀	Estimated median f₀ (MAD)	F₁	Estimated median f₁ (MAD)	F₂	Estimated median f₂ (MAD)	D	Estimated median d (MAD)
R1	0.1	0.01 (0.01)	0.01	0.001 (0.0003)	0.01	0.001 (0.0007)	0.2	0.11 (0.015)	−1.0	−1.0 (0.01)
R1	0.01	0.08 (0.11)	0.01	0.235 (0.348)	0.01	0.11 (0.163)	0.2	0.32 (0.474)	−1.0	−0.39 (1.14)
R1n1000	0.01	0.075 (0.1)	0.01	0.19 (0.281)	0.01	0.11 (0.163)	0.2	0.28 (0.415)	−1.0	−0.24 (1.21)
R2	0.1	0.09 (0.03)	0.01	0.008 (0.01)	0.01	0.008 (0.003)	0.5	0.51 (0.052)	−1.0	−0.99 (0.04)
R2	0.01	0.06 (0.08)	0.01	0.015 (0.022)	0.01	0.02 (0.03)	0.5	0.41 (0.46)	−1.0	−0.98 (0.27)
R3	0.1	0.09 (0.03)	0.01	0.008 (0.01)	0.01	0.01 (0.003)	0.8	0.8 (0.03)	−1.0	−1.0 (0.03)
R3	0.01	0.05 (0.05)	0.01	0.035 (0.037)	0.01	0.043 (0.033)	0.8	0.76 (0.267)	−1.0	−0.99 (0.06)

AFR1	0.2	0.11 (0.15)	0.04	0.01 (0.015)	0.04	0.015 (0.021)	0.2	0.16 (0.185)	−1.0	−0.94 (0.27)
AFR1n1000	0.2	0.1 (0.13)	0.04	0.01 (0.015)	0.04	0.01 (0.013)	0.2	0.165 (0.185)	−1.0	−0.98 (0.19)
AFR2	0.25	0.23 (0.06)	0.003	0.001 (0.001)	0.05	0.05 (0.015)	0.5	0.51 (0.059)	−0.81	−0.83 (0.07)

Model name	P	Estimated median p (MAD)		F₀	Estimated median f₀ (MAD)		F₁	Estimated median f₁ (MAD)		F₂	Estimated median f₂ (MAD)		D	Estimated median d (MAD)
		DST	DSQ		DST	DSQ		DST	DSQ		DST	DSQ		DST	DSQ
D1	0.1	0.1 (0.06)	0.11 (0.06)	0.01	0.008 (0.004)	0.008 (0.003)	0.2	0.12 (0.089)	0.13 (0.059)	0.2	0.31 (0.252)	0.33 (0.222)	1.0	−0.05 (0.8)	−0.03 (0.86)
D1	0.01	0.1 (0.03)	0.11 (0.03)	0.01	0.002 (0.003)	0.005 (0.006)	0.2	0.14 (0.089)	0.14 (0.074)	0.2	0.31 (0.267)	0.34 (0.208)	1.0	−0.01 (0.53)	−0.03 (0.59)
D1n1000	0.01	0.11 (0.01)	0.11 (0.03)	0.01	0.003 (0.003)	0.005 (0.006)	0.2	0.15 (0.059)	0.15 (0.059)	0.2	0.3 (0.208)	0.34 (0.163)	1.0	<0.01 (0.49)	−0.01 (0.5)
D2	0.1	0.1 (0.01)	0.1 (0.03)	0.01	0.006 (0.007)	0.008 (0.01)	0.5	0.47 (0.089)	0.49 (0.059)	0.5	0.555 (0.378)	0.51 (0.341)	1.0	0.41 (1.12)	0.66 (1.13)
D2	0.01	0.045 (0.01)	0.05 (0.04)	0.01	0.003 (0.004)	0.006 (0.006)	0.5	0.48 (0.074)	0.48 (0.044)	0.5	0.74 (0.385)	0.69 (0.46)	1.0	0.05 (0.41)	0.12 (0.5)
D3	0.1	0.08 (0.04)	0.09 (0.06)	0.01	0.015 (0.016)	0.008 (0.01)	0.8	0.8 (0.044)	0.8 (0.03)	0.8	0.71 (0.43)	0.81 (0.222)	1.0	0.81 (0.91)	0.88 (0.44)
D3	0.01	0.02 (0.02)	0.025 (0.03)	0.01	0.008 (0.004)	0.008 (0.004)	0.8	0.8 (0.03)	0.8 (0.03)	0.8	1.00 (0.0)	0.92 (0.119)	1.0	0.62 (0.15)	0.64 (0.2)

