Multivariate analysis
Math81
SAS-Problem4-NewResults-Key.docxSAS output for Cluster Analysis
Cluster Analysis
Wards Method
Ward's clustering method selected
The CLUSTER Procedure
Ward's Minimum Variance Cluster Analysis
Eigenvalues of the Correlation Matrix
Eigenvalue Difference Proportion Cumulative
1 2.72811968 1.21578483 0.3897 0.3897
2 1.51233485 0.11736187 0.2160 0.6058
3 1.39497299 0.50298170 0.1993 0.8051
4 0.89199129 0.54521262 0.1274 0.9325
5 0.34677866 0.24649107 0.0495 0.9820
7 variables used for clustering
6 0.10028759 0.07477267 0.0143 0.9964
Cluster variables standardized since each is measured on a different scale
7 0.02551493 0.0036 1.0000
The data have been standardized to mean 0 and variance 1
Root-Mean-Square Total-Sample Standard Deviation 1
Root-Mean-Square Distance Between Observations 3.741657
Default option in SAS is to use Euclidean distance as measure of similarity for computing "closeness" of cities
Statistics computed at each cluster stage, based on ANOVA for comparing mean DV across clusters, where DV=linear combination of 7 cluster variables
Table shows history of clustering algorithm, from weak solution (all cities in their own cluster) to strong solution (all cities in single cluster)
Cluster History
T
i
Stage at which 2 cities are clustered together
NCL --------Clusters Joined--------- Freq SpRSq RSq Ps F PsT2 e
40 Jacksonville New Orleans 2 0.0005 1.00 52.4 .
39 Des Moines Omaha 2 0.0005 .999 51.2 .
38 Nashville Richmond 2 0.0007 .998 46.4 .
37 Atlanta Memphis 2 0.0010 .997 40.4 .
36 Washington Indianapolis 2 0.0013 .996 35.4 .
35 Baltimore St. Louis 2 0.0013 .995 32.8 .
34 Louisville Columbus 2 0.0015 .993 30.7 .
33 Denver Salt Lake City 2 0.0018 .991 28.8 .
32 CL34 Cincinnati 3 0.0026 .989 25.5 1.8
31 Hartford Albany 2 0.0027 .986 23.6 .
30 CL36 Kansas City 3 0.0027 .983 22.5 2.1
29 Minneapolis Milwaukee 2 0.0027 .981 21.8 .
28 Little Rock CL38 3 0.0027 .978 21.4 3.8
27 CL31 Pittsburgh 3 0.0030 .975 21.0 1.1
26 CL40 Miami 3 0.0035 .971 20.4 7.2
25 Wilmington Norfolk 2 0.0038 .968 19.9 .
24 San Francisco Albuquerque 2 0.0045 .963 19.3 .
23 Dallas Houston 2 0.0046 .959 18.9 .
22 CL28 CL37 5 0.0053 .953 18.4 3.6
21 CL25 CL30 5 0.0059 .947 18.0 2.3
20 CL24 CL33 4 0.0061 .941 17.7 1.9
19 CL39 Wichita 3 0.0064 .935 17.5 11.9
18 CL32 Charleston 4 0.0069 .928 17.4 3.3
17 CL27 Seattle 4 0.0071 .921 17.4 2.5
16 Detroit Cleveland 2 0.0085 .912 17.3 .
15 CL29 Buffalo 3 0.0092 .903 17.3 3.4
14 CL21 CL35 7 0.0097 .893 17.4 3.2
13 CL16 Philadelphia 3 0.0148 .878 16.9 1.7
12 CL17 Providence 5 0.0160 .863 16.5 3.8
11 CL22 CL18 9 0.0184 .844 16.2 6.2
10 CL19 CL15 6 0.0276 .817 15.3 5.9
9 CL11 CL14 16 0.0286 .788 14.9 6.3
8 CL26 CL23 5 0.0306 .757 14.7 10.6
7 Phoenix CL20 5 0.0333 .724 14.9 8.1
6 CL12 CL13 8 0.0467 .677 14.7 5.4
5 CL9 CL8 21 0.0660 .611 14.2 9.5
4 CL6 CL10 14 0.0760 .535 14.2 6.3
3 CL5 CL4 35 0.1400 .395 12.4 11.0
Pseudo stats point towards 4 clusters (subjective)
2 CL7 CL3 40 0.1700 .225 11.3 10.7
1 CL2 Chicago 41 0.2253 .000 . 11.3
Total number of cities in cluster joined at that stage
Look for stage with
(1) large drop in Rsq,
(2) large pseudo F
(3) followed by large pseudo Tsq at next stage
Cluster Analysis
Scree Plot
Plot of _SPRSQ_*_NCL_. Legend: A = 1 obs, B = 2 obs, etc.
