AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 84:93-97 (1991)
Notes and Comments An Historical Note on the t-Test for Differences in Sexual Dimorphism Between Populations
where the d, are Xll - X12he., the differences between the cell means across the 2 columns of the table), and the W,are weights given as:
Lyle W. Konigsberg Department of Anthropology University of Tennessee Knoxville, Tennessee 37996-0720
Greene (1989) and Relethford and Hodges (1985)have recently brought to the attention of physical anthro ologists a t-test for different levels of sexua dimorphism between two populations. One of the major advantages of this test is that it can be calculated directly from summary statistics. However, a few issues must be addressed before the test can be readil accepted. One critical issue is the relationsti of this t-test to standard statistical proce ures. A second major concern is the possibility of multivariate extensions, a subject recently addressed by Van Vark et al. (1989). A review of the relevant statistical literature demonstrates that Brandt (1933) and Yates (1934) were the first to derive and ublish a more general form of the test out[ned by Greene (1989) and Relethford and Hod es (1985).What Greene and Relethford and odges have derived is the test for interaction in a two-way unbalanced ANOVA (see Herr (1986)for a complete historical review). Greene and Relethford and Hodges present the s ecial2 x 2 case of the more general test or interaction in an r x 2 unbalanced ANOVA, which has been presented in both the r x 2 and 2 x 2 cases in a number of standard statistical texts (e.g., Steel and Torrie 1960: 270-274; Snedecor and Cochran 1967:483485; Winer 1971:402-403). Brandt (1933:170) and Yates (193456) noted that the interaction sum of squares for an r x 2 table could be given as:
r
cp
nil ni2
w. =
nil
(2)
+ ni2
where nil and ni2are the sample sizes in the two columns for the ith sample (i = 1 to r). Unfortunately, Brandt’s equation (his equation number 7) contained an error, and both Brandt’s and Yates’ notations were rather different from that used above (which follows Snedecor and Cochran 1967:485). The error (or within-cell) sum of squares is: r
SS(E)=
C [(nil
-
1)s;
+ (niz - 1)s;I
(3)
i=l
where the s2 are the within-cell variances. Dividing the two sums of squares b their degrees of freedom and formin an $-ratio ves a test of interaction wit r - 1 and - 2r de ees of freedom. Restricting r equal to 2 K s do Greene and Relethford and Hodges), Equation 1becomes:
a
8
a
and substituting w1 and w2 and with a little further algebra:
P
r
i=l
C wi i=l
@
1991 WILEY-LISS, INC.
Equation 5 when divided by the within-cell mean square (i.e., e uation 3 divided by N - 4 degrees of free om) ylelds an F-statistic which is the square of Greene’s and Relethford and Hodges’ t-test. Taking the less restrictive case of r > 2, equation 1 makes it possible to construct the more general omnibus F-test for differing levels of sexual dimorphism across multiple (more than two) populations. Because both Greene and Relethford and Hodges suggest that the primary advantage of their t-test is that it can be calculated from summary statistics, we might ask whether
8
94
L.W. KONIGSBERG ET AL
this would also be true for a multivariate extension of their t-test. Indeed, if the pooled within-group covariance matrix and cell sample sizes are available, or the pooled within-group correlation matrix, cell sample sizes, and within-group standard deviations are available, then the error sums of squares and cross products matrix (SSCP ) can be calculated using the same methotgiven in both Greene and Relethford and Hodges. If additional1 the within-group means are available, d e n the SSCP matrix due to interaction is given by a matrix extension of equation l:
male and female sample sizes within subSam les, J is an r by r matrix of ones (=&’) and SCPa, is the sums of squares and crossroducts matrix due to interaction calcufated in equation 6. The SSCP matrices calculated in equations 7 and 8 should be compared to an error SSCP matrix which is the sum of the ooled within-group SSCP matrix and the SCP matrix due to interactions (calculated in equation 6). Equation 7
b
sp