AFD1	0.05	0.08 (0.11)	0.08 (0.11)	0.04	0.01 (0.014)	0.01 (0.014)	0.2	0.1 (0.082)	0.11 (0.096)	0.2	0.23 (0.282)	0.27 (0.334)	1.0	−0.11 (0.84)	−0.17 (0.9)
AFD1n1000	0.05	0.09 (0.12)	0.07 (0.09)	0.04	0.01 (0.014)	0.01 (0.014)	0.2	0.11 (0.089)	0.12 (0.089)	0.2	0.27 (0.297)	0.28 (0.311)	1.0	−0.15 (0.73)	−0.33 (0.7)
AFD2	0.25	0.17 (0.13)	0.16 (0.12)	0.003	0.035 (0.037)	0.03 (0.03)	0.5	0.47 (0.119)	0.49 (0.074)	0.5	0.48 (0.341)	0.41 (0.356)	1.0	0.28 (1.52)	0.67 (1.53)

Model name	P	Estimated median p (MAD)	F₀	Estimated median f₀ (MAD)	F₁	Estimated Median f₁ (MAD)	F₂	Estimated median f₂ (MAD)	D	Estimated median d (MAD)
D1	0.1	0.08 (0.07)	0.01	0.01 (0.003)	0.2	0.14 (0.074)	0.2	0.19 (0.163)	1.0	0.68 (1.32)
D1	0.01	0.035 (0.01)	0.01	0.004 (0.005)	0.2	0.19 (0.059)	0.2	0.27 (0.163)	1.0	0.33 (0.76)
D1n1000	0.01	0.035 (0.01)	0.01	0.004 (0.005)	0.2	0.19 (0.044)	0.2	0.27 (0.119)	1.0	0.39 (0.62)
D2	0.1	0.09 (0.03)	0.01	0.008 (0.012)	0.5	0.5 (0.044)	0.5	0.48 (0.193)	1.0	0.96 (0.77)
D2	0.01	0.015 (0.01)	0.01	0.01 (0.003)	0.5	0.5 (0.03)	0.5	0.54 (0.563)	1.0	0.42 (0.82)
D3	0.1	0.1 (0.03)	0.01	0.01 (0.015)	0.8	0.8 (0.03)	0.8	0.8 (0.148)	1.0	0.98 (0.39)
D3	0.01	0.015 (0.01)	0.01	0.008 (0.003)	0.8	0.8 (0.015)	0.8	0.84 (0.237)	1.0	0.68 (0.33)
AFD1	0.05	0.11 (0.1)	0.04	0.015 (0.015)	0.2	0.22 (0.104)	0.2	0.22 (0.178)	1.0	0.26 (0.91)
AFD1n1000	0.05	0.11 (0.07)	0.04	0.02 (0.015)	0.2	0.21 (0.096)	0.2	0.21 (0.148)	1.0	0.45 (0.79)
AFD2	0.25	0.18 (0.12)	0.003	0.04 (0.052)	0.5	0.43 (0.074)	0.5	0.43 (0.222)	1.0	1.08 (0.99)

