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Note On Correlation Coefficients Derived from Cumulative Distributions with Reference to Glaciological Studies

Published online by Cambridge University Press:  30 January 2017

J.T. Andrews ,
B.D. Faiiey  and
D. Alford
J.T. Andrews
Affiliation:
Institute of Arctic and Alpine Research, University of Colorado, Boulder, Colorado 80302, U.S.A.
B.D. Faiiey
Affiliation:
Institute of Arctic and Alpine Research, University of Colorado, Boulder, Colorado 80302, U.S.A.
D. Alford
Affiliation:
Institute of Arctic and Alpine Research, University of Colorado, Boulder, Colorado 80302, U.S.A.
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Abstract

In many areas of glaciology, cumulative degree days, either positive or negative, are regressed against another cumulative value, such as ablation or lake-ice growth. Very strong functional relationships are frequently found with high correlation coefficients. This note shows that, if pairs of random numbers are cumulated, the resulting correlation coefficients are extremely high with a Fisher transformed mean of r = 0.986 and standard error of ±0.001 (based on 50 individual computations of r which in turn were based on 20 cumulated pairs of random numbers between 0 and 99). These results indicate that caution must be exercised in the physical interpretation of data of this kind.

Résumé

Résumé

Dans beaucoup de domaines de la glaciologie, des degrés-jours cumulés, positifs ou négatifs, sont analysés par régression en fonction d’autres valeurs cumulées, telles que l’ablation ou la croissance de la glace de lac. Des relations fonctionnelles très étroites sont fréquemment trouvées avec de hauts coefficients de corrélations. Cette note montre que, en cumulant les valeurs d’un couple de nombres choisis au hasard, les coefficients de corrélation résultants sont extrêmement élevés avec une moyenne transformée de Fisher de r = 0,986 et une erreur standard de ±0,001 (basé sur 50 calculs individuels de r qui est à son tour basé sur 20 jours cumulés de nombres choisis au hasard entre 0 et 99). Ces résultats mettent en évidence les précautions à prendre dans l’interprétation physique de données de cette sorte.

Zusammenfassung

Zusammenfassung

In vielen Gebieten der Glaziologie werden kumulative Zeitspannen, entweder positiv oder negativ, in Regression gegenüber anderen kumulativen Werten wie etwa der Ablation oder dem Wachstum des See-Eises gesetzt. Häufig ergeben sich dabei sehr starke funktionale Beziehungen mit hohen Korrelationskoeffizienten. Hier wird gezeigt, dass bei einer Kumulierung von Paaren zufälliger Zahlen die sich ergebenden Korrelationskoeffizienten extrem hoch sind mit einem nach Fisher transformierten Mittel von r = 0,986 und einem mittleren Fehler von ±0,001. (Diese Werte beruhen auf 50 einzelnen Berechnungen von r, die ihrerseits auf 20 kumulative Paare von zufälligen Zahlen zwischen 0 und 99 gestützt sind.) Diese Ergebnisse zeigen, dass bei der physikalischen Interpretation von Daten dieser Art besondere Vorsicht gehoten ist.

Type
Research Article
Information
Journal of Glaciology , Volume 10 , Issue 58 , 1971 , pp. 145 - 147
Copyright
Copyright © International Glaciological Society 1971

Introduction

Measurement of the components of the surface-energy flux as they influence the growth or melt of ice in its various forms has yet to become a standardized technique. A number of indices of this energy flux, as it influences glaciological processes, have been proposed. Perhaps the most widely used of these is the “cumulative degree-day” index which is based upon the easily, if not always accurately, measured parameter of air temperature. In terms of this index, the cumulative increase er decrease of the glaciological parameter in question is plotted as some function of the cumulative departures of some aspect (i.e. the term is not standardized) of the daily temperature regime from a selected reference temperature, normally 0°C. The departure of the daily mean temperature is most often used, although other aspects of the diurnal temperature cycle have been used (Reference Gartska, Gartska, Love, Goodell and BertieGartska and others, 1958). The strong functional relationship which generally results from these plots is taken as an indication that air temperature is, in fact, a valid index of the energy-exchange process as it influences the rate of change of the ice property in question.

The impetus for this short note was provided by the extremely high correlation coefficients, r, which several of us at the Institute of Arctic and Alpine Research had obtained in plotting a variety of bivariate cumulative frequency distributions involving the relationship of air temperature to ice accretion or ablation. The cumulative relationships commonly produced r values of >0.95. At the same time, values of r for non-cumulative values were frequently non-significant. We undertook, therefore, a study of the statistical significance of cumulative data by determining values of r produced by a bivariate cumulative frequency-distribution analysis of pairs of random numbers. It is the purpose of this note to discuss our findings and comment briefly on the "cumulative degree-days" concept as an index of the surface-energy flux.

