Two-stage Cluster Sampling When Clusters are of Unequal Size

1
Cluster Analysis
2

• First used by Tryon (1939) encompasses a
  number of different algorithms and methods for
  grouping objects of similar kind into respective
  categories.

3
???? ??
????????????,?????????(???)????? ????????????????
??,???????????????????(homogeneity),??????????????
????
4
???????????????????????,???????????????
5

• ??????
• ????
• ???
• ???
• ???
• ???
• ???

6

• ???????
• ???????????????????????????????????,??????????????
  ,?????????????
• ??????????????????????,?????????N?????????????????
  ?????????

7
??????

• ????????????,??????????????(Euclidean Distance)
• ??N????,??????M???,??X?NM?????,???????????

8
dij
????????????,????????????????????????,??????0,????
?1?
9
??????

• ??????????????????,???????????,??????????????????
• ??????????????????(matching coefficient)???

10
Ex ?i?j???????(1???????,0????????)
11
??????
12

• ?????????
• ???? (non-hierarchical)????????????????????,?????
  ??

a. ?????? (sequential threshold)
?????,???????????,????????,???????????????????????
????????????????????,???????????????????,????????
?
13
b.??????(paralleled threshold)
???????????????????????,???????,???????????????,??
???????????????,?????(???)???????????
c.?????(optimizing partitioning) ????????
(???????????) ???,????????,????? (criterion
measure) ????????
14
d.????(K-means Method) ??????????????,??????????
??K???,????????????????,??????????????????????????
??????????????,??????????????????????,????????????
??????
15
??? (hierarchical)????? ??????????,?????????????
???????,?????????????????
????????,?????????????,?????
16
??????????????????,??????????,????????????,???????
??????????????????????? ???????????????????????,?
????????????,????????,??????????
17
?K-means???????
1.?????????K????? 2.?????????????(???)??(????????)
,?????????????????????????????????????????????????
??? 3.??????,?????????????????????
18
Ex????????????????
??????????????????,????1,2?????3,4?,??????????????
???
???1,2?     ?? ?3,4?
19
X2
X2
????????????????????,??????????????
?D21?1,2?(12-2)2(8-6)2104
??????????4????3,4?????,??????????2????3,4??????
,?????????3,4???????????1???2,3,4?,????????
20
???1?      ???2,3,4?
X112         X1
X28          X2
????????????1?????2,3,4??????
21
????????1????1????????2,3,4????2,3,4??????,?????
??????,???K2???,??????1?????2,3,4??
22
Two-stage Cluster Sampling When Clusters are of
Unequal Size

• Desired Sample Proportion pn/N
• a Desired of Clusters Selected in the 1st
  Stage
• A Total of Clusters
• b Sample Size within Each Cluster Selected
• Ni of Elements in Cluster i

23
Simple Two-stage Cluster Sampling

• The First-stage Prob. p1a/A
• The Second-stage Prob. p2p?(a/A)
• Sample size in cluster I, ni p2Ni

24
Probability Proportional to Size
where
25
Example

• Draw a sample of 1,000 households from a city
  that contains about 200,000 households
  distributed among 2000 blocks of unequal but
  known size.
• The desired sample proportion 1/200
• The desired of clusters selected in the 1st
  stage100
• How do we conduct the two-stage cluster sampling?

26
What is Cluster Analysis?

• Cluster Analysis is a class of statistical
  techniques that can be applied to data that
  exhibit natural groupings.
• CA is an interdependence technique that makes no
  distinction between dependent and independent
  variables.
• There is NO statistical significance testing in
  CA.
• CA is more a group of different algorithms that
  put objects into clusters following well-defined
  similarity rules.

27
What is A Cluster?

• A cluster is a group of relatively homogeneous
  cases and observations.
• Clusters exhibit high internal homogeneity and
  high external heterogeneity.

28
A Cluster Diagram Drinkers Perceptions of
Alcohol
29
Characteristics of CA

• Cluster Analysis is a tool of discovery.
• It discovers structures in data but does NOT
  explain why they exist.
• CA is used when we do not have an a priori
  hypothesis, but when we are in the exploratory
  phase.

