Multivariate Analysis -Overviefuchs/Introduction And Summary.pdf · •z4 =Diastolic blood pressure (DIAP) •z5 =QRS wave measurements (QRS) ... 860 722 1 1750 135 90 79 500 384

Multivariate Analysis - Overview

•In general: - Analysis of Multivariate data, i.e. each observation has two or more variables as predictor variables,

•Analyses

–Analysis of Treatment Means (Single (multivariate) sample, two-samples,etc.)

– Study interrelationships – correlations and predictions (regression)

– Other specific methods (discriminant analysis, principal components, clustering)

• Limitations:

–Many parameters estimated – large sample sizes

– Whenever testing hypotheses, assumption of normality (almost overall)

•We’ll focus on the multivariate methods and applications with somewhat limited mathematical emphasis (without proofs)

•Textbook: Applied Multivariate Analysis - Fifth Edition, Richard Johnson and Dean Wichern

Sweat data – One sample testing means Table 5.1

•Perspiration from 20 healthy women analyzed

•Three components:

•X1 =sweat rate

•X2 =sodium content

•X3 =potasium content

•Null hypothesis: simultaneousely

�µ1 = 4

�µ2 = 50

�µ3 = 10

or

[ 4 ]

µ =[ 50]

[10]

or

�µ’ = [4, 50,10]

Sweat data – One sample testing means Table 5.1Sweat rate Sodium Potasium

3.7 48.5 9.3

5.7 65.1 8

3.8 47.2 10.9

3.2 53.2 12

3.1 55.5 9.7

4.6 36.1 7.9

2.4 24.8 14

7.2 33.1 7.6

6.7 47.4 8.5

5.4 54.1 11.3

3.9 36.9 12.7

4.5 58.8 12.3

3.5 27.8 9.8

4.5 40.2 8.4

1.5 13.5 10.1

8.5 56.4 7.1

4.5 71.6 8.2

6.5 52.8 10.9

4.1 44.1 11.2

5.5 40.9 9.4


µ0’ = [4, 50,10]One-Sample T: Sweatrate, Sodium, Potasium

Variable N Mean StDev SE Mean 90% CI

Sweatrate 20 4.64000 1.69687 0.37943 (3.98391, 5.29609)

Sodium 20 45.4000 14.1347 3.1606 ( 39.9349, 50.8651)

Potasium 20 9.96500 1.90464 0.42589 (9.22858, 10.70142)

One-Sample T: Sweatrate, Sodium, Potasium

Variable N Mean StDev SE Mean 95% CI

Sweatrate 20 4.64000 1.69687 0.37943 (3.84584, 5.43416)

Sodium 20 45.4000 14.1347 3.1606 (38.7848, 52.0152)

Potasium 20 9.96500 1.90464 0.42589 (9.07360, 10.85640)


µ0’ = [4, 50,10]

0.401847 -0.001580.257969Potasium

=20*.487=9.74(Xbar-Miu0)’S(-1)n*(Xbar-Miu0)’

F(3,17,.9)=3.20F(3,17,.9)=2.44=2.905F=(n-p)/[(n-1)*p]*T2

-0.001580.006067-0.02209Sodium

0.257969-0.022090.586155Sweat rate

PotasiumSodiumSweat rateS-matrix(-1)

3.627658-5.64-1.80905Potasium

-5.64199.788410.01Sodium

-1.8090510.012.879368Sweat rate

PotasiumSodiumSweat rateS-matrix

Xbar-Miu0-0.035-4.60.64

Miu010504

Means9.96545.44.64

Turtle Carapaces- Two Samples – Testing means –Table 6.7

•Jolicoeur and Mosimann studied relationship of size and shape for painted turtled. Measures on the caprapaces of 24 male and 24 female turtles

•Three components:

•X1 =Length

•X2 =Width

•X3 =Height

•Null hypothesis: µ1 = µ2 (vectors of size 3)


