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![Page 1: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/1.jpg)
Quantitative Trait Loci, QTL
An introduction to quantitative geneticsand common methods for mapping of
loci underlying continuous traits:
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Why study quantitative traits?
• Many (most) human traits/disorders are complex in the sense that they are governed by several genetic loci as well as being influenced by environmental agents;
• Many of these traits are intrinsically continuously varying and need specialized statistical models/methods for the localization and estimation of genetic contributions;
• In addition, in several cases there are potential benefits from studying continuously varying quantities as opposed to a binary affected/unaffected response:
![Page 3: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/3.jpg)
For example:
• in a study of risk factors the underlying quantitative phenotypes that predispose disease may be more etiologically homogenous than the disease phenotype itself;
• some qualitative phenotypes occur once a threshold for susceptibility has been exceeded, e.g. type 2 diabetes, obesity, etc.;
• in such a case the binary phenotype (affected/unaffected) is not as informative as the actual phenotypic measurements;
![Page 4: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/4.jpg)
A pedigree representation
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Variance and variability
• methods for linkage analysis of QTL in humans rely on a partitioning of the total variability of trait values;
• in statistical theory, the variance is the expected squared deviation round the mean value,
• it can be estimated from data as:
• the square root of the variance is called the standard deviation;
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A simple model for the phenotype
Y = X + ewhere• Y is the phenotypic value, i.e. the trait value;• X is the genotypic value, i.e. the mean or
expected phenotypic value given the genotype;• e is the environmental deviation with mean 0.• We assume that the total phenotypic variance is
the sum of the genotypic variance and the environmental variance, V (Y ) = V (X ) + V (e), i.e. the environmental contribution is assumed independent of the genotype of the individual;
![Page 7: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/7.jpg)
Distribution of Y : a single biallelic locus
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A single biallelic locus: genetic effects
Genotype
Genotypic value
• a is the homozygous effect,• k is the dominance coeffcient• k = 0 means complete additivity,• k = 1 means complete dominance (of A2),• k > 1 if A2 is overdominant.
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Example: The pygmy gene, pg
• From data we have the following mean values of weight:
X++ = 14g, X+pg = 12g, Xpgpg = 6g,
• 2a = 14 -6 = 8 implies a =4,
• (1 + k)a = 12 - 6 = 6 implies k = 0.5.
Data suggest recessivity (although not complete) of the pygmy gene.
![Page 10: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/10.jpg)
Decomposition of the genotypic value, X
• Xij is the mean of Y for AiAj-individuals;• when k = 0 the two alleles of a biallelic locus be
haves in a completely additive fashion: X is a linear function of the number of A2-alleles;
• we can then think of each allele contributing a purely additive effect to X ;
• this can be generalized to k ≠ 0 by decomposition of X into additive contributions of alleles together with deviations resulting from dominance;
• the generalization is accomplished using least-squares regression of X on the gene content;
![Page 11: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/11.jpg)
Least-squares linear regression
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![Page 12: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/12.jpg)
Model 1
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![Page 13: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/13.jpg)
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![Page 14: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/14.jpg)
Interpretations
• in the linear regression
is the heritable component of the genotype,
δis the non-heritable part;
• the sum of an individuals additive allelic effects, αi+αj is called the breeding value and is denoted Λij
• under random mating αican be interpreted as the average excess of allele Ai
• this is defined as the difference between the expected phenotypic value when one allele (e.g. the paternally transmitted) is fixed at Ai and the population average, μ;
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![Page 15: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/15.jpg)
Linear Regression
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![Page 16: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/16.jpg)
Graphically
![Page 17: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/17.jpg)
Linear Regression Model solving
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![Page 18: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/18.jpg)
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![Page 19: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/19.jpg)
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![Page 20: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/20.jpg)
average excesses
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![Page 21: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/21.jpg)
Interpretations under random mating
• α= a [1+ k (p1-p2)] ;
α= - p2 α;
α= p1 α, Population parameters for k≠0• α is called the average effect of allelic substitution: substitute A1 A2for a randomly chosen A1 –
allele• then the expected change in X is,
(X12 -X11) p1 + (X22 -X12) p2 ;
• which equals α. (simple calculations).
