Statistical Issues in the Analysis of Quantitative Traits in Combined Crosses

Corresponding author: Fei Zou, Department of Statistics, 1210 W. Dayton St., Madison, WI 53706., feizou{at}stat.wisc.edu (E-mail)

Communicating editor: Z-B. ZENG

	ABSTRACT

TOP ABSTRACT SIMULATION STUDY OF BIAS... THRESHOLD AND POWER CALCULATIONS SIMULATION STUDY OF THRESHOLDS... CONCLUSION APPENDIX LITERATURE CITED

We consider some practical statistical issues in QTL analysis where several crosses originate in multiple inbred parents. Our results show that ignoring background polygenic variation in different crosses may lead to biased interval mapping estimates of QTL effects or loss of efficiency. Threshold and power approximations are derived by extending earlier results based on the Ornstein-Uhlenbeck diffusion process. The results are useful in the design and analysis of genome screen experiments. Several common designs are evaluated in terms of their power to detect QTL.

QUANTITATIVE trait analysis has many applications in plant and animal breeding and in human genetics. Mapping quantitative trait loci (QTL) that influence agriculturally important traits such as grain yield in rice or milk production in cows can help scientists produce specimens with more desirable qualities. Complex human diseases, like breast cancer and diabetes, are known to have genetic etiologies. Animal models may be useful in studying their origins.

Most existing statistical methods have been developed for experimental designs with a single cross from two inbred parents (LANDER and BOTSTEIN 1989 ; HALEY and KNOTT 1992 ; ZENG 1993, ZENG 1994 ; JANSEN and STAM 1994 ). DOERGE et al. 1997 provided a comprehensive review of methodologies for detecting and locating genes affecting quantitative traits in experimental breeding populations. However, quantitative traits are often influenced by several genes with large effects (major QTL) and many genes with relatively small effects (polygenes). In animal science, where outbred parental populations are available, the polygenic effect has been taken into consideration (FERNANDO and GROSSMAN 1989 ). In horticulture, less attention has been paid to genes with small effects, perhaps because researchers are able to rely on simple crosses such as F₂ or backcross (BC).

The effects of polygenes on standard approaches to major QTL mapping are not well understood. With a single cross, the progeny have identical relationships given the QTL genotypes, resulting in a compound symmetry structure (YANDELL 1997 , Ch. 25). Thus, unbiased estimates of QTL effects are still obtained when the polygenic effect is ignored, even though the power to detect the QTL is influenced by the magnitude of the polygenic effect. The situation is more complicated with several crosses, since the correlations may not be the same among all individuals. To avoid this difficulty, researchers may analyze data for each cross separately and then compare and combine the results in some fashion. Hence, some power to detect the QTL may be lost and estimates of QTL effects may be less precise.

Recently, methods were proposed to analyze all crosses simultaneously. BERNARDO 1994 used Wright's relationship matrix A to accommodate differential correlations when analyzing diallel crosses. However, when closely related crosses are from a small number of inbred lines, it is more reasonable to treat the polygenic effect as fixed. REBAI et al. 1994A extended the regression method of HALEY and KNOTT 1992 to several F₂'s from a diallel design of multiple inbred lines with all effects fixed. ELSTON 1990 proposed models for discriminating among modes of inheritance, including one-locus, two-locus, polygenic, and mixed major locus/polygenic inheritance when considering the F₁ and the reciprocal backcrosses derived from two inbred lines. The polygenic effect is treated as fixed and different phenotypic means and variances are used for different crosses. However, flanking marker information is not utilized and an estimate of the QTL position is not provided.

In this article, we consider an arbitrary number of crosses from multiple inbred lines. While we were preparing this manuscript, LIU and ZENG 2000 proposed a fixed-effect model to analyze combined crosses from multiple inbred lines (with or without overlapping inbred lines). Our model includes both QTL and polygenic effects and is a special case of their heteroscedastic model in the sense that the fixed effect and the variance component identify the polygenic effect. For this reason, we refer readers to LIU and ZENG 2000 for the analysis of combined lines. Our focus is the practical implications of the polygenic effects for QTL mapping, specifically bias and efficiency. Furthermore, we calculate threshold values for controlling the genome-wise type I error rate. Theoretical approximations were developed to address threshold and power (LANDER and BOTSTEIN 1989 ; DUPUIS and SIEGMUND 1999 ; REBAI et al. 1994B , REBAI et al. 1995 ) in some standard designs. However, these methods are either impractical or inappropriate with combined crosses. Our general formulas are widely applicable and easy to implement.

