The SAS System

Skull Maximum
Breadth
of Skull
(mm)		 Basibregmatic
Height of
Skull (mm)		 Basialveolar
Length of
Skull (mm)		 Nasal Height
of Skull
(mm)		 Time Period LMaxBreadth LBasHeight LBasLength LNasHeight
1 131 138 89 49 4000 B.C. 4.87520 4.92725 4.48864 3.89182
2 125 131 92 48 4000 B.C. 4.82831 4.87520 4.52179 3.87120
3 131 132 99 50 4000 B.C. 4.87520 4.88280 4.59512 3.91202
4 119 132 96 44 4000 B.C. 4.77912 4.88280 4.56435 3.78419
5 136 143 100 54 4000 B.C. 4.91265 4.96284 4.60517 3.98898
6 138 137 89 56 4000 B.C. 4.92725 4.91998 4.48864 4.02535
7 139 130 108 48 4000 B.C. 4.93447 4.86753 4.68213 3.87120
8 125 136 93 48 4000 B.C. 4.82831 4.91265 4.53260 3.87120
9 131 134 102 51 4000 B.C. 4.87520 4.89784 4.62497 3.93183
10 134 134 99 51 4000 B.C. 4.89784 4.89784 4.59512 3.93183
11 129 138 95 50 4000 B.C. 4.85981 4.92725 4.55388 3.91202
12 134 121 95 53 4000 B.C. 4.89784 4.79579 4.55388 3.97029
13 126 129 109 51 4000 B.C. 4.83628 4.85981 4.69135 3.93183
14 132 136 100 50 4000 B.C. 4.88280 4.91265 4.60517 3.91202
15 141 140 100 51 4000 B.C. 4.94876 4.94164 4.60517 3.93183
16 131 134 97 54 4000 B.C. 4.87520 4.89784 4.57471 3.98898
17 135 137 103 50 4000 B.C. 4.90527 4.91998 4.63473 3.91202
18 132 133 93 53 4000 B.C. 4.88280 4.89035 4.53260 3.97029
19 139 136 96 50 4000 B.C. 4.93447 4.91265 4.56435 3.91202
20 132 131 101 49 4000 B.C. 4.88280 4.87520 4.61512 3.89182
21 126 133 102 51 4000 B.C. 4.83628 4.89035 4.62497 3.93183
22 135 135 103 47 4000 B.C. 4.90527 4.90527 4.63473 3.85015
23 134 124 93 53 4000 B.C. 4.89784 4.82028 4.53260 3.97029
24 128 134 103 50 4000 B.C. 4.85203 4.89784 4.63473 3.91202
25 130 130 104 49 4000 B.C. 4.86753 4.86753 4.64439 3.89182
26 138 135 100 55 4000 B.C. 4.92725 4.90527 4.60517 4.00733
27 128 132 93 53 4000 B.C. 4.85203 4.88280 4.53260 3.97029
28 127 129 106 48 4000 B.C. 4.84419 4.85981 4.66344 3.87120
29 131 136 114 54 4000 B.C. 4.87520 4.91265 4.73620 3.98898
30 124 138 101 46 4000 B.C. 4.82028 4.92725 4.61512 3.82864
31 124 138 101 48 3300 B.C. 4.82028 4.92725 4.61512 3.87120
32 133 134 97 48 3300 B.C. 4.89035 4.89784 4.57471 3.87120
33 138 134 98 45 3300 B.C. 4.92725 4.89784 4.58497 3.80666
34 148 129 104 51 3300 B.C. 4.99721 4.85981 4.64439 3.93183
35 126 124 95 45 3300 B.C. 4.83628 4.82028 4.55388 3.80666
36 135 136 98 52 3300 B.C. 4.90527 4.91265 4.58497 3.95124
37 132 145 100 54 3300 B.C. 4.88280 4.97673 4.60517 3.98898
38 133 130 102 48 3300 B.C. 4.89035 4.86753 4.62497 3.87120
39 131 134 96 50 3300 B.C. 4.87520 4.89784 4.56435 3.91202
40 133 125 94 46 3300 B.C. 4.89035 4.82831 4.54329 3.82864
41 133 136 103 53 3300 B.C. 4.89035 4.91265 4.63473 3.97029
42 131 139 98 51 3300 B.C. 4.87520 4.93447 4.58497 3.93183
43 131 136 99 56 3300 B.C. 4.87520 4.91265 4.59512 4.02535