As an example of the multivariate test of different levels of sexual dimorphism, Table 1 contains the summar statistics for low density lip0 rotein (LD ) and apolipoprotein-B (apo- ) levels in 604 baboons measured on two different diets at the Southwest Foundation for Biomedical Research, San Antonio, Texas. The two diets were a basal diet and a high cholesterol, saturated fat diet (HCSF),and the baboons were classified into one of three groups on the basis of subspecies (Papio hamadryas anubis, P.h. cynocephalus, or anubistcynocephalus hybrid). Each animal was measured on each of the two diets, with LDL and apo-B quantified in milligrams per deciliter. Using the sample sizes, standard deviations, and pooled within-group correlation matrix from Table 1,the pooled within-group SSCP matrix can be written as SSCPE= TRT, where R is the pooled within-group correlation matrix and T is a diagonal matrix with the square roots of the pooled sums of squares of the traits on the main diagonal. Applying equation 6 to calculate the interaction SSCP matrix, and taking the ratio of the determinant of SSCPE to the determinant of SSCP, + SSCPIyields a Wilks’ lambda for the sex-by-subspecies interaction equal to 0.9820 (Table 2). When converted to Rao’s R and compared to an F-value with 8 and 1190 de rees of freedom, a probability of 0.2108 is o tained, indicating that there is no si ificant sex-by-subs ecies interaction. T e inter retation of t is result is that sexual dimor ism in LDL and apo-B does not differ signi icantly across the subspecies. Assuming no interaction, the approximate F-value for the sex main effect is 10.4060 (p < 0.0001) and for the subs ecies effect is 7.5486 (p < 0.0001). The hig ly significant F-tests for sex and for subspecies indicate that there is significant sexual dimorphism as well as significant differences between the subs ecies. All of the approximate F-values an Wilk‘s lambdas discussed
E
li
where D is an r by p Eiatrix of differences bxtween male means (M) and female means (F), with r equal to the number of malet female Sam les and p equal to the number of traits, w is t e r by 1vector of weights, 1, is a p by 1vector of ones, 1, is an r by 1 vector of ones, and o is the Hadamard product operator (element-by-element matrix multiplication). The ratio of the determinant of the SSCPEto the determinant of the sum of the SSCP and SSCPI matrices yields Wilks’ lambrfa. Lambda can be converted to an approximate F-value (Rao’sR, see e.g., Morrison, 1990 or Tatsuoka, 1971) to test whether or not the interaction is significant. The test has been used reviously by Key and Jantz (1981:250), w o noted for their particular case “that the site”sex interaction is not significant, indicatin that there is not a site difference in sexual imorphism.” Ifthe interaction is not significant h e . , the levels of sexual dimorphism are homogenous across samples), then the SSCP matrix due to sex can be calculated as:
K
R f
SSCP,,,,,
D‘ww’D
= ____
1:w
(7)
while the SSCP matrix due to differences across samples is: sscP,,,m,,,,
= XLSi
+ xlfFlkrn
xm
(8)
where is equal to mlbsm and X, is equal to flL@F,m and fare the r by one vectors of
%
8“
E
Pg
E
B
95
NOTES AND COMMENTS
TABLE 1. Summary statistics for low density lipoprotein (LDL) and apolipoprotein B (ApoB) levels measured on two diets in 604 baboons Sex
Male
Female
Subspecies
N
Anubis
78
Hybrid
90
Cynocephalus
22
Anubis
259
Hybrid
111
Cynocephalus
44
LDL basal
LDL HCSF'
46.44872 (17.1983) 43.2333 (17.4749) 40.9091 (20.8599) 55.0232 (17.1913) 48.5135 (20.7596) 48.5227 (23.6992)
97.4231 39.3833 (41.4364) (13.9573) 106.1000 37.6755 (54.5311) (12.9743) 87.5454 35.9909 (40.6644) (16.0057) 105.7490 49.0575 (14.7880) (44.1428) 101.9189 42.9811 (51.2467) (13.9412) 114.7045 42.1795 (65.2397) (13.4013) Pooled within-crrom - - correlation matrix LDL ApoB HCSF basal
LDL basal LDL-basal LDL-HCSF ApoB-basal APOB-HCSF
ApoB basal
1.000000 0.487002 0.681163 0.476761
0.487002 1.000000 0.326327 0.804516
0.681163 0.326327 1.000000 0.399609
ApoB HCSF 57.9590 (21.8897) 58.1111 (22.4638) 47.8772 (16.8267) 67.7031 (21.4590) 59.3252 (20.8992) 58.4773 (24.0234)
ApoB HCSF 0.476761 0.804516 0.399609 1.000000
'High cholesterol saturated fat diet. 'Means and standard deviations, with standard deviations in parentheses. All values reported in mg. per dl.