	SIMULATION STUDY OF BIAS AND EFFICIENCY

TOP ABSTRACT SIMULATION STUDY OF BIAS... THRESHOLD AND POWER CALCULATIONS SIMULATION STUDY OF THRESHOLDS... CONCLUSION APPENDIX LITERATURE CITED

If one combines different crosses simultaneously but ignores the different relationships among individuals, substantial bias may result. In this section, we show the effect of polygenes on the QTL estimates. We examine two crosses, BC1 and F₂, from common inbred parents P1 and P2. Although the design is simple, it illustrates the key issues. The additive effect of a single major QTL is set to 0 (i.e., no QTL) and 5, respectively, with no dominance effect. Five markers are located at 0, 20, 40, 60, and 80 cM. The major QTL is located at 30 cM. The environmental errors are identically distributed for BC1 and F₂ and are sampled from N(0, 25). One hundred individuals from BC1 and F₂ are simulated without background polygenes or with 10 background polygenes. The 10 background polygenes are in coupling phase and have common additive effects (i.e., allele substitution effect _k, k = 1, 2, ... , 10) 1 or 2 (see FERNANDO et al. 1994 ). This leads to expected polygenic differences between F₂ and BC1 of 5 and 10, respectively. We fit the model using LIU and ZENG 2000 , hereafter called "model P." In addition, we employed traditional interval mapping by ignoring the polygenic effects, hereafter called "model N." For each parameter combination, 100 simulated datasets were analyzed. The results are presented in Table 1.

We observe that when there are no polygenes both models consistently estimate the QTL effects. However, model P gives more accurate estimates than model N when there are polygenic effects. The bias of model N increases as the expected polygenic differences between BC1 and F₂ increase. In summary, our simulations indicate that when analyzing combined crosses, the polygenic model produces more precise and less biased estimates than the traditional interval mapping method.

	THRESHOLD AND POWER CALCULATIONS

TOP ABSTRACT SIMULATION STUDY OF BIAS... THRESHOLD AND POWER CALCULATIONS SIMULATION STUDY OF THRESHOLDS... CONCLUSION APPENDIX LITERATURE CITED

On the basis of the simulations in the above section, fitting combined crosses (LIU and ZENG 2000 ) has many advantages. Calculating thresholds and power is an important practical issue in the design and analysis of such studies. The usual pointwise significance level based on the chi-square approximation is inadequate because the entire genome is tested for the presence of a QTL. LANDER and BOTSTEIN 1989 showed that with an infinitely dense map, the LOD score may be approximated in large samples by an Ornstein-Uhlenbeck diffusion process for BC. DUPUIS and SIEGMUND 1999 derived a similar result for F₂. These approximations provide formulas for the threshold and power.

For more general models (LIU and ZENG 2000 ), no such approximation is available. CHURCHILL and DOERGE 1994 used a randomization idea to calculate the threshold. The approach is applicable for all designs, with a dense or sparse map. However, the method is computationally intensive. In addition, since the thresholds depend on the observed data, it is unclear how to compare various designs. REBAI et al. 1994B , REBAI et al. 1995 gave an upper bound for the threshold for BC and F₂ based on DAVIES 1977 , DAVIES 1987 . The calculation is formidable, even for an F₂ population, and is not exact. PIEPHO 2001 proposed an efficient numerical method to compute the thresholds in REBAI et al. 1994B , REBAI et al. 1995 for general designs.

Our approach extends the Ornstein-Uhlenbeck large sample approximations. It is quite simple and practically useful. Calculating the threshold and power under different map distances can be accomplished with closed-form expressions arising from the Ornstein-Uhlenbeck setup. Simulations shown below indicate this works well with realistic sample sizes.

