Statistical Science

Statistical issues in the search for genes affecting quantitative traits in experimental populations

R. W. Doerge,B. S. Weir, and Z-B. Zeng

Full-text: Open access

Abstract

This article reviews key contributions in the area of statistics as applied to the use of molecular marker technology and quantitative genetics in the search for genes affecting quantitative traits responsible for specific diseases and economically important agronomic traits. Since an exhaustive literature review is not possible, the limited scope of this work is to encourage further statistical work in this vast field by first reviewing human and domestic species literature, and then concentrating on the statistical developments for experimental breeding populations. Substantial gains have been made over the years by both plant and animal breeders toward a long-term goal of locating genes affecting quantitative traits (quantitative trait loci, QTLs) for the eventual characterization and manipulation of these genes in order to develop improved agronomically important traits. Our main concern is that the care and expense that are required in generating both genetic marker data and quantitative trait data should be accompanied by equal care in the statistical analysis of the data. Through an example using an $F_2$ male genetic map of mouse chromosome 10, and quantitative trait values measured on weight gain, we implement much of the reviewed methodology for the purpose of detecting or locating a QTL having an effect on weight gain.

Article information

Source
Statist. Sci. Volume 12, Number 3 (1997), 195-219.

Dates
First available: 22 August 2002

Permanent link to this document
http://projecteuclid.org/euclid.ss/1030037909

Digital Object Identifier
doi:10.1214/ss/1030037909

Citation

Doerge, R. W.; Zeng, Z-B.; Weir, B. S. Statistical issues in the search for genes affecting quantitative traits in experimental populations. Statistical Science 12 (1997), no. 3, 195--219. doi:10.1214/ss/1030037909. http://projecteuclid.org/euclid.ss/1030037909.