T A B L E 2. Results o f multiple analyses o f variance from summary statistics for low density lipoprotein (LDL) and apolipoprotein B (ApoB) levels measured on two diets in 604 baboons Effect
Wilk's Lamba
Rao's
R
df
Prob
Sex*Subspecies Sex Subspecies
0.9820 0.9348 0.9060
1.3581 10.4061 7.5486
81190 4; 597 8;1194
< 0.0001 < 0.0001
0.2108
above are identical with those calculated bility to the multiple population and multifrom the raw data usin a standard statisti- variate case. This is not to suggest that this test, or for that matter the Greene or Relethcal acka e (SYSTAT, ilkinson, 1988). &e ta&les of means, sample sizes, and ford and Hodges tests are in any way new. To pooled within-group covariance matrices (or the contrary, as has been shown here, these standard deviations and ooled within- statistics have been presented previously as oup correlation matrices) requently pub- tests for interaction in an unbalanced Khed in physical anthro ological mono- ANOVA or MANOVA. While the reco ition graphs (see e.g. LittlewooLP 1972; Howells of this fact may serve to popularize t e test, 1973; Heathcote 19861, provide a large body the more important issue is that we continue of summar statistical data which can be to publish (or make available on request) at used in mu tivariate tests of differing levels least the minimal summary statistics from of sexual dimorphism. Thus, while Greene which others can test various univariate or and Relethford and Hod es provided a multivariate hypotheses. Without making rather specific test limite to two popula- this information available we severely limit tions, two sexes, and one variable, the more the future usefulness of our research and our general version of the test has wide applica- hard-earned data.
b
P
8"
P
cf
96
L.W. KONIGSBERG ET AL. ACKNOWLEDGMENTS
I thank John Blangero, Susan R. Frankenberg, David L. Greene, John H. Relethford, and Gerritt N. van Vark for their comments on an earlier draft. This work was supported by NIH grant HL28972 and NIH contract HV53030 to the Southwest Foundation for Biomedical Research, and funds from the Southwest Foundation for Biomedical Research. The calculations reported in this note were performed using programs written in FORTRAN and GAUSS, and are available on request. LITERATURE CITED Brandt AE (1933) The analysis of variance in a ‘2 x s’ table with disproportionate frequencies. J . Am. Statist. Assoc. 28:164-173. Greene DL (1989)Comparison of t-tests for differences in sexual dimorphism between populations. Am. J. Phys. Anthropol. 79:121-125. Heathcote GM (1986) Exploratory Human Craniometry of Recent Eskaleutian Regional Groups from the Western Arctic and Subarctic of North America. BAR International Series No. 301. Oxford: BAR. Herr DG (1986)On the history ofANOVA in unbalanced, factorial designs: the first 30 years. Am. Statist. 40965-270. Howells WW (1973) Cranial Variation in Man: A Study by Multivariate Analysis of Patterns of Difference
Among Recent Human Populations. Papers of the Peabody Museum of Archaeology and Ethnology, Vol. 67. Cambridge: Harvard University. Key P, and Jantz RL (1981) A multivariate analysis of temporal change in Arikara craniometrics: A methodological approach. Am. J . Phys. Anthropol. 55.247259. Littlewood RA (1972)Physical Anthropology of the Eastern Highlands of New Guinea. Seattle: Univ. of Washington Press. Morrison DF (1990) Multivariate Statistical Methods, 3rd ed. New York: McGraw-Hill. Relethford JH, and Hodges D C (1985) A statistical test for differences in sexual dimorphism between populations. Am. J. Phys. Anthropol. 66:5541. Snedecor GW, and Cochran WG (1967) Statistical Methods, 6th ed. Ames: Iowa State Univ. Press. Steel RGD, and Torrie J H (1960) Principles and Procedures of Statistics. New York: McGraw-Hill. Tatsuoka MM (1971)Multivariate Analysis: Techniques for Educational and Psychological Research, New York: John Wiley. Van Vark GN, Van Der Sman PGM, Dijkema J , and Buikstra J E (1989) Some multivariate tests for differences in sexual dimorphism between human populations. Ann. Hum. Biol. 16:301-310. Wilkinson, L (1988) Systat: The System for Statistics. Evanston, IL: Systat, Inc. Winer BJ (1971) Statistical Principles in Experimental Designs, 2nd ed. New York McGraw-Hill. Yates F (1934) The analysis of multiple classifications with unequal numbers in the different classes. J. Am. Statist. Assoc. 29:5146.