Length Width Height Gender Length Width Height Gender

98 81 38 female 93 74 37 male

103 84 38 female 94 78 35 male

103 86 42 female 96 80 35 male

105 86 42 female 101 84 39 male

109 88 44 female 102 85 38 male

123 92 50 female 103 81 37 male

123 95 46 female 104 83 39 male

133 99 51 female 106 83 39 male

133 102 51 female 107 82 38 male

133 102 51 female 112 89 40 male

134 100 48 female 113 88 40 male

136 102 49 female 114 86 40 male

138 98 51 female 116 90 43 male

138 99 51 female 117 90 41 male

141 105 53 female 117 91 41 male

147 108 57 female 119 93 41 male

149 107 55 female 120 89 40 male

153 107 56 female 120 93 44 male

155 115 63 female 121 95 42 male

155 117 60 female 125 93 45 male

158 115 62 female 127 96 45 male

159 118 63 female 128 95 45 male


ANOVA: Length, Width, Height versus Gender

Gender fixed 2 female, male

Analysis of Variance for Length

Source DF SS MS F P

Gender 1 5474.8 5474.8 17.19 0.000

Error 47 14965.9 318.4

Total 48 20440.7

S = 17.8444 R-Sq = 26.78% R-Sq(adj) = 25.23%

Analysis of Variance for Width

Source DF SS MS F P

Gender 1 2208.0 2208.0 18.71 0.000

Error 47 5548.0 118.0

Total 48 7756.0

S = 10.8647 R-Sq = 28.47% R-Sq(adj) = 26.95%

Analysis of Variance for Height

Source DF SS MS F P

Gender 1 1420.8 1420.8 34.47 0.000

Error 47 1937.2 41.2

Total 48 3358.0

S = 6.42005 R-Sq = 42.31% R-Sq(adj) = 41.08%


Means

Gender N Length Width Height

female 25 134.52 101.72 51.480

male 25 113.38 88.29 40.708

SSCP Matrix for Gender (B) MANOVA for Gender

Length Width Height s = 1 m = 0.5 n = 21.5

Length 5475 3477 2789 Test DF

Width 3477 2208 1771 Criterion Statistic F Num Denom P

Height 2789 1771 1421 Wilks' 0.410 21.570 3 44 0.000

Determinant =-426 Lawley-Hotelling 1.43798 21.570 3 44 0.000

SSCP Matrix for Error (W) Pillai's 0.58983 21.570 3 44 0.000

Length Width Height

Length 14966 8841 5189

Width 8841 5548 3131

Height 5189 3131 1937

Determinant = 6.067E+08

SSCP Matrix for Error (B+W)

Length Width Height

Length 20441 12318 7978

Width 12318 7756 4902

Height 7978 4902 3358

Determinant = 1.481E+09 Determinant (W)/Determinant (B+W)=.6067/1.481=.410

Notations for MANOVA-

Exact F-distributions for Wilk’s Lambda

~ )( )( : 2 )

~ )( )( : 2 )

~ )( )( : 1)

~ )( )( : 1)

)1,min(way}-One {),min(

way)-Onein ( E of

way)-Onein 1-( (H) Hypothesis of

MANOVAin Denote

)1(2,2

11

)1(2,211

1,

11

,1

pvpp

pv

vqqv

pvpp

pv

vqqv

FFqd

FFpc

FF qb

FF pa

gpInqps

N-gdfv

gdfq

−+

−+

Λ

Λ−

−

−

Λ

Λ−

−+

−+

Λ

Λ−

Λ

Λ−

==

==

==

==

−==

=

=


Hotelling T2Means

Gender N Length Width Height

female 25 134.52 101.72 51.480

male 25 113.38 88.29 40.708

Diff. 22.67 14.29 11.33

SSCP Matrix for Error (W)

Length Width Height

Length 14966 8841 5189

Width 8841 5548 3131

Height 5189 3131 1937

Spooled = SSCP Matrix for Error (W)/(n1+n2-2)

