Behavior Genetics

, 39:580 | Cite as

Rank-Based Inverse Normal Transformations are Increasingly Used, But are They Merited?

  • T. Mark BeasleyEmail author
  • Stephen Erickson
  • David B. Allison
Original Research


Many complex traits studied in genetics have markedly non-normal distributions. This often implies that the assumption of normally distributed residuals has been violated. Recently, inverse normal transformations (INTs) have gained popularity among genetics researchers and are implemented as an option in several software packages. Despite this increasing use, we are unaware of extensive simulations or mathematical proofs showing that INTs have desirable statistical properties in the context of genetic studies. We show that INTs do not necessarily maintain proper Type 1 error control and can also reduce statistical power in some circumstances. Many alternatives to INTs exist. Therefore, we contend that there is a lack of justification for performing parametric statistical procedures on INTs with the exceptions of simple designs with moderate to large sample sizes, which makes permutation testing computationally infeasible and where maximum likelihood testing is used. Rigorous research evaluating the utility of INTs seems warranted.


Blom Inverse normal transformation Robustness Type 1 error rate 



We thank Christian Dina for asking cogent questions that inspired this commentary and for making useful comments on an earlier draft and also thank Jay Conover, Roger Berger, Brian Hicks, Rui Feng, Michael C. Neale, Goncalo Abecasis, Bernard S. Gorman and Alfred A. Bartolucci for their helpful advice or comments on earlier drafts. This article is supported in part by NIH grants P30DK056336, U54CA100949, R01ES09912, and T32HL072757.


  1. Akritas MG (1990) The rank transform method on some two factor designs. J Am Stat Assoc 85:73–78. doi: 10.2307/2289527 CrossRefGoogle Scholar
  2. Allison DB, Neale MC, Zannolli RZ, Schork NJ, Amos CI, Blangero J (1999) Testing the robustness of the likelihood ratio test in a variance-component quantitative trait loci (QTL) mapping procedure. Am J Hum Genet 65:531–544. doi: 10.1086/302487 PubMedCrossRefGoogle Scholar
  3. Allison DB, Cui X, Page GP, Sabripour M (2006) Microarray data analysis: from disarray to consolidation and consensus. Nat Rev Genet 7(1):55–65. doi: 10.1038/nrg1749 PubMedCrossRefGoogle Scholar
  4. Almasy L, Blangero J (1998) Multipoint quantitative trait linkage analysis in general pedigrees. Am J Hum Genet 62:1198–1211. doi: 10.1086/301844 PubMedCrossRefGoogle Scholar
  5. Amos CI (1994) Robust variance-components approach for assessing genetic linkage in pedigrees. Am J Hum Genet 54:535–543PubMedGoogle Scholar
  6. Analysis System 130 (2003) Method and apparatus for analysis of data from biomolecular arrays, US Patent 6516276, (
  7. Anokhin AP, Heath AC, Ralano A (2003) Genetic influences on frontal brain function: WCST performance in twins. NeuroReport 14(15):1975–1978. doi: 10.1097/00001756-200310270-00019 PubMedCrossRefGoogle Scholar
  8. Ashton GC, Borecki IB (1987) Further evidence for a gene influencing spatial ability. Behav Genet 17(3):243–256. doi: 10.1007/BF01065504 PubMedCrossRefGoogle Scholar
  9. Barnard GA (1957) Mathematical gazette, 41(338), 298–300. Review of: Tafeln zum Vergleich Zweier Stichproben mittels X-Test und Zeichentest tables for comparing two samples by X-test and sign test by B. L. van der Waerden; E. Nievergelt. doi: 10.2307/3610142
