666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666...
Transcript of 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666...
![Page 1: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/1.jpg)
The Lander-Green Algorithm in Practice
Biostatistics 666Lecture 23
![Page 2: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/2.jpg)
Last Lecture:Lander-Green Algorithm
More general definition for I, the "IBD vector"Probability of genotypes given “IBD vector”Transition probabilities for the “IBD vectors”
∑ ∏∑ ∏==
−=1 12
11 )|()|()(...I
m
iii
I
m
iii IGPIIPIPL
m
![Page 3: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/3.jpg)
Lander-Green Recipe
1. List all meiosis in the pedigree • There should be 2n meiosis for n non-founders
2. List all possible IBD patterns• Total of 22n possible patterns by setting each
meiosis to one of two possible outcomes
3. At each marker location, score P(G|I)• Evaluate using founder allele graph
![Page 4: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/4.jpg)
Lander-Green Recipe
4. Build transition matrix for moving along chromosome
• Patterned matrix, built from matrices for individual meiosis
⎥⎦
⎤⎢⎣
⎡
−−
=⊗⊗
⊗⊗+⊗
nn
nnn
TTTT
T)1(
)1(1
θθθθ
![Page 5: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/5.jpg)
Lander-Green Recipe
5. Run Markov chain• Start at first marker, m=1
• Build a vector listing P(Gfirst marker|I) for each I
• Move along chromosome• Multiply vector by transition matrix
• Combine with information at the next marker• Multiply each component of the vector by P(Gcurrent marker|I)
• Repeat previous two steps until done
![Page 6: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/6.jpg)
Pictorial Representation
Forward recurrence
Backward recurrence
At an arbitrary location
![Page 7: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/7.jpg)
Today:Lander-Green Algorithm in practice
Refining the Lander-Green algorithm• Speed up transition step• Reducing size of inheritance space
Common applications of the algorithm• Non-parametric linkage analysis• Parametric linkage analysis • Information content calculation (time permitting)
![Page 8: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/8.jpg)
Markov Chain Calculations
P(X1,…,Xm|Im) for each Im define a vectorP(Im|Im-1) for each pair Im-1 , Im defines a matrixP(Xm|Im) for each Im define another vector
∑−
−−−=1
)|()|()|,...,()|,...,( 11111mI
mmmmmmmm IXPIIPIXXPIXXP
![Page 9: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/9.jpg)
As Matrix Operations …
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
====
======
−−
−−
−−−−
)11|(...
)00|(
)11|11(...)11|00(.........
)00|11(...)00|00()11|(),...,00|(
11
11
11..111..1
mm
mm
mmmm
mmmm
mmmm
IXP
IXP
IIPIIP
IIPIIPIXPIXP o
Given all the ingredients are available, how complex is this operation?
![Page 10: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/10.jpg)
First Refinement …
Speeding up the transitions in the Markov Chain
Divide and Conquer Algorithm (Idury and Elston 1997)
Fast Fourier Transforms(Kruglyak and Lander 1998)
![Page 11: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/11.jpg)
Matrix Multiplication Bottleneck
At each location we track 22n IBD patterns
To move along genome we consider• 22n * 22n transition probabilities
How much computation is required in a nuclear family with 5 offspring?
![Page 12: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/12.jpg)
Elston-Idury Algorithm0000000100100011010001010110011110001001101010111100110111101111
T⊗2n =
(1- θ) T⊗2n-1 + θ T⊗2n-1
00000001001000110100010101100111
00000001001000110100010101100111
(1- θ) T⊗2n-1 + θ T⊗2n-1
10001001101010111100110111101111
10001001101010111100110111101111
00000001001000110100010101100111
Replace one matrix multiplication with 4 smaller ones …
![Page 13: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/13.jpg)
Operations required …
Multiplication by full transition matrix:
Multiplication by smaller transition matrix• Per matrix• Two of these operations needed• Multiplication by (1-θ) and θ
![Page 14: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/14.jpg)
Elston-Idury Algorithm
Matrix multiplication is an expensive operation
Replaces multiplication by a matrix with 22n*22n
elements with …
.. multiplication by 2 matrices each with 22n-1*22n-1
elements and 3 * 22n additions and multiplications
Can be applied recursively!• Overall cost becomes 3 * 2n * 22n instead of 22n * 22n
![Page 15: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/15.jpg)
Second Refinement …
Reducing number of inheritance vectors
![Page 16: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/16.jpg)
IBD Space
Default recipe is inefficient
Check resulting number of IBD states for:• Sibling pair• Half-sibling pair• Uncle nephew pair
Many ad-hoc solutions …… but a general strategy for reducing IBD space?
