Multivariate normality tests of planktonic foraminiferal data

Mathematical Geology, Vol. 11, No. 3, 1979

Multivariate Normality Tests of Planktonic Foraminiferal Data ~ Bjfrn A. M a l m g r e n 2

Data of a microfossil group, the planktonic foraminifera, have been tested to determine the conformity of various real data distributions to univariate and multivariate normality and the effects that standard transformations have upon the distributions. Studies of two bivariate samples, one trivariate sample, and two quadrivariate samples of size data indicate that distributions frequently deviate greatly from multivariate normality. Univariate distributions are generally positively skewed and show a tendency for leptokurtosis. A logarithmic transformation improved both univariate and multivariate distributions but the number of distributions conformable to normality increased only slightly--from zero to one in the multivariate case andfrom one tofour in the univariate case (totally 15 distributions). Aresine (p/lO0) ~ transformations of percentage data in two samples including 16 and 23 species, respectively, decreased highly significant deviationsfrom multivariate normality but distributions remained greatly non-normal. Although markedly positively skewed and leptokurtic univariate distributions were improved in most instances, the number of normal distributions (two) did not change. It follows that neither of the transformations caused significant increases in the number of normal distributions but if it is assumed that the consequences of non-normality are less severe as the deviation from normality decreases, the transformations are justified. KEY WORDS: multivariate normality, paleontology.

INTRODUCTION During the last few years there have been increasing applications of multivariate methods to the study of planktonic foraminifera. Multivariate methods have been used for determining associations of species and for the analyses of variation in morphology. Most multivariate methods assume that data are multivariate normally distributed but that they are sensitive to deviations from normality in different ways. It is the purpose of this article to test the null hypothesis of a multivariate normal distribution of planktonic foraminiferal data and the effects that standard transformations have upon the data and tests. Two kinds of data are analyzed: dimensional data measured from morphologic characteristics of foraminiferal tests, and percentage frequencies of foraminiferal species within 1 Manuscript received 3 January 1978; revised 26 October 1978. 2 Geologiska Institutionen, University of Stockholm, Box 6801, S- 113 86 Stockholm, Sweden.

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total assemblages. The transformations tested are the routinely applied logarithmic transformation (base 10) of dimensional data and arcsine (p/lO0) l/z (p = a percentage frequency) transformation of percentage data. The purpose of nonlinear transformations such as these is threefold: (1), to stabilize the variance, (2), to obtain additivity, and (3), to make the observations follow a normal distribution (Koch and Link, 1971). Percentage data follow the binomial distribution (Sokal and Rohlf, 1969) but the distributions are tested for normality since the normal distribution is the theoretical requirement for parametric statistical methods. The arcsine transformation of percentage data also serves to "open up" data arrays subject to the constant-sum constraint. This results in strong negative expected correlations between variables with large sample variances (Chayes, 1962, 1971). Reyment (1971) emphasized that the question of multivariate normality is one where the interests of theoretical statisticians and paleontologists tend to drift apart. Whereas normality is desirable from a statistical point of view, deviations from normality may reveal significant information about the geological history of a sample. In this report, the analyses are carried out from only a purely statistical point of view. The possible consequences of nonnormality upon various biometric methods are not discussed. MATERIALS AND METHODS Five samples of dimensional data measured on different species are analyzed from different areas and geological stages (Table 1). The tests comprise two bivariate cases, one trivariate case, and two quadrivariate cases (Fig. 1). The data on the Globigerinoides-Orbulina lineage are from Scott (1966, 1967) and the data on Globoconusa daubjergensis are from Malmgren (1974). Table 1. Samples of Dimensional Data Tested for Multivariate Normafity

Species

Age

Site

Number of Number of variables specimens

GlobigerinoidesOrbulina lineage

Miocene

Clifden Section S 167/645 (New Zealand)

2

521

Globoconusa daubjergensis

Danian

Core Limhamn D104 1.00 m (Sweden)

2

250

Globorotalia truncatulinoides

Quaternary

DSDP Leg 3, 3-15-1-6, 1.50 m (the South Atlantic Ocean)

3

300

Heterohelix striata

Maestrichtian "Dania" quarry (Denmark)

Globigerina bulloides

Quaternary

Eltanin 45-74, 1.20 m (the Indian Ocean)

4

350

4

403

Multivariate Normality Tests of Planktonic Foraminiferal Data

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