44 138 134 98 49 3300 B.C. 4.92725 4.89784 4.58497 3.89182
45 130 136 104 53 3300 B.C. 4.86753 4.91265 4.64439 3.97029
46 131 128 98 45 3300 B.C. 4.87520 4.85203 4.58497 3.80666
47 138 129 107 53 3300 B.C. 4.92725 4.85981 4.67283 3.97029
48 123 131 101 51 3300 B.C. 4.81218 4.87520 4.61512 3.93183
49 130 129 105 47 3300 B.C. 4.86753 4.85981 4.65396 3.85015
50 134 130 93 54 3300 B.C. 4.89784 4.86753 4.53260 3.98898
51 137 136 106 49 3300 B.C. 4.91998 4.91265 4.66344 3.89182
52 126 131 100 48 3300 B.C. 4.83628 4.87520 4.60517 3.87120
53 135 136 97 52 3300 B.C. 4.90527 4.91265 4.57471 3.95124
54 129 126 91 50 3300 B.C. 4.85981 4.83628 4.51086 3.91202
55 134 139 101 49 3300 B.C. 4.89784 4.93447 4.61512 3.89182
56 131 134 90 53 3300 B.C. 4.87520 4.89784 4.49981 3.97029
57 132 130 104 50 3300 B.C. 4.88280 4.86753 4.64439 3.91202
58 130 132 93 52 3300 B.C. 4.86753 4.88280 4.53260 3.95124
59 135 132 98 54 3300 B.C. 4.90527 4.88280 4.58497 3.98898
60 130 128 101 51 3300 B.C. 4.86753 4.85203 4.61512 3.93183
61 137 141 96 52 1850 B.C. 4.91998 4.94876 4.56435 3.95124
62 129 133 93 47 1850 B.C. 4.85981 4.89035 4.53260 3.85015
63 132 138 87 48 1850 B.C. 4.88280 4.92725 4.46591 3.87120
64 130 134 106 50 1850 B.C. 4.86753 4.89784 4.66344 3.91202
65 134 134 96 45 1850 B.C. 4.89784 4.89784 4.56435 3.80666
66 140 133 98 50 1850 B.C. 4.94164 4.89035 4.58497 3.91202
67 138 138 95 47 1850 B.C. 4.92725 4.92725 4.55388 3.85015
68 136 145 99 55 1850 B.C. 4.91265 4.97673 4.59512 4.00733
69 136 131 92 46 1850 B.C. 4.91265 4.87520 4.52179 3.82864
70 126 136 95 56 1850 B.C. 4.83628 4.91265 4.55388 4.02535
71 137 129 100 53 1850 B.C. 4.91998 4.85981 4.60517 3.97029
72 137 139 97 50 1850 B.C. 4.91998 4.93447 4.57471 3.91202
73 136 126 101 50 1850 B.C. 4.91265 4.83628 4.61512 3.91202
74 137 133 90 49 1850 B.C. 4.91998 4.89035 4.49981 3.89182
75 129 142 104 47 1850 B.C. 4.85981 4.95583 4.64439 3.85015
76 135 138 102 55 1850 B.C. 4.90527 4.92725 4.62497 4.00733
77 129 135 92 50 1850 B.C. 4.85981 4.90527 4.52179 3.91202
78 134 125 90 60 1850 B.C. 4.89784 4.82831 4.49981 4.09434
79 138 134 96 51 1850 B.C. 4.92725 4.89784 4.56435 3.93183
80 136 135 94 53 1850 B.C. 4.91265 4.90527 4.54329 3.97029
81 132 130 91 52 1850 B.C. 4.88280 4.86753 4.51086 3.95124
82 133 131 100 50 1850 B.C. 4.89035 4.87520 4.60517 3.91202
83 138 137 94 51 1850 B.C. 4.92725 4.91998 4.54329 3.93183
84 130 127 99 45 1850 B.C. 4.86753 4.84419 4.59512 3.80666
85 136 133 91 49 1850 B.C. 4.91265 4.89035 4.51086 3.89182
86 134 123 95 52 1850 B.C. 4.89784 4.81218 4.55388 3.95124
87 136 137 101 54 1850 B.C. 4.91265 4.91998 4.61512 3.98898
88 133 131 96 49 1850 B.C. 4.89035 4.87520 4.56435 3.89182
89 138 133 100 55 1850 B.C. 4.92725 4.89035 4.60517 4.00733
90 138 133 91 46 1850 B.C. 4.92725 4.89035 4.51086 3.82864