Model name	P	Estimated median p (MAD)		F₀	Estimated median f₀ (MAD)		F₁	Estimated median f₁ (MAD)		F₂	Estimated median f₂ (MAD)		D	Estimated median d (MAD)
		ASP	AST		ASP	AST		ASP	AST		ASP	AST		ASP	AST
A1	0.1	0.1 (0.04)	0.1 (0.01)	0.01	0.05 (0.007)	0.04 (0.03)	0.1	0.49 (0.044)	0.42 (0.163)	0.2	0.875 (0.185)	0.91 (0.129)	−0.05	0.0 (0.67)	0.02 (0.47)
A1	0.01	0.05 (0.07)	0.05 (0.06)	0.01	0.045 (0.052)	0.025 (0.034)	0.1	0.11 (0.089)	0.49 (0.133)	0.2	0.47 (0.563)	0.93 (0.104)	−0.05	0.0 (0.62)	0.03 (0.47)
A1n1000	0.01	0.01 (0.01)	0.03 (0.03)	0.01	0.01 (0.015)	0.015 (0.022)	0.1	0.1 (0.03)	0.49 (0.074)	0.2	0.45 (0.519)	0.94 (0.089)	−0.05	0.0 (0.67)	−0.01 (0.33)
A2	0.1	0.09 (0.04)	0.07 (0.04)	0.01	0.045 (0.007)	0.04 (0.03)	0.2	0.79 (0.297)	0.45 (0.119)	0.5	0.87 (0.193)	0.95 (0.074)	−0.22	0.66 (0.99)	−0.12 (0.31)
A2	0.01	0.1 (0.0)	0.008 (0.01)	0.01	0.01 (0.013)	0.045 (0.007)	0.2	0.50 (0.267)	0.93 (0.059)	0.5	0.89 (0.163)	0.96 (0.059)	−0.22	0.01 (0.99)	0.89 (0.39)
A3	0.1	0.1 (0.01)	0.1 (0.01)	0.01	0.01 (0.014)	0.035 (0.037)	0.5	0.55 (0.43)	0.50 (0.133)	0.8	0.86 (0.208)	0.87 (0.193)	0.24	0.54 (0.82)	0.30 (0.67)
A3	0.01	0.02 (0.02)	0.01 (0.0)	0.01	0.005 (0.006)	0.008 (0.003)	0.5	0.48 (0.133)	0.44 (0.178)	0.8	0.52 (0.638)	0.19 (0.282)	0.24	0.0 (0.85)	−0.07 (0.8)
AFA1	0.1	0.1 (0.09)	0.11 (0.09)	0.03	0.023 (0.033)	0.02 (0.03)	0.13	0.11 (0.089)	0.16 (0.089)	0.23	0.325 (0.348)	0.32 (0.282)	0.0	0.0 (0.38)	−0.08 (0.55)
AFA1n1000	0.1	0.1 (0.07)	0.11 (0.06)	0.03	0.015 (0.022)	0.02 (0.03)	0.13	0.1 (0.074)	0.15 (0.074)	0.23	0.235 (0.252)	0.28 (0.267)	0.0	0.0 (0.2)	−0.08 (0.46)
AFA2	0.5	0.14 (0.2)	0.415 (0.14)	0.003	0.04 (0.044)	0.01 (0.015)	0.25	0.15 (0.148)	0.45 (0.423)	0.5	0.46 (0.534)	0.85 (0.219)	<0.01	0.0 (0.1)	0.0 (0.68)