30
How does CA differ

• From Discriminant Analysis
• A dependence technique
• Predict the probability that an object will fall
  into one of two or more mutually exclusive
  categories based on several independent
  variables.
• Find a linear combination of independent
  variables.
• Find natural groupings based on distances among
  objects.

31

• From Factor Analysis
• Similar to cluster analysis in that it is an
  interdependence technique.
• Primary difference lies in the focus on objects
  and variables.
• Factor analysis reduces variables to a few
  factors. Cluster analysis reduces objects to a
  few clusters.

32
Cluster Analysis Methods

• Three Cluster Analysis Methods
• Joining (Tree Clustering)
• Two-way Joining
• K-means Testing

33
Joining (Tree Clustering)

• A type of hierarchical clustering --
  agglomerative
• Each unit is a cluster.
• Dendogram ?
• Many other methods

34
The first level shows all samples xi as singleton
clusters. Increase levels, more samples are
clustered together in a hierarchical manner.
35
It is based on sets where each cluster level may
contain sets that are subclusters as shown in the
Venn diagram.
36
Two-way Joining Hartigan (1975)

• Two-way Joining tries to cluster both variables
  and objects.
• Only useful if you think clustering along BOTH
  lines will be useful.
• Very rare in application.

37
k-Means Clustering

• Begin with a preconception about the number of
  clusters (k).
• Thought of as ANOVA in reverse.
• ANOVA evaluates between group var. against within
  group var. when computing stat. signif. of
  hypothesis that groups are different.
• In k-Means the computer will try to move objects
  in and out of the groups to get the most
  significant ANOVA results.

38
Its all about distance

• Distance Measures
• Euclidean Distance
• Squared Euclidean Distance
• Manhattan Distance
• Chebychev Distance
• Power Distance

39
EQUATION Euclidean Distance

• Basic equation for determining distance measure.
• Distance (x,y) Si (xi yi)21/2
• A standard formula for determining the distance
  between two points on a plane

40
Fairly simple, right?
41
In other words, how do we get from this
42
To this
43
To this
44
How to Determine Clusters.

• Use a computer.
• Call a professional.

45

• Clusters in the
• Real World

46
Why is Cluster Analysis Important?

• Relatively new/evolving technique
• Highly useful for market segmentation
• Segmentation identifying groupings of customers
  using statistical multi-variate analysis, often
  based on perceptions and attitudes as well as
  demographics and behavior.
• Segmentation helpful to small companies
  attempting to carve out a niche
• Large companies trying to tailor their
  products/services to different segments

47
In addition to segmentation, clusters are used to

• Design products and establish brands
• Target direct mail
• Make decisions about customer conversion and
  retention
• Decide on marketing cost levels

48
Ex Luxury Car Customers

• Demographic examples easier to illustrate
• Demographics
• Gender
• Education
• Age
• 149 customers (objects) of a luxury car dealership

49
Using SPSS for Clustering

• Chose TwoStep Cluster Analysis
• Basically, the agglomerative technique
  (dendogram).
• Step One Creates very small (individual)
  sub-clusters.
• Step Two Cluster sub-clusters into desired
  number of clusters.
• Automatically finds optimum number of clusters.

50
Two-Step CA Output
What are these clusters?
51
Two-Step CA Output
52
(No Transcript)
53
(No Transcript)
54
What does this mean?

• Cluster 5
• Age 36 - 65
• Education High School graduate or above
• Gender Female
• Could have used k-Means, would have generated
  different results.
• Clustering is a powerful marketing research tool.

55
Claritas Clustering Experts

• Example Claritas Corporation
• Claritas founded the U.S. geodemographic industry
  when it launched the first PRIZM segmentation
  system in 1974.
• PRIZM (Potential Rating Index for Zip Markets)
  categorizes every U.S. neighborhood into 1 of 62
  clusters.
• Descriptive Names
• Money and Brains
• Young Literati
• Shotguns and Pickups

56
Money and Brains

• Sophisticated Urban Fringe Couples
• Cluster is a mix of family types singles,
  married couples with children and married couples
  without children. These families own their homes
  in upscale neighborhoods near cities. Dual
  incomes provide luxuries, travel and
  entertainment.
• Demographics
• Affluent
• Age Groups 55-64, 65
• Predominantly White, High Asian

57
Clusters Work!

• At a conservative estimate, more than 20,000
  companies in the United States and Canada alone
  used clusters as part of their marketing
  information mix last year.

