The SAS System

crudeoil vanadium iron beryllium Saturated
hydrocarbons		 Aromatic hydrocabons Zones of sandstone Lvanadium Liron Lberyllium LSathydroCarb LAromhydroCarb
1 3.9 51.0 0.20 7.06 12.19 Wilhelm 1.36098 3.93183 -1.60944 1.95445 2.50062
2 2.7 49.0 0.07 7.14 12.23 Wilhelm 0.99325 3.89182 -2.65926 1.96571 2.50389
3 2.8 36.0 0.30 7.00 11.30 Wilhelm 1.02962 3.58352 -1.20397 1.94591 2.42480
4 3.1 45.0 0.08 7.20 13.01 Wilhelm 1.13140 3.80666 -2.52573 1.97408 2.56572
5 3.5 46.0 0.10 7.81 12.63 Wilhelm 1.25276 3.82864 -2.30259 2.05540 2.53607
6 3.9 43.0 0.07 6.25 10.42 Wilhelm 1.36098 3.76120 -2.65926 1.83258 2.34373
7 2.7 35.0 0.00 5.11 9.00 Wilhelm 0.99325 3.55535 . 1.63120 2.19722
8 5.0 47.0 0.07 7.06 6.10 SubMuli 1.60944 3.85015 -2.65926 1.95445 1.80829
9 3.4 32.0 0.20 5.82 4.69 SubMuli 1.22378 3.46574 -1.60944 1.76130 1.54543
10 1.2 12.0 0.00 5.54 3.15 SubMuli 0.18232 2.48491 . 1.71199 1.14740
11 8.4 17.0 0.07 6.31 4.55 SubMuli 2.12823 2.83321 -2.65926 1.84214 1.51513
12 4.2 36.0 0.50 9.25 4.95 SubMuli 1.43508 3.58352 -0.69315 2.22462 1.59939
13 4.2 35.0 0.50 5.69 2.22 SubMuli 1.43508 3.55535 -0.69315 1.73871 0.79751
14 3.9 41.0 0.10 5.63 2.94 SubMuli 1.36098 3.71357 -2.30259 1.72811 1.07841
15 3.9 36.0 0.07 6.19 2.27 SubMuli 1.36098 3.58352 -2.65926 1.82294 0.81978
16 7.3 32.0 0.30 8.02 12.92 SubMuli 1.98787 3.46574 -1.20397 2.08194 2.55878
17 4.4 46.0 0.07 7.54 5.76 SubMuli 1.48160 3.82864 -2.65926 2.02022 1.75094
18 3.0 30.0 0.00 5.12 10.77 SubMuli 1.09861 3.40120 . 1.63315 2.37676
19 6.3 13.0 0.50 4.24 8.27 Upper 1.84055 2.56495 -0.69315 1.44456 2.11263
20 1.7 5.6 1.00 5.69 4.64 Upper 0.53063 1.72277 0.00000 1.73871 1.53471
21 7.3 24.0 0.00 4.34 2.99 Upper 1.98787 3.17805 . 1.46787 1.09527
22 7.8 18.0 0.50 3.92 6.09 Upper 2.05412 2.89037 -0.69315 1.36609 1.80665
23 7.8 25.0 0.70 5.39 6.20 Upper 2.05412 3.21888 -0.35667 1.68455 1.82455
24 7.8 26.0 1.00 5.02 2.50 Upper 2.05412 3.25810 0.00000 1.61343 0.91629
25 9.5 17.0 0.05 3.52 5.71 Upper 2.25129 2.83321 -2.99573 1.25846 1.74222
26 7.7 14.0 0.30 4.65 8.63 Upper 2.04122 2.63906 -1.20397 1.53687 2.15524
27 11.0 20.0 0.50 4.27 8.40 Upper 2.39790 2.99573 -0.69315 1.45161 2.12823
28 8.0 14.0 0.30 4.32 7.87 Upper 2.07944 2.63906 -1.20397 1.46326 2.06306
29 8.4 18.0 0.20 4.38 7.98 Upper 2.12823 2.89037 -1.60944 1.47705 2.07694
30 10.0 18.0 0.10 3.06 7.67 Upper 2.30259 2.89037 -2.30259 1.11841 2.03732
31 7.3 15.0 0.05 3.76 6.84 Upper 1.98787 2.70805 -2.99573 1.32442 1.92279
32 9.5 22.0 0.30 3.98 5.02 Upper 2.25129 3.09104 -1.20397 1.38128 1.61343
33 8.4 15.0 0.20 5.02 10.12 Upper 2.12823 2.70805 -1.60944 1.61343 2.31451
34 8.4 17.0 0.20 4.42 8.25 Upper 2.12823 2.83321 -1.60944 1.48614 2.11021
35 9.5 25.0 0.50 4.44 5.95 Upper 2.25129 3.21888 -0.69315 1.49065 1.78339
36 7.2 22.0 1.00 4.70 3.49 Upper 1.97408 3.09104 0.00000 1.54756 1.24990
37 4.0 12.0 0.50 5.71 6.32 Upper 1.38629 2.48491 -0.69315 1.74222 1.84372
38 6.7 52.0 0.50 4.80 3.20 Upper 1.90211 3.95124 -0.69315 1.56862 1.16315
39 9.0 27.0 0.30 3.69 3.30 Upper 2.19722 3.29584 -1.20397 1.30563 1.19392
40 7.8 29.0 1.50 6.72 5.75 Upper 2.05412 3.36730 0.40547 1.90509 1.74920
41 4.5 41.0 0.50 3.33 2.27 Upper 1.50408 3.71357 -0.69315 1.20297 0.81978
42 6.2 34.0 0.70 7.56 6.93 Upper 1.82455 3.52636 -0.35667 2.02287 1.93586
43 5.6 20.0 0.50 5.07 6.70 Upper 1.72277 2.99573 -0.69315 1.62334 1.90211
44 9.0 17.0 0.20 4.39 8.33 Upper 2.19722 2.83321 -1.60944 1.47933 2.11986
45 8.4 20.0 0.10 3.74 3.77 Upper 2.12823 2.99573 -2.30259 1.31909 1.32708
46 9.5 19.0 0.50 3.72 7.37 Upper 2.25129 2.94444 -0.69315 1.31372 1.99742
47 9.0 20.0 0.50 5.97 11.17 Upper 2.19722 2.99573 -0.69315 1.78675 2.41323
48 6.2 16.0 0.05 4.23 4.18 Upper 1.82455 2.77259 -2.99573 1.44220 1.43031
49 7.3 20.0 0.50 4.39 3.50 Upper 1.98787 2.99573 -0.69315 1.47933 1.25276
50 3.6 15.0 0.70 7.00 4.82 Upper 1.28093 2.70805 -0.35667 1.94591 1.57277
51 6.2 34.0 0.07 4.84 2.37 Upper 1.82455 3.52636 -2.65926 1.57691 0.86289
52 7.3 22.0 0.00 4.13 2.70 Upper 1.98787 3.09104 . 1.41828 0.99325
53 4.1 29.0 0.70 5.78 7.76 Upper 1.41099 3.36730 -0.35667 1.75440 2.04898
54 5.4 29.0 0.20 4.64 2.65 Upper 1.68640 3.36730 -1.60944 1.53471 0.97456
55 5.0 34.0 0.70 4.21 6.50 Upper 1.60944 3.52636 -0.35667 1.43746 1.87180
56 6.2 27.0 0.30 3.97 2.97 Upper 1.82455 3.29584 -1.20397 1.37877 1.08856

