A CRITICAL REVIEW OF THE MEAN

MEASURE OF DIVERGENCE AND

MAHALANOBIS DISTANCES USING

ARTIFICIAL DATA AND NEW

APPROACHES TO THE ESTIMATION OF

BIODISTANCES EMPLOYING

NONMETRIC TRAITS

Fitch Laboratory, British School at Athens,

52 Souidias Street, 106 76 Athens, GREECE Dr. E. Nikita

AIM

Review the two most common distance measures for the calculation of biodistances based on nonmetric traits:

- Mean measure of divergence (MMD)

- Tetrachoric Mahalanobis D2 distance (TMD)

Propose two new approaches for the estimation of biodistances from nonmetric traits:

- Direct application of the Mahalanobis distance to ordinally recorded data (OMD)

- Approximation of the covariances of the Mahalanobis distance by Pearson correlation coefficients (RMD)

DENTAL NONMETRIC TRAITS

Upper incisors –

shovel-shaped

Lower molars –

tuberculum sextum

Upper molars –

Carabelli’s trait

Lower premolars –

Tome’s root

CONTINUOUS VARIATION TURNED

ORDINAL AND THEN BINARY

BIODISTANCE MEASURES IN SKELETAL

ANALYSIS OF NONMETRIC TRAITS

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CORRECTION FOR SMALL SAMPLE SIZES

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Mahalanobis-type distances

LIMITATIONS

Highly correlated and nondiagnostic traits

should not be analysed using the MMD

Serious reduction of the data being

analysed

TMD should be used cautiously when the

amount of missing data is large

Problematic in archaeological samples

ADDITIONAL CONSIDERATIONS

How effective are these measures in relation to

the information being lost from a statistical point

of view by the use of binary dichotomies for the

coding of an underlying continuous phenomenon

– the genetic variation that gives rise to the

expression of nonmetric traits?

MATERIALS

Artificial datasets that follow the multivariate normal

distribution of known variance-covariance. Two variance-

covariance matrices were used, one (CT) derived from

cranial data from North African populations, and the other

(DT) from dental data recorded on Cretan EBA samples.

For the transformation of each continuous dataset to a

binary one, the threshold value was the mean value. For the

transformation to four-rank ordinal data, the threshold

values were the first quartile, the median and the third

quartile.

The effect of missing values was examined by using 20%

missing values randomly affecting all traits.

BIODISTANCE COMPARISONS –

POPULATION DATA

DT matrix, (o) MD, (+) TMD, (x) RMD, () OMD,

and () MMD

• TMD tends to coincide

with the MD

• RMD and OMD differ

slightly from the MD

• MMD tends to exhibit

the greatest

divergence from the

remaining measures

BIODISTANCE COMPARISONS –

SAMPLE DATA

o All distances perform

satisfactorily, although the

OMD appears to have the

best performance

o TMD seems to deteriorate

when the number of traits is

large

Correlation coefficients between MD and

OMD (), RMD (o), TMD (+), and MMD (x) for

different datasets without missing values and

with r = 30 (A) and r = 10 (B) traits.

All examined biodistances

recover correct phenotypic Fst

values when populations are being

compared

When samples are being

analysed, all biodistances except

for RMD and TMD, give accurate

Fst values, provided that a

correction for small sample sizes

has been adopted

• Matrix CT, r = 10 (A) and 30 (B)

• The distances have been calculated in the

absence (+), (x), (Δ) and in the presence (o) of

missing values

• Scaled Fst values are shown by (x) and (Δ)

BIODISTANCE COMPARISONS –

FIXATION INDEX

For the assessment of the statistical significance of

the various biodistances, p-values can be calculated

either using permutations or using test statistics,

whenever these are available in the literature.

P-VALUE COMPARISONS

P-VALUE COMPARISONS Continuous data – CMD/MD

• Sjøvold’s test statistic

gives identical results to

the permutation method

• Rao’s test statistic gives

results divergent from

the permutation method

• The chi-squared test

statistic gives results

convergent to

permutations when the

number of traits is small Matrix CT, r = 30 and 10 (inset). p-values calculated from

permutations (o) and the test statistics: Sjovold (+), Rao

(), and chi-squared (x)

Ordinal data – COMD/OMD

• Sjøvold’s test statistic gives results

identical to the permutation method

provided that there are no missing

values

• In the absence of missing values, Rao’s

test statistic and the chi-squared test

statistic yield reliable results when the

number of traits is small

• The presence of missing values affects

the performance of all test statistics

Matrix DT, r = 30 and 10 (inset) without missing values

(A), and with 20% missing values (B). p-values

calculated from permutations (o) and the test statistics:

Sjovold (+), Rao (), and chi-squared (x)

P-VALUE COMPARISONS

Binary data – MMD

Satisfactory convergence

between the p-values

estimated from the test statistic

proposed for MMD and those

of the permutation methods,

regardless of the number of

traits considered or whether or

not missing values are present

P-values using permutations (o) and the test

statistic proposed for MMD () on matrix CT

with r=10 (A) r=30 (B). Insets correspond to

the same datasets with 20% missing values

P-VALUE COMPARISONS

Binary data – TMD/RMD

The p-values obtained from all

test statistics used for

Mahalanobis-type distances never

converge with those obtained

from permutations, regardless of

the number of traits, sample sizes,

presence or absence of missing

values, or whether the traits are

strongly correlated or not.

P-values for the (A) TMD and (B) RMD using

matrix CP, r = 10. p-values calculated from

permutations (o) and the test statistics: Sjøvold

(+), Rao (), and chi-squared (x)

P-VALUE COMPARISONS

CONCLUSIONS

All biodistances give fairly accurate results when the

number of traits is small.

When the number of traits is large, problems arise with

the TMD and RMD, while MMD and COMD are not

affected

COMD exhibits the best performance under all conditions

The presence of missing values does not affect

significantly OMD and MMD, but it may affect TMD when

the number of traits is large.

AKNOWLEDGEMENTS

British School at Athens – Fitch Laboratory

Professor Lyle Konigsberg

Professor Torstein Sjovold

An anonymous reviewer of the AJPA