58
Web Sources

• http//cwis.livjm.ac.uk/bus/busrmccl/ae230/lect10.
  ppt
• http//www.clusterbigip1.claritas.com/claritas/Def
  ault.jsp?main3submenusegsubcatsegprizm
• http//www.clusterbigip1.claritas.com/claritas/Def
  ault.jsp?main3submenusegsubcatsegprizmne
• http//www.insightsc.ie/newsletter7.htm
• http//www.directionsmag.com/article.asp?article_i
  d12
• http//fun.supereva.it/scoleri.freeweb/cern/biogra
  fie/hawking.jpg
• http//www.statsoft.com/textbook/stcluan.html
• http//www-db.stanford.edu/ullman/mining/cluster1
  .pdf
• http//www.snr.missouri.edu/multivariate/ClusterAn
  alysis.pdf

59
Print Sources

• Recent Developments in Clustering and Data
  Analysis. Edited by Chikio Hayashi, Edwin Diday,
  Michel Jambou, Noboru Ohsumi. Academic Press,
  Inc. 1988.
• Finding Groups in Data An Introduction to
  Cluster Analysis. Leonard Kaufman, Peter J.
  Rousseeuw. John Wiley and Sons, Inc. 1990.
• Marketing Research An Aid to Decision Making.
  Dr. Alan T. Shao. South-Western. 2002.
• Exploring Marketing Research. William G. Zikmund.
  South-Western. 2003.

60
Ex 7 Hypothetical Data
Subject Id. Income (1000) Education (years)
S1 5 5
S2 6 6
S3 15 14
S4 16 15
S5 25 20
S6 30 19
61
Similarity Matrix (Euclidean Distances)
Id S1 S2 S3 S4 S5 S6
S1 0 2 181 221 625 821
S2 2 0 145 181 557 745
S3 181 145 0 2 136 250
S4 221 181 2 0 106 212
S5 625 557 136 106 0 26
S6 821 745 250 212 26 0
d(S1, S3) ? (15-5)2 (19-5)2 181 d(S1, S2)
? 2 ???? (?????) ???
62
Centroid Method Five ClustersData For Five
Clusters
Cluster Cluster Members Income (1000) Education (years)
1 S1S2 (5,5) (6,6) 5.5 56/2 5.5 56/2
2 S3 15 14
3 S4 16 15
4 S5 25 20
5 S6 30 19
63
Similarity Matrix (Euclidean Distances)
Id S1 S2 S3 S4 S5 S6
S1 S2 0 162.5 200.5 590.5 782.5
S3 162 0 2 135.96 250
S4 200.5 2 0 106 212
S5 590.5 135.96 106 0 26
S6 782.5 250 212 26 0
d(S1 S2 , S3) ? (5.5-15)2 (5.5-14)2 ?
162.5 d( S3, S4) ? 2 ???? (?????) ???
64
Centroid Method Four ClustersData For Four
Clusters
Cluster Cluster Members Income (1000) Education (years)
1 S1S2 (5,5) (6,6) 5.5 56/2 5.5 56/2
2 S3 S4 (15,14) (16,15) 15.5 1516/2 14 .5 1415/2
3 S5 25 20
4 S6 30 19
65
Similarity Matrix (Euclidean Distances)
Id S1 S2 S3S4 S5 S6
S1 S2 0 181 590.5 782.5
S3 S4 181 0 120.5 230.5
S5 590.5 120.5 0 26
S6 782.5 230.5 26 0
d(S1 S2 , S5) ? (5.5-25)2 (5.5-20)2 ?
590.5 d( S5, S6) ? 26 ???? (?????) ???
66
Centroid Method Three ClustersData For Three
Clusters
Cluster Cluster Members Income (1000) Education (years)
1 S1S2 (5,5) (6,6) 5.5 56/2 5.5 56/2
2 S3 S4 (15,14) (16,15) 15.5 1516/2 14 .5 1415/2
3 S5 S6 (25,20) (30,19) 27.5 2530/2 19.5 1415/2
67
Similarity Matrix (Euclidean Distances)
Id S1 S2 S3S4 S5 S6
S1 S2 0 181 680
S3 S4 181 0 169
S5 S6 680 169 0
d(S1 S2 , S5 S6) ? (5.5-27.5)2 (5.5-19.5)2
? 680 d( S3 S4, S5 S6) ? 169 ????
(?????) ???
68
Exhibit 7-1SAS Output for cluster analysis on
data in Table 7.1
1
???????????