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 56 DF Total 55
Variables 5 DF Within Classes 53
Classes 3 DF Between Classes 2

Class Level Information
Zones Variable
Name		 Frequency Weight Proportion Prior
Probability
SubMuli SubMuli 11 11.0000 0.196429 0.333333
Upper Upper 38 38.0000 0.678571 0.333333
Wilhelm Wilhelm 7 7.0000 0.125000 0.333333

Within Covariance Matrix Information
Zones Covariance
Matrix Rank		 Natural Log of the
Determinant of the
Covariance Matrix
SubMuli 5 4.86504
Upper 5 4.02474
Wilhelm 5 -5.77102
Pooled 5 4.30089

The SAS System

The DISCRIM Procedure
Test of Homogeneity of Within Covariance Matrices

Chi-Square DF Pr > ChiSq
49.126781 30 0.0153

Since the Chi-Square value is significant at the 0.1 level, the within covariance matrices 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 Zones
From Zones SubMuli Upper Wilhelm
SubMuli 4.86504 16.42265 162.37251
Upper 18.93261 4.02474 123.32096
Wilhelm 10.52910 45.17244 -5.77102

The SAS System

The DISCRIM Procedure
Classification Summary for Calibration Data: WORK.CRUDEOILSAMPLE
Resubstitution Summary using Quadratic Discriminant Function

Number of Observations and Percent Classified
into Zones
From Zones SubMuli Upper Wilhelm Total
SubMuli 10
90.91		 1
9.09		 0
0.00		 11
100.00
Upper 1
2.63		 37
97.37		 0
0.00		 38
100.00
Wilhelm 0
0.00		 0
0.00		 7
100.00		 7
100.00
Total 11
19.64		 38
67.86		 7
12.50		 56
100.00
Priors 0.33333
 0.33333
 0.33333

Error Count Estimates for Zones
  SubMuli Upper Wilhelm Total
Rate 0.0909 0.0263 0.0000 0.0391
Priors 0.3333 0.3333 0.3333  