Two inbred strains:
In this section, we consider combined crosses from two inbred parents (P1 and P2), BC1, F₂, and BC2. Our goal is to extend DUPUIS and SIEGMUND 1999 . Generalizing the results to other designs (LIU and ZENG 2000 ) is straightforward. In the sequel, we assume an equispaced marker map. Suppose

where X is the design matrix and e is the random error. n₁, n₂, n₃ are the number of observations for BC1, F₂, and BC2, respectively, with n observations in total. The submatrix X₁ corresponds to the covariates identifying crosses or other measurements that do not involve the QTL effects and X₂ corresponds to the QTL effects. Suppose the allele from parent P1 is q and from P2 is Q. The possible QTL genotypes are qq, Qq, and QQ. Ignoring other covariate effects, we let

where x_1ki = 1 (or 0) if individual i in cross k has genotype Qq (or else), and x_2ki = 1 (or 0) if individual i in cross k has genotype QQ (or else). The random error e is normally distributed with mean 0 and Var(e_ki) = ²_k, i = 1, 2, ... , n_k, k = 1, 2, 3. In general Var(e) = ²₃G with G^-1 = diag(₁, ... , ₁; ₂; ... , ₂; 1, ... , 1), where _k = for k = 1, 2. In the following, we assume that _k is known. If _k is unknown, then consistent maximum-likelihood (ML) estimates may be substituted and the result still holds. Without loss of generality, assume that ²₃ = 1. At locus d, the hypothesis of no QTL effect is H₀: b₂ = 0 vs. H₁: b₂ 0, or equivalently, H₀: Hb = 0 vs. H₁: Hb 0, where

From the normal regression theory, the likelihood ratio statistic is

where = G^-1/2y and (d) = G^-1/2X(d). Under H₀,

where (_{1 2})' is the maximum-likelihood estimate of b₂ = (_{1 2})' and A₂₂ is given in (A2) in the Appendix

The distribution of 2 LR(d) depends on ₁ and ₂, which are correlated. Thus, we cannot directly apply DUPUIS and SIEGMUND 1999 to derive the threshold and power. In the Appendix, we propose an orthogonal transformation such that 2 LR(d) is partitioned into the sum of squares of two uncorrelated random variables Z₁ and Z₂. This makes the asymptotic distribution transparent. Letting ₁ = ₁₁, ₂ = ₁₁ + ₂₂ (see Appendix for ₁ and _i), Z_i = _i, i = 1, 2. It is shown in the Appendix that Z₁ and Z₂ are asymptotically independent and distributed N (0, 1). Thus,

which depends on two uncorrelated normal variates and is asymptotically ²₂. Note that _i, _i, and Z_i, i = 1, 2 all depend on the locus d. In the sequel, when necessary, we use ₁(d), ₂(d), _i(d), and Z_i(d), i = 1, 2 to emphasize their dependence on d.

To demonstrate the Ornstein-Uhlenbeck equivalence, the covariances at different loci d₁ and d₂ are proved to be

and

This means that for large n, Z₁(d) and Z₂(d) are approximately independent Ornstein-Uhlenbeck processes with mean zero and covariance 1 - ß₁r + O(r²) and 1 - ß₂r + O(r²), respectively. Adapting the argument in DUPUIS and SIEGMUND 1999 , the tail distribution of 2 LR under the null hypothesis satisfies

(1)

where = the distance between markers (in Morgans), C = the number of chromosomes, and L = total length of the genome (in Morgans). The definition of v(x) can be found in SIEGMUND 1985 . For dense maps, v = 1. Similarly, the power is given by

when a QTL is located at a marker locus. Here

For a QTL between markers, the noncentrality parameters ^*₁ and ^*₂ are ^*₁ exp(-ß₁₁) and ^*₂ exp(-ß₂₁), respectively, where the distance between the QTL and the marker is ₁. In the case of an F₂ population, the formulas above reduce to those in DUPUIS and SIEGMUND 1999 .

General Results:
The derivations above can be generalized to more complicated models, including those in LIU and ZENG 2000 . Advanced crosses, such as F_x, x 2, and models with covariates are also possible. Our framework can be modified for a wide variety of designs.

