Composite Method for QTL Mapping

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Composite Method for QTL Mapping

• Zeng (1993, 1994)
• Limitations of single marker analysis
• Limitations of interval mapping 
• The test statistic on one interval can be
  affected by QTL located at other intervals (not
  precise)
• Only two markers are used at a time (not
  efficient)
• Strategies to overcome these limitations
• Equally use all markers at a time (time
  consuming, model selection, test statistic)
• One interval is analyzed using other markers to
  control genetic background

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Foundation of composite interval mapping

• Interval mapping Only use two flanking markers
  at a time to test the existence of a QTL
  (throughout the entire chromosome)
• Composite interval mapping Conditional on other
  markers, two flanking markers are used to test
  the existence of a QTL in a test interval
• Note An understanding of the foundation of
  composite interval mapping needs a lot of basic
  statistics. Please refer to A. Stuart and J. K.
  Ords book, Kendalls Advanced Theory of
  Statistics, 5th Ed, Vol. 2. Oxford University
  Press, New York.

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• Assume a backcross and one marker
• Aa aa ? Aa aa Mean
• Frequency ½ ½ 1
• Value 1 0 ½
• Deviation ½ -½
• Variance ?2 (½)2½ (-½)2½ ¼
• Two markers, A and B
• AaBb aabb ? AaBb Aabb aaBb
  aabb
• Frequency ½(1-r) ½r ½r ½(1-r)
• Value (A) 1 1 0 0
• Value (B) 1 0 1 0
• Covariance ?AB (1-2r)/4 Correlation 1 - 2r

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• Conditional variance
• ?2BA ?2B - ?2AB /?2A
• ¼ - (1-2r)/42/(¼)
• r(1-r)
• For general markers, j and k, we have
• Covariance ?jk (1 - 2rjk)/4
• Correlation 1 - 2rjk
• Conditional variance
• ?2kj ?2k - ?2kj /?2j
• ¼ - (1-2rjk)/42/(¼)
• rjk(1-rjk)

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• Three markers, j, k and l
• Covariance between markers j and k conditional on
  marker l
• ?jkl?jk - ?jl ?kl /?2l
• (1-2rjk)-(1-2rjl)(1-2rkl)/4
• 0 For order -j-l-k- or -k-l-j-
• rkl(1-rkl)(1-2rjk) For order -j-k-l- or
  -l-k-j-
• rjl(1-rjl)(1-2rjk) For order -l-j-k- or
  -k-j-l-
•  
• Note (1-2rjk)(1-2rjl)(1-2rkl) for order jlk or
  klj

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• Three markers, j, k and l 
• Variance of markers j conditional on markers k
  and l
• ?2jkl ?2jk - ?jlk /?2lk
• ?2jl - ?jkl /?2kl
• ?2jk For order -j-k-l-
• ?2jl For order -k-l-j-
  rkj(1-rkj)rjl(1-rjl)/rkl(1-rkl) For order
  -k-j-l-
• In general, the variance of markers j
  conditional on all other markers is 
• ?2js_ ?2j(j-1)(j1) , s_ is denotes a set
  that
• includes all markers except markers (j-1) and
  (j1).

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Important conclusions

•   Conditional on an intermediate marker, the
  covariance between two flanking markers is
  expected to be zero.
•     This conclusion is the foundation for
  composite interval mapping which aims to
  eliminate the effect of genome background on the
  estimation of QTL parameters

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• Four markers, j lt k, l lt m
• Covariance between markers j and k conditional on
  markers l and m
• ?jklm
• ?jkl ?jml ?kml /?2ml
• ?jkm ?jlm ?klm /?2lm
• 0 For order -j-l-k-m- or -j-l-m-k-
• ?jkl For order -j-k-l-m-
• ?jkm For order -l-m-j-k-
• rlj(1- rlj)rkm(1- rkm)(1- 2rjk)/rlm(1-
  rlm) For order -l-j-k-m-

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• In general, for -(l-1)-l-j-k-m-(m1)-, we have 
• ?jk(l-1)lm(m1) ?jklm(m1) ?jklm,
• which says that
• The covariance between markers j and (j1)
  conditional on all other markers is
• ?j(j1)s_ ?j(j1)(j-1)(j1)
• (s_ is denotes a set that includes all markers
  except markers j and (j1).