Literature

A number of workers in the field of glaciology have proposed empirical equations relating ice accretion or ablation to air temperatures. Negative degree-days have been related to the rate of lake-ice formation (Reference Andrews and McCloughanAndrews and McCloughan, 1961; Reference RagleRagle, 1963; Reference JonesJones, 1969), with the formation of sea ice (Reference ZuboyZubov, 1945), with the thermal expansion and contraction of a lake-ice cover (Reference Wilson, Wilson, Zumberge and MarshallWilson and others, 1954) and with river-ice formation (Reference StevensStevens, 1940). Negative degree-days form the basis for equations predicting the rate of front penetration into the soil (Reference CarlsonCarlson, 1952; Reference Carlson and KerstenCarlson and Kersten. 1953; Reference AldrichAldrich. 1956). Positive degree-days, as an index of the energy required for ice ablation, have been used by Reference ClydeClyde (1929), Reference Church and MeinzerChurch (1942), Reference ZingsZingg (1951), Reference SchyttSchytt (1955), Reference Gartska, Gartska, Love, Goodell and BertieGartska and others (1958), Reference KeelerKeeler (1964), and Reference Outcalt and MacPhailOutcalt and MacPhail (1965). The general application of the technique has been discussed by Reference Ingersoll, Ingersoll, Zobel and IngersollIngersoll and others (1949) and Reference Linsley, Linsley, Kohler and PaulhausLinsley and others (1949). In every case, a significant statistical functional relationship was found to exist between cumulative air temperature, expressed in either positive or negative degree-days, and the rate of ice accretion or ablation. At the same time, there is little agreement among the various workers as to the quantitative relationship between the two. As reported by Reference Gartska and ChowGartska (1964), values of the point melt rate of snow vary by an order of magnitude among the various studies. There is a similar lack of agreement among the empirical equations derived for ice-growth prediction derived from cumulative degree-day studies.

Data and Analysis

To check the statistical significance of cumulative data, 50 groups of random numbers, each comprising 20 pairs, were summed and the data regressed for y on x (y = a + bx). The random numbers ranged in value from 0 to 99, which we feel is realistic in terms of the length of time and spread for actual data. Footnote *. The regressions were computed using an Olivetti 101 Programma and checked on a CDC 6400 series computer. The 50 runs provided distributions for r, a and b. The respective means and standard deviations are shown in Table I. Footnote †.The results show, as expected, that a→o and b→1. Correlation coefficients are extremely high, with a sample mean of 0.986. On the other hand, if the random numbers are run as non-cumulative samples, the mean value of r is 0.236 (N = 10 and r value not significant at the 95% level).

Table I. Descriptive statistics for sample distkiuutions of r, a and b

‡ Established from tables of Fisher’s Z (Reference Arkin and ColtonArkin and Colton, 1963, p. 127).

Some idea of the statistical significance of r values derived from cumulative distributions can be gamed by computing the standard error of the mean, σm.

For an N of 50, the standard error of Fisher’s Z is 0.0403. Re-converting back to r provides confidence limits which lie between 0.988 and 0.984.

Conclusions

The degree-day index has been used extensively as a substitute for more detailed energy-flux data. While we do not question the probable physical relationship that exists between air temperature and ice accretion/ablation, we consider that the computed values of r for this, or any other cumulative distribution have to be examined very critically. Our study suggests that virtually any two parameters can be plotted as cumulative functions of each other and produce high correlation coefficients, irrespective of the existence of any physical relationship. At the 95% level, the confidence limits for pairs of cumulative random numbers are 0.984 and 0.988. Any value of r less than this can probably be explained largely in terms of the “smoothing” introduced by the analytical technique itself. This would suggest that the cumulative degree-day index of net surface-energy flux should not be used in correlation studies.

We did not analyze in detail the significance of the correlation produced by plotting cumulative values of one parameter against incremental changes in the other. One instinctively feels that less procedural bias is introduced by this second method but only an extended analysis similar to that described here will determine if this is correct. Calculation indicates that r based on random numbers is ≈ 0.2 for such procedures.

A cursory check of the relevant literature has failed to produce any results similar to those presented here, although we suspect that the findings are not unknown amongst statisticians.

Acknowledgements

The authors would like to thank Dr Roger G. Barry for his contributions to discussions on this topic and his critical review of a draft of this manuscript. This paper was prepared in part under research for contract DA-ARO-D-31-124-G1163 (U.S. Army Research Office. Durham, U.S.A.).

MS. received 17 July 1970

Footnotes

page note * Values for r will be sensitive to the range of incremented values. If the numbers are large (> 100) and tin-record short (N ≈ 10), r values could be less than those presented here.

page note † Samples involving high r values tend to be non-normally distributed. Thus, observations of r were transformed to Fisher’s Z (Reference Arkin and ColtonArkin and Colton, 1963, p. 16), the distribution of which is normal even when the population value of r is large and the sample small.

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Figure 0

Table I. Descriptive statistics for sample distkiuutions of r, a and b