![Page 17: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/17.jpg)
Improvements:Reducing the inheritance space
Kruglyak et al (1996)• Founder symmetry
Gudbjartsson et al (2000)• Founder couple symmetry
Abecasis et al (2001)• Arbitrary symmetries depending on genotypes
Approaches to avoid consideration of inheritance vectors that always produce equivalent founder allele graphs
![Page 18: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/18.jpg)
Founder Symmetry
Allele ordering for founders is unknowable• Grand-paternal allele?• Grand-maternal allele?
Arbitrarily assume outcome of meiosis for one sibling
Inheritance space becomes 22n-f
![Page 19: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/19.jpg)
Founder Couple Symmetry
Maternal / paternal origin for ungenotyped couples is unknowable• Except if male-female recombination rates differ
Arbitrarily assume outcome of meiosis for one grandchild
Inheritance space becomes 22n-f-c
![Page 20: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/20.jpg)
Example Application of Inheritance Vector SymmetriesAssume that allele frequency p1= 0.1
Consider a first cousin pair sharing genotype 1/1
Try the following:• Enumerate reduced set of inheritance vectors• Calculate probability for each one• Calculate probability that the pair is IBD=0• Calculate probability that the parents are IBD=1
![Page 21: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/21.jpg)
Reduced Inheritance Spaces
Greatly speed up calculations
Each state examined considered now represents collection inheritance vectors• Vectors in the collection are indistinguishable
Requires changes to transition matrices
![Page 22: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/22.jpg)
Uses of the Lander Green Algorithm
Non-parametric linkage analysis
Parametric linkage analysis
Information content calculation • Time permitting!
![Page 23: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/23.jpg)
Nonparametric Linkage Analysis
Model-free
Does not require specification of a trait model
Test for evidence of excess IBD sharing among affected individuals
![Page 24: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/24.jpg)
Non-parametric Analysis for Arbitrary Pedigrees
Must rank general IBD configurations• Low scores correspond to no linkage• High scores correspond to linkage
Multiple possible orderings are possible• Especially for large pedigrees
Under linkage, probability for vectors with high scores should increase
![Page 25: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/25.jpg)
Nonparametric Linkage Statistic
Statistic S(I) which ranks IBD vectorsThen, following Whittemore and Halpern (1995)
[ ]
)1,0(~)(
)()(
)()(
)|()()(
22
NGSZ
GPGS
GPGS
GIPISGS
G
G
I
σµ
µσ
µ
−=
−=
=
=
∑
∑
∑
![Page 26: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/26.jpg)
Nonparametric Linkage Statistic
Original definition not useful for multipoint data…Kruglyak et al (1996) proposed:
[ ]
)1,0(~)(
)()(
)()(
)|()()(
22
NGSZ
IPIS
IPIS
GIPISGS
I
I
I
σµ
µσ
µ
−=
−=
=
=
∑
∑
∑
![Page 27: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/27.jpg)
The Pairs Statistic
Sum of IBD sharing for all affected pairs
∑∈
=pairs) (affected),(
)|,()(ba
pairs IbaIBDIS
( )∑
∑−=
=
Iuniformpairs
Iuniformpairs
IPIS
IPIS
)()(
)()(
22 µσ
µ
![Page 28: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/28.jpg)
The Spairs Statistic
Total allele sharing among affected relatives
Sibpair: A-B A-C B-CSPairs = 2 + 1 + 1 = 4
12 34
13 13 14
A B C
![Page 29: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/29.jpg)
Example:Pedigree with 4 affected individuals
![Page 30: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/30.jpg)
What is Spairs(I) for this Descent Graph?
A B
C D E F
G H
![Page 31: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/31.jpg)
The NPL Score
Non-parametric linkage score
Variance will always be ≤ 1 so using standard normal as reference gives conservative test.