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References

  • Abler, B. S. B., Edwards, M. D. and Stuber, C. W. (1991). Isoenzy matic identification of quantitative trait loci in crosses of elite maize inbreds. Crop Science 31 267-274. Andersson, L., Haley, C. S., Ellegen, H., Knott, S. A., Johansson, M., Andersson, K., Andersson-Ekliund, L., EdforsLilja, I., Fredholm, M., Hansson, I., Hakansson, J. and
  • Lundstrom, K. (1994). Genetic mapping of quantitative trait loci for growth and fatness in pigs. Science 263 1771- 1774.
  • Bailar, J. C. (1995). A larger perspective. Amer. Statist. 49 10-11. Barendse, W., Armitage, S. M., Kossarek, L. M., Shalom, A., Kirkpatrick, B. W., Ry an, A. M., Clay ton, D., Li, L., Neibergs, H. L., Zhang, N., Grosse, W. M., Weiss, J., Creighton, P., McCarthy, F., Ron, M., Teale, A. J., Fries, R., McGraw, R. A., Moore, S. S., Georges, M., Soller, M.,
  • Womack, J. E. and Hetzel, D. J. S. (1994). A genetic linkage map of the bovine genome. Nature Genetics 6 227-235.
  • Basten, C. J., Weir, B. S. and Zeng, Z.-B. (1995-1996). QTL CARTOGRAPHER: A Reference Manual and Tutorial for QTL Mapping. Dept. Statistics, North Carolina State University, Raleigh.
  • Beckmann, J. S. and Soller, M. (1988). Detection of linkage between marker loci and loci affecting quantitative traits in crosses between segregating populations. Theoretical and Applied Genetics 76 228-236.
  • (1993). Localization of the gene for familial dy sautomia on chromosome 9 and definition of DNA markers for genetic diagnosis. Nature Genetics 4 160-163.
  • Bovenhuis, H. and Weller, J. I. (1994). Mapping and analysis of dairy cattle quantitative trait loci by maximum likelihood methodology using milk protein genes as genetic markers. Genetics 137 267-280. Bubeck, D. M., Goodman, M. M., Beavis, W. D. and Grant, D.
  • (1993). Quantitative trait loci controlling resistance to grapy leaf spot in maize. Crop Science 33 838-847. Buetow, K. H. and Chakravarti, A. (1987a). Multipoint gene mapping using seriation. I. General methods. American Journal of Human Genetics 41 180-188. Buetow, K. H. and Chakravarti, A. (1987b). Multipoint gene mapping using seriation. II. Analy sis of simulated and empirical data. American Journal of Human Genetics 41 189-201. Bull, J. K., Cooper, M., DeLacy, I. H., Basford, K. E. and
  • Woodruff, D. R. (1992). Utility of repeated checks for hierarchical classification of data from plant breeding trials. Field Crops Research 30 70-95. Carbonell, E. A., Gerig, T. M., Balansard, E. and Asins,
  • M. J. (1992). Interval mapping in the analysis of nonadditive quantitative trait loci. Biometrics 48 305-315.
  • Castle, W. E. (1921). An improved method of estimating the number of genetic factors concerned in cases of blending inheritance. Science 54 541-553.
  • Churchill, G. A. and Doerge, R. W. (1994). Empirical threshold values for quantitative trait mapping. Genetics 138 963-971.
  • Cockerham, C. C. (1986). Modifications in estimating the number of genes for a quantitative character. Genetics 114 659-664.
  • Coe, E. H., Hoisington, D. A. and Nuffer, M. G. (1993). Linkage map of corn (maize) (Zea may s L.) 2n = 20. In Genetic Maps, 6th ed. (S. J. O'Brien, ed.). Cold Spring Harbor Laboratory Press, Cold Spring Harbor, NY.
  • Comstock, R. E. and Enfield, F. D. (1981). Gene number estimation when multiplicative genetic effects are assumedgrowth in flour beetles and mice. Theoretical and Applied Genetics 59 373-379.
  • Cooper, M. and DeLacy, I. H. (1994). Relationships among analytical methods used to study genoty pic variation and genoty pe-by-environmental interaction in plant breeding multi-environment experiments. Theoretical and Applied Genetics 88 561-572. Copeland, N. G., Jenkins, N. A., Gilbert, D. J., Eppig, J. T., Maltais, L. J., Miller, J. C., Dietrich, W. F., Lincoln, S. E., Steen, R. G., Stein, L. D., Nadeau, J. H. and Lander,
  • E. S. (1993). A genetic linkage map of the mouse: current applications and future prospects. Science 262 57-66.
  • Corana, A., Marchesi, M., Martini, C. and Ridella, S. (1987). Minimizing multimodal functions of continuous variables with the "simulated annealing" algorithm. ACM Trans. Math. Software 13 262-280. Darvasi, A., Weinreb, A., Minke, V., Weller, J. I. and Soller,