As before, let the model be

where X₁ is an n x p submatrix not involving the QTL effects, X₂ is an n x m matrix corresponding to the m QTL effects, and e is the random error. Following the procedure in the Appendix, we first compute A₂₂ using (A2) and then derive the orthogonal transformation matrix P from (A3). Note that both A₂₂ and P involve only the design matrix X and not ß or the correlation parameters. Next, 2 LR(d) can be partitioned into the sum of squares of m asymptotically independent N(0, 1) random variables Z₁, ... , Z_m, where Z = (Z₁, ... , Z_m)' = P₂. To calculate Cov(Z_j(d₁), Z_j(d₂)), j = 1, 2, ... , m, we find D in (A4) on the basis of the specific designs. It is straightforward to establish

where ß_j is the jth diagonal element of -PA₂₂DA'₂₂P'. Now, the tail distribution of 2 LR under the null hypothesis is approximately

where ²_m is a ² random variable with m degrees of freedom and ß = (ß₁ + ß₂ + ... + ß_m)/m.

The formula for power may also be obtained. However, it is quite complicated and is omitted here.

	SIMULATION STUDY OF THRESHOLDS AND POWER

TOP ABSTRACT SIMULATION STUDY OF BIAS... THRESHOLD AND POWER CALCULATIONS SIMULATION STUDY OF THRESHOLDS... CONCLUSION APPENDIX LITERATURE CITED

We investigated the performance of (1) with different marker distances and different polygenic backgrounds. Thresholds for the log-likelihood were based on interval mapping with combined BC1, F₂, and BC2 crosses. n₁ = n₂ = n₃ = 100, giving 300 observations in total and chromosome length = 100 cM. The marker interval lengths are set at 10, 5, and 2 cM, respectively. Different polygenic effects are sampled, as reflected by models a–d (see legend of Table 2 for details; Table 3). The approximations from (1) with v(a{2ß}^1/2) are always smaller than the empirical thresholds derived in the simulations. However, as the interval length decreases, our approximations are more similar to the empirical thresholds. In general, the dense map assumption (v = 1) produces conservative thresholds. Since more markers are likely to be typed around promising loci (LANDER and KRUGLYAK 1995 ), the stringent thresholds based on a dense map should be used even with a sparse map. Also, the approximations provide conservative control of the genome-wise type I error rate. Note that (1) gives upper and lower bounds for the threshold with v = 1 (assuming a dense map) and with v(a{2ß}^1/2) (using the true map distances), respectively.

Next, we evaluate the power with different proportions of BC1, F₂, and BC2. The power is calculated for dominant (₁ = ₂) and additive (₂ = 2₁) models. We compare our results with those of DUPUIS and SIEGMUND 1999 for the dominant model. We use the same values of the noncentrality parameter. In theory, as the proportion of F₂ approaches 1, our power approximation should agree with DUPUIS and SIEGMUND 1999 . Other noncentrality parameters in DUPUIS and SIEGMUND 1999 show the same pattern and are omitted. For the additive model, the comparisons are qualitatively similar.

Fig 1 and Fig 2 exhibit the power curves. When the polygenes are in linkage equilibrium and have only additive effects, the phenotypic variation due to polygenes and environment satisfies ²_BC1 = ²_BC2 = ²_P + ²_e and _F2 = 2²_P + ²_e, respectively, where ²_P is the total polygenic variation in the BC population and ²_P is the environmental variation. For this reason, we take ₁ 1 and choose ^-1₂ = 1, 0.75, 0.67, 0.57, 0.5, which correspond to ²_P = 0, ²_e/2, ²_e, 3²_e, or ²_P >> ²_e, respectively. We also evaluate the power by using F₂'s only, which quantifies the loss in power when discarding data from the BCs (see Fig 1A).

View larger version (27K): In this window In a new window Download PPT slide   Figure 1. (a) Additive QTL model. (b) Dominant QTL model. Power curves for equal BC1 and BC2 ratios under dense maps. The horizontal line is the power calculated from DUPUIS and SIEGMUND's (1999) formula (i.e., the proportion of F2 is 1). The lowest curve in a is the power when only available F2's are used by discarding BCs. Thus power is reduced because there are fewer individuals.

View larger version (32K): In this window In a new window Download PPT slide   Figure 2. Power curves for different ratios of BC1, BC2, and F2 with dominant QTL under dense maps. Dominant model with (a) 2 = 1, (b) 2 = 0.75, (c) 2 = 0.67, (d) 2 = 0.5. a–d use the same graphical symbols. k1 is the sample proportion of BC1 and 2 = 2BC2/2F2.

