Lecture 15: Linkage Analysis VII

Date: 10/14/02 Correction: power calculation Lander-Green Algorithm (Titles on updated or added slides highlighted)

Sample Size Calculation

What is the sample size needed in order to achieve a particular statistical power for an estimate?

We shall assume the relevant statistic is distributed as chi-square statistic.

Sample Size Calculation (cont.)

is the statistical power is the critical value to reject H0 with significance level

. c is the non-centrality parameter, usually the expectation of

the log-likelihood ratio test statistic under particular HA and experimental conditions.

df is the degrees of freedom

22,P

1

cdf

2

Sample Size Calculation (cont.)

12

,, 2

dfc

freedom. of degrees

and parameter ity noncentral with square-chi

central-nonfor valuecritical- is

E

2

12

,,

0

2

df

Gnc

df

Modeling

Test your modeling skills. Propose a model for the following family ascertainment situation.

What if you knew that probands were detected independently and with the same probability in each family, except all secondary probands are more easily detected (second, third, etc all to the same degree) than the first proband in a family.

The model formulation and calculation of pr probabilities for families with 3 affected are now posted to the website.

Lander-Green Algorithm

Like the Elston-Stewart algorithm, the Lander-Green algorithm models the pedigree and data as a Hidden Markov Model (HMM), except that the hidden states are the so-called inheritance vectors.

Like the Elston-Stewart algorithm, the Lander-Green algorithm assumes that there is no interference.

LG – (Dis)Advantages

The Lander-Green algorithm is linear in the number of loci and exponential in the number of members in the pedigree.

Recall that the Elston-Stewart algorithm is complementary, linear in the number of members, but exponential in the number of loci.

Simulation methods (MCMC in particular) are used to deal with pedigrees with both high numbers of members and loci.

LG – Inheritance Vector

The inheritance vector is a vector defined for each locus i in the dataset.

It is a binary vector with two components for each non-founder individual in the pedigree. Thus, it is of length 2(n – f).

The entry in the inheritance vector is 0 if the individual’s allele at that position is grandmaternal. If grandpaternal, it is 1. There are 22(n – f) possible inheritance vectors for each locus.

LG – Inheritance Vector (cont)

The inheritance vector holds information about the number of crossovers that occurred to produce each non-founder in the population.

Thus, it is appropriate for estimating recombination fractions as is our goal here with the LG algorithm.

LG – Inheritance Vector Example

4

AA aa

aA aA aaaa

Aa

1 2

3 5 6

7 89

Aaaa

Gamete v

4M 0|1

4P 0|1

5M 0|1

5P 0|1

7M 1

7P 0|1

8M 0

8P 0|1

9M 0

9P 0|1

LG – Simplification by Conditioning

Fortunately, conditional on the inheritance vectors, the genotypes of each offspring are independent.

Of course, conditional on the genotype, the phenotype probabilities are independent.

Thus, we can calculate the probability for each individual in the pedigree independently of the others once we condition on the inheritance vectors.

LG – Hidden States

The inheritance vector constitutes the unknown hidden state for each allele. We must define transition probabilities among the hidden states (from locus-to-locus).

Begin, by considering the transition probability between loci within a single individual, where the inheritance vector is of length 2.

Therefore, the hidden state at each locus is a binary vector of length 2.

LG – Initial State

We must define the initial state of the first marker locus.

Prior to viewing the genotypes, all inheritance vectors are equally likely.

Assume the initial state of the inheritance vector at marker 1 is uniform over {(0,0), (1,0), (0,1), (1,1)}, where we list the maternal status first. In other words, marker 1 has ¼ probability of being in each of these possible states.

Because of the assumption of no interference, the transition probabilities from the state at locus i to the state at locus i+1 are given by:

LG – Pairwise Transition Probabilities

22

22

22

22

111)1,1(

111)1,0(

111)0,1(

111)0,0(

)1,1()1,0()0,1()0,0(

iiiiii

iiiiii

iiiiii

iiiiii

where i is the recombination fraction between locus i and locus i+1.

LG – Switch in Notation

From this point on, assume there are n non-founders (rather than n – f).

The reason for this change is simplification of the equations.

LG – Inheritance Vector Transition Probabilities

The transition probabilities between inheritance vectors defined on full pedigrees with n relevant members, are given by

wvdni

wvdivw ,2, 1P

where d(v,w) is the Hamming distance between inheritance vectors v and w, i.e. the number of discordances between them.

LG – Forward Variable

i

i

i

i

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iii

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vbvOOb

byOb

byOOb

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,,,,P

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111

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1111

1111

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1

LG – Backward Variable

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,,,P

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111112

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ilii

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bvvOOb

b

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LG – i(v,w)

wviii

iii

ii

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O

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,11

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P,P)(

P,P)(

P

,,P

,,P,

transition probability

penetrance parameter

LG – Baum’s Lemma

Baum’s Lemma: Let

v

OvOvQ ',Plog,P',

If

,', QQ

then

OO P'P

LG – Proof of Baum’s Lemma

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LG – Jensen’s Inequality

function concave a is

when EE xfxfxf

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v

v

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LG – EM Algorithm

We maximize Q(’) over ’ to maximize the likelihood P(O|) conditional on the current parameter estimates .

This may sound familiar. It is the M step of the EM algorithm, and the EM algorithm is how we maximize over a pedigree.

Details are shown below. Maximization is the difficult step. We show it first.

LG - Maximization

viii

i

v ii

v

v

vvOv

vOvOvvvOvQ

vOvOvvvOvQ

OvOvQ

',Plog,P

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',

PP',PPlog,P',

',Plog,P',

1

22111121

22111121

Key step: by conditional independence, this probabilitybecomes a product of conditional probabilities.

LG - Maximization

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LG – EM Agorithm (M Step)

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LG – EM Algorithm (E Step)

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,,

,,P,,,E

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wviiii

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sum over all pairs ofinheritance vectors

the usualconditionalprobabilitiesneeded tocalculateexpectation

Heterogeneity in Recombination Fraction

Allow for two recombination fraction parameters in each interval.

Allow for one recombination fraction in each interval and a universal constant relating male and female recombination fractions.

Use nested models to test for evidence of sex-based differences.

Model Misspecification

Penetrance parameters, allele frequencies may be incorrectly specified.

The model is robust to misspecification such that the false positive rate for linkage is unaffected by misspecification of these parameters.

Model Misspecification and Ascertainment

When ascertainment is made independent of disease state and marker loci, the method remains robust to misspecification in both.

When ascertainment is made with respect to disease state, then the method is robust to misspecification of the disease parameters.

Effects on Power

Power in two-point linkage analysis is largely unaffected as long as the dominance is specified correctly.

Multipoint linkage analysis is much more sensitive to misspecification of the model. However, there is more information when model parameters are jointly estimated along with position.