  • M. (1993). Detecting marker-qtl linkage and estimating QTL gene effect and map location using a saturated map. Genetics 134 943-951.
  • Darvasi, A. and Weller, J. I. (1992). On the use of the moments method of estimation to obtain approximate maximum likelihood estimates of linkage between a genetic marker and a quantitative locus. Heredity 68 43-46.
  • Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39 1-38.
  • Doerge, R. W. (1993). Statistical methods for locating quantitative trait loci with molecular markers. Ph.D. dissertation, Dept. Statistics, North Carolina State Univ., Raleigh.
  • Doerge, R. W. and Churchill, G. A. (1996). Permutation tests for multiple loci affecting a quantitative character. Genetics 142 285-294.
  • Doerge, R. W. and Reba¨i, A. (1996). Significance thresholds for QTL interval mapping tests. Heredity 76 459-464.
  • Dupuis, J. (1994). Statistical problems associated with mapping complex and quantitative traits from genomic mismatch scanning data. Ph.D. dissertation, Dept. Statistics, Stanford Univ.
  • Edwards, M. D., Stuber, C. W. and Wendel, J. F. (1987). Molecular-marker-facilitated investigations of quantitativetrait loci in maize. I. Numbers, genomic distribution and ty pes of gene action. Genetics 116 113-125.
  • Ehm, M. G., Kimmel, M. and Cottingham, R. W. (1996). Error detection for genetic data, using likelihood methods. American Journal of Human Genetics 58 225-234.
  • Eisenberger, I. (1964). Genesis of bimodal distributions. Technometrics 6 357-363.
  • Elston, R. C. and Stewart, J. (1973). The analysis of quantitative traits for simple genetic models from parental, F1 and backcross data. Genetics 73 695-711.
  • Falk, C. T. (1992). Preliminary ordering of multiple linked loci using pairwise linkage data. Genetic Epidemiology 9 367- 375.
  • Feingold, E., Brown, P. O. and Siegmund, D. (1993). Gaussian models for genetic linkage analysis using complete highresolution maps of identity by descent. American Journal of Human Genetics 53 234-251.
  • Feng, Z. (1990). Statistical inference using maximum likelihood estimation and the generalized likelihood ratio under nonstandard conditions. Ph.D. dissertation, Dept. Plant Breeding and Biometry, Cornell Univ.
  • Fisher, R. A. (1935). The Design of Experiments, 3rd ed. Oliver and Boy d, London.
  • Fox, P. N. and Rosielle, A. A. (1982). Reducing the influence of environmental main-effects on pattern analysis of plant breeding environments. Euphy tica 31 645-656.
  • Freeman, G. H. (1973). Statistical methods for the analysis of genoty pe-environment interactions. Heredity 31 339-354.
  • Freeman, G. H. (1990). Modern statistical methods for analyzing genoty pe-by-environment interactions. In Genoty pe-by Environment Interaction and Plant Breeding (M. S. Kang, ed.) 118-125. Louisiana State University, Baton Rouge.
  • Georges, M., Drinkwater, R. and King, T. et al. (1993). Microsatellite mapping of a gene affecting horn development in Bos taurus. Nature Genetics 4 206-210.
  • Ghosh, J. K. and Sen, P. K. (1985). On the asy mptotic performance of the log likelihood ratio statistic for the mixture model and related results. In Proceedings of the Berkeley Conference in Honor of Jerzy Ney man and Jack Kiefer (L. M. LeCam and R. A. Olshen, eds.) 2 789-807. Wadsworth, Belmont, CA.
  • Good, P. (1994). Permutation Tests: A Practical Guide to Resampling for Testing Hy potheses. Springer, New York. Graner, A., Jahoor, A., Schondelmaier, J., Seidler, H., Pillen, K., Fishbeck, G., Wenzel, G. and Herrmann, R. G.
  • (1991). Construction of an RFLP map of barley. Theoretical and Applied Genetics 83 250-256.
  • Guffy, R. D., Stuber, C. W. and Edwards, M. D. (1989). Dissecting and enhancing heterosis in corn using molecular markers. In Proceedings of the 25th Illinois Corn Breeders School 99-120. Champaign, IL.
  • Haldane, J. B. S. (1919). The combination of linkage values and the calculation of distance between the loci of linked factors. Journal of Genetics 8 299-309.
  • Haley, C. S. and Knott, S. (1992). A simple method for mapping quantitative trait loci in line crosses using flanking markers. Heredity 69 315-324.
  • Haley, C. S., Knott, S. A. and Elsen, J-M. (1994). Mapping quantitative trait loci in crosses between outbred lines using least squares. Genetics 136 1195-1207.