Hotelling T2 =(xbar1-xbar2)’ [Spooled (1/n1+1/n2)]-1(xbar1-xbar2)=65.66

F= Hotelling T2*(n1+n2-p-1)/[(n1+n2-2)p]= 65.66*[44/(3*46)]=65.66*0.319=20.94

df=p,n1+n2-p-1=3,44

Wilks' 0.410 21.570 3 44 0.000

11.333 14.292 22.667

Amitriptyline Data –Multivariate Regression Analysis - Table7.6

•Amitriptyline – drug for depression. Several side effects: irregular heartbeat, abnormal BP, etc. Data on 17 patients admited after amitriptyline overdose Two dependent variables and 5 predictor variables:

•Y1 =Total TCAD plasma level (TOT)

•Y2 =Amount of amitriptyline in TCAD plasma level (AMI)

•z1 =Gender 1=female, 0=male

•z2 =Amount of amitriptyline taken at time of overdose(AMT)

•z3 =PR wave measurements (PR)

•z4 =Diastolic blood pressure (DIAP)

•z5 =QRS wave measurements (QRS)

•Analysis: Model to predict Y1 and Y2 from the predictor variables

•Multivariate Linear Regression Models


Y1-Tot -

TCAD Y2-Ami

z1-

Gender

z2-Amt

Anti-

depress z3-PR

z4-Diap

(Diastolic

BP) z5-QRS

3389 3149 1 7500 220 0 140

1101 653 1 1975 200 0 100

1131 810 0 3600 205 60 111

596 448 1 675 160 60 120

896 844 1 750 185 70 83

1767 1450 1 2500 180 60 80

807 493 1 350 154 80 98

1111 941 0 1500 200 70 93

645 547 1 375 137 60 105

628 392 1 1050 167 60 74

1360 1283 1 3000 180 60 80

652 458 1 450 160 64 60

860 722 1 1750 135 90 79

500 384 0 2000 160 60 80

781 501 0 4500 180 0 100

1070 405 0 1500 170 90 120

1754 1520 1 3000 180 0 129


•Multivariate Linear Regression ModelsRegression Analysis: Y1 versus Z1, Z2, Z3, Z4, Z5

The regression equation is

Y1 = - 2879 + 676 Z1 + 0.285 Z2 + 10.3 Z3 + 7.25 Z4 + 7.60 Z5

Predictor Coef SE Coef T P

Constant -2879.5 893.3 -3.22 0.008

Z1 675.7 162.1 4.17 0.002

Z2 0.28485 0.06091 4.68 0.001

Z3 10.272 4.255 2.41 0.034

Z4 7.251 3.225 2.25 0.046

Z5 7.598 3.849 1.97 0.074

S = 281.232 R-Sq = 88.7% R-Sq(adj) = 83.6%

Analysis of Variance

Source DF SS MS F P

Regression 5 6835932 1367186 17.29 0.000

Residual Error 11 870008 79092

Total 16 7705940


•Multivariate Linear Regression ModelsRegression Analysis: Y2 versus Z1, Z2, Z3, Z4, Z5

The regression equation is

Y2 = - 2729 + 763 Z1 + 0.306 Z2 + 8.90 Z3 + 7.21 Z4 + 4.99 Z5

Predictor Coef SE Coef T P

Constant -2728.7 928.8 -2.94 0.014

Z1 763.0 168.5 4.53 0.001

Z2 0.30637 0.06334 4.84 0.001

Z3 8.896 4.424 2.01 0.070

Z4 7.206 3.354 2.15 0.055

Z5 4.987 4.002 1.25 0.239

Analysis of Variance

Source DF SS MS F P

Regression 5 6669669 1333934 15.60 0.000

Residual Error 11 940709 85519

Total 16 7610378

Radiotheraphy-Principal Components Table1.5

•Data of the 98 average ratings over the course of the treatment for 98 patients undergoing radiotherapy. Six components:

•X1 =number of symptoms (as nausea, sore throat)

•X2 =amount of activity (on a 1-5 scale)

•X3 =amount of sleep (on a 1-5 scale)

•X4 =amount of food consumed (on a 1-3 scale)

•X5 =appetite (on a 1-5 scale)

•X6 =skin reaction (on a 0-3 scale)

•Analysis: Finding a single (or several) measures – linear combinations of the 6 components, to represent patients’response to therapy.