  10. Basrak B, Klaassen CA, Beekman M, Martin NG, Boomsma DI (2004) Copulas in QTL mapping. Behav Genet 34(2):161–171. doi: 10.1023/ PubMedCrossRefGoogle Scholar
  11. Beasley TM (2002) Multivariate aligned rank test for interactions in multiple group repeated measures designs. Multiv Behav Res 37:197–226. doi: 10.1207/S15327906MBR3702_02 CrossRefGoogle Scholar
  12. Beasley TM, Zumbo BD (2003) Comparison of aligned Friedman rank and parametric methods for testing interactions in split-plot designs. Comput Stat Data Anal 42(4):569–593CrossRefGoogle Scholar
  13. Berry WD (1993) Understanding regression assumptions. Sage, Newbury ParkGoogle Scholar
  14. Blair RC, Sawilowsky SS, Higgins JJ (1987) Limitations of the rank transform statistic in test for interactions. Comm Stat-Simul Comp 16(113):3–1145Google Scholar
  15. Bliss CI (1967) Statistics in biology. McGraw-Hill, New YorkGoogle Scholar
  16. Blom G (1958) Statistical estimates and transformed beta-variables. Wiley, New YorkGoogle Scholar
  17. Blonigen DM, Carlson SR, Krueger RF, Patrick CJ (2003) A twin study of self-reported psychopathic personality traits. Pers Individ Dif 35:179–197. doi: 10.1016/S0191-8869(02)00184-8 CrossRefGoogle Scholar
  18. Box GEP, Cox DR (1964) An analysis of transformations. J R Stat Soc B 26:211–252Google Scholar
  19. Bradley JV (1968) Distribution-free statistical tests. Prentice-Hall, New YorkGoogle Scholar
  20. Bradley JV (1978) Robustness? Br J Math Stat Psychol 31:144–152Google Scholar
  21. Chen WM, Abecasis GR (2006) Estimating the power of variance component linkage analysis in large pedigrees. Genet Epidemiol 30:471–484. doi: 10.1002/gepi.20160 PubMedCrossRefGoogle Scholar
  22. Chernoff H, Savage IR (1958) Asymptotic normality and efficiency of certain nonparametric tests. Ann Math Stat 29:972–994. doi: 10.1214/aoms/1177706436 CrossRefGoogle Scholar
  23. Cockerham CC (1954) An extension of the concept of partitioning hereditary variance for analysis of covariances among relatives when epistasis is present. Genetics 39:859–882PubMedGoogle Scholar
  24. Conover WJ (1973) Rank tests for one sample, two samples, and k samples without the assumption of a continuous distribution function. Ann Stat 1(6):1105–1125. doi: 10.1214/aos/1176342560 CrossRefGoogle Scholar
  25. Conover WJ (1980) Practical nonparametric statistics, 2nd edn. Wiley, New YorkGoogle Scholar
  26. Conover WJ, Iman RL (1981) Rank transformations as a bridge between parametric and nonparametric statistics. Am Stat 35:124–133. doi: 10.2307/2683975 CrossRefGoogle Scholar
  27. Diao G, Lin DY (2005) A powerful and robust method for mapping quantitative trait loci in general pedigrees. Am J Hum Genet 77:97–111. doi: 10.1086/431683 PubMedCrossRefGoogle Scholar
  28. Dixon AL, Liang L, Moffatt MF, Chen W, Heath S, Wong KC, Taylor J, Burnett E, Gut I, Farrall M, Lathrop GM, Abecasis GR, Cookson WO (2007) A genome-wide association study of global gene expression. Nat Genet 39(10):1202–1207. doi: 10.1038/ng2109 PubMedCrossRefGoogle Scholar
  29. Etzel CJ, Shete S, Beasley TM, Fernandez JR, Allison DB, Amos CI (2003) Effect of box–cox transformation on power of Haseman–Elston and maximum-likelihood variance components tests to detect quantitative trait loci. Hum Hered 55:108–116. doi: 10.1159/000072315 PubMedCrossRefGoogle Scholar
