Theor Appl Genet (1992) 85:110-111 !~NN ~gN',
N
9 Springer-Verlag 1992
Correction: Detection of linkage between quantitative trait loci and restriction fragment length polymorphism using inbred lines S.P. Simpson AFRC Institute of Animal Physiology and Genetics Research, Edinburgh Research Station, Edinburgh EH25 9PS, UK Received April 20, 1992; Accepted May 7, I992 Communicated by L. D. Van Vleck
It has been drawn to my attention that Table 2 and Table 3 in Simpson (1989) are in error. This arose as a consequence of poor convergence during m a x i m u m likelihood estimation for small gene effects, d, and for recombination rates, r, close to 0.5, which led to incorrect interpretation of the results. Table 1 presented here gives the mean m a x i m u m likelihood ratio test statistic for 200 observations based on 100 replicates using the N A G algorithm E04JAF (Numerical Algorithms G r o u p 1990), which converges well, even for small values of d and r close to 0.5, and the expected t-statistic from comparing marker genotype means. Table 2 and Table 3 should replace those published. Table 2 shows that the numbers of inbred lines required to detect linkage are similar for the
t-test and the m a x i m u m likelihood ratio test. For small values of d and r close to 0.5 there appear to be large differences between the numbers of lines needed for the two tests, however the standard error of the estimate of the mean of the likelihood ratio test statistic is large compared with the mean, and the estimates of the n u m bers of lines needed in these cases will have a wide confidence interval. Table 3 shows that the powers of the two tests are similar. To validate these results, data sets were generated using sample sizes given in Table 2 for a range of values of d and r and replicated 100 times, and the resulting powers of 5% significance tests were close the the expected value of 0.5.
Table 1. Mean likelihood ratio test statistics and expected t-statistics Standardised differ- Recombination r a t e - r c ence between means d = (m 1- m2) Selfing 0.05 d = 0.25 )~zG tb d=0.50 Zz t d=0.75 Xz t d=1.00 Z2 t d=2.00 )~2 t d=3.00 xz t
2.58 1.44 8.27 2.86 18.39 4.24 29.57 5.56 83.35 10.03 123.44 13.14
Brother-sister mating
0.10
0.20
0.30
0.05
0.10
0.20
0.30
2.45 1.17 6.57 2.32 10.75 3.41 18.27 4.42 54.07 7.56 79.77 9.43
1.28 0.75 3.22 1.48 5.49 2.15 7.95 2.76 20.71 4.50 31.12 5.40
1.23 0.44 1.56 0.86 2.60 1.25 3.63 1.59 8.83 2.54 11.74 3.01
2.50 1.22 5.92 2.41 12.12 3.54 22.14 4.60 56.41 7.94 86.52 9.97
1.84 0.88 3.65 1.73 7.51 2.52 10.75 3.24 29.28 5.35 42.24 6.47
1.25 0.48 1.69 0.94 2.94 1.36 4.14 1.74 9.85 2.78 12.46 3.30
1.48 0.25 1.31 0.49 1.59 0.71 1.54 0.91 3.14 1.44 4.37 1.69
a Z2 denotes likelihood ratio test statistic b t denotes comparison of marker genotype means ~ Proportion recombinant is R=2r/(l +2r) for selfing and R=4r/(l +6r) for brother-sister mating
111 Table 2, Total number of inbred lines needed to detect linkage at 5% significance level Standardised difference between means d = (m 1 - m2)
d=0.25
~2a tb
d=0.50
~2
d=0.75
t Z2 t
d=l.00
~2
d=2.00 d=3.00
t Z2 t Z2 t
Recombination rate
r~
Selfing
Brother-sister mating
0.05
0.10
0.20
0.30
0.05
0.10
0.20
0.30
485 369 106 94 44 43 27 25 9 8 6 4
531 558 138 143 79 66 44 39 14 13 10 9
2,704 1,355 346 352 171 166 110 101 39 38 25 26
3,278 3,990 1,362 1,041 479 495 292 303 98 119 72 85
512 517 156 132 69 61 36 36 14 12 9 8
915 995 290 257 118 121 79 73 27 27 19 18
3,126 3,352 1,118 874 395 415 244 254 87 99 67 71
1,602 12,227 2,513 3,195 1,304 1,522 1,421 937 359 372 228 268
. Z2 denotes likelihood ratio test statistic b ~ denotes comparison of marker genotype means ~ Proportion recombinant is R = 2r/(1 + 2r) for selfing and R =4r/(1 + 6 r) for brother-sister mating
Table 3. Power of a 5% test using a total of 200 inbred lines Standardised difference between means d =(m I -m2)