‚
‚
0.25 ˆ
‚
‚
‚ A
‚
Scree plot: look for stage at which curve starts to level off (or where big drop occurs): 4 clusters
‚
S ‚
e 0.20 ˆ
m ‚
i ‚
- ‚
P ‚ A
a ‚
r ‚
t 0.15 ˆ
i ‚ A
a ‚
l ‚
‚
R ‚
- ‚
S 0.10 ˆ
q ‚
u ‚
a ‚ A
r ‚
e ‚ A
d ‚
0.05 ˆ A
‚
‚ A
‚ A A A
‚ A
‚ A A
‚ A A A A A A A A A A A A
0.00 ˆ A A A A A A A A A A A A A A A A
‚
Šƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒ
1 6 11 16 21 26 31 36 41
Number of Clusters
Cluster Analysis Listing of 4 Cluster Solution
SAS output for 4 Cluster solution
------------------------------------------ CLUSTER=1 ------------------------------------------
CITY SO2 TEMP FACTORIES POPLN WINDSPD RAIN DAYS
Atlanta 24 61.5 368 497 9.1 48.34 115
Baltimore 47 55.0 625 905 9.6 41.31 111
Charleston 31 55.2 35 71 6.5 40.75 148
Values of cluster variables for each city
Cincinnati 23 54.0 462 453 7.1 39.04 132
Cities in Cluster 1
Columbus 26 51.5 266 540 8.6 37.01 134
Dallas 9 66.2 641 844 10.9 35.94 78
Houston 10 68.9 721 1233 10.8 48.19 103
Indianapolis 28 52.3 361 746 9.7 38.74 121
Jacksonville 14 68.4 136 529 8.8 54.47 116
Kansas City 14 54.5 381 507 10.0 37.00 99
Little Rock 13 61.0 91 132 8.2 48.52 100
Louisville 30 55.6 291 593 8.3 43.11 123
Memphis 10 61.6 337 624 9.2 49.10 105
Miami 10 75.5 207 335 9.0 59.80 128
Nashville 18 59.4 275 448 7.9 46.00 119
New Orleans 9 68.3 204 361 8.4 56.77 113
Norfolk 31 59.3 96 308 10.6 44.68 116
Richmond 26 57.8 197 299 7.6 42.59 115
St. Louis 56 55.9 775 622 9.5 35.89 105
Washington 29 57.3 434 757 9.3 38.89 111
Wilmington 36 54.0 80 80 9.0 40.25 114
------------------------------------------ CLUSTER=2 ------------------------------------------
CITY SO2 TEMP FACTORIES POPLN WINDSPD RAIN DAYS
Cities in Cluster 2
Albany 46 47.6 44 116 8.8 33.36 135
Buffalo 11 47.1 391 463 12.4 36.11 166
Cleveland 65 49.7 1007 751 10.9 34.99 155
Des Moines 17 49.0 104 201 11.2 30.85 103
Detroit 35 49.9 1064 1513 10.1 30.96 129
Hartford 56 49.1 412 158 9.0 43.37 127
Milwaukee 16 45.7 569 717 11.8 29.07 123
Minneapolis 29 43.5 699 744 10.6 25.94 137
Omaha 14 51.5 181 347 10.9 30.18 98
Philadelphia 69 54.6 1692 1950 9.6 39.93 115
Pittsburgh 61 50.4 347 520 9.4 36.22 147
Providence 94 50.0 343 179 10.6 42.75 125
Seattle 29 51.1 379 531 9.4 38.79 164
Wichita 8 56.6 125 277 12.7 30.58 82
Cities in Cluster 3
----------------------------------------- CLUSTER=3 -------------------------------------------
CITY SO2 TEMP FACTORIES POPLN WINDSPD RAIN DAYS
Albuquerque 11 56.8 46 244 8.9 7.77 58
Denver 17 51.9 454 515 9.0 12.95 86
Phoenix 10 70.3 213 582 6.0 7.05 36
Salt Lake City 28 51.0 137 176 8.7 15.17 89
San Francisco 12 56.7 453 716 8.7 20.66 67
----------------------------------------- CLUSTER=4 -------------------------------------------
CITY SO2 TEMP FACTORIES POPLN WINDSPD RAIN DAYS
Chicago 110 50.6 3344 3369 10.4 34.44 122
Only City in Cluster 4