Model name	P	Estimated median p (MAD)		F₀	Estimated median f₀ (MAD)		F₁	Estimated median f₁ (MAD)		F₂	Estimated median f₂ (MAD)		D	Estimated median d (MAD)
		DST	DSQ		DST	DSQ		DST	DSQ		DST	DSQ		DST	DSQ
A1	0.1	0.06 (0.06)	0.06 (0.07)	0.01	0.01 (0.003)	0.01 (0.003)	0.1	0.1 (0.03)	0.1 (0.03)	0.2	0.17 (0.178)	0.17 (0.17)	−0.05	0.01 (0.56)	0.01 (0.89)
A1	0.01	0.01 (0.01)	0.01 (0.01)	0.01	0.01 (0.003)	0.01 (0.0)	0.1	0.09 (0.059)	0.09 (0.059)	0.2	0.15 (0.193)	0.17 (0.222)	−0.05	−0.12 (0.81)	−0.35 (0.7)
A1n1000	0.01	0.01 (0.01)	0.01 (0.01)	0.01	0.01 (0.0)	0.01 (0.0)	0.1	0.1 (0.03)	0.1 (0.044)	0.2	0.165 (0.23)	0.17 (0.23)	−0.05	−0.30 (0.64)	−0.45 (0.67)
A2	0.1	0.11 (0.04)	0.12 (0.03)	0.01	0.006 (0.006)	0.006 (0.006)	0.2	0.18 (0.104)	0.18 (0.089)	0.5	0.43 (0.208)	0.45 (0.156)	−0.22	−0.08 (0.53)	−0.12 (0.53)
A2	0.01	0.1 (0.03)	0.11 (0.03)	0.01	0.002 (0.003)	0.004 (0.005)	0.2	0.15 (0.089)	0.15 (0.074)	0.5	0.335 (0.259)	0.35 (0.208)	−0.22	−0.01 (0.5)	−0.03 (0.56)
A3	0.1	0.1 (0.01)	0.1 (0.03)	0.01	0.006 (0.006)	0.006 (0.009)	0.5	0.48 (0.089)	0.5 (0.059)	0.8	0.77 (0.319)	0.78 (0.282)	0.24	0.14 (0.59)	0.22 (0.47)
A3	0.01	0.045 (0.01)	0.05 (0.04)	0.01	0.003 (0.004)	0.005 (0.007)	0.5	0.48 (0.059)	0.49 (0.044)	0.8	0.79 (0.311)	0.745 (0.378)	0.24	0.04 (0.37)	0.1 (0.43)

AFA1	0.1	0.07 (0.09)	0.06 (0.08)	0.03	0.01 (0.012)	0.01 (0.007)	0.13	0.09 (0.074)	0.09 (0.074)	0.23	0.195 (0.259)	0.19 (0.248)	0.0	−0.08 (0.77)	−0.12 (0.82)
AFA1n1000	0.1	0.08 (0.11)	0.06 (0.08)	0.03	0.01 (0.007)	0.01 (0.007)	0.13	0.09 (0.059)	0.1 (0.074)	0.23	0.17 (0.222)	0.17 (0.222)	0.0	−0.11 (0.65)	−0.18 (0.71)
AFA2	0.5	0.1 (0.14)	0.05 (0.07)	0.003	0.045 (0.058)	0.045 (0.052)	0.25	0.25 (0.252)	0.38 (0.237)	0.5	0.49 (0.489)	0.545 (0.489)	<0.01	−0.04 (0.8)	0.0 (0.92)

Model name	P	Estimated median p (MAD)		F₀	Estimated median f₀ (MAD)		F₁	Estimated median f₁ (MAD)		F₂	Estimated median f₂ (MAD)
		DST	DSQ		DST	DSQ		DST	DSQ		DST	DSQ
U1	0.1	0.12 (0.09)	0.12 (0.09)	0.01	0.008 (0.007)	0.008 (0.004)	0.2	0.09 (0.074)	0.1 (0.059)	0.01	0.25 (0.319)	0.31 (0.267)
U1	0.01	0.1 (0.03)	0.11 (0.04)	0.01	0.003 (0.004)	0.005 (0.006)	0.2	0.14 (0.089)	0.14 (0.074)	0.01	0.31 (0.267)	0.34 (0.208)
U2	0.1	0.11 (0.03)	0.11 (0.03)	0.01	0.005 (0.007)	0.005 (0.007)	0.5	0.5 (0.089)	0.51 (0.044)	0.01	0.07 (0.104)	0.03 (0.044)
U2	0.01	0.045 (0.01)	0.05 (0.03)	0.01	0.002 (0.003)	0.005 (0.007)	0.5	0.49 (0.059)	0.49 (0.044)	0.01	0.63 (0.549)	0.58 (0.563)
U3	0.1	0.11 (0.03)	0.11 (0.03)	0.01	0.006 (0.009)	0.006 (0.009)	0.8	0.81 (0.03)	0.8 (0.03)	0.01	0.025 (0.037)	0.01 (0.015)
U3	0.01	0.02 (0.02)	0.03 (0.03)	0.01	0.008 (0.006)	0.008 (0.006)	0.8	0.8 (0.03)	0.8 (0.03)	0.01	0.89 (0.163)	0.73 (0.4)
U3n1000	0.01	0.02 (0.01)	0.025 (0.01)	0.01	0.008 (0.003)	0.01 (0.003)	0.8	0.8 (0.03)	0.8 (0.015)	0.01	0.86 (0.208)	0.64 (0.43)
U4	0.35	0.36 (0.07)	0.36 (0.09)	0.01	0.003 (0.004)	0.01 (0.015)	0.9	0.9 (0.03)	0.9 (0.015)	0.01	0.001 (0.001)	0.0 (0.0)