In Fig 1, the proportions of BC1 and BC2 are assumed equal. When the QTL is dominant, power is gained by using BC populations unless there is no polygenic effect (i.e., ₂ is close to 1). The larger the polygenic effects, the greater is the gain with BCs. However, when the QTL is additive, F₂'s tend to have more information for detecting a QTL than do BCs, unless _P >> _e (i.e., the polygenic effects are very large). Note that when the proportion of F₂ approaches 1, our results again match those of Dupuis and Siegmund.

In Fig 2, we allow the proportions of BC1 and BC2 to be unequal with a dominant QTL. In this case, BC1 is more powerful than F₂, which is expected. When the QTL is additive, both BC1 and BC2 individuals have identical contributions in detecting the QTL, so only the total proportion of BC1 and BC2 influences the power, as shown in Fig 1A.

	CONCLUSION

TOP ABSTRACT SIMULATION STUDY OF BIAS... THRESHOLD AND POWER CALCULATIONS SIMULATION STUDY OF THRESHOLDS... CONCLUSION APPENDIX LITERATURE CITED

In this article, we addressed some important practical issues in the analysis of closely related crosses derived from multiple inbred lines when both QTL and polygenes influence a trait. We showed that biased and inefficient estimates of the QTL effects may occur if the polygenic effect is ignored. We derived simple and general approximations for the threshold and power to detect a QTL, allowing different designs to be compared.

Based on our power calculations, we find that the F₂ population is more robust in detecting QTL than the two backcross populations. This confirms LIU and ZENG 2000 . Thus if the goal is to detect the QTL, then using a large F₂ population is highly recommended. However, scientists may not be able to produce enough F₂ individuals or may for other reasons use different crosses. In this situation, analyzing all the data simultaneously is preferred. This strategy improves the power to detect major QTL. In addition, this is an opportunity to detect potential polygenic effects. The derivation of the threshold approximation is easily extended to other designs beyond the combination of BC1, F₂, and BC2. However, to our knowledge, the theoretical computation of thresholds involving multiple QTL is an open problem.

	ACKNOWLEDGMENTS

We thank anonymous reviewers for their critical reading of this manuscript. This research was supported in part by the U.S. Department of Agriculture Hatch project through the University of Wisconsin, College of Agricultural and Life Sciences.

Manuscript received August 14, 2000; Accepted for publication April 18, 2001.

	APPENDIX

TOP ABSTRACT SIMULATION STUDY OF BIAS... THRESHOLD AND POWER CALCULATIONS SIMULATION STUDY OF THRESHOLDS... CONCLUSION APPENDIX LITERATURE CITED

In this section, for combined crosses from two inbred parents, we prove that the likelihood ratio 2 LR(d) can be partitioned into the sum of the squares of two asymptotically independent Ornstein-Uhlenbeck processes through an orthogonal transformation. Define

(A1)

Note that B does not depend on locus d. Let

(A2)

Define

(A3)

where ₁ = , and ₁ = c₃/, ₂ = .

Making the orthogonal transformation

gives

For the same reason,

This indicates that Z₁ and Z₂ are approximately independent N(0, 1). Furthermore,

Now, let (d₁) and (d₂) be the corresponding transformed incidence matrices at loci d₁ and d₂. Note that the first and second columns of depend only on the proportions of BC1, F₂, and BC2 and not on d₁ and d₂. The third and fourth columns depend on d₁ and d₂. Also x_1ki(d) = 0, 1, or 0 and x_2ki(d) = 0, 0, or 1 if individual i in cross K's genotype at locus d is qq, Qq, or QQ, respectively.

For BC1,

where r is the recombination fraction between loci d₁ and d₂. Enumerating the probabilities of (x_1ki(d₁), x_1ki(d₂)), (x_1ki(d₁), x_2ki(d₂)), and (x_2ki(d₁), x_2ki(d₂)) for BC1, F₂, and BC2, and using the fact that (d) = G^-1/2X(d), we obtain

(A4)

with

Thus,

Therefore,

and

	LITERATURE CITED

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