The SAS System

X 1 row 3 cols (numeric)

4000 3300 1850

P 3 rows 3 cols (numeric)

0.5773502692 0.5773502692 0.5773502692
0.6125845862 0.1612064701 -0.773791056
0.5398210735 -0.80042435 0.2606032769

The SAS System

The DISCRIM Procedure

Observations 90 DF Total 89
Variables 4 DF Within Classes 87
Classes 3 DF Between Classes 2

Class Level Information
TimePeriod Variable
Name		 Frequency Weight Proportion Prior
Probability
1850 B.C. _1850_B_C_ 30 30.0000 0.333333 0.333333
3300 B.C. _3300_B_C_ 30 30.0000 0.333333 0.333333
4000 B.C. _4000_B_C_ 30 30.0000 0.333333 0.333333

Within Covariance Matrix Information
TimePeriod Covariance
Matrix Rank		 Natural Log of the
Determinant of the
Covariance Matrix
1850 B.C. 4 11.22162
3300 B.C. 4 11.03211
4000 B.C. 4 11.47659
Pooled 4 11.50244

The SAS System

The DISCRIM Procedure
Test of Homogeneity of Within Covariance Matrices

Chi-Square DF Pr > ChiSq
21.048436 20 0.3943

Since the Chi-Square value is not significant at the 0.1 level, a pooled covariance matrix will be used in the discriminant function.
Reference: Morrison, D.F. (1976) Multivariate Statistical Methods p252.


The SAS System

The DISCRIM Procedure

Generalized Squared Distance to TimePeriod
From TimePeriod 1850 B.C. 3300 B.C. 4000 B.C.
1850 B.C. 0 0.68856 0.95669
3300 B.C. 0.68856 0 0.11112
4000 B.C. 0.95669 0.11112 0

Linear Discriminant Function for TimePeriod
Variable Label 1850 B.C. 3300 B.C. 4000 B.C.
Constant   -914.96753 -907.01411 -907.25409
MaxBreadth Maximum Breadth of Skull (mm) 5.45980 5.35068 5.29059
BasHeight Basibregmatic Height of Skull (mm) 5.06085 5.00721 5.04903
BasLength Basialveolar Length of Skull (mm) 3.13509 3.26866 3.27286
NasHeight Nasal Height of Skull (mm) 2.32484 2.33918 2.38241

The SAS System

The DISCRIM Procedure
Classification Summary for Calibration Data: WORK.EGYPTIANSKULLS
Resubstitution Summary using Linear Discriminant Function

Number of Observations and Percent Classified
into TimePeriod
From TimePeriod 1850 B.C. 3300 B.C. 4000 B.C. Total
1850 B.C. 20
66.67		 5
16.67		 5
16.67		 30
100.00
3300 B.C. 11
36.67		 9
30.00		 10
33.33		 30
100.00
4000 B.C. 8
26.67		 8
26.67		 14
46.67		 30
100.00
Total 39
43.33		 22
24.44		 29
32.22		 90
100.00
Priors 0.33333
 0.33333
 0.33333

Error Count Estimates for TimePeriod
  1850 B.C. 3300 B.C. 4000 B.C. Total
Rate 0.3333 0.7000 0.5333 0.5222
Priors 0.3333 0.3333 0.3333  

The SAS System

The GLM Procedure

Class Level Information
Class Levels Values
TimePeriod 3 4000 B.C. 3300 B.C. 1850 B.C.