• Simple statistics
• Mean Std Dev
  Skewness Kurtosis Bimodality
• INCOME 16.1667 9.9883 0.2684
  -1.4015 0.2211
• EDUC 13.1667 6.3692
  -0.4510 -1.8108 0.2711
• Root-Mean-Square Total-Sample Standard Deviation
  8.376555

69
Root-Mean-Square Total-Sample Standard
Deviation8.376555 (RMSSTD)
RMSSTO?????????????(?????????)

• Step Number
  Frequency RMS STD
• Number of
  of New of New Semipartial
  Centroid
• Clusters Clusters Joined Cluster
  Cluster R-Squared R-Squared
  Distance
• 1 5 S1 S2
  2 0.707107 0.001425
  0.998575 1.4142
• 2 4 S3 S4
  2 0.707107 0.001425
  0.997150 1.4142
• 3 3 S5 S6
  2 2.549510 0.018527
  0.978622 5.0990
• 4 2 CL4 CL3
  4 5.522681 0.240855 0.737767
  13.0000
• 5 1 CL5 CL2
  6 8.376555 0.737767 0.000000
  19.7041

?????,?R2????
70

• CLUSTER1 CLUSTER2
  CLUSTER3
• OBS SID INCOME EDUC OBS SID INCOME EDUC
  OBS SID INCOME EDUC
• 1 S1 5 5 3
  S3 15 14 5
  S5 25 20
• 2 S2 6 6 4
  S4 16 15 6
  S6 30 19

71
Exhibit 7.2Non-hierarchical Clustering On Data

• ReplaceFULL Radius0 Maxclusters3 Maxiter20
  Converge0.02
• Initial Seeds
• Cluster INCOME EDUC
• -------- -----------------------------------
• 1 5.0000 5.0000
• 2 30.0000 19.0000
• 3 16.0000 15.0000

??????????S1, S6, S4
72
Exhibit 7-2 (continued)

• Minimum Distance Between Seeds 14.56022
  
• Iteration Change in Cluster Seeds
• 1 2
  3
• -------------------------------------------------
  -
• 1 0.707107 2.54951 0.707107
• 2 0 0
  0
• Statistics for Variables
• Variable Total STD Within STD
  R-Squared RSQ/(1-RSQ)
• -------------- -----------------------------------
  -------------------------------------------
• INCOME 9.988327 2.121320
  0.972937 35.950617
• EDUC 6.369197 0.707107
  0.992605 134.222222
• OVER-ALL 8.376555 1.581139
  0.978622 45.777778

73
Exhibit 7-2 (continued)

• Pseudo
  F Statistic 68.67
• Approximate Expected Over-All R-Squared .
  