  30. Farrell P, Rogers-Stewart K (2006) Comprehensive study of tests for normality and symmetry: extending the Spiegelhalter test. J Stat Comp Simul 76(9):803–816. doi: 10.1080/10629360500109023 CrossRefGoogle Scholar
  31. Feir-Walsh BJ, Toothaker LE (1974) An empirical comparison of the ANOVA F-test, normal scores test and Kruskal–Wallis test under violation of assumptions. Educ Psychol Measur 34:789–799. doi: 10.1177/001316447403400406 CrossRefGoogle Scholar
  32. Fisher RA, Yates F (1938) Statistical tables for biological, agricultural, and medical research, 1st edn. Oliver & Boyd, EdinburghGoogle Scholar
  33. George VT, Elston RC (1987) Testing the association between polymorphic markers and quantitative traits in pedigrees. Genet Epidemiol 4(3):193–201. doi: 10.1002/gepi.1370040304 PubMedCrossRefGoogle Scholar
  34. Good PI (1999) Resampling methods. A practical guide to data analysis. Birkhauser, BostonGoogle Scholar
  35. Good PI (2004) Efficiency comparisons of rank and permutation tests by statistics in medicine 2001; 20:705–731. Statistics in Medicine, 23(5), 857. doi: 10.1002/sim.1738
  36. Hájek J, Sidák F (1967) Theory of rank tests. Academic Press and Academia, PragueGoogle Scholar
  37. Harter HL (1961) Expected values of normal order statistics. Biometrika 48:151–165Google Scholar
  38. Headrick TC, Rotou O (2001) An investigation of the rank transformation in multiple regression. Comput Stat Data Anal 38:203–215. doi: 10.1016/S0167-9473(01)00034-2 CrossRefGoogle Scholar
  39. Headrick TC, Sawilowsky SS (2000) Properties of the rank transformation in factorial analysis of covariance. Comm Stat-Simul Comp 29:1059–1087. doi: 10.1080/03610910008813654 CrossRefGoogle Scholar
  40. Headrick TC, Vineyard G (2001) An empirical investigation of four tests of interaction in the context of factorial analysis of covaraince. Mult Linear Regress View 27:3–15Google Scholar
  41. Hettmansperger TP, McKean JW (1977) A robust alternative based on ranks to least squares in analyzing linear models. Technometrics 19:275–284. doi: 10.2307/1267697 CrossRefGoogle Scholar
  42. Hicks BM, Krueger RF, Iacono WG, McGue M, Patrick CJ (2004) Family transmission and heritability of externalizing disorders: a twin-family study. Arch Gen Psychiatry 61:922–928. doi: 10.1001/archpsyc.61.9.922 PubMedCrossRefGoogle Scholar
  43. Hicks BM, Bernat E, Malone SM, Iacono WG, Patrick CJ, Krueger RF, McGue M (2007) Genes mediate the association between P3 amplitude and externalizing disorders. Psychophysiology 44(1):98–105. doi: 10.1111/j.1469-8986.2006.00471.x PubMedCrossRefGoogle Scholar
  44. Higgins JJ, Tashtoush S (1994) An aligned rank transform test for interaction. Nonlinear World 1:201–211Google Scholar
  45. Hodges JL, Lehmann EL (1962) Rank methods for combination of independent experiments in analysis of variance. Ann Math Stat 33:482–497. doi: 10.1214/aoms/1177704575 CrossRefGoogle Scholar
  46. Hora SC, Conover WJ (1984) The F-statistic in the two-way layout with rank-score transformed data. J Am Stat Assoc 79:668–673. doi: 10.2307/2288415 CrossRefGoogle Scholar
  47. Jaeckel LA (1972) Estimating regression coefficients by minimizing the dispersion of the residuals. Ann Math Stat 43:1449–1458. doi: 10.1214/aoms/1177692377 CrossRefGoogle Scholar