  • Hartigan, J. A. (1985). A failure of likelihood asy mptotics for normal distributions. In Proceedings of the Berkeley Conference in Honor of Jerzy Ney man and Jack Kiefer (L. M. LeCam and R. A. Olshen, eds.) 2 807-810. Wadsworth, Belmont, CA.
  • Hilbert, P., Lindpaintner, K., Beckmann, J. S. et al. (1991). Chromosomal mapping of two genetic loci associated with blood-pressure regulation in hereditary hy pertensive rats. Nature 353 521-529.
  • Horvat, S. and Medrano, J. F. (1995). Interval mapping of high growth (hg), a major locus that increases weight gain in mice. Genetics 139 1737-1748. The Huntington's Disease Collaborative Research Group
  • (1993). A novel gene containing a trinucleotide repeat that is expanded and unstable on Huntington's disease chromosomes. Cell 72 971-983.
  • Jacob, H. J., Lindpaintner, K., Lincoln, S. E. et al. (1991). Genetic mapping of a gene causing hy pertension in the stroke-prone spontaneously hy pertensive rat. Cell 67 213- 224.
  • Jansen, R. C. (1992). A general mixture model for mapping quantitative trait loci by using molecular markers. Theoretical and Applied Genetics 85 252-260.
  • Jansen, R. C. (1993). Interval mapping of multiple quantitative trait loci. Genetics 135 205-211.
  • Jansen, R. C. (1994). Controlling the ty pe I and ty pe II errors in mapping quantitative trait loci. Genetics 138 871-881.
  • Jansen, R. C. and Stam, P. (1994). High resolution of quantitative traits into multiple loci via interval mapping. Genetics 136 1447-1455. Jeunemaitre, X., Lifton, R. P., Hunt, S. C., Williams, R. R.
  • and Lalouel, J-M. (1992). Absence of linkage between the angiotensin converting enzy me and human essential hy pertension. Nature Genetics 1 72-75.
  • Jiang, C. and Zeng, Z-B. (1995). Multiple trait analysis of genetic mapping for quantitative trait loci. Genetics 140 1111- 1127.
  • Kang, M. S. and Gauch, H. G., Jr., eds. (1996). Genoty pe-by Environment Interaction. CRC Press, Boca Raton, FL.
  • Kao, C-H. and Zeng, Z-B. (1997). General formulae for obtaining the MLEs and the asy mptotic variance-covariance matrix in mapping quantitative trait loci when using the EM algorithm. Biometrics. To appear.
  • Kearsey, M. J. and Hy ne, V. (1994). QTL analysis: a simple `marker-regression' approach. Theoretical and Applied Genetics 89 698-702. Keim, P., Diers, B. W., Olson, T. C. and Shoemaker, R. C.
  • (1990). RFLP mapping in soy bean: association between marker loci and variation in quantitative traits. Genetics 126 735-742. Kerem, B-S., Rommens, J. M., Buchanan, J. A., Markiewicz, D., Cox, T. K., Chakravarti, A., Buchwald, M. and Tsui,
  • L-C. (1989). Identification of the cy stic fibrosis gene: genetic analysis. Science 245 1073-1080.
  • Knapp, S. J., Bridges, W. C. and Birkes, D. (1990). Mapping quantitative trait loci using molecular marker linkage maps. Theoretical and Applied Genetics 79 583-592.
  • Knott, S. A. and Haley, C. S. (1992). Aspects of maximum likelihood methods for the mapping of quantitative trait loci in line crosses. Genetical Research 60 139-151.
  • Kosambi, D. D. (1944). The estimation of map values from recombination values. Annals of Eugenics 12 172-175.
  • Krugly ak, L. and Lander, E. S. (1995). A nonparametric approach for mapping quantitative trait loci. Genetics 139 1421-1428.
  • Lande, R. (1981). The minimum number of genes contributing to quantitative variation between and within populations. Genetics 99 541-553.
  • Lander, E. S. and Botstein, D. (1989). Mapping Mendelian factors underlying quantitative traits using RFLP linkage maps. Genetics 121 185-199.
  • Lander, E. S. and Botstein, D. (1994). Corrigendum. Genetics 36 705.
  • Lander, E. S. and Green, P. (1987). Construction of multilocus genetic linkage maps in humans. Proc. Nat. Acad. Sci. U.S.A. 84 2363-2367.
  • Lathrop, G., Lalouel, J., Julier, C. and Ott, J. (1985). Multilocus linkage analysis in humans: detection of linkage and estimation of recombination. American Journal of Human Genetics 37 482-498. Lincoln, S., Daly, M. and Lander, E. (1992a). Constructing genetic maps with MAPMAKER/EXP 3.0, 3rd ed. Technical report, Whitehead Institute, Boston, MA. Lincoln, S., Daly, M. and Lander, E. (1992b). Mapping genes controlling quantitative traits with MAPMAKER/QTL 1.1, 2nd ed. Technical report, Whitehead Institute, Boston, MA.