•Principal components


Symptoms Activity Sleep Eat Appetite

Skin

Reaction

0.889 1.389 1.555 2.222 1.945 1

2.813 1.437 0.999 2.312 2.312 2

1.454 1.091 2.364 2.455 2.909 3

0.294 0.941 1.059 2 1 1

2.727 2.545 2.819 2.727 4.091 0

3.937 1.25 1.937 2.937 3.749 1

2.786 1.714 2.357 2.071 2 2

5.231 2.692 1.077 1.846 2.539 1

1.15 1.1 0.95 2 1 1

6.5 2.562 1.749 2.562 2.499 1

0.8 1 2.2 2.267 2.466 2

4.6 2 3 2.5 3.4 1

3.5 1.286 2.714 1.286 1.252 3

3.444 2.556 2.388 2.389 3 1

4.071 1 1 2.357 1.572 1

3.692 1 2.538 2.154 2.615 1

5.167 3 1 2.667 3.666 0

0.5 1 1 2 1 0

2.385 1.923 2.539 2.154 2.461 1

2.1 1.3 1.3 1.8 2.6 1

5 3.25 3.125 2.375 3.375 0

4.571 1.214 3.286 2.571 3.572 1

2.733 1.133 2.6 1.933 1.667 1

4.235 2.294 2.706 2.176 1.883 1

0 1 1.941 2 2 0


•X1 =number of symptoms (as nausea, sore throat) X2 =amount of activity (on a 1-5 scale)

•X3 =amount of sleep (on a 1-5 scale) X4 =amount of food consumed (on a 1-3 scale)

•X5 =appetite (on a 1-5 scale) X6 =skin reaction (on a 0-3 scale)

•Principal components and Factor Analysis•Principal Component Analysis: Symptoms, Activity, Sleep, Eat, Appetite, SkinRea

Eigenanalysis of the Correlation Matrix

Eigenvalue 2.8643 1.0764 0.7776 0.6503 0.3880 0.2433

Proportion 0.477 0.179 0.130 0.108 0.065 0.041

Cumulative 0.477 0.657 0.786 0.895 0.959 1.000

Variable PC1 PC2 PC3 PC4 PC5 PC6

Symptoms 0.445 -0.027 0.339 0.551 0.601 0.146

Activity 0.429 -0.292 0.499 0.061 -0.687 0.076

Sleep 0.359 0.380 -0.628 0.421 -0.332 0.212

Eat 0.463 -0.021 -0.125 -0.666 0.207 0.533

Appetite 0.521 -0.074 -0.203 -0.201 0.103 -0.794

SkinReaction 0.056 0.874 0.430 -0.179 -0.053 -0.116

Radiotheraphy- Factor Analysis Table1.5•Principal Component Factor Analysis of the Correlation Matrix

Unrotated Factor Loadings and Communalities

Variable Factor1 Factor2 Factor3 Factor4 Communality

Symptoms 0.753 -0.028 0.299 0.444 0.855

Activity 0.727 -0.303 0.440 0.049 0.815

Sleep 0.607 0.394 -0.554 0.340 0.946

Eat 0.783 -0.022 -0.110 -0.537 0.914

Appetite 0.882 -0.076 -0.179 -0.162 0.842

SkinReaction 0.095 0.907 0.379 -0.144 0.996

Variance 2.864 1.076 0.777 0.650 5.368

% Var 0.477 0.179 0.130 0.108 0.895

Rotated Factor Loadings and Communalities - Varimax Rotation

Variable Factor1 Factor2 Factor3 Factor4 Communality

Symptoms 0.125 -0.858 0.314 -0.065 0.855

Activity 0.392 -0.806 -0.091 0.052 0.815

Sleep 0.220 -0.121 0.936 -0.086 0.946

Eat 0.927 -0.188 0.121 -0.066 0.914

Appetite 0.739 -0.393 0.369 0.072 0.842

SkinReaction 0.015 -0.008 0.071 -0.995 0.996

Variance 1.625 1.590 1.138 1.014 5.368

% Var 0.271 0.265 0.190 0.169 0.895

Hemophilia Data- Discriminant Analysis- Table 11.8

•To construct a procedure for detecting potential hemophilia A carriers, blood samples assayed and measurements made on two variables:

•X1 =log10(AHF activity) where AHF=antihemophilic factor

•X2 = log10(AHF-like antigen)

•Measurements taken on two groups of women:– A group of n1=24 women who do not carry the hemophilic gene –Normal group– A group n2=22 women from known hemophilia A carriers (daughters of hemophiliacs, mothers with more than one hemophiliac son, mothers with one hemophiliac son and other hemophilic relatives) –Obligatory carriers

–New cases to be classified–Classification and Discrimination – Discriminant Analysis

Hemophilia Data- Discriminant Analysis- Table 11.8

Noncarriers Obligatory Carriers New cases Requiring Classification

Group

log(AHF

activity)

log(AHF

antigen) Group

log(AHF

activity)

log(AHF

antigen) Group

log(AHF

activity)

log(AHF

antigen)

1 -0.0056 -0.1657 2 -0.3478 0.1151 3 -0.112 -0.279

1 -0.1698 -0.1585 2 -0.3618 -0.2008 3 -0.059 -0.068

1 -0.3469 -0.1879 2 -0.4986 -0.086 3 0.064 0.012

1 -0.0894 0.0064 2 -0.5015 -0.2984 3 -0.043 -0.052

1 -0.1679 0.0713 2 -0.1326 0.0097 3 -0.05 -0.098

1 -0.0836 0.0106 2 -0.6911 -0.339 3 -0.094 -0.113

1 -0.1979 -0.0005 2 -0.3608 0.1237 3 -0.123 -0.143

1 -0.0762 0.0392 2 -0.4535 -0.1682 3 -0.011 -0.037

1 -0.1913 -0.2123 2 -0.3479 -0.1721 3 -0.21 -0.09

1 -0.1092 -0.119 2 -0.3539 0.0722 3 -0.126 -0.019

1 -0.5268 -0.4773 2 -0.4719 -0.1079

1 -0.0842 0.0248 2 -0.361 -0.0399

1 -0.0225 -0.058 2 -0.3226 0.167

1 0.0084 0.0782 2 -0.4319 -0.0687

1 -0.1827 -0.1138 2 -0.2734 -0.002

1 0.1237 0.214 2 -0.5573 0.0548

1 -0.4702 -0.3099 2 -0.3755 -0.1865

1 -0.1519 -0.0686 2 -0.495 -0.0153

1 0.0006 -0.1153 2 -0.5107 -0.2483

1 -0.2015 -0.0498 2 -0.1652 0.2132

1 -0.1932 -0.2293 2 -0.2447 -0.0407

1 0.1507 0.0933 2 -0.4232 -0.0998

Hemophilia Data- Discriminant Analysis- Table 11.8Discriminant Analysis: GroupDisc versus logActivity, logAntigen

Linear Method for Response: GroupDisc

Predictors: logActivity, logAntigen

Group 1 2

Count 30 45

Summary of classification

True Group

Put into Group 1 2

1 27 8

2 3 37

Total N 30 45

N correct 27 37

Proportion 0.900 0.822

N = 75 N Correct = 64 Proportion Correct = 0.853

Squared Distance Between Groups

1 2

1 0.00000 4.57431

2 4.57431 0.00000

Hemophilia Data- Discriminant Analysis- Table 11.8Discriminant Analysis: GroupDisc versus logActivity, logAntigen