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MARKER and QTL

• Assume a backcross and one QTL
• Qq x qq ? Aa aa mean
• Frequency ½ ½ 1
• Value a 0 ½a
•   Variance ?2 1/4a2
•  One marker k and one QTL u
•  AaQq x aaqq ? AaQq Aaqq aaQq aaqq
• Frequency ½(1-r) ½r ½r ½(1-r)
• Value (A) 1 1 0 0
• Value (Q) a 0 a 0
• Covariance ?ku (1-2ruk)a/4
• Correlation 1-2rku

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• Two markers, j and k, and one trait, y, including
  many QTLs
•  
• Covariance between trait y and marker j
  conditional on marker k
•  ?yjk
• ?yj - ?yk?jk /?2k
• ?u1(1-2ruj)-(1-2ruk)(1-2rjk)au/4
• rjk(1-rjk)?u?j(1-2ruj)au ?jltultkruk(1-ruk)(1-
  2rju)au
• For order -u-j-u-k-
• rjk(1-rjk)?u?k(1-2rku)au ?kltultjrku(1-rku)(1-
  2ruj)au
• For order -k-u-j-u-
•  

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• Covariance between trait y and marker j
  conditional on markers k and l
• ?yjkl
• ?yj/k - ?yk/j ?jl/k /?2l/k
• ?yj/l - ?yk/l ?jk/l /?2k/l
• ?yj/k For order -j-k-l-
• ?yj/l For order -j-l-k-
• rjk(1- rjk)/rlk(1- rlk)?lltu?j rlu(1- rlu)(1-
  2ruj)au rlj(1- rlj)/rlk(1- rlk)?jltultk
  ruk(1- ruk)(1- 2rju)au For order -l-j-k-

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• In general, for order --(j-1)-j-(j1)--, we
  have
•  ?yjs_ ?yj(j-1)(j1)
• Partial regression coefficient
• byjs_
• ?yjs_/?2js_
• ?yj(j-1)(j1)/ ?2j(j-1)(j1)
• ?(j-1)ltu?j r(j-1)u(1- r(j-1)u)(1-
  2ruj)/r(j-1)j(1- r(j-1)j)au
• ?jltult(j1) ru(j1)(1- ru(j1))(1-
  2rju)/rj(j1)(1- rj(j1))au

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• Two summations
• The first is for all QTL located between
  markers (j-1) and j
• The second is for all QTL located between
  markers j and (j1).

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• Important conclusion
• The partial regression coefficient depends only
  on those QTL which are located between markers
  (j-1) and (j1)

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• Suppose there is only one QTL between markers
  (j-1) and j, we have
• byjs_ r(j-1)u(1- r(j-1)u)(1-
  2ruj)/r(j-1)j(1- r(j-1)j)au.
• An estimate of byjs_ is a biased estimate of
  au.

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• Properties of composite interval mapping
• In the multiple regression analysis, assuming
  additivity of QTL effects between loci (i.e.,
  ignoring interactions), the expected partial
  regression coefficient of the trait on a marker
  depends only on those QTL which are located on
  the interval bracketed by the two neighboring
  markers, and is unaffected by the effects of QTL
  located on other intervals.
• Conditioning on unlinked markers in the multiple
  regression analysis will reduce the sampling
  variance of the test statistic by controlling
  some residual genetic variation and thus will
  increase the power of QTL mapping.

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• Conditioning on linked markers in the multiple
  regression analysis will reduce the chance of
  interference of possible multiple linked QTL on
  hypothesis testing and parameter estimation, but
  with a possible increase of sampling variance.
• Two sample partial regression coefficients of the
  trait value on two markers in a multiple
  regression analysis are generally uncorrelated
  unless the two markers are adjacent markers.

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Composite model for interval mapping and
regression analysis
zi QTL genotype xik marker genotype

• yi ? a zi ?km-2bkxik ei
• Expected means
• Qq ? a ?kbkxik a XiB
• qq ? ?kbkxik XiB
• Xi (1, xi1, xi2, , xi(m-2))1x(m-1)
• B (?, b1, b2, , bm-2)T

M1 x1 M1m1 1 ?b1 m1m1 0 ?
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Likelihood function

• L(y,M?) ?i1n?1if1(yi) ?0if0(yi)
• log L(y,M?) ?i1n log?1if1(yi) ?0if0(yi)
• f1(yi) 1/(2?)½?exp-½(y-?1)2, ?1 aXiB
• f0(yi) 1/(2?)½?exp-½(y-?0)2, ?0 XiB
• Define
• ?1i ?1if1(yi)/?1if1(yi) ?0if0(yi) (1)
• ?0i ?0if1(yi)/?1if1(yi) ?0if0(yi) (2)

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• a ?i1n?1i(yi-a-XiB)/ ?i1n?1i (3)
• ?1 (Y-XB)/c
• B (XX)-1X(Y-?1a) (4)
• ?2 1/n (Y-XB)(Y-XB) a2 c (5)
• (?i1n2?1i ?i1n3?0i)/(n2n3) (6)
• Y yinx1, ? ?1inx1, c ?i1n?1i

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Hypothesis test

• H0 a0 vs H1 a?0
• L0 ?i1nf(yi) ? B (XX)-1XY,
  ?21/n(Y-XB)(Y-XB)
• L1 ?i1n?1if1(yi) ?0if0(yi)
• LR -2(lnL0 lnL1)
• LOD -(logL0 logL1)

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Example
Interval mapping Composite interval mapping
LR
Testing position