  • Luo, Z. W. and Kearsey, M. J. (1989). Maximum likelihood estimation of linkage between a marker gene and a quantitative trait locus. Heredity 63 401-408.
  • Luo, Z. W. and Kearsey, M. J. (1991). Maximum likelihood estimation of linkage between a marker gene and a quantitative trait locus. II. Application to backcross and doubled haploid populations. Heredity 66 117-124.
  • Luo, Z. W. and Kearsey, M. J. (1992). Interval mapping of quantitative trait loci in an F2 population. Heredity 69 236-242.
  • Luo, Z. W. and Woolliams, J. A. (1993). Estimation of genetic parameters using linkage between a marker gene and a locus underlying a quantitative character in F2 populations. Heredity 70 245-253.
  • Martinez, O. and Curnow, R. N. (1992). Estimating the locations and the size of the effects of quantitative trait loci using flanking markers. Theoretical and Applied Genetics 85 480-488.
  • McMillan, I. and Robertson, A. (1974). The power of methods for the detection of major genes affecting quantitative characters. Heredity 32 349-356.
  • McPeek, M. S. and Speed, T. P. (1995). Modeling interference in genetic recombination. Genetics 139 1031-1044. Medrano, J. F., Pomp, D., Tay lor, B. A. and Bradford, G.
  • E. (1992). The high growth gene (hg) in mice is located on chromosome 10 linked to Igf1. In Advances in Gene Technology: Feeding the World in the 21st Century. The 1992 Miami Bio/Technology Winter Sy mposium (W. J. Whelan, ed.) 1 12. Moreno-Gonzalez, J. (1992a). Estimates of marker-associated QTL effects in Monte Carlo backcross generations using multiple regression. Theoretical and Applied Genetics 85 423- 434. Moreno-Gonzalez, J. (1992b). Genetic models to estimate additive and non-additive effects of marker-associated QTL using multiple regression techniques. Theoretical and Applied Genetics 85 435-444.
  • Morton, N. E. (1995). LODs past and present. Genetics 140 7-12.
  • B. E. and Shapiro, J. A. (1996). QTL analysis of the production of acy lsugars responsible for pest resistance in the wild tomato Ly copersicon pennellii. Theoretical and Applied Genetics 92 709-719. Nienhuis, J., Helentjaris, T., Slocum, M., Ruggero, B. and
  • Schaefer, A. (1987). Restriction fragment length poly morphism analysis of loci associated with insect resistance in tomato. Crop Science 27 797-803.
  • NIH/CEPH Collaborative Mapping Group. (1992). A comprehensive genetic linkage map of the human genome. Science 258 148-162.
  • O'Brien, S. J., ed. (1993). Genetic Maps, 6th ed. Cold Spring Harbor Laboratory Press, Cold Spring Harbor, NY.
  • Ott, J. (1991). Analy sis of Human Genetic Linkage. Johns Hopkins Univ. Press. Paterson, A. H., Damon, S., Hewitt, J. D., Zamir, D., Rabinowitch, H. D., Lincoln, S. E., Lander, E. S. and Tanksley, S.
  • D. (1991). Mendelian factors underlying quantitative traits in tomato: comparison across species, generations, and environments. Genetics 127 181-197. Paterson, A. H., Lander, E. S., Hewitt, J. D., Peterson, S., Lin
  • coln, S. E. and Tanksley, S. D. (1988). Resolution of quantitative traits into Mendelian factors by using a complete map of restriction fragment length poly morphisms. Nature 335 721-726.
  • Plomin, R., McClearn, G. E. and Gora-Maslak, G. (1991). Quantitative trait loci and psy chopharmacology. Journal of Psy chopharmacology 5 1-9.
  • Rafalski, J. A. and Tingey, S. V. (1993). Genetic diagnostics in plant breeding: RAPDs, microsatellites and machines. Trends in Genetics 9 275-280.
  • Reba¨i, A., Goffinet, B. and Mangin, B. (1994). Approximate thresholds of interval mapping tests for QTL detection. Genetics 138 235-240.
  • Reba¨i, A., Goffinet, B. and Mangin, B. (1995). Comparing power of different methods of QTL detection. Biometrics 51 87-99. Reiter, R. S., Cors, J. G., Sussman, M. R. and Gabelman, W.
  • H. (1991). Genetic analysis of tolerance to low-phosphorus stress in maize using restriction fragment length poly morphisms. Theoretical and Applied Genetics 82 561-568.
  • Sax, K. (1923). The association of size differences with seed-coat pattern and pigmentation in Phaseolus vulgaris. Genetics 8 552-560.
  • Schuler, G. D., Boguski, M. S., Stewart, E. A. et al. (1996). A gene map of the human genome. Science 274 540-546.