Model name	P	F₁/F₀	Estimated median f₁/f₀ (MAD)		F₂/F₀	Estimated median f₂/f₀ (MAD)		F₂/F₁	Estimated median f₂/f₁ (MAD)
			ASP	AST		ASP	AST		ASP	AST
R1	0.1	1	1.0 (0.74)	1.0 (0.66)	20	21.67 (3.95)	20.27 (4.84)	20	20.22 (9.54)	20.89 (8.85)
R1	0.01	1	1.0 (0.69)	0.78 (1.15)	20	1.09 (1.55)	19.74 (4.75)	20	1.0 (0.8)	18.6 (26.71)
R1n1000	0.01	1	1.0 (0.67)	0.76 (1.13)	20	1.0 (1.36)	19.8 (2.11)	20	1.16 (0.89)	21.33 (22.36)
R2	0.1	1	1.0 (0.37)	1.0 (0.89)	50	51.0 (8.9)	51.25 (18.16)	50	51.0 (17.05)	51.0 (42.4)
R2	0.01	1	1.59 (1.62)	1.0 (0.74)	50	8.0 (10.41)	50.0 (2.97)	50	2.0 (2.85)	51.0 (26.19)
R3	0.1	1	1.0 (0.74)	1.6 (1.85)	80	90.0 (29.65)	82.0 (34.1)	80	83.67 (40.46)	60.33 (61.28)
R3	0.01	1	1.6 (2.19)	1.0 (0.74)	80	2.5 (3.35)	83.0 (10.38)	80	6.11 (8.73)	80.0 (34.59)
AFR1	0.2	1	2.2 (2.92)	0.75 (0.98)	5	9.8 (13.05)	4.88 (6.85)	5	4.13 (5.11)	5.33 (2.69)
AFR1n1000	0.2	1	2.22 (2.97)	0.67 (0.83)	5	10.44 (14.0)	4.5 (6.33)	5	3.55 (4.33)	5.0 (2.58)
AFR2	0.25	16.67	10.0 (5.93)	6.0 (2.97)	166.67	65.67 (36.08)	33.33 (13.1)	10	8.41 (2.05)	5.6 (0.89)

D1	0.1	20	16.8 (8.43)	15.37 (9.18)	20	20.0 (15.12)	17.45 (17.58)	1	1.14 (1.14)	0.98 (0.79)
D1	0.01	20	47.0 (43.88)	22.5 (5.19)	20	50.0 (64.2)	22.22 (28.66)	1	1.14 (1.24)	1.01 (0.44)
D1n1000	0.01	20	50.0 (47.15)	20.67 (2.35)	20	62.71 (73.82)	20.0 (14.66)	1	1.11 (1.2)	1.01 (0.19)
D2	0.1	50	31.0 (29.65)	22.22 (19.01)	50	30.83 (35.63)	22.22 (31.46)	1	1.09 (1.25)	0.86 (0.74)
D2	0.01	50	95.0 (66.72)	59.0 (21.87)	50	113.13 (167.72)	62.75 (92.22)	1	1.96 (1.42)	1.1 (1.58)
D3	0.1	80	50.0 (49.42)	27.71 (27.15)	80	49.0 (66.42)	22.88 (33.47)	1	1.08 (1.31)	0.84 (0.71)
D3	0.01	80	95.0 (44.48)	84.0 (38.55)	80	113.75 (168.65)	70.0 (103.78)	1	1.91 (1.85)	1.01 (1.49)
AFD1	0.05	5	5.0 (4.45)	5.5 (1.73)	5	10.0 (10.04)	10.0 (5.93)	1	1.88 (0.88)	1.68 (1.01)
AFD1n1000	0.05	5	5.0 (2.8)	5.33 (0.99)	5	9.0 (5.07)	9.33 (3.95)	1	1.82 (0.74)	1.55 (0.84)
AFD2	0.25	166.67	8.83 (5.07)	4.75 (2.8)	166.67	14.92 (18.16)	3.0 (4.45)	1	1.9 (2.0)	1.17 (1.68)