![Page 22: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/22.jpg)
: Average effect of allelic substitution
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![Page 23: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/23.jpg)
α is a function of p2 and k :
![Page 24: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/24.jpg)
Partitioning the genetic variance
• the variance, V (X ), of the genotypic values in a population is called the genetic variance:
• is the additive genetic variance, i.e. variance associated with
additive allelic effects;• dominance genetic
variance, i.e. due to dominance deviations;
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![Page 25: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/25.jpg)
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![Page 26: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/26.jpg)
V (X); VA; VD are functions of p2 and k:
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![Page 27: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/27.jpg)
Example: The Booroola gene, (Lynch and Walsh, 1998)
![Page 28: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/28.jpg)
In summary• The homozygous effect a, and the dominance co
efficient k are intrinsic properties of allelic products.
• The additive effect αi, and the average excess αi* are properties of alleles in a particular population.
• The breeding value is a property of a particular individual in reference to a particular population. It is the sum of the additive effects of an individual's alleles.
• The additive genetic variance, VA, , is a property of a particular population. It is the variance of the breeding values of individuals in the population.
![Page 29: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/29.jpg)
Multilocus traits
• Do the separate locus effects combine in an additive way, or do there exist non-linear interaction between different loci: epistasis?
• Do the genes at different loci segregate independently?
• Do the gene expression vary with the environmental context: gene by environment interaction?
• Are specic genotypes associated with particular environments: covariation of genotypic values and environmental effects?
![Page 30: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/30.jpg)
Example: epistasis
Average length of vegetative internodes in the lateral branch(in mm) of teosinte. Table from Lynch and Walsh (1998).
![Page 31: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/31.jpg)
Two independently segregating loci
• Extending the least-squares decomposition of X :
• Λk is the breeding value of the k'th locus,
δk is the dominance deviation of the k'th locus,
ε is a residual term due to epistasis;• if the loci are independently segregating
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![Page 32: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/32.jpg)
Neglecting V (ε)
• the epistatic variance components contributing to V (ε) are often small compared to VA and VD;
• in linkage analysis it is this often assumed that V (ε) = 0;
• note however: the relative magnitude of the variance components provide only limited insight into the physiological mode of gene action;
• epistatic interactions, can greatly inflate the additive and/or dominance components of variance;
![Page 33: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/33.jpg)
Resemblance between relatives
A model for the trait values of two relatives:
Yk = Xk + ek, k = 1 , 2,
where for the k’th relative• Yk is the phenotypic value,• Yk is the genotypic value,• ek is the mean zero environmental deviation.• the ek’s are assumed to be mutually independent
and also independent of k. Hence, the covariance of the trait values of two relatives is given by the genetic covariance, C(X1; X2), i.e.
C(Y1; Y2) = C(X1; X2)
![Page 34: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/34.jpg)
A (preliminary) formula for C(X1 ,X 2)
For a single locus trait
C(X1; X2) = c1VA + c2VD
• c1 and c2 are constants determined by the type of relationship between the two relatives.
• same formula applies for multilocus traits if no epistatic variance components are included in the model, i.e. V (ε) = 0.
• in this latter case and are given by summation of the corresponding locus-specific contributions.
![Page 35: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/35.jpg)
Joint distribution of sibling trait values
Single biallelic, dominant (k =1 ) model. Correlation 0.46.
![Page 36: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/36.jpg)
Measures of relatedness
• N = the number of alleles shared IBD by two relatives at a given locus;
• the kinship coefficient, θ , is given by2 θ = E(N) / 2;
i.e. twice the kinship coefficient equals the expected proportion of alleles shared IBD at the locus.
• The coefficient of fraternity, Δ, is defined as
Δ = P(N = 2).