  48. James GS (1959) The Behrens–Fisher distribution and weighted means. J R Stat Soc [Ser A] 21:73–80Google Scholar
  49. Keselman HJ, Rogan JC, Feir-Walsh BJ (1977) An evaluation of some nonparametric and parametric tests for location equality. Br J Math Stat Psychol 30:213–221Google Scholar
  50. Knoke JD (1991) Nonparametric analysis of covariance for comparing change in randomized studies with baseline values subject to error. Biometrics 47(2):523–533. doi: 10.2307/2532143 PubMedCrossRefGoogle Scholar
  51. Knoll J, Ejeta G (2008) Marker-assisted selection for early-season cold tolerance in sorghum: QTL validation across populations and environments. Theor Appl Genet 116(4):541–553. doi: 10.1007/s00122-007-0689-8 PubMedCrossRefGoogle Scholar
  52. Kohr RL, Games PA (1974) Robustness of the analysis of variance, the Welch procedure, and a Box procedure to heterogeneous variances. J Exp Educ 43:61–69Google Scholar
  53. Kraja AT, Corbett J, Ping A, Lin RS, Jacobsen PA, Crosswhite M, Borecki IB, Province MA (2007) Rheumatoid arthritis, item response theory, Blom transformation, and mixed models. BMC Proc 1(Suppl. 1):S116PubMedCrossRefGoogle Scholar
  54. Kruskal WH, Wallis WA (1952) Use of ranks in one-criterion variance analysis. J Am Stat Assoc 47:583–621. doi: 10.2307/2280779 CrossRefGoogle Scholar
  55. Li M, Boehnke M, Abecasis GR, Song PX (2006) Quantitative trait linkage analysis using Gaussian copulas. Genetics 173(4):2317–2327. doi: 10.1534/genetics.105.054650 PubMedCrossRefGoogle Scholar
  56. Mann HB, Whitney DR (1947) On a test of whether one of two random variables is stochastically larger than the other. Ann Math Stat 18:50–60. doi: 10.1214/aoms/1177730491 CrossRefGoogle Scholar
  57. Mansouri H, Chang G-H (1995) A comparative study of some rank tests for interaction. Comput Stat Data Anal 19:85–96. doi: 10.1016/0167-9473(93)E0045-6 CrossRefGoogle Scholar
  58. Maritz JS (1982) Distribution-free statistical methods. Chapman and Hall, LondonGoogle Scholar
  59. Martin LJ, Crawford MH (1998) Genetic and environmental components of thyroxine variation in Mennonites from Kansas and Nebraska. Hum Biol 70(4):745–760PubMedGoogle Scholar
  60. McSweeney M, Penfield D (1969) The normal scores test for the c-sample problem. Br J Math Stat Psychol 20:187–204Google Scholar
  61. Mehta T, Tanik M, Allison DB (2004) Toward sound epistemological foundations of statistical methods for high dimensional biology. Nat Genet 36:943–947. doi: 10.1038/ng1422 PubMedCrossRefGoogle Scholar
  62. Micceri T (1989) The unicorn, the normal curve, and other improbable creatures. Psychol Bull 105:156–166. doi: 10.1037/0033-2909.105.1.156 CrossRefGoogle Scholar
  63. Nanda NJ, Rommelse Arias-VásquezA, Altink ME, Buschgens CJM, Fliers E, Asherson P, Faraone SV, Buitelaar JK, Sergeant JA, Oosterlaan J, Franke B (2008) Neuropsychological endophenotype approach to genome-wide linkage analysis identifies susceptibility loci for ADHD on 2q21.1 and 13q12.11. Am J Hum Gen 9:9–105Google Scholar
  64. Neave HR, Wothington PL (1989) Distribution-free tests. Routledge, New YorkGoogle Scholar
  65. Peng B, Yu RK, DeHoff KL, Amos CI (2007) Normalizing a large number of quantitative traits using empirical normal quantile transformation. BMC Proc, (Suppl I), p S156Google Scholar
  66. POLY: Computer program for polygenic analysis and power analysis (2003) []