d=0.25
•2a tb
d=0.50
Z2
d=0.75
t Z2 t
d = 1.00
Z2
t Z2 t d ~ 3.00 ~(2 z d = 200
Recombination r a t e - r c Brother-sister mating
Selfing 0.05
0.10
0.20
0.30
0.05
0.10
0.20
0.30
0.24 0.30 0.81 0.82 0.98 0.99 0.98 1.00 1.00 1.00 1.00 1.00
0.20 0.21 0.66 0.64 0.92 0.92 0.96 0.99 1.00 1.00 i .00 1.00
0.09 0.11 0.32 0.31 0.53 0.58 0.79 0.79 1.00 0.99 1.00 1.00
0.07 0.06 0.10 0.13 0.25 0.24 0.35 0.36 0.87 0.72 0.87 0.85
0.25 0.23 0.62 0.67 0.97 0.94 1.00 1.00 1.00 1.00 1.00 1.00
0.20 0.14 0.37 0.41 0.72 0.71 0.85 0.90 1.00 1.00 1.00 1.00
0.07 0.07 0.12 0.15 0.29 0.28 0.41 0.41 0.84 0.79 0.98 0.91
0.12 0.04 0.09 0.07 0.13 0.11 0.13 0.15 0.32 0.30 0.45 0.40
" Z2 denotes likelihood ratio test statistic b t denotes comparison of marker genotype means Proportion recombinant is R = 2 r/(1 + 2r) for selling and R = 4 r/(1 + 6 r) for brother-sister mating
I n t h e l i g h t of t h e s e results, t h e c o n c l u s i o n o n t h e r e l a t i v e p o w e r of t h e t w o tests m u s t b e revised, a n d it is n o w c o n c l u d e d t h a t c o m p a r i s o n of m a r k e r g e n o t y p e m e a n s a n d t h e l i k e l i h o o d r a t i o test h a v e s i m i l a r p o w e r . A l t h o u g h t h e test s t a t i s t i c for c o m p a r i s o n of m a r k e r g e n o t y p e m e a n s is easier to c a l c u l a t e t h a n t h e m a x i m u m l i k e l i h o o d r a t i o test statistic, m a x i m u m l i k e l i h o o d estim a t i o n is a useful t o o l for e s t i m a t i o n of t h e r e c o m b i n a t i o n r a t e a n d t h e size of t h e effect of t h e q u a n t i t a t i v e t r a i t locus.
Acknowledgements. I
am indebted to Drs. C. S. Haley and R. Thompson for bringing these problems to my attention.
References Numerical Algorithms G r o u p (1990) The N A G Fortran Library Manual Mark 14. N A G Ltd Oxford Simpson SP (1989) Detection of linkage between quantitative trait loci and restriction fragment length polymorphism using inbred lines, Theor Appl Genet 77:815-819
N
9 Springer-Verlag 1992
Correction: Detection of linkage between quantitative trait loci and restriction fragment length polymorphism using inbred lines S.P. Simpson AFRC Institute of Animal Physiology and Genetics Research, Edinburgh Research Station, Edinburgh EH25 9PS, UK Received April 20, 1992; Accepted May 7, I992 Communicated by L. D. Van Vleck
It has been drawn to my attention that Table 2 and Table 3 in Simpson (1989) are in error. This arose as a consequence of poor convergence during m a x i m u m likelihood estimation for small gene effects, d, and for recombination rates, r, close to 0.5, which led to incorrect interpretation of the results. Table 1 presented here gives the mean m a x i m u m likelihood ratio test statistic for 200 observations based on 100 replicates using the N A G algorithm E04JAF (Numerical Algorithms G r o u p 1990), which converges well, even for small values of d and r close to 0.5, and the expected t-statistic from comparing marker genotype means. Table 2 and Table 3 should replace those published. Table 2 shows that the numbers of inbred lines required to detect linkage are similar for the
t-test and the m a x i m u m likelihood ratio test. For small values of d and r close to 0.5 there appear to be large differences between the numbers of lines needed for the two tests, however the standard error of the estimate of the mean of the likelihood ratio test statistic is large compared with the mean, and the estimates of the n u m bers of lines needed in these cases will have a wide confidence interval. Table 3 shows that the powers of the two tests are similar. To validate these results, data sets were generated using sample sizes given in Table 2 for a range of values of d and r and replicated 100 times, and the resulting powers of 5% significance tests were close the the expected value of 0.5.
