Transcript Complex” inheritance - CSC's mainpage — CSC

Variance components-based
linkage analysis
Rationale of variance components-based
linkage analysis
The pattern of phenotypic similarity among pedigree
members should be reflected by the pattern of IBD
sharing among them at chromosomal loci influencing
the trait of interest.
Variance components approach:
multivariate normal distribution (MVN)
In variance components analysis, the phenotype is generally
assumed to follow a multivariate normal distribution:
f x  
1
2  ž 
n
1
2

exp 

1
x   ' ž
2
1
x    

n
1
1
'
ln f x    ln 2   ž  x    ž
2
2
2
no. of individuals
(in a pedigree)
nn covariance
matrix
phenotype
vector
1
x   
mean
phenotype
vector
Modeling the resemblance among relative
heritability analysis
    2
linkage analysis
2
ˆ
    2   q

2
e
2
e
2
g
2
g
Identity-by-state (IBS) vs. identity-by-descent (IBD)
12
34
12
13
11
23
12
12
13
14
12
13
13
12
12
12
IBD
IBS
(also IBS)
(not IBD)
?
?
(both or
neither IBD)
If IBD then necessarily IBS (assuming absence of mutation event).
If IBS then not necessarily IBD (unless a locus is 100% informative, i.e.
has an infinite number of alleles, each with infinitesimally small allele
frequency).
Probabilistic inference of IBD
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12
13
11
23
12
12
13
14
12
13
13
12
12
12
IBD
1
0
0.5
1
NIBD 1
2
1.5
1

0
0.25
0.5
0.5

Matrix of estimated allele sharing
among relatives
expected  2
P
M
S1
S2
S3
P
M
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S1
S2
S3
13
13
13
ˆ

estimated
P
M
S1
S2
S3
1
0
0.5
0.5
0.5
P
1
0.5
0.5
0.5
1
0.5
0.5
 M
1
0.5
S2
1
S3
S1
P
M
S1
S2
S3
1
0
0.5
0.5
0.5
1
0.5
0.5
0.5
1
0.75
0.75
1
0.75
1
Variance components-based lod score
LH1 | data
lod  log10
LH 0 | data
 log10
2
2
2
max
L

,

,

 e g q ,| data
2
2
2
 e , g , q
2
2
2
max
L

,

,

 e g q  0 | data
2
2
 e , g


Asymptotically,2ln10lod ~ 0.521.
Power in variance components-based
linkage analysis
• Power in VC-based linkage analysis depends on
several factors:
– locus-specific (additive) heritability:
• h2locus = Varlocus / Varall factors
• Power scales linearly with square of h2locus.
– residual (additive) heritability
– pedigree complexity and size
– information on transmission pattern (informativeness of
marker genotypes)
Sample size requirements to detect linkage to
a QTL with a lod score of ≥ 3 and 80% power
(under random ascertainment)
Number of Individuals
100,000
Pedigree
Sibship (2)
Sibship (4)
10,000
1,000
100
0
0.1
0.2
0.3
Heritability due to QTL
0.4
0.5
Pros and cons of
variance-components-based linkage
analysis
+ no need to specify inheritance model (but typically assumes
“additivity”
+ robust to allelic heterogeneity at a locus
+ modeling flexibility (for some phenomena)
+ computationally feasible even on large pedigrees
-
not always well-behaved statistically (depending on phenotypic
distribution and ascertainment)
generally less powerful than penetrance-model-based linkage
analysis under suitable model
“quick but dirty”
Covariate selection
•
•
•
In linkage studies, covariates should be chosen in such a way that
power to detect linkage (with genes of interest) is maximized. While the
specific effects of these covariates may be of interest, this is clearly
secondary to the goal of gene localization.
Covariates may be regressed out first, followed by linkage analysis of
the residuals, or regression takes place simultaneous with linkage
assessment.
Any combination of a given phenotype and a set of covariates ought to
be considered a different trait.
– Only few combinations ought to be analyzed, in order to avoid complications
due to unnecessary multiple testing.
– Interpretation of linkage results must take into account the chosen
covariates.
Inclusion of covariates
Measured covariates can easily be incorporated as
“fixed effects” in the multivariate normal model of the
phenotype, by making the expected phenotype
different for different individuals as a function of the
measured covariates.
n
1
1
'
ln f x    ln 2   ž  x    ž
2
2
2
1
x   
   overall  Y 
i   overall    jYij
j
Inclusion of covariates
If covariates are not of interest in and of themselves,
one can “regress them out” before pedigree analysis.
Xˆ i  ˆ 0   ˆ jYij
j
ˆ  Y
X
Xi  Xˆ i  eˆi
ˆ  eˆ
XX
Then use residuals as phenotype of interest in
pedigree analysis.
Inference regarding covariates
in heritability analysis
H0: measured covariate Y does not influence phenotype.
H1: measured covariate Y does influence phenotype.
L H 0 
  2 ln
L H 1 

2
2
ˆ
ˆ
L  a , u ,   0
 2 ln
2
2 ˆ
ˆ
ˆ
L  , , 
Be cautious in interpretation.

a
u


Choice of covariates in gene mapping:
“rules of thumb”
• Covariates ought to be included in the likelihood model if they
are known to influence the phenotype of interest and if their own
genetic regulation does not overlap the genetic regulation of the
target phenotype.
• Typical examples include sex and age.
• In the analysis of height, information on nutrition during
childhood should probably be included during analysis.
However, known growth hormone levels probably should not be.
Choice of covariates
phenotypic
variance due to
2
3
A
genes
covariates
5
C1
B
4
total phenotypic
variance
Choice of covariates:
special case of treatment/medication
•
Covariates are accounted for before or during analysis to reduce the
phenotypic variance due to these covariates, thus simplifying the
etiology of the trait and increasing power to detect the remaining
etiological factors.
•
In contrast, the goal of accounting for treatment/medication is not to
reduce phenotypic variance, but rather to increase it to the level at
which it would have been w/o treatment/medication.
•
Accounting for treatment/medication by means of covariates achieves
just the opposite, giving the same phenotypic mean to both treated and
untreated individuals.
probability density
Before treatment/medication
of affected individuals
phenotype
unaffected
affected
probability density
After (partially effective) treatment /
medication of affected individuals
Accounting for
medication with
covariates adjusts in
the wrong direction!
true effect of
medication
phenotype
apparent
effect of
unaffected covariate
affected
How to deal with treatment/medication
•
censor treated individuals
•
infer phenotype before treatment, or integrate over possible
range of phenotype before treatment, based on information
on treatment efficacy etc.
•
dichotomize phenotype
All these approaches have drawbacks
and are not entirely satisfactory.