  67. Pratt JW (1964) Robustness of some procedures for the two-sample location problem. J Am Stat Assoc 59:665–680. doi: 10.2307/2283092 CrossRefGoogle Scholar
  68. Przybyla-Zawislak BD, Thorn BT, Alia SF et al (2005) Identification of rat hippocampal mRNAs altered by the mitochondrial toxicant, 3-NPA. Ann N Y Acad Sci 1053:162–173. doi: 10.1196/annals.1344.014 PubMedCrossRefGoogle Scholar
  69. Pulli K, Karma K, Norio R, Sistonen P, Göring HH, Järvelä I (2008) Genome-wide linkage scan for loci of musical aptitude in Finnish families: evidence for a major locus at 4q22. J Med Genet 45(7):451–456. doi: 10.1136/jmg.2007.056366 PubMedCrossRefGoogle Scholar
  70. Ray WD, Pitman A (1961) An exact distribution of the Fisher–Behrens–Welch statistic for testing the difference between the means of two normal populations with unknown variances. J R Stat Soc [Ser A] 23:377–384Google Scholar
  71. Salter KC, Fawcett RF (1993) The ART test of interaction: a robust and powerful test of interaction in factorial models. Comm Stat-Simul Comp 22:137–153CrossRefGoogle Scholar
  72. Scuteri A, Sanna S, Chen WM, Uda M, Albai G, Strait J, Najjar S, Nagaraja R, Orru M, Usala G, Dei M, Lai S, Maschio A, Busonero F, Mulas A, Ehret GB, Fink AA, Weder AB, Cooper RS, Galan P, Chakravarti A, Schlessinger D, Cao A, Lakatta E, Abecasis GR (2007) Genome-wide association scan shows genetic variants in the FTO gene are associated with obesity-related traits. PLOS Genetics 3(7):e115. doi: 10.1371/journal.pgen.0030115 PubMedCrossRefGoogle Scholar
  73. Servin B, Stephens M (2007) Imputation-based analysis of association studies: candidate regions and quantitative traits. PLOS Genetics 3(7):e114. doi: 10.1371/journal.pgen.0030114 PubMedCrossRefGoogle Scholar
  74. Shete S, Beasley TM, Etzel CJ, Fernández JR, Chen J, Allison DB, Amos CI (2004) Effect of Winsorization on power and type 1 error of variance components and related methods of QTL detection. Behav Genet 34:153–159. doi: 10.1023/B:BEGE.0000013729.26354.da PubMedCrossRefGoogle Scholar
  75. Silverman EK, Province MA, Campbell EJ, Pierce JA, Rao DC (1990) Biochemical intermediates in alpha 1-antitrypsin deficiency: residual family resemblance for total alpha 1-antitrypsin, oxidized alpha 1-antitrypsin, and immunoglobulin E after adjustment for the effect of the Pi locus. Genet Epidemiol 7(2):137–149. doi: 10.1002/gepi.1370070204 PubMedCrossRefGoogle Scholar
  76. SOLAR: Sequential Oligogenic Linkage Analysis Routines (2008) []
  77. Sprent P, Smeeton NC (2001) Applied nonparametric statistical methods, 3rd edn. Chapman & Hall, LondonGoogle Scholar
  78. Stuart A (1954) Asymptotic relative efficiencies of distribution-free tests of randomness against normal alternatives. J Am Stat Assoc 49:147–157. doi: 10.2307/2281041 CrossRefGoogle Scholar
  79. Thompson GL (1991) A note on the rank transform for interactions. Biometrika 78:697–701. doi: 10.1093/biomet/78.3.697 CrossRefGoogle Scholar
  80. Thompson GL (1993) A correction note on the rank transform for interactions. Biometrika 80:711Google Scholar
  81. Toothaker LE, Newman D (1994) Nonparametric competitors to the two way ANOVA. J Educ Behav Stat 19:237–273Google Scholar
  82. Tukey JW (1962) The future of data analysis. Ann Math Stat 33:1–67. doi: 10.1214/aoms/1177704711 CrossRefGoogle Scholar