The SAS System

The GLM Procedure

Class Level Information
Class Levels Values
Zones 3 SubMuli Upper Wilhelm

Number of Observations Read 56
Number of Observations Used 56

The SAS System

The GLM Procedure
 
Dependent Variable: vanadium vanadium

Source DF Sum of Squares Mean Square F Value Pr > F
Model 2 135.6731502 67.8365751 19.17 <.0001
Error 53 187.5752427 3.5391555    
Corrected Total 55 323.2483929      

R-Square Coeff Var Root MSE vanadium Mean
0.419718 30.43941 1.881264 6.180357

Source DF Type I SS Mean Square F Value Pr > F
Zones 2 135.6731502 67.8365751 19.17 <.0001

Source DF Type III SS Mean Square F Value Pr > F
Zones 2 135.6731502 67.8365751 19.17 <.0001

Contrast DF Contrast SS Mean Square F Value Pr > F
Linear Zones 1 16.0539075 16.0539075 4.54 0.0378
Quadratic Zones 1 121.8922528 121.8922528 34.44 <.0001

The SAS System

The GLM Procedure
 
Dependent Variable: iron iron

Source DF Sum of Squares Mean Square F Value Pr > F
Model 2 3186.681172 1593.340586 20.01 <.0001
Error 53 4221.158113 79.644493    
Corrected Total 55 7407.839286      

R-Square Coeff Var Root MSE iron Mean
0.430177 32.99651 8.924376 27.04643

Source DF Type I SS Mean Square F Value Pr > F
Zones 2 3186.681172 1593.340586 20.01 <.0001

Source DF Type III SS Mean Square F Value Pr > F
Zones 2 3186.681172 1593.340586 20.01 <.0001

Contrast DF Contrast SS Mean Square F Value Pr > F
Linear Zones 1 807.408687 807.408687 10.14 0.0024
Quadratic Zones 1 2452.387285 2452.387285 30.79 <.0001

The SAS System

The GLM Procedure
 
Dependent Variable: beryllium beryllium

Source DF Sum of Squares Mean Square F Value Pr > F
Model 2 0.98442037 0.49221018 5.88 0.0049
Error 53 4.43566535 0.08369180    
Corrected Total 55 5.42008571      

R-Square Coeff Var Root MSE beryllium Mean
0.181625 84.73086 0.289295 0.341429

Source DF Type I SS Mean Square F Value Pr > F
Zones 2 0.98442037 0.49221018 5.88 0.0049

Source DF Type III SS Mean Square F Value Pr > F
Zones 2 0.98442037 0.49221018 5.88 0.0049

Contrast DF Contrast SS Mean Square F Value Pr > F
Linear Zones 1 0.05823286 0.05823286 0.70 0.4079
Quadratic Zones 1 0.93800448 0.93800448 11.21 0.0015

The SAS System

The GLM Procedure
 
Dependent Variable: SathydroCarb Saturated hydrocarbons

Source DF Sum of Squares Mean Square F Value Pr > F
Model 2 48.8034220 24.4017110 22.67 <.0001
Error 53 57.0404334 1.0762346    
Corrected Total 55 105.8438554      

R-Square Coeff Var Root MSE SathydroCarb Mean
0.461089 19.57721 1.037417 5.299107

Source DF Type I SS Mean Square F Value Pr > F
Zones 2 48.80342197 24.40171098 22.67 <.0001

Source DF Type III SS Mean Square F Value Pr > F
Zones 2 48.80342197 24.40171098 22.67 <.0001

Contrast DF Contrast SS Mean Square F Value Pr > F
Linear Zones 1 1.98247317 1.98247317 1.84 0.1805
Quadratic Zones 1 47.30483521 47.30483521 43.95 <.0001

The SAS System

The GLM Procedure
 
Dependent Variable: AromhydroCarb Aromatic hydrocabons

Source DF Sum of Squares Mean Square F Value Pr > F
Model 2 209.2941996 104.6470998 16.41 <.0001
Error 53 338.0228861 6.3777903    
Corrected Total 55 547.3170857      

R-Square Coeff Var Root MSE AromhydroCarb Mean
0.382400 39.25392 2.525429 6.433571

Source DF Type I SS Mean Square F Value Pr > F
Zones 2 209.2941996 104.6470998 16.41 <.0001

Source DF Type III SS Mean Square F Value Pr > F
Zones 2 209.2941996 104.6470998 16.41 <.0001

Contrast DF Contrast SS Mean Square F Value Pr > F
Linear Zones 1 178.9559227 178.9559227 28.06 <.0001
Quadratic Zones 1 34.3918597 34.3918597 5.39 0.0241

