, Volume 112, Issue 1, pp 105–125 | Cite as

Why the null matters: statistical tests, random walks and evolution

  • H. David Sheets
  • Charles E. Mitchell


A number of statistical tests have been developed to determine what type of dynamics underlie observed changes in morphology in evolutionary time series, based on the pattern of change within the time series. The theory of the ‘scaled maximum’, the ‘log-rate-interval’ (LRI) method, and the Hurst exponent all operate on the same principle of comparing the maximum change, or rate of change, in the observed dataset to the maximum change expected of a random walk. Less change in a dataset than expected of a random walk has been interpreted as indicating stabilizing selection, while more change implies directional selection. The ‘runs test’ in contrast, operates on the sequencing of steps, rather than on excursion. Applications of these tests to computer generated, simulated time series of known dynamical form and various levels of additive noise indicate that there is a fundamental asymmetry in the rate of type II errors of the tests based on excursion: they are all highly sensitive to noise in models of directional selection that result in a linear trend within a time series, but are largely noise immune in the case of a simple model of stabilizing selection. Additionally, the LRI method has a lower sensitivity than originally claimed, due to the large range of LRI rates produced by random walks. Examination of the published results of these tests show that they have seldom produced a conclusion that an observed evolutionary time series was due to directional selection, a result which needs closer examination in light of the asymmetric response of these tests.