( ))|()(
/)(
GIPIZZ
SIZ
INPL
pairs
∑=
−= σµ
![Page 32: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/32.jpg)
Accurately Measuring NPL Evidence for Linkage
For a single marker…
Estimating variance of statistic over all possible genotype configurations is not practical for multipoint analysis
One possibility is to evaluate the empirical variance of the statistic over families in the sample…
( )∑∑∈
−=*
22 )()|()(Ii G
uniformpairs iPiGPGS µσ
![Page 33: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/33.jpg)
Kong and Cox Method
A probability distribution for IBD states• Under the null and alternative
Null• All IBD states are equally likely
Alternative• Increase (or decrease) in probability is proportional to S(I)
"Generalization" of the MLS method
![Page 34: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/34.jpg)
Kong and Cox Method
)0()ˆ(log
)|()|()(
)(1)()|(
10 ==
=
⎟⎠⎞
⎜⎝⎛ −
+=
∏ ∑
δδ
δδ
σµδδ
LLLOD
IPIGPL
ISIPIP
families I
![Page 35: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/35.jpg)
Note:Alternative NPL Statistics
Any arbitrary statistic can be used
Vectors with high scores must be more common when linkage exists
Statistics have been defined that• Focus on the most common allele among affecteds• Count number of founder alleles among affecteds• Evaluate linkage for quantitative traits
![Page 36: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/36.jpg)
Many Alternative NPL Statistics!
McPeek (1999) Genetic Epidemiology 16:225–249
![Page 37: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/37.jpg)
Parametric Linkage Analysis
X phenotype data (affected/normal)I inheritance vector (meiosis outcomes)
Calculate P(X|I) based on…
Trait locus allele frequencies• p and q
Penetrances for each genotype• f11, f12, f22
![Page 38: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/38.jpg)
Parametric Linkage Analysis
∑ ∑ ∏∏=1 2
),|()(...)|(a a j
ji
if
IXPaPIXP a
Sum over all allele states for each founder• Due to incomplete penetrance
Once P(X|I) is available, the trait “plugs into” the calculation as if it was a marker locus
![Page 39: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/39.jpg)
Likelihood Ratio Test
Evaluate evidence for linkage as…
Is a particular set of meiotic outcomes likely for a given trait model?
∑∈
=
*)()|(
)|()(
Iiuniform
observed
iPiXPIXPILR
![Page 40: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/40.jpg)
Allowing for uncertainty…
Weighted sum over possible meiotic outcomes…
∑∑
∑
∈
∈
∈
=
=
*
*
*
)()|(
)|()|(
)|()(
Iiuniform
Ii
Ii
iPiXP
GiPiXP
GiPiLRLR
![Page 41: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/41.jpg)
Genotype Data Informativeness
Based on the Shannon entropy measure:
Ranges between 0 and 1.Randomness in distribution of conditional probabilities.
0
2
1
log
EEI
PPE ii
−=
−= ∑
![Page 42: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/42.jpg)
Some Exemplar Entropies
1
2 2
2
/ 2 2/
2/ 2/ 2
2 2
2
/ 2 2/
2/ 2/1
1 2
3
/ 2 2/
2/ 2/
Information = 1 Information = 0.5
(with 4 inheritance vectors)
Information = 0
![Page 43: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/43.jpg)
Example of Multipoint Information Content
![Page 44: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/44.jpg)
More on Information Content…
The theoretical maximum is 1.0• All probability concentrated on one inheritance vector
The practical maximum is lower• It will depend on which individuals are genotyped
Useful in a comparative manner• Identifies regions where study conclusions are less certain
![Page 45: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/45.jpg)
Today
Refinements of Lander-Green algorithm
Non-parametric linkage analysis
Parametric linkage analysis
![Page 46: 666.23 -- Lander-Green in Practicecsg.sph.umich.edu/abecasis/class/666.23.pdfBiostatistics 666 Lecture 23. Last Lecture: Lander-Green Algorithm zMore general definition for I, the](https://reader033.fdocuments.us/reader033/viewer/2022051812/603044ea7caaa96bbd1771b7/html5/thumbnails/46.jpg)
Reference
Kruglyak, Daly, Reeve-Daly, Lander (1996)Am J Hum Genet 58:1347-63
Whittemore and Halpern (1994)Biometrics 50:109-117