A1	0.1	10	10.2 (1.19)	10.5 (4.98)	20	18.0 (5.19)	19.53 (10.3)	2	1.88 (0.8)	1.84 (0.88)
A1	0.01	10	9.4 (3.11)	12.35 (4.97)	20	17.0 (11.86)	22.22 (29.32)	2	1.88 (0.76)	1.72 (0.75)
A1n1000	0.01	10	10.0 (1.98)	10.8 (1.78)	20	18.2 (13.39)	20.0 (25.87)	2	1.89 (1.32)	1.77 (0.55)
A2	0.1	20	18.8 (10.91)	10.47 (4.16)	50	20.0 (19.71)	22.22 (19.45)	2.5	1.11 (1.14)	2.13 (0.95)
A2	0.01	20	49.0 (46.26)	21.67 (4.02)	50	56.0 (71.17)	22.22 (13.19)	2.5	1.19 (1.17)	1.03 (0.19)
A3	0.1	50	39.67 (33.31)	18.8 (15.12)	80	47.0 (56.19)	24.19 (32.3)	1.6	1.12 (1.24)	1.41 (0.8)
A3	0.01	50	92.75 (61.9)	55.0 (18.29)	80	111.25 (164.94)	81.0 (97.11)	1.6	1.96 (1.43)	1.44 (1.69)
AFA1	0.1	4.33	5.0 (3.9)	4.89 (1.91)	7.67	10.0 (9.14)	8.89 (5.35)	1.77	1.89 (0.73)	1.83 (0.99)
AFA1n1000	0.1	4.33	5.0 (2.67)	4.5 (1.07)	7.67	9.0 (3.56)	8.4 (3.56)	1.77	1.8 (0.36)	1.89 (0.87)
AFA2	0.5	83.33	5.0 (4.94)	8.58 (12.19)	166.67	10.0 (10.91)	24.54 (36.38)	2	1.85 (0.67)	1.78 (1.16)

U1	0.1	20	16.0 (10.5)	25.0 (17.3)	1	15.0 (21.99)	1.19 (1.66)	0.05	1.05 (1.56)	0.07 (0.1)
U1	0.01	20	46.0 (43.0)	23.5 (6.67)	1	49.0 (61.68)	22.22 (30.87)	0.05	1.14 (1.24)	0.75 (0.51)
U2	0.1	50	40.0 (50.41)	118.75 (146.41)	1	29.33 (39.78)	1.0 (1.48)	0.02	0.59 (0.87)	0.01 (0.01)
U2	0.01	50	99.17 (72.15)	71.67 (33.36)	1	113.75 (168.65)	17.75 (26.32)	0.02	1.96 (2.09)	0.27 (0.4)
U3	0.1	80	80.0 (107.49)	272.5 (375.99)	1	33.33 (47.94)	1.0 (1.48)	0.0125	0.38 (0.57)	0.0 (0.0)
U3	0.01	80	98.0 (47.07)	112.5 (67.95)	1	116.25 (172.35)	11.13 (16.49)	0.0125	1.92 (2.85)	0.16 (0.24)