• Cubic Clustering
  Criterion .
• WARNING The two above values are invalid for
  correlated variables.
• Cluster Means
• Cluster INCOME EDUC
• --------- -----------------------------------
• 1 5.5000 5.5000
• 2 27.5000 19.5000
• 3 15.5000 14.5000

???????(?????)
74
Exhibit 7.4 Hierarchical Cluster Analysis For
Food Data

• SINGLE LINKAGE CLUSTER ANALYSIS
• 
  
  
• SIMPLE STATISTICS
  
• 
  
  
• MEAN STD DEV SKEWNESS KURTOSIS
  BIMODALITY
• 
  
  
• CALORIES 207.407 101.208
  0.542 -0.675 0.478
  
• PROTEIN 19.000 4.252
  -0.824 1.327
  0.357
• FAT 13.481 11.257
  0.790 -0.624
  0.589
• CALCIUM 43.963 78.034
  3.159 11.345 0.746
  
• IRON 2.381 1.461
  1.230 1.469
  0.518

75
Exhibit 7.4 (continued)
(?????)

• COMPLETE LINKAGE CLUSTER ANALYSIS
• 
  
  
• NUMBER
  FREQUENCY RMS STD
  
• OF CLUSTERS
  OF NEW OF NEW
  SEMIPARTIAL MAXIMUM
• CLUSTERS JOINED
  CLUSTER CLUSTER R-SQUARED
  R-SQUARED DISTANCE
• 10 CL15 CANNED CRABMEAT
  4 11.32324
  0.003476 0.985594 50.6665
• 9 CL17 ROAST LAMB
  SHOUL 3 12.59929
  0.003226 0.982367 55.6611
• 8 CL14 CANNED SHRIMP
  3 16.10565
  0.005231 0.977136 71.1677
• 7 CL13 ROAST BEEF
  6 14.34190
  0.009755 0.967381 80.9343
• 6 CL10 CL8
  7
  22.14096 0.023782 0.943599
  108.1758
• 5 CL9 CL11
  11
  20.22234 0.039103 0.904496
  141.7814
• 4 CL6 CL12
  9
  30.07489 0.048662 0.855835
  154.4447
• 3 CL7 CL5
  17
  38.73570 0.220433 0.635402
  262.5666
• 2 CL4 CANNED
  SARDINES 10 51.36181
  0.192623 0.442779 364.8934
• 1 CL3 CL2
  27
  57.40958 0.442779 0.000000
  433.7617

76
Exhibit 7.4 (continued)

• ROOT-MEAN-SQUARE TOTAL-SAMPLE STANDARD DEVIATION
  57.4096
• 
  
  
• NUMBER
  FREQUENCY RMS STD
  
• OF CLUSTERS
  OF NEW OF NEW
  SEMIPARTIAL MINIMUM
• CLUSTERS JOINED
  CLUSTER CLUSTER R-SQUARED
  R-SQUARED DISTANCE
• 
  
  
• 10 CANNED CANNED
  2 11.16786
  0.001455 0.973438 35.3159
  MACKEREL SALMON
• 9 CL14
  ROAST LAMB 3 12.59929
  0.003226 0.970211
  35.4131
• 
  SHOULDER
• 8 CL11
  CANNED 12 16.80697
  0.014701 0.955510
  39.5267
• 
  CRABMEAT
• 7 CL15
  CL9 8
  20.48901 0.028341 0.927169
  40.1627
• 6 CL7
  CL8 20
  40.04817 0.285060 0.642109
  40.2746
• 5 CL12
  CANNED 3 16.10565
  0.005231 0.636878
  44.8504
• 
  SHRIMP
• 4 CL6
  ROAST BEEF 21 43.49500
  0.085924 0.550954
  45.7642
• 3 CL4
  CL5 24
  48.72189 0.189548 0.361406
  48.7139
• 2 CL3
  CL10 26
  50.53988 0.106595 0.254811
  62.2624
• 1 CL2
  CANNED 27 57.40958
  0.254811 0.000000
  211.5691

77
Exhibit 7.4 (continued)
(???)

• CENTROID HIERARCHICAL CLUSTER ANALYSIS
• NUMBER
  FREQUENCY RMS STD
  
• OF CLUSTERS
  OF NEW OF NEW SEMIPARTIAL
  CENTROID
• CLUSTERS JOINED
  CLUSTER CLUSTER R-SQUARED R-SQUARED
  DISTANCE
• 
  