![Page 37: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/37.jpg)
Some examples
• Siblings
(z0; z1; z2) = (1/4; 1/2; 1/4) implying E(N) = 1.Thus θ= 1/4 and Δ = 1/4:
• Parent-offspring
(z0; z1; z2) = (0; 1; 0) implying E(N) = 1.Thus θ = 1/4 and Δ = 0:
• Grandparent - grandchild
(z0; z1; z2) = (1/2; 1/2; 0) implying E(N) = 1=2.Thus θ = 1/8 and Δ = 0:
![Page 38: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/38.jpg)
Covariance formula for a single locus
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![Page 39: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/39.jpg)
A single locus; perfect marker data
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![Page 40: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/40.jpg)
Covariance formula for multiple loci
n independently segregating loci assuming noepistatic interaction, i.e. putting V (ε) = 0
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![Page 41: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/41.jpg)
Covariance formula for multiple loci
n independently segregating loci assuming noepistatic interaction, i.e. putting V (ε) = 0
ly.respective ons,contributi
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![Page 42: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/42.jpg)
Covariance... continued
);MD|2(
and
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Define for every pair of relatives
For two related individuals we then have,
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![Page 43: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/43.jpg)
Haseman-Elston method
• Uses pairs of relatives of the same type: most often sib pairs;
• for each relative pair calculate the squared phenotypic difference: Z = (Y1 –Y2)2;
• given MDx regress the Z's on the expected proportion of alleles IBD, π(x) = E [Nx |MDx]/2, at the test locus;
• a slope coefficient β< 0, if statistically significant, is considered as evidence for linkage;
![Page 44: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/44.jpg)
HE: an example
0.5Proportion of marker alleles identical by decent
Solid line is the tted regression line;
Dotted line indicates true underlying relationship
![Page 45: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/45.jpg)
HE: motivation
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Assume strictly additive gene action at each locus,i.e.VD = 0. Then, for a putative QTL at x,
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![Page 46: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/46.jpg)
HE: linkage test
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![Page 47: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/47.jpg)
HE: examples with simulated data
simulated data from n = 200 sib-pairs;top to bottom: h2 = 0:50; 0:33; 0:25.
![Page 48: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/48.jpg)
Heritability and power
• for a given locus we may define the locus-specific heritability as the proportion of the total variance 'explained' by that particular site, e.g. (in the narrow-sense),
• the locus-specific heritability is the single most important parameter for the power of QTL linkage methods;
• heritabilities below 10% leads, in general, to unrealistically large sample sizes.
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![Page 49: Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649e525503460f94b4811a/html5/thumbnails/49.jpg)
HE: two-point analysis)(2
21
~~genotypes]marker |)[( mYYE
~
)(mwhere is the expected proportion of marker alleles shared IBD.
• depends on the type of relatives considered;• for sib pairs• recombination fraction (θ) and effect size (VA;l ) are confounded and cannot be separately estimated;
;)21(2~
,2
lAV
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HE: in summary
Simple, transparent and comparatively robust but:
• poor statistical power in many settings;• different types of relatives cannot be mixed;• parents and their offspring cannot be used in HE;• assumptions of the statistical model not generally
satisfied;
• Remedy:• use one of several suggested extensions of HE;• alternatively, use VCA instead
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Mathematically:
Yi=+Tai+gi+qi+ei
where is the population mean, a are the “environmental” predictor variables, q is the major trait locus, g is the polygenic effect, and e is the residual error.
Polygenes Independent environment
QTL
Trait value
VCA
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VCA: an additive model
RDRAxDx
xAx
x
RDRAxDxA
DA
p
i ii
n
ll
p
iii
VVVVYYC
eVVVVV
eVVVYV
zYE
eXzY
,,,)(
2,)(
21
,,,,
1
11
2)MD|,(
)(
)()(
;)(
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VCA: major assumptionThe joint distribution of the phenotypic values in a pedigree is assumed to be multivariate normal with the given mean values, variances and covariances;
• the multivariate normal distribution is completely specified by the mean values, variances and covariances;• the likelihood, L, of data can be calculated and we can estimate the variance components VA;x; VD;x ; VA;R; VD;R;
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VCA: linkage test
The linkage test of
H0 : VA;x = VD;x = 0
uses the LOD score statistic
When the position of the test locus, x, is varied over a chromosomal region the result can be summarized in a LOD score curve.
)0(L
model) L(fulllogLOD
,,10
xDxA
x VV
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VCA vs HE: LOD score proles
From Pratt et al.; Am. J. Hum. Genet. 66:1153-1157, (2000)
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Linkage methods for QTL
• Fully parametric linkage approach is difficult;• Model-free tests comprise the alternative choice; • We will discuss
Haseman-Elston Regression (HE);Variance Components Analysis (VCA);
Both can be viewed as two-step procedures:1. use polymorphic molecular markers to extract information on inheritance patterns;2. evaluate evidence for a trait-influencing locus at specified locations;
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Similarities and differences
• HE and VCA are based on estimated IBD-sharing given marker data;
• both methods require specification of a statistical model!('model-free' means 'does not require specification of genetic model')
• similarity in IBD-sharing is used to evaluate trait similarity using eitherlinear regression (HE) orvariance components analysis (VCA);