Number of Observations Read 90
Number of Observations Used 90

The SAS System

The GLM Procedure
 
Dependent Variable: MaxBreadth Maximum Breadth of Skull (mm)

Source DF Sum of Squares Mean Square F Value Pr > F
Model 2 150.200000 75.100000 3.66 0.0298
Error 87 1785.400000 20.521839    
Corrected Total 89 1935.600000      

R-Square Coeff Var Root MSE MaxBreadth Mean
0.077599 3.412936 4.530104 132.7333

Source DF Type I SS Mean Square F Value Pr > F
TimePeriod 2 150.2000000 75.1000000 3.66 0.0298

Source DF Type III SS Mean Square F Value Pr > F
TimePeriod 2 150.2000000 75.1000000 3.66 0.0298

Contrast DF Contrast SS Mean Square F Value Pr > F
Linear TimePeriod 1 150.1983368 150.1983368 7.32 0.0082
Quadratic TimePeriod 1 0.0016632 0.0016632 0.00 0.9928

The SAS System

The GLM Procedure
 
Dependent Variable: BasHeight Basibregmatic Height of Skull (mm)

Source DF Sum of Squares Mean Square F Value Pr > F
Model 2 20.600000 10.300000 0.47 0.6293
Error 87 1924.300000 22.118391    
Corrected Total 89 1944.900000      

R-Square Coeff Var Root MSE BasHeight Mean
0.010592 3.526383 4.703019 133.3667

Source DF Type I SS Mean Square F Value Pr > F
TimePeriod 2 20.60000000 10.30000000 0.47 0.6293

Source DF Type III SS Mean Square F Value Pr > F
TimePeriod 2 20.60000000 10.30000000 0.47 0.6293

Contrast DF Contrast SS Mean Square F Value Pr > F
Linear TimePeriod 1 2.69719335 2.69719335 0.12 0.7278
Quadratic TimePeriod 1 17.90280665 17.90280665 0.81 0.3708

The SAS System

The GLM Procedure
 
Dependent Variable: BasLength Basialveolar Length of Skull (mm)

Source DF Sum of Squares Mean Square F Value Pr > F
Model 2 190.288889 95.144444 3.84 0.0251
Error 87 2153.000000 24.747126    
Corrected Total 89 2343.288889      

R-Square Coeff Var Root MSE BasLength Mean
0.081206 5.071572 4.974648 98.08889

Source DF Type I SS Mean Square F Value Pr > F
TimePeriod 2 190.2888889 95.1444444 3.84 0.0251

Source DF Type III SS Mean Square F Value Pr > F
TimePeriod 2 190.2888889 95.1444444 3.84 0.0251

Contrast DF Contrast SS Mean Square F Value Pr > F
Linear TimePeriod 1 174.0152807 174.0152807 7.03 0.0095
Quadratic TimePeriod 1 16.2736082 16.2736082 0.66 0.4196

The SAS System

The GLM Procedure
 
Dependent Variable: NasHeight Nasal Height of Skull (mm)

Source DF Sum of Squares Mean Square F Value Pr > F
Model 2 2.0222222 1.0111111 0.10 0.9007
Error 87 840.2000000 9.6574713    
Corrected Total 89 842.2222222      

R-Square Coeff Var Root MSE NasHeight Mean
0.002401 6.160534 3.107647 50.44444

Source DF Type I SS Mean Square F Value Pr > F
TimePeriod 2 2.02222222 1.01111111 0.10 0.9007

Source DF Type III SS Mean Square F Value Pr > F
TimePeriod 2 2.02222222 1.01111111 0.10 0.9007

Contrast DF Contrast SS Mean Square F Value Pr > F
Linear TimePeriod 1 0.16496882 0.16496882 0.02 0.8963
Quadratic TimePeriod 1 1.85725341 1.85725341 0.19 0.6621

The SAS System

The GLM Procedure
Multivariate Analysis of Variance

Characteristic Roots and Vectors of: E Inverse * H, where
H = Type III SSCP Matrix for TimePeriod
E = Error SSCP Matrix
Characteristic Root Percent Characteristic Vector V'EV=1
MaxBreadth BasHeight BasLength NasHeight
0.18696913 92.61 -0.01731044 -0.00316643 0.01599700 0.00492103
0.01491287 7.39 -0.01243028 0.01722156 -0.00745836 0.01318513
0.00000000 0.00 0.00150700 -0.00942315 0.00123927 0.03287969
0.00000000 0.00 0.01189934 0.01182951 0.01252783 0.00000000

MANOVA Test Criteria and F Approximations for
the Hypothesis of No Overall TimePeriod Effect
H = Type III SSCP Matrix for TimePeriod
E = Error SSCP Matrix

S=2 M=0.5 N=41
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.83010268 2.05 8 168 0.0436
Pillai's Trace 0.17221185 2.00 8 170 0.0489
Hotelling-Lawley Trace 0.20188200 2.11 8 117.7 0.0405
Roy's Greatest Root 0.18696913 3.97 4 85 0.0053
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
NOTE: F Statistic for Wilks' Lambda is exact.