Average values of cluster variables for those cities in the cluster
SAS output for Summary of 4 Cluster solution
Cluster Analysis
Summary Statistics for Clusters
Standard deviation of cluster variables for those cities in the cluster
----------------------------------------- CLUSTER=1 -------------------------------------------
Variable N Mean Std Dev Minimum Maximum
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
SO2 21 23.5238095 12.7891323 9.0000000 56.0000000
TEMP 21 59.6761905 6.4495662 51.5000000 75.5000000
FACTORIES 21 332.5238095 214.9636293 35.0000000 775.0000000
POPLN 21 518.2857143 283.3011018 71.0000000 1233.00
WINDSPD 21 8.9571429 1.1474195 6.5000000 10.9000000
RAIN 21 44.1138095 6.8955591 35.8900000 59.8000000
DAYS 21 114.5714286 14.5827687 78.0000000 148.0000000
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
Use this information to describe characteristics of each cluster:
e.g., cities in Cluster 2 have lowest mean TEMP and highest mean DAYS
----------------------------------------- CLUSTER=2 -------------------------------------------
Variable N Mean Std Dev Minimum Maximum
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
SO2 14 39.2857143 26.3422111 8.0000000 94.0000000
TEMP 14 49.7000000 3.3270454 43.5000000 56.6000000
FACTORIES 14 525.5000000 457.4788352 44.0000000 1692.00
POPLN 14 604.7857143 531.8548354 116.0000000 1950.00
WINDSPD 14 10.5285714 1.2243859 8.8000000 12.7000000
RAIN 14 34.5071429 5.2941135 25.9400000 43.3700000
DAYS 14 129.0000000 24.4823327 82.0000000 166.0000000
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
------------------------------------------ CLUSTER=3 ------------------------------------------
Variable N Mean Std Dev Minimum Maximum
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
SO2 5 15.6000000 7.4363970 10.0000000 28.0000000
TEMP 5 57.3400000 7.7209455 51.0000000 70.3000000
FACTORIES 5 260.6000000 185.7533311 46.0000000 454.0000000
POPLN 5 446.6000000 229.0519592 176.0000000 716.0000000
WINDSPD 5 8.2600000 1.2700394 6.0000000 9.0000000
RAIN 5 12.7200000 5.6069243 7.0500000 20.6600000
DAYS 5 67.2000000 21.7186556 36.0000000 89.0000000
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
No standard deviation since only sample of 1 city in this cluster
----------------------------------------- CLUSTER=4 -------------------------------------------
Variable N Mean Std Dev Minimum Maximum
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
SO2 1 110.0000000 . 110.0000000 110.0000000
TEMP 1 50.6000000 . 50.6000000 50.6000000
FACTORIES 1 3344.00 . 3344.00 3344.00
POPLN 1 3369.00 . 3369.00 3369.00
WINDSPD 1 10.4000000 . 10.4000000 10.4000000
RAIN 1 34.4400000 . 34.4400000 34.4400000
DAYS 1 122.0000000 . 122.0000000 122.0000000
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
Is Chicago an outlier? Extreme values for almost all cluster variables