  
• 10 CL15 CANNED
  4 11.32324
  0.003476 0.985594 44.5633
• 
  CRABMEAT
• 9 CL16 ROAST
  LAMB 3 12.59929
  0.003226 0.982367 45.5370
• 
  SHOULDER
• 8 CL14 CANNED
  SHRIMP 3 16.10565 0.005231
  0.977136 57.9815
• 7 CL13 CL10
  12 16.80697
  0.026857 0.950279 65.6901
• 6 CL12 ROAST
  BEEF 6 14.34190
  0.009755 0 940524 70.8222
• 5 CL6 CL9
  9 24.36751
  0.039727 0.900797
  92.2533
• 4 CL8 CL11
  5 26.85628
  0.026158 0.874639 96.6423
  
• 3 CL7 CL4
  17 31.36108
  0.113709 0.760930 117.4906
  
• 2 CL5 CL3
  26 50.53988
  0.506119 0.254811 191.9655
  
• 1 CL2 CANNED
  27 57.40958
  0.254811 0.000000 336.7134
• SARDINES

78
Exhibit 7.4 (continued)
(???)

• WARD'S MINIMUM VARIANCE CLUSTER ANALYSIS
• 
  
  
• NUMBER
  FREQUENCY RMS STD
  BETWEEN-
• OF CLUSTERS
  OF NEW OF NEW SEMIPARTIAL
  CLUSTER
• CLUSTERS JOINED
  CLUSTER CLUSTER R-SQUARED R-SQUARED
  SUM OF
  
  
  SQUARES
• 10 CL14 CANNED
  4 11.32324 0.003476
  0.985908 1489.42
• CRABMEAT
• 9 CL16 CL20
  8 7.75641
  0.003541 0.982367 1517.12
• 8 CL15 CANNED
  3 16.10565 0.005231
  0.977136 2241.24
• SHRIMP
• 7 CL12 ROAST BEEF
  6 14.34190 0.009755
  0.967381 4179.83
• 6 CL10 CL8
  7 22.14096
  0.023782 0.943599 10189.5
• 5 CL11 CL9
  11 20.22234
  0.039103 0.904496 16754.1
• 4 CL6 CL13
  9 30.07489
  0.048662 0.855835 20849.7
• 3 CL5 CL4
  20 36.22080
  0.158726 0.697109 68007.8
• 2 CL3 CANNED
  21 47.72546 0.240715
  0.456394 103137
• SARDINES
• 1 CL7 CL2
  27 57.40958
  0.456394 0.000000 195548

79
Exhibit 7.5 Non-Hierarchical Analysis For
Food-Nutrient Data

• INITIAL SEEDS (??????)
• CLUSTER CALORIES PROTEIN
  FAT CALCIUM IRON
• --------------------------------------------------
  -------------------------------------------------
• 1 331.111 19.000
  27.556 8.778 2.467
  
• 2 161.667 20.500
  7.500 14.250 1.925
• 3 100.000 14.800
  3.400 114.000 3.000

80
Exhibit 7.5 (continued)

• MINIMUM DISTANCE BETWEEN SEEDS 117.4876
• ITERATION CHANGE IN CLUSTER SEEDS
• 1
  2 3
• ----------------------- --------------------------
  ----------------
• 1 10.8475
  6.46446 0.3
• 2 0
  6.85281 12.7855
• 3 0
  0 0

81

• CLUSTER SUMMARY
• MAXIMUM
• 
  DISTANCE
• CLUSTER RMS STD
  FROM SEED TO NEAREST CENTROID
• NUMBER FREQUENCY DEVIATION OBSERVATION
  CLUSTER DISTANCE
• --------------------------------------------------
  --------------------------------------------------
  ------------
• 1 8 20.8936
  78.8882 2 168.5
• 2 12 16.3651
  70.9576 3 117.9
• 3 6 27.8059
  79.6672 2 117.9
• ????? ?2?????? ??? ?????
  
• ??? ???
  