since they provide working tools for those not versed in, or interested in, advanced statistical methods. The other major contribution shown here, and in earlier papers, is the utility of methods designed tb work’with summary statistics. While the extension of statistical methJohn H. Relethford ods for use with summary statistics, rather Injury Control and Disability Prevention than original data, are not new to statistiPrograms, Division of Epidemiology, New cians, they may be new to those of us that York State Department of Health, Albany, deal with applications. Relethford and HodgNY 12237; Department of Anthropology, es’ and Greene’s methods are useful since State University of New York at Albany, they are presented to work directly with Albany, Ny 12222 basic univariate summary statistics (sample Konigsberg’s paper is an excellent addi- size, mean, and standard deviation). Konigstion to the growing literature on methodolo- berg’s method shows how multivariate exg e s for assessing variation in sexual dimor- tension can be obtained with the addition of phism among human and nonhuman the entire within-group correlation matrix. primate populations. His paper provides the Researchers wishing to compare the results needed extension of the earlier methods of of their own data analyses with previous Relethford and Hodges, Greene, and others. studies need not always need to have access As with these pa ers, Konigsberg points out to the original data (which is often difficult that the metho s are not new, but rather or, over long periods of time, impossible). I different formulations and computations join Konigsberg in urging that we, when geared for the practicing biological anthroReceived for publication March 14, 1990, revision accepted pologist. Such contributions are valuable June 1,1990,
Reply to Konigsberg: “An Historical Note on the t-test for Differences in Sexual Dimorphism Between Populations”
2
Notes and Comments An Historical Note on the t-Test for Differences in Sexual Dimorphism Between Populations
where the d, are Xll - X12he., the differences between the cell means across the 2 columns of the table), and the W,are weights given as:
Lyle W. Konigsberg Department of Anthropology University of Tennessee Knoxville, Tennessee 37996-0720
Greene (1989) and Relethford and Hodges (1985)have recently brought to the attention of physical anthro ologists a t-test for different levels of sexua dimorphism between two populations. One of the major advantages of this test is that it can be calculated directly from summary statistics. However, a few issues must be addressed before the test can be readil accepted. One critical issue is the relationsti of this t-test to standard statistical proce ures. A second major concern is the possibility of multivariate extensions, a subject recently addressed by Van Vark et al. (1989). A review of the relevant statistical literature demonstrates that Brandt (1933) and Yates (1934) were the first to derive and ublish a more general form of the test out[ned by Greene (1989) and Relethford and Hod es (1985).What Greene and Relethford and odges have derived is the test for interaction in a two-way unbalanced ANOVA (see Herr (1986)for a complete historical review). Greene and Relethford and Hodges present the s ecial2 x 2 case of the more general test or interaction in an r x 2 unbalanced ANOVA, which has been presented in both the r x 2 and 2 x 2 cases in a number of standard statistical texts (e.g., Steel and Torrie 1960: 270-274; Snedecor and Cochran 1967:483485; Winer 1971:402-403). Brandt (1933:170) and Yates (193456) noted that the interaction sum of squares for an r x 2 table could be given as:
r
cp
nil ni2
w. =
nil
(2)
+ ni2
where nil and ni2are the sample sizes in the two columns for the ith sample (i = 1 to r). Unfortunately, Brandt’s equation (his equation number 7) contained an error, and both Brandt’s and Yates’ notations were rather different from that used above (which follows Snedecor and Cochran 1967:485). The error (or within-cell) sum of squares is: r
SS(E)=
C [(nil
-
1)s;
+ (niz - 1)s;I
(3)
i=l
where the s2 are the within-cell variances. Dividing the two sums of squares b their degrees of freedom and formin an $-ratio ves a test of interaction wit r - 1 and - 2r de ees of freedom. Restricting r equal to 2 K s do Greene and Relethford and Hodges), Equation 1becomes:
a
8
a
and substituting w1 and w2 and with a little further algebra:
P
r
i=l
C wi i=l
@
1991 WILEY-LISS, INC.
Equation 5 when divided by the within-cell mean square (i.e., e uation 3 divided by N - 4 degrees of free om) ylelds an F-statistic which is the square of Greene’s and Relethford and Hodges’ t-test. Taking the less restrictive case of r > 2, equation 1 makes it possible to construct the more general omnibus F-test for differing levels of sexual dimorphism across multiple (more than two) populations. Because both Greene and Relethford and Hodges suggest that the primary advantage of their t-test is that it can be calculated from summary statistics, we might ask whether
8
94
L.W. KONIGSBERG ET AL
this would also be true for a multivariate extension of their t-test. Indeed, if the pooled within-group covariance matrix and cell sample sizes are available, or the pooled within-group correlation matrix, cell sample sizes, and within-group standard deviations are available, then the error sums of squares and cross products matrix (SSCP ) can be calculated using the same methotgiven in both Greene and Relethford and Hodges. If additional1 the within-group means are available, d e n the SSCP matrix due to interaction is given by a matrix extension of equation l:
male and female sample sizes within subSam les, J is an r by r matrix of ones (=&’) and SCPa, is the sums of squares and crossroducts matrix due to interaction calcufated in equation 6. The SSCP matrices calculated in equations 7 and 8 should be compared to an error SSCP matrix which is the sum of the ooled within-group SSCP matrix and the SCP matrix due to interactions (calculated in equation 6). Equation 7
b
sp
As an example of the multivariate test of different levels of sexual dimorphism, Table 1 contains the summar statistics for low density lip0 rotein (LD ) and apolipoprotein-B (apo- ) levels in 604 baboons measured on two different diets at the Southwest Foundation for Biomedical Research, San Antonio, Texas. The two diets were a basal diet and a high cholesterol, saturated fat diet (HCSF),and the baboons were classified into one of three groups on the basis of subspecies (Papio hamadryas anubis, P.h. cynocephalus, or anubistcynocephalus hybrid). Each animal was measured on each of the two diets, with LDL and apo-B quantified in milligrams per deciliter. Using the sample sizes, standard deviations, and pooled within-group correlation matrix from Table 1,the pooled within-group SSCP matrix can be written as SSCPE= TRT, where R is the pooled within-group correlation matrix and T is a diagonal matrix with the square roots of the pooled sums of squares of the traits on the main diagonal. Applying equation 6 to calculate the interaction SSCP matrix, and taking the ratio of the determinant of SSCPE to the determinant of SSCP, + SSCPIyields a Wilks’ lambda for the sex-by-subspecies interaction equal to 0.9820 (Table 2). When converted to Rao’s R and compared to an F-value with 8 and 1190 de rees of freedom, a probability of 0.2108 is o tained, indicating that there is no si ificant sex-by-subs ecies interaction. T e inter retation of t is result is that sexual dimor ism in LDL and apo-B does not differ signi icantly across the subspecies. Assuming no interaction, the approximate F-value for the sex main effect is 10.4060 (p < 0.0001) and for the subs ecies effect is 7.5486 (p < 0.0001). The hig ly significant F-tests for sex and for subspecies indicate that there is significant sexual dimorphism as well as significant differences between the subs ecies. All of the approximate F-values an Wilk‘s lambdas discussed
E
li