Table 1. Mean likelihood ratio test statistics and expected t-statistics Standardised differ- Recombination r a t e - r c ence between means d = (m 1- m2) Selfing 0.05 d = 0.25 )~zG tb d=0.50 Zz t d=0.75 Xz t d=1.00 Z2 t d=2.00 )~2 t d=3.00 xz t
2.58 1.44 8.27 2.86 18.39 4.24 29.57 5.56 83.35 10.03 123.44 13.14
Brother-sister mating
0.10
0.20
0.30
0.05
0.10
0.20
0.30
2.45 1.17 6.57 2.32 10.75 3.41 18.27 4.42 54.07 7.56 79.77 9.43
1.28 0.75 3.22 1.48 5.49 2.15 7.95 2.76 20.71 4.50 31.12 5.40
1.23 0.44 1.56 0.86 2.60 1.25 3.63 1.59 8.83 2.54 11.74 3.01
2.50 1.22 5.92 2.41 12.12 3.54 22.14 4.60 56.41 7.94 86.52 9.97
1.84 0.88 3.65 1.73 7.51 2.52 10.75 3.24 29.28 5.35 42.24 6.47
1.25 0.48 1.69 0.94 2.94 1.36 4.14 1.74 9.85 2.78 12.46 3.30
1.48 0.25 1.31 0.49 1.59 0.71 1.54 0.91 3.14 1.44 4.37 1.69
a Z2 denotes likelihood ratio test statistic b t denotes comparison of marker genotype means ~ Proportion recombinant is R=2r/(l +2r) for selfing and R=4r/(l +6r) for brother-sister mating
111 Table 2, Total number of inbred lines needed to detect linkage at 5% significance level Standardised difference between means d = (m 1 - m2)
d=0.25
~2a tb
d=0.50
~2
d=0.75
t Z2 t
d=l.00
~2
d=2.00 d=3.00
t Z2 t Z2 t
Recombination rate
r~
Selfing
Brother-sister mating
0.05
0.10
0.20
0.30
0.05
0.10
0.20
0.30
485 369 106 94 44 43 27 25 9 8 6 4
531 558 138 143 79 66 44 39 14 13 10 9
2,704 1,355 346 352 171 166 110 101 39 38 25 26
3,278 3,990 1,362 1,041 479 495 292 303 98 119 72 85
512 517 156 132 69 61 36 36 14 12 9 8
915 995 290 257 118 121 79 73 27 27 19 18
3,126 3,352 1,118 874 395 415 244 254 87 99 67 71
1,602 12,227 2,513 3,195 1,304 1,522 1,421 937 359 372 228 268
. Z2 denotes likelihood ratio test statistic b ~ denotes comparison of marker genotype means ~ Proportion recombinant is R = 2r/(1 + 2r) for selfing and R =4r/(1 + 6 r) for brother-sister mating
Table 3. Power of a 5% test using a total of 200 inbred lines Standardised difference between means d =(m I -m2)
d=0.25
•2a tb
d=0.50
Z2
d=0.75
t Z2 t
d = 1.00
Z2
t Z2 t d ~ 3.00 ~(2 z d = 200
Recombination r a t e - r c Brother-sister mating
Selfing 0.05
0.10
0.20
0.30
0.05
0.10
0.20
0.30
0.24 0.30 0.81 0.82 0.98 0.99 0.98 1.00 1.00 1.00 1.00 1.00
0.20 0.21 0.66 0.64 0.92 0.92 0.96 0.99 1.00 1.00 i .00 1.00
0.09 0.11 0.32 0.31 0.53 0.58 0.79 0.79 1.00 0.99 1.00 1.00
0.07 0.06 0.10 0.13 0.25 0.24 0.35 0.36 0.87 0.72 0.87 0.85
0.25 0.23 0.62 0.67 0.97 0.94 1.00 1.00 1.00 1.00 1.00 1.00
0.20 0.14 0.37 0.41 0.72 0.71 0.85 0.90 1.00 1.00 1.00 1.00
0.07 0.07 0.12 0.15 0.29 0.28 0.41 0.41 0.84 0.79 0.98 0.91
0.12 0.04 0.09 0.07 0.13 0.11 0.13 0.15 0.32 0.30 0.45 0.40
" Z2 denotes likelihood ratio test statistic b t denotes comparison of marker genotype means Proportion recombinant is R = 2 r/(1 + 2r) for selling and R = 4 r/(1 + 6 r) for brother-sister mating
I n t h e l i g h t of t h e s e results, t h e c o n c l u s i o n o n t h e r e l a t i v e p o w e r of t h e t w o tests m u s t b e revised, a n d it is n o w c o n c l u d e d t h a t c o m p a r i s o n of m a r k e r g e n o t y p e m e a n s a n d t h e l i k e l i h o o d r a t i o test h a v e s i m i l a r p o w e r . A l t h o u g h t h e test s t a t i s t i c for c o m p a r i s o n of m a r k e r g e n o t y p e m e a n s is easier to c a l c u l a t e t h a n t h e m a x i m u m l i k e l i h o o d r a t i o test statistic, m a x i m u m l i k e l i h o o d estim a t i o n is a useful t o o l for e s t i m a t i o n of t h e r e c o m b i n a t i o n r a t e a n d t h e size of t h e effect of t h e q u a n t i t a t i v e t r a i t locus.
Acknowledgements. I
am indebted to Drs. C. S. Haley and R. Thompson for bringing these problems to my attention.
References Numerical Algorithms G r o u p (1990) The N A G Fortran Library Manual Mark 14. N A G Ltd Oxford Simpson SP (1989) Detection of linkage between quantitative trait loci and restriction fragment length polymorphism using inbred lines, Theor Appl Genet 77:815-819
