Abstract
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.
Similar content being viewed by others
References
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.
Bone, E. & A. Farres, 2001. Trends and rates of microevolution in plants. Genetica 112-113: 165–182.
Bookstein, F.L., 1987. Random walk and the existence of evolutionary rates. Paleobiology 13: 446–464.
Bookstein, F.L., 1988. Random walk and the biometrics of morphological characters. Evol. Biol. 23: 369–398.
Bury, K.V., 1975. Statistical Models in Applied Science. Wiley, New York.
Carroll, R.L., 2000. Towards a new evolutionary synthesis. Trends Ecol. Evol. 15: 27–32.
Charlesworth, B., 1984a. Some quantitative methods for studying evolutionary patterns in single characters. Paleobiology 10: 308–318.
Charlesworth, B., 1984b. The cost of phenotypic evolution. Paleobiology 10: 319–327.
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.
Efron, B., 1982. The Jackknife, the Bootstrap and Other Resampling Plans. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania.
Efron, B. & R.J. Tibshirani, 1993. An Introduction to the Bootstrap. Chapman and Hall, New York, N.Y.
Endler, J.A., 1986. Natural Selection in the Wild. Princeton University Press, Princeton.
Freund, J.E. & R.E. Walpole, 1980. Mathematical Statistics. Prentice Hall, Englewood Cliffs, N.Y.
Frey, K.J. & J.B. Holland, 1999. Nine cycles of recurrent selection for increased groat-oil content in oat. Crop Sci. 39: 1636–1641.
Gingerich, P.D., 1993. Quantification and comparison of evolutionary rates. Am. J. Sci. 293–A: 453–478.
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.
Hastings, H.M. & G. Sugihara, 1993. Fractals: A User' Guide for the Natural Sciences. Oxford University Press, Oxford.
Hayami, I. & T. Ozawa, 1975. Evolutionary models of lineage zones. Lethaia 8: 1–14.
Hendry, A.P. & M.T. Kinnison, 1999. The pace of modern life: measuring rates of contemporary microevolution. Evolution 53: 1637–1653.
Hurst, H.E., 1951. Long-term storage capacity of reservoirs. Trans. Am. Soc. Civ. Engrs. 116: 770–808.
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.
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.
Lande, R., 1976. Natural selection and random genetic drift in phenotypic evolution. Evolution 37: 1210–1226.
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.
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.
Lynch, M., 1990. The rate of morphological evolution in mammals from the standpoint of the neutral expectation. Am. Nat. 136: 727–741.
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.
Mandelbrot, B.B. & J.R. Wallis, 1969. Some long-run properties of geophysical records. Water Resour. Res. 5: 321–340.
Middleton, G.V., 2000. Data Analysis in the Earth Sciences Using Matlab. Prentice Hall, Upper Saddle River, N.J.
Press, WH, B.P. Flannery, S.A. Teukolsky & W.T. Vetterling, 1988. Numerical Recipes in C. Cambridge University Press, New York.
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.
Raup D.M. & R.E. Crick, 1981. Evolution of single characters in the Jurassic ammonite Kosmoceras. Paleobiology 7: 200–215.
Reif, F., 1965. Fundamentals of Statistical and Thermal Physics. McGraw-Hill, N.Y.
Rohlf, F.J., 2000. Statistical power comparisons among alternative morphometric methods. Am. J. Phys. Anthrop. 111: 463–478.
Roopnarine, P.D., G. Byars & P. Fitzgerald, 1999. Anagenetic evolution, stratophenetic patterns, and random walk models. Paleobiology, 25: 41–57.
Turelli, M.T., J.H. Gillespie & R. Lande, 1988. Rate tests on quantitative characters during macroevolution and microevolution. Evolution 42: 1085–1089.
Zuch, E.L. (ed.), 1987. Data Acquisition and Conversion Handbook. Datel Corp., Mansfield, MA.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sheets, H.D., Mitchell, C.E. Why the null matters: statistical tests, random walks and evolution. Genetica 112, 105–125 (2001). https://doi.org/10.1023/A:1013308409951
Issue Date:
DOI: https://doi.org/10.1023/A:1013308409951