82
?????(??)???,?????RMSSTD.????,???? Within
SD/Total SD

• VARIABLE TOTAL STD WITHIN STD
  R-SQUARED RSQ/(1-RSQ)
• -------------------------------------------------
  --------------------------------------------------
  -------
• CALORIES 103.06085
  39.89286 0.86216
  6.25453
• PROTEIN 4.29257
  3.58590 0.35798
  0.55758
• FAT 11.44357
  4.52989 0.85584
  5.93681
• CALCIUM 44.70188
  22.76009 0.76150
  3.19291
• IRON 1.49005
  1.51663 0.04688
  0.04919
• OVER-ALL 50.53988
  20.71299 0.84547
  5.47135
• PSEUDO F STATISTIC 62.92
• APPROXIMATE EXPECTED OVER-ALL R-SQUARED
  0.78678
• CUBIC
  CLUSTERING CRITERION 2.186

STATISTICS FOR VARIABLES
83
Exhibit 7.5 (continued)

• CLUSTER MEANS
• CLUSTER CALORIES PROTEIN FAT
  CALCIUM IRON
• --------------------------------------------------
  ---------------------------------------------
• 1 341.875
  18.750 28.875 8.750
  2.437
• 2 174.583
  21.083 8.750 11.833
  2.083
• 3 98.333
  14.667 3.167 101.333
  2.883

Cluster 1?????? Cluster 2??????? Cluster 3?????
84
Exhibit 7.5 (continued)
?????????(????,?????,??) (8 Cases)

• CLUSTER1
• OBS NAME CLUS DISTA
  CALORIES PROTEIN FAT CALCIUM IRON
• 1 BRAISED BEEF 1
  2.4357 340 2 0 28
  9 2.6
• 2 ROAST BEEF 1
  78.8882 420 15
  39 7 2.0
• 3 BEEF STEAK 1
  33.2744 375 19
  32 9 2.6
• 4 ROST LAMB LEG 1 77.3963
  265 20 20
  9 2.6
• 5 ROAST LAMB 1
  42.0616 300 18
  25 9 2.3
• 6 SMOKED HAM 1 2.4311
  340 20 28
  9 2.5
• 7 PORK ROAST 1
  1.9132 340 19
  29 9 2.5
• 8 PORK SIMMERED 1 13.1779
  355 19 30
  9 2.4

85
Exhibit 7.5 (continued)
?????????(????,?????,??) (12 Cases)

• CLUSTER2
• OBS NAME CLUSTER
  DISTANCE CALORIES PROTEIN FAT CALCIUM IRON
• 9 HAMBURGER 2
  70.9576 245 21
  17 9 2.7
• 10 CANNED BEEF 2
  7.8135 180 22
  10 17 3.7
• 11 BROILED CHICKEN 2
  59.9964 115 20 3
  8 1.4
• 12 CANNED CHICKEN 2
  6.3070 170 25 7
  12 1.5
• 13 BEEF HEART 2
  16.4369 160 26
  5 14 5.9
• 14 BEEF TONGUE 2
  31.3971 205 18
  14 7 2.5
• 15 VEAL CUTLET 2
  10.9841 185 23
  9 9 2.7
• 16 BAKED BLUEFISH 2
  42.0215 135 22
  4 25 0.6
• 17 FRIED HADDOCK 2
  40.2403 135 16
  5 15 0.5
• 18 BROILED MACKEREL 2
  26.7634 200 19 13
  5 1.0
• 19 FRIED PERCH 2
  21.2850 195 16
  11 14 1.3
• 20 CANNED TUNA 2
  7.9719 170 25
  7 7 1.2

86
Exhibit 7.5 (continued)
???????????? (6 Cases)

• CLUSTER3
• OBS NAME CLUSTER
  DISTANCE CALORIES PROTEIN FAT CALCIUM IRON
• 21 RAW CLAMS 3
  34.7046 70 11
  1 82 6.0
• 22 CANNED CLAMS 3
  60.5092 45 7
  1 74 5.4
• 23 CANNED CRABMEAT 3
  63.9273 90 14
  2 38 0.8
• 24 CANNED MACKEREL 3
  79.6672 155 16
  9 157 1.8
• 25 CANNED SALMON 3
  61.7127 120 17
  5 159 0.7
• 26 CANNED SHRIMP 3
  14.8809 110 23
  1 98 2.6