Hurst exponent hypothesis testing log-rate-interval microevolution random walks stochastic models 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bell, M.A., J. Baumgartner & E. Olson, 1985. Patterns of temporal change in single morphological characters of a Miocene stickleback fish. Paleobiology 11: 258–271.Google Scholar
  2. Bone, E. & A. Farres, 2001. Trends and rates of microevolution in plants. Genetica 112-113: 165–182.Google Scholar
  3. Bookstein, F.L., 1987. Random walk and the existence of evolutionary rates. Paleobiology 13: 446–464.Google Scholar
  4. Bookstein, F.L., 1988. Random walk and the biometrics of morphological characters. Evol. Biol. 23: 369–398.Google Scholar
  5. Bury, K.V., 1975. Statistical Models in Applied Science. Wiley, New York.Google Scholar
  6. Carroll, R.L., 2000. Towards a new evolutionary synthesis. Trends Ecol. Evol. 15: 27–32.Google Scholar
  7. Charlesworth, B., 1984a. Some quantitative methods for studying evolutionary patterns in single characters. Paleobiology 10: 308–318.Google Scholar
  8. Charlesworth, B., 1984b. The cost of phenotypic evolution. Paleobiology 10: 319–327.Google Scholar
  9. Clyde, W.C. & P.D. Gingerich, 1994. Rates of evolution in the dentition of early Eocene Cantius: comparison of size and shape. Paleobiology 20: 506–522.Google Scholar
  10. Efron, B., 1982. The Jackknife, the Bootstrap and Other Resampling Plans. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania.Google Scholar
  11. Efron, B. & R.J. Tibshirani, 1993. An Introduction to the Bootstrap. Chapman and Hall, New York, N.Y.Google Scholar
  12. Endler, J.A., 1986. Natural Selection in the Wild. Princeton University Press, Princeton.Google Scholar
  13. Freund, J.E. & R.E. Walpole, 1980. Mathematical Statistics. Prentice Hall, Englewood Cliffs, N.Y.Google Scholar
  14. Frey, K.J. & J.B. Holland, 1999. Nine cycles of recurrent selection for increased groat-oil content in oat. Crop Sci. 39: 1636–1641.Google Scholar
  15. Gingerich, P.D., 1993. Quantification and comparison of evolutionary rates. Am. J. Sci. 293–A: 453–478.Google Scholar
  16. Gingerich, P.D., 1994. Rates of evolution in divergent species lineages as a test of character displacement in the fossil record: tooth size in Paleocene Plesiadapis (Mamalia, proprimates). Palaeovertebrata 25: 193–204.Google Scholar
  17. Hastings, H.M. & G. Sugihara, 1993. Fractals: A User' Guide for the Natural Sciences. Oxford University Press, Oxford.Google Scholar
  18. Hayami, I. & T. Ozawa, 1975. Evolutionary models of lineage zones. Lethaia 8: 1–14.Google Scholar
  19. Hendry, A.P. & M.T. Kinnison, 1999. The pace of modern life: measuring rates of contemporary microevolution. Evolution 53: 1637–1653.Google Scholar
  20. Hurst, H.E., 1951. Long-term storage capacity of reservoirs. Trans. Am. Soc. Civ. Engrs. 116: 770–808.Google Scholar
  21. Kingsolver, J.G., H.E. Hoekstra, J.M. Hoekstra, D. Berrigan, S.N. Vignieri, C.E. Hill, A. Hoang, P. Gilbert & P. Beerli, 2000. The strength of phenotypic selection in natural populations. Am. Nat. 157: 245–261.Google Scholar
  22. Kucera, M. & B.A. Malmgren, 1998. Differences between evolution of mean form and evolution of new morphotypes: an example from late Cretaceous planktonic foraminifera. Paleobiology 41: 49–63.Google Scholar
  23. Lande, R., 1976. Natural selection and random genetic drift in phenotypic evolution. Evolution 37: 1210–1226.Google Scholar
  24. Lambert, R.J., D.E. Alexander, E.L. Mollring & B. Wiggens, 1997. Selection for increased oil concentration in maize kernals and associated changes in several kernal plants. Maydica 42: 39–43.Google Scholar
  25. Lucas, H.L., 1964. Stochastic elements in biological models; their sources and significance, pp. 335–383 in Stochastic Models in Medicine and Biology, edited by J. Gurland. University of Wisconsin Press, Madison, Wisconsin.Google Scholar
  26. Lynch, M., 1990. The rate of morphological evolution in mammals from the standpoint of the neutral expectation. Am. Nat. 136: 727–741.Google Scholar
  27. Malmgren, B.A. & J.P. Kennett, 1981. Phyletic gradualsim in a late Cenozoic planktonic foraminiferal lineage; DSDP site 284, southwest Pacific. Paleobiology 7: 230–240.Google Scholar
  28. Mandelbrot, B.B. & J.R. Wallis, 1969. Some long-run properties of geophysical records. Water Resour. Res. 5: 321–340.Google Scholar
  29. Middleton, G.V., 2000. Data Analysis in the Earth Sciences Using Matlab. Prentice Hall, Upper Saddle River, N.J.Google Scholar
  30. Press, WH, B.P. Flannery, S.A. Teukolsky & W.T. Vetterling, 1988. Numerical Recipes in C. Cambridge University Press, New York.Google Scholar
  31. Raup, D.M., 1977. Stochastic models in evolutionary paleontology, pp. 59–78 in Patterns of Evolution: As Illustrated by the Fossil Record, edited by A. Hallam. Elsevier, Amsterdam.Google Scholar
  32. Raup D.M. & R.E. Crick, 1981. Evolution of single characters in the Jurassic ammonite Kosmoceras. Paleobiology 7: 200–215.Google Scholar
  33. Reif, F., 1965. Fundamentals of Statistical and Thermal Physics. McGraw-Hill, N.Y.Google Scholar
  34. Rohlf, F.J., 2000. Statistical power comparisons among alternative morphometric methods. Am. J. Phys. Anthrop. 111: 463–478.Google Scholar
  35. Roopnarine, P.D., G. Byars & P. Fitzgerald, 1999. Anagenetic evolution, stratophenetic patterns, and random walk models. Paleobiology, 25: 41–57.Google Scholar
  36. Turelli, M.T., J.H. Gillespie & R. Lande, 1988. Rate tests on quantitative characters during macroevolution and microevolution. Evolution 42: 1085–1089.Google Scholar
  37. Zuch, E.L. (ed.), 1987. Data Acquisition and Conversion Handbook. Datel Corp., Mansfield, MA.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • H. David Sheets
    • 1
  • Charles E. Mitchell
    • 2
  1. 1.Department of PhysicsCanisius CollegeBuffaloUSA
  2. 2.Department of GeologySUNY at BuffaloAmherstUSA

Personalised recommendations