1. 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

2. 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. Ord’s book, Kendall’s Advanced Theory of Statistics, 5 th Ed, Vol. 2. Oxford University Press, New York.

3. Assume a backcross and one marker Aa × aa  AaaaMean Frequency½½1 “Value”10½ “Deviation”½ -½ Variance  2 = (½) 2 ×½ + (-½) 2 ×½ = ¼ Two markers, A and B: AaBb × aabb  AaBb Aabb aaBb aabb Frequency ½(1-r) ½r½r ½(1-r) “Value” (A)1 10 0 “Value” (B)1 01 0 Covariance  AB = (1-2r)/4 Correlation = 1 - 2r

4. Conditional variance:  2 B | A =  2 B -  2 AB /  2 A = ¼ - [(1-2r)/4] 2 /(¼) = r(1-r) For general markers, j and k, we have Covariance  jk = (1 - 2r jk )/4 Correlation = 1 - 2r jk Conditional variance:  2 k|j =  2 k -  2 kj /  2 j = ¼ - [(1-2r jk )/4] 2 /(¼) = r jk (1-r jk )

5. Three markers, j, k and l Covariance between markers j and k conditional on marker l:  jk|l =  jk -  jl  kl /  2 l = [(1-2r jk )-(1-2r jl )(1-2r kl )]/4 = 0For order -j-l-k- or -k-l-j- r kl (1-r kl )(1-2r jk )For order -j-k-l- or -l-k-j- r jl (1-r jl )(1-2r jk )For order -l-j-k- or -k-j-l- Note: (1-2r jk )=(1-2r jl )(1-2r kl ) for order jlk or klj

6. Three markers, j, k and l Variance of markers j conditional on markers k and l  2 j|kl =  2 j|k -  jl|k /  2 l|k =  2 j|l -  jk|l /  2 k|l =  2 j|k For order -j-k-l-  2 j|l For order -k-l-j- [r kj (1-r kj )r jl (1-r jl )]/[r kl (1-r kl )] For order -k-j-l- In general, the variance of markers j conditional on all other markers is  2 j|s_ =  2 j|(j-1)(j+1), s_ is denotes a set that includes all markers except markers (j-1) and (j+1).

7. 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

8. Four markers, j < k, l < m Covariance between markers j and k conditional on markers l and m:  jk|lm =  jk|l –  jm|l  km|l /  2 m|l =  jk|m –  jl|m  kl|m /  2 l|m = 0 For order -j-l-k-m- or -j-l-m-k-  jk|l For order -j-k-l-m-  jk|m For order -l-m-j-k- [r lj (1- r lj )r km (1- r km )(1- 2r jk )]/[r lm (1- r lm )] For order -l-j-k-m-

9. In general, for -(l-1)-l-j-k-m-(m+1)-, we have  jk|(l-1)lm(m+1) =  jk|lm(m+1) =  jk|lm, which says that The covariance between markers j and (j+1) conditional on all other markers is  j(j+1)|s_ =  j(j+1)|(j-1)(j+1) (s_ is denotes a set that includes all markers except markers j and (j+1).

10. MARKER and QTL Assume a backcross and one QTL Qq x qq  Aa + aamean Frequency½½1 Valuea0½a Variance  2 = 1/4a 2 One marker k and one QTL u: AaQq x aaqq  AaQq Aaqq aaQqaaqq Frequency½(1-r) ½r ½r½(1-r) Value ( A )1 1 00 Value (Q)a 0 a0 Covariance  ku = (1-2r uk )a/4 Correlation = 1-2r ku

11. Two markers, j and k, and one trait, y, including many QTLs Covariance between trait y and marker j conditional on marker k  yj|k =  yj -  yk  jk /  2 k =  u=1 [(1-2r uj )-(1-2r uk )(1-2r jk )]a u /4 = r jk (1-r jk )  u  j (1-2r uj )a u +  j<u<k r uk (1-r uk )(1-2r ju )a u For order -u-j-u-k- r jk (1-r jk )  u  k (1-2r ku )a u +  k<u<j r ku (1-r ku )(1-2r uj )a u For order -k-u-j-u-

