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Application of the Kolmogorov-Smirnov test to seasonal phenomena may be inappropriate.

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AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 68:393-394 (1985)
Application of the Kolmogorov-Smirnov Test to Seasonal
Phenomena May Be Inappropriate
JOHN M. McCULLOUGH
Department of Human Genetics, University of Newcastle-upon-Tyne,
Newcastle-upon-Tyne, U.K. NE2 4AA and Department ofdnthropology,
University of Utah, Salt Lake City, Utah 84112
KEY WORDS
Seasonality, Statistics
ABSTRACT
The Kolmogorov-Smirnov one-sample test is sometimes used
to test seasonality in births or other annually cyclic phenomena. However, it
is shown that the maximum deviation from expected (D) may differ by a factor
of almost two if the cycle is initiated in different months. Thus, different results
are possible from the same data. Unless there is a logical initiation point in a
cycle, it is probably better to use other nonparametric statistical tests.
The Kolmogorov-Smirnov one-sample test The Eskimo (Condon, 19821, the Yucatec
(Kolmogorov, 1933) has occasionally been Maya from 1920 to 1928 (McCullough, 1985)
used to test for presence of seasonality in and a ten percent sample of social class I
biological events, especially in births (Miura births for England and Wales, July 1963 to
and Richter, 1981; Stoeckel and Choudhury, June 1964 (James, 1971). Birth frequencies
1972;Zelnick, 1969).While the requirements were compared with expected frequencies for
and assumptions of the Kolmogorov-Smirnov each month, corrected for days in month. If
test are generous, allowing a test of any the- each study had statistical initiation in any
oretical distribution, there is one major as- month but January, results would be somesumption which may be violated, specifically what different. The ratio of maximum to
minimum value of D ranges from 1.77 for
that of range.
The Kolmogorov-Smirnov one-sample test England and Wales to 1.99 for Yucatan. Erassumes that theoretical and actual frequen- ror probability levels for Yucatan are consiscies will be accumulated from one end of an tently significant, those for England vary
ordinal or interval range to another. While slightly, while the Eskimo values are never
most probability distributions may be used significant by the Kolmogorov-Smirnov test,
to generate expected frequencies (i.e., uni- whereas by x2 test, all three samples’ distriform, Poisson, etc.),the range is still required. butions are shown t o be significantly differIn tests of seasonality, the distibution is ent from a uniform distribution. In these
accumulated over a time series (ordinarily in examples significance level shifts very little;
months), and because truly seasonal phe- in marginal cases the value of D may stradnomena are expected, in the short run, to be dle the error threshold as in the English and
cyclic, the test may theoretically begin at Welsh case. Slakter (1965), has already
any time during the year. While most studies pointed out that the x2 distribution probably
begin in January as a convenience, this is has a higher level of validity. If our assertion
not required, nor is there any a priori reason is correct, the Kolmogorov-Smirnov test is
why any calendar following the annual cycle also conservative to type I1 (p) error.
The presence of seasonality in small popucould not be used. This makes the starting
point arbitrary, and if the Kolmogorov-Smir- lations is probably best tested using the x2
nov test is used, initiating the cumulation at test (e.g., McCullough, 1985; Condon, 1982)
different months will (except in the trivial Wilcoxin’s “T” test (e.g., Malina and Himes,
case of exact concordance of actual with ex- 1977) or where normality may be assumed,
pected cumulations) necessarily lead to some- one-way analysis of variance or t test (Adair
what different results.
To test this assertion, the birth distributions of three samples are shown (Table 1): Received April 29, 1985; accepted June 7,1985
0 1985 ALAN R. LISS, INC
394
J.M. McCULLOUGK
TABLE 1. Maximum deviations DJof expected from theoretical cumulative distributions
From three seDarate sarndesl
Beginning
month
Jan.
Feb.
Mar.
Apr.
May
Jun.
Jul.
Aug.
Sep.
Oct.
Nov.
Dec.
Eskimo
(Condon, 1982)
(n = 228)
Ticul Yucatecs
(McCullough, 19851
(n = 3.266)
- ,0549
-.0643’*
-.0868
-.0501
,0718
,0856
- .0589**
,0888
,0546
-.0470
,0469
,0608
-.0499
-.0530
-.0572**
- .0640**
- .0601*
*
-.0332**
.0550**
.0644*
.0515**
.0346**
-.0430**
- .0547**
Social Class I
England and Wales
July 1965 to June 1969
(James, 1971)
(n = 4.107)
-.0410**
- .04i 2 * *
-.0325**
- .0258**
.0237*
.0296**
.0387**
.0413**
.0370**
.0342**
.0306**
- .0233*
’Ratio of highest to lowest value of D is 1.89:1.99:1.77.
*p < .05.
**p < .01.
and Pollitt, 1983). For very large national
samples the methods utilized by James (1984)
seem most appropriate.
ACKNOWLEDGMENTS
This paper was written during a sabbatical
leave from the University of Utah and work
was partially supported by a grant from the
Research Committee, University of Utah. I
thank Karin A. Engstrom for bibliographic
assistance and Professor D.F. Roberts for
other kindnesses while I was a Visiting Fellow in the Department of Human Genetics,
University of Newcastle-upon-Tyne.
LITERATURE CITED
Adair, LS, and Pollitt, E (1983)Seasonal variation in pre
and p o s t . p ~ m maternal body measurements and in.
fant birth weights. Am. J. Phys. Anthropol. 62325221
Condon,RG (1982) Inuit natality rhythms in the Central
Canadian Arctic. J. Biosoc. Sci. 14:167-177.
James, WH (1971) Social class and season of birth. J.
Biosoc. Sci. 3:309-320.
James, WH (1984) Seasonality in the sex ratio of US
Black births. Ann. Hum. Biol. 11(1):67-69.
Kolmogorov, AN (1933) Sulla determinazione empirica
di una legge di distribuzione. G. Inst. Ital. Altuari
4:83-91.
McCullough, Jh4 (1985) Seasonality ofbirth in the humid
tropics: Is it the heat or the work? Unpublished
manuscript.
Malina, RM, and Himes, JH (1977)Seasonality of births
in a rural Zapotec municipio, 1945-1970. Hum. Biol.
49:125-137.
Miura, T, and Richter, J (1981) Changes in the seasonal
distribution of births in Gorlitz, Germany, during the
period between 1675 and 1816. Hum. Biol. 53:15-22.
Slakter, MJ (1965) Comparison of the Pearson Chi-square
and Kolmogorov’s goodness-of-fittests with respect to
validity. J. Amer. Stat. Assoc. 60:854-858.
J~ and ChoudhurJ’, AKMA
variation in births in rural East Pakistan. J. Biosoc.
Sci. 4:107-116.
Zelnick. M (19691Socioeconomic and seasonal variations
~~~~~.
in births: A replication. mil bank^ m memorial Fund
Quarterly 47:159-165.
Stoeckell
~
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