Abstract
Empirical power laws are frequently used to relate parameters in complex hydrological and hydrometeorological processes. The validity of power laws relating two parameters with a common variable may be compromised by spurious influences of the common variable. Theoretical results are presented that allow both the spurious self-correlation coefficient and the slope of a spurious self-correlation to be determineda priori. Raising a common variable to a higher power in either parameter amplifies the spurious effects.
Power law regression equations are not single-valued analytical functions and must not be treated as such. Because of the strong influence of a common variable on the correlation coefficient, the transfer of a common variable from one side of a power-law regression equation to another (by cross-multiplying) may severely distort the results. Examples from lake hydrology are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benson, M. A. 1965: Spurious correlation in hydraulics and hydrology. J. Hydraulics Div., Am. Soc. Civ. Eng. 91, 35–42
Burling, R. W. 1959: The spectrum of waves at short fetches. Dtsch. Hydrogr. Z. 12, 45–64
Chayes, F. 1949: On ratio correlation in petrography. J. Geol. 57, 237–254
Faller, A. J.; Caponi, E. A. 1978: Laboratory studies of winddriven Langmuir circulations. J. Geophys. Res. 83, 3617–3633
Hasselmann, K.; Barnett, T. P.; Bouws, E.; Carlson, H.; Cartwright, D. E.; Enke, K.; Ewing, J. A.; Gienapp, H.; Hasselmann, D. E.; Kruseman, P.; Meerburg, A.; Muller, P.; Olbers, D. J.; Richter, K.; Sell, W.; Walden, H. 1973: Measurements of wind wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Herausgegeben vom Deutsch. Hydrogr. Institut., Reihe A, no. 12, 95 pp
Hawkins, R. H.; Pole, R. A. 1989: Standardization of peak-volume transformations. Water Resources Bull. 25, 377–380
Hicks, B. B. 1978: Some limitations of dimensional analysis and power laws. Boundary-Layer Meteorology 14, 567–569
Jackson, D. A.; Somers, K.M. 1991: The spectre of ‘spurious’ correlations. Oecologia. 86, 147–151
Kenney, B. C. 1982: Beware of spurious self-correlations! Water Resources Res. 18, 1041–1048
Kenney, B. C. 1991: Comments on ‘Some misconceptions about the spurious correlation problem in the ecological literature’ by Y. T. Prairie and D. F. Bird. Oecologia 86, 152
Kite, G. 1989: Some statistical observations. Water Resources Bull. 25, 483–490
Kitaigorodskii, S. A.; Strekalov, S. S. 1962: Contribution to an analysis of the spectra of wind-caused wave action. Izv., Geophys. Ser. Acad. Sci., USSR 9, 1221–1228
Liu, P. C. 1985: In search of universal parametric correlations for wind waves. In: The Ocean Surface. Y. Toba and H. Mitsuyasu (eds.), Reidel
Mitsuyasu, H. 1966: Interactions between water waves and wind (I). Rep. Res. Inst. Appl. Mech., Kyushu Univ. 14, 67–88
Molfino, M. E.; Cruise, J. F. 1990: An additional analysis of peak-volume relations and standardization procedures. Water Resources Bull. 26, 687–692
Neville, A. M.; Kennedy, J. B. 1964: Basic statistical methods for engineers and scientists. International Textbook Company, Scranton, Pa.
Pearson, K. 1897: On a form of spurious correlation which may arise when indices are used in the measurement of organs. Proc. Roy. Soc., London 60, 489–502
Phillips, O. M. 1977: The Dynamics of the Upper Ocean. 2nd. Edition, Cambridge Univ. Press, Cambridge
Reed, J. L. 1921: On the correlation between any two functions and its application to the general case of spurious correlation. J. Wash. Acad. Sci. 11, 449–455
Schlichting, H. 1960: Boundary layer theory. 4th Edition, New York: McGraw-Hill
Waythomas, C. F.; Williams, G. P. 1988. Sediment yield and spurious correlation - Toward a better portrayal of the annual suspended-sediment load of rivers. Geomorphology 1, 309–316
Yalin, M. S.; Kamphuis, J. W. 1971: Theory of dimensions and spurious correlation. J. Hydraul. Res. I.A.H.R. 9, 249–265
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kenney, B.C. On the validity of empirical power laws. Stochastic Hydrol Hydraul 7, 179–194 (1993). https://doi.org/10.1007/BF01585597
Issue Date:
DOI: https://doi.org/10.1007/BF01585597