  83. Tzou GG, Everson DO, Bulls RC, Olson DP (1991) Classification of beef calves as protein-deficient or thermally stressed by discriminant analysis of blood constituents. J Anim Sci 69:864–873PubMedGoogle Scholar
  84. Valdar W, Solberg LC, Gauguier D, Cookson WO, Rawlins JN, Mott R, Flint J (2006) Genetic and environmental effects on complex traits in mice. Genetics 174(2):959–984. doi: 10.1534/genetics.106.060004 PubMedCrossRefGoogle Scholar
  85. van den Oord EJ, Simonoff E, Eaves LJ, Pickles A, Silberg J, Maes H (2000) An evaluation of different approaches for behavior genetic analyses with psychiatric symptom scores. Behav Genet 30(1):1–18. doi: 10.1023/A:1002095608946 PubMedCrossRefGoogle Scholar
  86. van der Waerden BL (1952) Order tests for the two-sample problem and their power. Proc Koninklijke Nederlandse Akademie van Wetenschappen. Ser A 55:453–458Google Scholar
  87. Vargha A, Delaney HD (1998) The Kruskal-Wallis test and stochastic homogeneity. J Educ Behav Stat 23:170–192Google Scholar
  88. Wang K, Huang J (2002) A score-statistic approach for the mapping of quantitative-trait loci with sibships of arbitrary size. Am J Hum Genet 70(2):412–424. doi: 10.1086/338659 PubMedCrossRefGoogle Scholar
  89. Welch BL (1947) The generalization of Student’s problem when several different population variances are involved. Biometrika 34:28–35Google Scholar
  90. Wilcox RR (1995) ANOVA: a paradigm for low power and misleading measures of effect size? Rev Educ Res 65:51–77Google Scholar
  91. Wilcoxon F (1945) Individual comparisons by ranking methods. Biometrics 1:80–83. doi: 10.2307/3001968 CrossRefGoogle Scholar
  92. Wu X, Cooper RS, Borecki I, Hanis C, Bray M, Lewis CE, Zhu X, Kan D, Luke A, Curb D (2002) A combined analysis of genomewide linkage scans for body mass index from the National Heart, Lung, and Blood Institute Family Blood Pressure Program. Am J Hum Genet 70(5):1247–1256. doi: 10.1086/340362 PubMedCrossRefGoogle Scholar
  93. Yang R, Yi N, Xu S (2006) Box-Cox transformation for QTL mapping. Genetica 128(1–3):133–143. doi: 10.1007/s10709-005-5577-z PubMedCrossRefGoogle Scholar
  94. Yuen KK (1974) The two-sample trimmed t for unequal population variances. Biometrika 61:165–170Google Scholar
  95. Zak M, Baierl A, Bogdan M, Futschik A (2007) Locating multiple interacting quantitative trait Loci using rank-based model selection. Genetics 176(3):1845–1854. doi: 10.1534/genetics.106.068031 PubMedCrossRefGoogle Scholar
  96. Zimmerman DW (1996) A note on homogeneity of variance of scores and ranks. J Exp Educ 64:351–362Google Scholar
  97. Zimmerman DW (2004) A note on preliminary tests of equality of variances. Br J Math Stat Psychol 57(1):173–181PubMedCrossRefGoogle Scholar
  98. Zumbo BD, Coulombe D (1997) Investigation of the robust rank-order test for non-normal populations with unequal variances: the case of reaction time. Can J Exp Psychol 51:139–150. doi: 10.1037/1196-1961.51.2.139 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • T. Mark Beasley
    • 1
    Email author
  • Stephen Erickson
    • 1
  • David B. Allison
    • 1
    • 2
    • 3
  1. 1.Department of Biostatistics, Section on Statistical GeneticsUniversity of Alabama at BirminghamBirminghamUSA
  2. 2.Department of Nutrition SciencesUniversity of Alabama at BirminghamBirminghamUSA
  3. 3.Clinical Nutrition Research CenterUniversity of Alabama at BirminghamBirminghamUSA

Personalised recommendations