12. Covariance between trait y and marker j conditional on markers k and l  yj|kl =  yj/k -  yk/j  jl/k /  2 l/k =  yj/l -  yk/l  jk/l /  2 k/l =  yj/k For order -j-k-l-  yj/l For order -j-l-k- [r jk (1- r jk )]/[r lk (1- r lk )]  l<u  j r lu (1- r lu )(1- 2r uj )a u + [r lj (1- r lj )]/[r lk (1- r lk )]  j<u<k r uk (1- r uk )(1- 2r ju )a u For order -l-j- k-

13. In general, for order -…-(j-1)-j-(j+1)-…-, we have  yj|s_ =  yj|(j-1)(j+1) Partial regression coefficient b yj|s_ =  yj|s_ /  2 j|s_ =  yj|(j-1)(j+1) /  2 j|(j-1)(j+1) =  (j-1)<u  j [r (j-1)u (1- r (j-1)u )(1- 2r uj )]/[r (j-1)j (1- r (j-1)j )]a u +  j<u<(j+1) [r u(j+1) (1- r u(j+1) )(1- 2r ju )]/[r j(j+1) (1- r j(j+1) )]a u

14. 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 (j+1).

15. Important conclusion: The partial regression coefficient depends only on those QTL which are located between markers (j-1) and (j+1)

16. Suppose there is only one QTL [between markers (j-1) and j], we have b yj|s_ = [r (j-1)u (1- r (j-1)u )(1- 2r uj )]/[r (j-1)j (1- r (j- 1)j )]a u. An estimate of b yj|s_ is a biased estimate of a u.

17. 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.

18. 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.

19. Composite model for interval mapping and regression analysis y i =  + a* z i +  k m-2 b k x ik + e i Expected means: Qq:  + a* +  k b k x ik = a* + X i B qq:  +  k b k x ik = X i B X i = (1, x i1, x i2, …, x i(m-2) ) 1x(m-1) B = ( , b 1, b 2, …, b m-2 ) T z i : QTL genotype x ik : marker genotype M 1 x 1 M 1 m 1 1  +b 1 m 1 m 1 0 

20. Likelihood function L(y,M|  ) =  i=1 n [  1|i f 1 (y i ) +  0|i f 0 (y i )] log L(y,M|  ) =  i=1 n log[  1|i f 1 (y i ) +  0|i f 0 (y i )] f 1 (y i ) = 1/[(2  ) ½  ]exp[-½(y-  1 ) 2 ],  1 = a*+X i B f 0 (y i ) = 1/[(2  ) ½  ]exp[-½(y-  0 ) 2 ],  0 = X i B Define  1|i =  1|i f 1 (y i )/[  1|i f 1 (y i ) +  0|i f 0 (y i )](1)  0|i =  0|i f 1 (y i )/[  1|i f 1 (y i ) +  0|i f 0 (y i )](2)

21. a* =  i=1 n  1|i (y i -a*-X i B)/  i=1 n  1|i (3) =  1 (Y-XB)´/c B = (X´X) -1 X´(Y-  1 a*) (4)  2 = 1/n (Y-XB)´(Y-XB) – a* 2 c (5)  = (  i=1 n2  1|i +  i=1 n3  0|i )/(n 2 +n 3 )(6) Y = {y i } nx1,  = {  1|i } nx1, c =  i=1 n  1|i

22. Hypothesis test H0: a*=0 vs H1: a*  0 L0 =  i=1 n f(y i )  B = (X´X) -1 X´Y,  2 = 1/n(Y-XB)´(Y-XB) L1=  i=1 n [  1|i f 1 (y i ) +  0|i f 0 (y i )] LR = -2(lnL0 – lnL1) LOD = -(logL0 – logL1)

23. Example LR Testing position Interval mapping Composite interval mapping