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 Zones
E = Error SSCP Matrix
Characteristic Root Percent Characteristic Vector V'EV=1
vanadium iron beryllium SathydroCarb AromhydroCarb
4.17841435 86.25 0.04520460 -0.00835607 0.33510578 -0.06343120 -0.02721843
0.66601376 13.75 -0.01708629 0.00561389 0.27026635 -0.12953980 0.05543973
0.00000000 0.00 -0.00739870 0.00239697 0.38022080 0.02723223 -0.00331487
0.00000000 0.00 0.04500856 0.01198739 -0.01143913 -0.00580153 -0.01157736
0.00000000 0.00 0.04773022 -0.00454769 -0.05266021 0.09044752 0.01348588

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

S=2 M=1 N=23.5
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.11591099 18.98 10 98 <.0001
Pillai's Trace 1.20665556 15.21 10 100 <.0001
Hotelling-Lawley Trace 4.84442811 23.43 10 70.804 <.0001
Roy's Greatest Root 4.17841435 41.78 5 50 <.0001
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 Zones
E = Error SSCP Matrix
Characteristic Root Percent Characteristic Vector V'EV=1
vanadium iron beryllium SathydroCarb AromhydroCarb
1.42921965 100.00 -0.04634853 0.01005049 -0.10386533 -0.02767747 0.05517612
0.00000000 0.00 0.01744996 -0.00427799 -0.25831367 0.16392059 -0.00759932
0.00000000 0.00 0.00495834 0.00102439 0.49592448 -0.01389307 0.00971742
0.00000000 0.00 0.06326601 0.00303710 0.04961471 -0.02253983 0.01576534
0.00000000 0.00 0.01463633 0.01191122 -0.08303571 0.03772290 -0.02638505

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

S=1 M=1.5 N=23.5
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.41165483 14.01 5 49 <.0001
Pillai's Trace 0.58834517 14.01 5 49 <.0001
Hotelling-Lawley Trace 1.42921965 14.01 5 49 <.0001
Roy's Greatest Root 1.42921965 14.01 5 49 <.0001

Characteristic Roots and Vectors of: E Inverse * H, where
H = Contrast SSCP Matrix for Quadratic Zones
E = Error SSCP Matrix
Characteristic Root Percent Characteristic Vector V'EV=1
vanadium iron beryllium SathydroCarb AromhydroCarb
3.49137381 100.00 0.04105271 -0.00711427 0.38100087 -0.08725933 -0.01599668
0.00000000 0.00 -0.02204002 0.00347824 0.16233885 -0.09071201 0.06233302
0.00000000 0.00 -0.00899377 0.00286002 0.39134067 0.02007714 0.00000000
0.00000000 0.00 0.04270061 -0.00839115 -0.09026910 0.11625030 0.00000000
0.00000000 0.00 0.05117388 0.01140885 0.00000000 0.00000000 0.00000000

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

S=1 M=1.5 N=23.5
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.22264903 34.22 5 49 <.0001
Pillai's Trace 0.77735097 34.22 5 49 <.0001
Hotelling-Lawley Trace 3.49137381 34.22 5 49 <.0001
Roy's Greatest Root 3.49137381 34.22 5 49 <.0001

The SAS System

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

Zones vanadium LSMEAN LSMEAN Number
SubMuli 4.44545455 1
Upper 7.22631579 2
Wilhelm 3.22857143 3

Least Squares Means for effect Zones
Pr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: vanadium
i/j 1 2 3
1   0.0002 0.3809
2 0.0002   <.0001
3 0.3809 <.0001  

Zones iron LSMEAN LSMEAN Number
SubMuli 33.0909091 1
Upper 22.2526316 2
Wilhelm 43.5714286 3

Least Squares Means for effect Zones
Pr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: iron
i/j 1 2 3
1   0.0023 0.0480
2 0.0023   <.0001
3 0.0480 <.0001  

Zones beryllium LSMEAN LSMEAN Number
SubMuli 0.17090909 1
Upper 0.43210526 2
Wilhelm 0.11714286 3

Least Squares Means for effect Zones
Pr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: beryllium
i/j 1 2 3
1   0.0290 0.9219
2 0.0290   0.0283
3 0.9219 0.0283  

Zones SathydroCarb LSMEAN LSMEAN Number
SubMuli 6.56090909 1
Upper 4.65815789 2
Wilhelm 6.79571429 3

Least Squares Means for effect Zones
Pr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: SathydroCarb
i/j 1 2 3
1   <.0001 0.8865
2 <.0001   <.0001
3 0.8865 <.0001  

Zones AromhydroCarb LSMEAN LSMEAN Number
SubMuli 5.4836364 1
Upper 5.7678947 2
Wilhelm 11.5400000 3

Least Squares Means for effect Zones
Pr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: AromhydroCarb
i/j 1 2 3
1   0.9422 <.0001
2 0.9422   <.0001
3 <.0001 <.0001