Model name	P	F		GHM analysis with imprinting							GHM analysis without imprinting
				Estimated median parameters (MAD)							Estimated median parameters (MAD)
				MOD	p	f₀	f_1,pat	f_1,mat	f₂	i	MOD	p	f₀	f₁	f₂
cpi (I = −1)	0.01	0; 0; 1; 1	ped 1	26.0	0.008	0.0002	0.0003	0.94	0.94	−1	9.26	0.1	0.01	0.54	0.5
				(2.66)	(0.003)	(0.0003)	(0.0004)	(0.089)	(0.089)	(0.0)	(2.42)	(0.0)	(0.0)	(0.615)	(0.638)

cpi (I = −1)	0.01	0; 0; 1; 1	ped 2	39.61	0.003	0.0	0.0	0.95	0.555	–0.1	39.49	0.002	0.0001	0.43	1.0
				(2.81)	(0.004)	(0.0)	(0.0)	(0.074)	(0.645)	(1.28)	(2.93)	(0.0)	(0.0001)	(0.163)	(0.0)

cmi (I = 1)	0.01	0; 1; 0; 1	ped 3	24.17	0.008	0.0003	0.9	0.001	0.7	0.93	18.81	0.05	0.0003	0.12	0.51
				(2.63)	(0.003)	(0.0004)	(0.148)	(0.002)	(0.445)	(0.13)	(2.78)	(0.01)	(0.0004)	(0.059)	(0.415)

ni	0.01	0; 0.5; 0.5; 1	ped 1	43.89	0.01	0.0002	0.71	0.76	0.94	0	43.86	0.01	0.0	0.2	0.9
				(4.0)	(0.0)	(0.0003)	(0.423)	(0.356)	(0.089)	(0.44)	(3.99)	(0.0)	(0.0)	(0.274)	(0.148)
			ped 2	34.45	0.008	0.0005	0.38	0.43	0.92	0	34.35	0.008	0.0005	0.48	0.83
				(3.48)	(0.003)	(0.0007)	(0.561)	(0.415)	(0.119)	(0.76)	(3.4)	(0.0)	(0.0004)	(0.356)	(0.252)
			ped 3	27.06	0.008	0.0004	0.001	0.44	0.67	−0.035	26.89	0.008	0.0003	0.48	0.85
				(3.17)	(0.003)	(0.0006)	(0.001)	(0.474)	(0.489)	(0.85)	(3.12)	(0.01)	(0.0004)	(0.334)	(0.222)

PERMALINK

Estimation of Trait-Model Parameters in a MOD Score Linkage Analysis

Markus Brugger

Susanne Rospleszcz

Konstantin Strauch

Abstract

Background/Aims

Methods

Results

Conclusion

Introduction

Trait Inheritance and Pedigree Analysis

Linkage Analysis

Mode of Inheritance and Trait-Model Parameters

Likelihood and Sample Space

Sampling Model-Based Likelihood

Sampling Model-Free Likelihood

Likelihood in a MOD Score Analysis

Summary of Conditions to Obtain Unbiased

Parameter Estimates from a MOD Score Analysis

Identifiability of Inheritance Parameters

Fig. 1.

Imprinting

Aims of the Present Study

Methods

Nomenclature

Data Generation

Fig. 2.

Table 1.

Table 2.

Data Analysis

Results

Table 3.

Recessive Models

Fig. 3.

Table A1.

Table A5.

Table A9.

Fig. 4.

Dominant Models

Fig. 5.

Table A2.

Table A6.

Table A10.

Fig. 6.

Fig. 7.

Additive Models

Fig. 8.

Table A3.

Table A7.

Table A11.

Overdominant Models

Fig. 9.

Table A4.

Table A8.

Table A12.

Fig. 10.

Penetrance Ratios for ASPs and ASTs

Table 4.

Summary of Trait-Model Parameter Estimation Results

Imprinting Models

Table 5.

NI Model

Imprinting Models

Summary of Imprinting Results

Discussion

Acknowledgements

References

ACTIONS

PERMALINK

RESOURCES

Similar articles

Cited by other articles

Links to NCBI Databases