  • Self, S. G. and Liang, K-Y. (1987). Asy mptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. J. Amer. Statist. Assoc. 82 605-610. Slocum, M. K., Figdore, S. S., Kennard, W. C., Suzuki, J. Y.
  • and Osborne, T. C. (1990). Linkage arrangement of restriction fragment length poly morphism loci in Brassica oleracea. Theoretical and Applied Genetics 80 57-64.
  • Speed, T. J., McPeek, M. S. and Evans, S. N. (1992). Robustness of the no-interference model for ordering genetic models. Proc. Nat. Acad. Sci. U.S.A. 89 3103-3106.
  • Soller, M., Brody, T. and Genizi, A. (1976). On the power of experimental designs for the detection of linkage between marker loci and quantitative loci in crosses between inbred lines. Theoretical and Applied Genetics 47 35-39.
  • Soller, M., Brody, T. and Genizi, A. (1979). The expected distribution of marker-linked quantitative effects in crosses between inbred lines. Heredity 43 179-190.
  • Stuart, A. and Ord, J. K. (1991). Kendall's Advanced Theory of Statistics. Oxford Univ. Press.
  • Stuber, C. W., Edwards, M. D. and Wendel, J. F. (1987). Molecular-marker-facilitated investigations of quantitativetrait loci in maize. II. Factors influencing yield and its component traits. Crop Science 27 639-648. Stuber, C. W., Lincoln, S. E., Wolff, D. W., Helentjaris, T.
  • and Lander, E. S. (1992). Identification of genetic factors contributing to heterosis in a hy brid from two elite inbred lines using molecular markers. Genetics 132 823-839. Tanksley, S. D., Ganal, M. W., Prince, J. P., de Vicente, M. C., Bonierbale, M. W., Broun, P., Fulton, T. M., Giovannoni, J. J., Grandillo, S., Martin, G. B., Messeguer, R., Miller, J. C., Miller, L., Paterson, A. H., Pineda, O., R ¨oder, M.
  • S., Wing, R. A., Wu, W. and Young, N. D. (1992). High density molecular linkage maps of tomato and potato genomes. Genetics 132 1141-1160.
  • Thoday, J. M. (1961). Location of poly genes. Nature 191 368- 370.
  • Thompson, E. A. (1984). Information gain in joint linkage analysis. IMA Journal of Mathematical Applied Medical Biology 1 31-49.
  • Van Ooijen, J. W. and Maliepaard, C. (1996). MapQTL (tm) Version 3.0 Software for the Calculation of QTL Positions on Genetic Maps. DLO-Centre for Plant Breeding and Reproduction Research, Wageningen, The Netherlands.
  • Weeks, D. and Lange, K. (1987). Preliminary ranking procedures for multilocus ordering. Genomics 1 236-242.
  • Weller, J. I. (1986). Maximum likelihood techniques for the mapping and analysis of quantitative trait loci with the aid of genetic markers. Biometrics 42 627-640.
  • Weller, J. I. (1987). Mapping and analysis of quantitative trait loci in Ly copersicon (tomato) with the aid of genetic markers using approximate maximum likelihood methods. Heredity 59 413-421.
  • Wright, A. J. and Mowers, R. P. (1994). Multiple regression for molecular-marker, quantitative trait data from large F2 populations. Theoretical and Applied Genetics 89 305-312.
  • Wu, W. R. and Li, W. M. (1994). A new approach for mapping quantitative trait loci using complete genetic marker linkage maps. Theoretical and Applied Genetics 89 535-539.
  • Zehr, B. E. (1990). Use of RFLP markers in maize as an aid in selection during inbreeding. Ph.D. dissertation, Dept. Agronomy, Univ. Illinois, Urbana.
  • Zeng, Z-B. (1992). Correcting the bias of Wright's estimates of the number of genes affecting a quantitative character-a further improved method. Genetics 131 987-1001.
  • Zeng, Z-B. (1993). Theoretical basis of precision mapping of quantitative trait loci. Proc. Nat. Acad. Sci. U.S.A. 90 10,972-10,976.
  • Zeng, Z-B. (1994). Precision mapping of quantitative trait loci. Genetics 136 1457-1468.
  • Zeng, Z-B., Houle, D. and Cockerham, C. C. (1990). How informative is Wright's estimator of the number of genes affecting a quantitative character? Genetics 126 235-247. Zhao, H., McPeek, M. S. and Speed, T. P. (1995a). Statistical analysis of chromatid interference. Genetics 139 1057-1065. Zhao, H., Speed, T. P. and McPeek, M. S. (1995b). Statistical analysis of crossover interference using the chi-square model. Genetics 139 1045-1056.
  • Zobel, R. W. (1990). A powerful statistical model for understanding genoty pe-by-environment interactions. In Genoty pe-by-Environment Interaction and Plant Breeding (M. S. Kang, ed.) 126-140. Louisiana State Univ., Baton Rouge.