where D is an r by p Eiatrix of differences bxtween male means (M) and female means (F), with r equal to the number of malet female Sam les and p equal to the number of traits, w is t e r by 1vector of weights, 1, is a p by 1vector of ones, 1, is an r by 1 vector of ones, and o is the Hadamard product operator (element-by-element matrix multiplication). The ratio of the determinant of the SSCPEto the determinant of the sum of the SSCP and SSCPI matrices yields Wilks’ lambrfa. Lambda can be converted to an approximate F-value (Rao’sR, see e.g., Morrison, 1990 or Tatsuoka, 1971) to test whether or not the interaction is significant. The test has been used reviously by Key and Jantz (1981:250), w o noted for their particular case “that the site”sex interaction is not significant, indicatin that there is not a site difference in sexual imorphism.” Ifthe interaction is not significant h e . , the levels of sexual dimorphism are homogenous across samples), then the SSCP matrix due to sex can be calculated as:
K
R f
SSCP,,,,,
D‘ww’D
= ____
1:w
(7)
while the SSCP matrix due to differences across samples is: sscP,,,m,,,,
= XLSi
+ xlfFlkrn
xm
(8)
where is equal to mlbsm and X, is equal to flL@F,m and fare the r by one vectors of
%
8“
E
Pg
E
B
95
NOTES AND COMMENTS
TABLE 1. Summary statistics for low density lipoprotein (LDL) and apolipoprotein B (ApoB) levels measured on two diets in 604 baboons Sex
Male
Female
Subspecies
N
Anubis
78
Hybrid
90
Cynocephalus
22
Anubis
259
Hybrid
111
Cynocephalus
44
LDL basal
LDL HCSF'
46.44872 (17.1983) 43.2333 (17.4749) 40.9091 (20.8599) 55.0232 (17.1913) 48.5135 (20.7596) 48.5227 (23.6992)
97.4231 39.3833 (41.4364) (13.9573) 106.1000 37.6755 (54.5311) (12.9743) 87.5454 35.9909 (40.6644) (16.0057) 105.7490 49.0575 (14.7880) (44.1428) 101.9189 42.9811 (51.2467) (13.9412) 114.7045 42.1795 (65.2397) (13.4013) Pooled within-crrom - - correlation matrix LDL ApoB HCSF basal
LDL basal LDL-basal LDL-HCSF ApoB-basal APOB-HCSF
ApoB basal
1.000000 0.487002 0.681163 0.476761
0.487002 1.000000 0.326327 0.804516
0.681163 0.326327 1.000000 0.399609
ApoB HCSF 57.9590 (21.8897) 58.1111 (22.4638) 47.8772 (16.8267) 67.7031 (21.4590) 59.3252 (20.8992) 58.4773 (24.0234)
ApoB HCSF 0.476761 0.804516 0.399609 1.000000
'High cholesterol saturated fat diet. 'Means and standard deviations, with standard deviations in parentheses. All values reported in mg. per dl.
T A B L E 2. Results o f multiple analyses o f variance from summary statistics for low density lipoprotein (LDL) and apolipoprotein B (ApoB) levels measured on two diets in 604 baboons Effect
Wilk's Lamba
Rao's
R
df
Prob
Sex*Subspecies Sex Subspecies
0.9820 0.9348 0.9060
1.3581 10.4061 7.5486
81190 4; 597 8;1194
< 0.0001 < 0.0001
0.2108
above are identical with those calculated bility to the multiple population and multifrom the raw data usin a standard statisti- variate case. This is not to suggest that this test, or for that matter the Greene or Relethcal acka e (SYSTAT, ilkinson, 1988). &e ta&les of means, sample sizes, and ford and Hodges tests are in any way new. To pooled within-group covariance matrices (or the contrary, as has been shown here, these standard deviations and ooled within- statistics have been presented previously as oup correlation matrices) requently pub- tests for interaction in an unbalanced Khed in physical anthro ological mono- ANOVA or MANOVA. While the reco ition graphs (see e.g. LittlewooLP 1972; Howells of this fact may serve to popularize t e test, 1973; Heathcote 19861, provide a large body the more important issue is that we continue of summar statistical data which can be to publish (or make available on request) at used in mu tivariate tests of differing levels least the minimal summary statistics from of sexual dimorphism. Thus, while Greene which others can test various univariate or and Relethford and Hod es provided a multivariate hypotheses. Without making rather specific test limite to two popula- this information available we severely limit tions, two sexes, and one variable, the more the future usefulness of our research and our general version of the test has wide applica- hard-earned data.