Characteristic Roots and Vectors of: E Inverse * H, where
H = Contrast SSCP Matrix for Linear TimePeriod
E = Error SSCP Matrix
Characteristic Root Percent Characteristic Vector V'EV=1
MaxBreadth BasHeight BasLength NasHeight
0.18220788 100.00 -0.01788239 -0.00234325 0.01562392 0.00554295
0.00000000 0.00 -0.00268462 0.02284829 0.00035429 0.00012569
0.00000000 0.00 -0.00283228 -0.00249704 -0.00185431 0.03533268
0.00000000 0.00 0.01621789 -0.00125357 0.01491115 0.00000000

MANOVA Test Criteria and Exact F Statistics for the
Hypothesis of No Overall Linear TimePeriod Effect
H = Contrast SSCP Matrix for Linear TimePeriod
E = Error SSCP Matrix

S=1 M=1 N=41
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.84587492 3.83 4 84 0.0066
Pillai's Trace 0.15412508 3.83 4 84 0.0066
Hotelling-Lawley Trace 0.18220788 3.83 4 84 0.0066
Roy's Greatest Root 0.18220788 3.83 4 84 0.0066

Characteristic Roots and Vectors of: E Inverse * H, where
H = Contrast SSCP Matrix for Quadratic TimePeriod
E = Error SSCP Matrix
Characteristic Root Percent Characteristic Vector V'EV=1
MaxBreadth BasHeight BasLength NasHeight
0.01967411 100.00 -0.00179429 0.01640847 -0.01460652 0.00879582
0.00000000 0.00 0.02367931 -0.00011291 0.00010051 -0.00006053
0.00000000 0.00 -0.00525911 -0.00700031 0.00431574 0.03466653
0.00000000 0.00 -0.00253798 0.01473385 0.01542813 0.00000000

MANOVA Test Criteria and Exact F Statistics for the
Hypothesis of No Overall Quadratic TimePeriod Effect
H = Contrast SSCP Matrix for Quadratic TimePeriod
E = Error SSCP Matrix

S=1 M=1 N=41
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.98070549 0.41 4 84 0.7987
Pillai's Trace 0.01929451 0.41 4 84 0.7987
Hotelling-Lawley Trace 0.01967411 0.41 4 84 0.7987
Roy's Greatest Root 0.01967411 0.41 4 84 0.7987

The SAS System

The GLM Procedure
Least Squares Means
Adjustment for Multiple Comparisons: Tukey

TimePeriod MaxBreadth LSMEAN LSMEAN Number
4000 B.C. 131.366667 1
3300 B.C. 132.366667 2
1850 B.C. 134.466667 3

Least Squares Means for effect TimePeriod
Pr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: MaxBreadth
i/j 1 2 3
1   0.6700 0.0256
2 0.6700   0.1771
3 0.0256 0.1771  

TimePeriod BasHeight LSMEAN LSMEAN Number
4000 B.C. 133.600000 1
3300 B.C. 132.700000 2
1850 B.C. 133.800000 3

Least Squares Means for effect TimePeriod
Pr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: BasHeight
i/j 1 2 3
1   0.7397 0.9852
2 0.7397   0.6381
3 0.9852 0.6381  

TimePeriod BasLength LSMEAN LSMEAN Number
4000 B.C. 99.1666667 1
3300 B.C. 99.0666667 2
1850 B.C. 96.0333333 3

Least Squares Means for effect TimePeriod
Pr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: BasLength
i/j 1 2 3
1   0.9967 0.0438
2 0.9967   0.0528
3 0.0438 0.0528  

TimePeriod NasHeight LSMEAN LSMEAN Number
4000 B.C. 50.5333333 1
3300 B.C. 50.2333333 2
1850 B.C. 50.5666667 3

Least Squares Means for effect TimePeriod
Pr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: NasHeight
i/j 1 2 3
1   0.9259 0.9990
2 0.9259   0.9094
3 0.9990 0.9094