b
P
8"
P
cf
96
L.W. KONIGSBERG ET AL. ACKNOWLEDGMENTS
I thank John Blangero, Susan R. Frankenberg, David L. Greene, John H. Relethford, and Gerritt N. van Vark for their comments on an earlier draft. This work was supported by NIH grant HL28972 and NIH contract HV53030 to the Southwest Foundation for Biomedical Research, and funds from the Southwest Foundation for Biomedical Research. The calculations reported in this note were performed using programs written in FORTRAN and GAUSS, and are available on request. LITERATURE CITED Brandt AE (1933) The analysis of variance in a ‘2 x s’ table with disproportionate frequencies. J . Am. Statist. Assoc. 28:164-173. Greene DL (1989)Comparison of t-tests for differences in sexual dimorphism between populations. Am. J. Phys. Anthropol. 79:121-125. Heathcote GM (1986) Exploratory Human Craniometry of Recent Eskaleutian Regional Groups from the Western Arctic and Subarctic of North America. BAR International Series No. 301. Oxford: BAR. Herr DG (1986)On the history ofANOVA in unbalanced, factorial designs: the first 30 years. Am. Statist. 40965-270. Howells WW (1973) Cranial Variation in Man: A Study by Multivariate Analysis of Patterns of Difference
Among Recent Human Populations. Papers of the Peabody Museum of Archaeology and Ethnology, Vol. 67. Cambridge: Harvard University. Key P, and Jantz RL (1981) A multivariate analysis of temporal change in Arikara craniometrics: A methodological approach. Am. J . Phys. Anthropol. 55.247259. Littlewood RA (1972)Physical Anthropology of the Eastern Highlands of New Guinea. Seattle: Univ. of Washington Press. Morrison DF (1990) Multivariate Statistical Methods, 3rd ed. New York: McGraw-Hill. Relethford JH, and Hodges D C (1985) A statistical test for differences in sexual dimorphism between populations. Am. J. Phys. Anthropol. 66:5541. Snedecor GW, and Cochran WG (1967) Statistical Methods, 6th ed. Ames: Iowa State Univ. Press. Steel RGD, and Torrie J H (1960) Principles and Procedures of Statistics. New York: McGraw-Hill. Tatsuoka MM (1971)Multivariate Analysis: Techniques for Educational and Psychological Research, New York: John Wiley. Van Vark GN, Van Der Sman PGM, Dijkema J , and Buikstra J E (1989) Some multivariate tests for differences in sexual dimorphism between human populations. Ann. Hum. Biol. 16:301-310. Wilkinson, L (1988) Systat: The System for Statistics. Evanston, IL: Systat, Inc. Winer BJ (1971) Statistical Principles in Experimental Designs, 2nd ed. New York McGraw-Hill. Yates F (1934) The analysis of multiple classifications with unequal numbers in the different classes. J. Am. Statist. Assoc. 29:5146.
since they provide working tools for those not versed in, or interested in, advanced statistical methods. The other major contribution shown here, and in earlier papers, is the utility of methods designed tb work’with summary statistics. While the extension of statistical methJohn H. Relethford ods for use with summary statistics, rather Injury Control and Disability Prevention than original data, are not new to statistiPrograms, Division of Epidemiology, New cians, they may be new to those of us that York State Department of Health, Albany, deal with applications. Relethford and HodgNY 12237; Department of Anthropology, es’ and Greene’s methods are useful since State University of New York at Albany, they are presented to work directly with Albany, Ny 12222 basic univariate summary statistics (sample Konigsberg’s paper is an excellent addi- size, mean, and standard deviation). Konigstion to the growing literature on methodolo- berg’s method shows how multivariate exg e s for assessing variation in sexual dimor- tension can be obtained with the addition of phism among human and nonhuman the entire within-group correlation matrix. primate populations. His paper provides the Researchers wishing to compare the results needed extension of the earlier methods of of their own data analyses with previous Relethford and Hodges, Greene, and others. studies need not always need to have access As with these pa ers, Konigsberg points out to the original data (which is often difficult that the metho s are not new, but rather or, over long periods of time, impossible). I different formulations and computations join Konigsberg in urging that we, when geared for the practicing biological anthroReceived for publication March 14, 1990, revision accepted pologist. Such contributions are valuable June 1,1990,
Reply to Konigsberg: “An Historical Note on the t-test for Differences in Sexual Dimorphism Between Populations”
2