Linear Discriminant Function for Groups

1 2

Constant -0.411 -3.970

logActivity -6.824 -26.143

logAntigen 1.270 18.394

Summary of Misclassified Observations

True Pred Squared

Observation Group Group Group Distance Probability

5** 1 2 1 2.7065 0.288

2 0.8962 0.712

32** 2 1 1 2.366 0.502

2 2.383 0.498

Prediction for Test Observations

Squared

Observation Pred Group From Group Distance Probability

1 1 1 4.260 0.998

2 16.607 0.002

Hemophilia Data- Discriminant Analysis- Table 11.8Discriminant Analysis: -Minitab

Distance and discriminant functions

Squared distance: The squared distance (also called the Mahalanobisdistance) of observation x to the center (mean) of group i is given by the general form:

di2 (x) = (x - mi)' Sp

-1 (x - mi) where: x = p-column vector with the values of this observation

mi = column vector of length p containing the means of the predictors calculated from the data in group i;

Sp = pooled covariance matrix, used in linear discriminant analysis;

The linear discriminant function =mi' Sp-1x - 0.5mi'Sp

-1mi + ln p where:

x = column vector of length p containing the values of the predictors for this observation (note, this column vector is stored as one row)

mi = column vector of length p containing the means of the predictors calculated from the data in group i

Sp = pooled covariance matrix

ln p = natural log of the prior probability

For a given x, the group with the smallest squared distance has the largest linear discriminant function.

Hemophilia Data- Discriminant Analysis- Table 11.8Discriminant Analysis: -Minitab

Distance and discriminant functions

Posterior probability- The posterior probability for group i given the data and is calculated by:

pi fi(x)/Σpi fi(x)

where:

pi = prior probability of group i

fi (x) = the joint density for the data in group i (with the population parameters replaced by sample estimates)

The largest posterior probability is equivalent to the largest value of ln [pi fi(x)], where (under normality):

ln [pi fi(x)] = -0.5 [di2(x) - 2 lnpi] - constant value

where:

di2 (x) = -2 [mi' Sp

-1x - 0.5mi'Sp-1mi + lnpi] + x' Sp

-1x

Bone and Skull - White leghorn fowls –

Canonical Correlation- Ex10.4

•To assess correlation between two sets of variables

•Head Measurements (X(1) )–X(1)1 =Skull length

–X(1)2 =Skull breadth

•Leg Measurements (X(1) )–X(2)1 =Femur length

–X(2)2 =Tibia length

–Create new variables which represent the most of the correlations between the two sets of variables –

–Cannonical Correlation

Bone and Skull - White leghorn fowls –

Canonical Correlation- Ex10.4

Skull

Lenghts

Skull

Breadth

Skull

Breadth

Tibia

Length

Skull

Lenghts 1 0.505 0.569 0.602Skull

Breadth 0.505 1 0.422 0.467

Skull

Breadth 0.569 0.422 1 0.926Tibia

Length 0.602 0.467 0.926 1

Universities – Cluster Analysis-Table 12-9

•Data on certain universities for certain variables used to compare or rank major universities. The variables:

•X1 =Average SAT for new freshmen

•X2 = Percent new freshmen in top 10% of high school class

•X3 =Percent of applicants accepted

•X4 =Student faculty ratio

•X5 =Estimated annual expenses

•X6 =Graduation rate (%)

•Analysis: Clustering observations (universities) based on linear combinations of variables (or specific variables). Definition of distances

•Cluster Analysis, Distance Methods and Ordination

Universities – Cluster Analysis-Table 12-9# SAT Top10 Accept SFRatio Expenses

1 Harvard 14.00 91 14 11 39.525

2 Princeton 13.75 91 14 8 30.220

3 Yale 13.75 95 19 11 43.514

4 Stanford 13.60 90 20 12 36.450

5 MIT 13.80 94 30 10 34.870

6 Duke 13.15 90 30 12 31.585

7 CalTech 14.15 100 25 6 63.575

8 Dartmouth 13.40 89 23 10 32.162

9 Brown 13.10 89 22 13 22.704

10 JohnsHopkins 13.05 75 44 7 58.691

11 Uchicago 12.90 75 50 13 38.380

12 UPenn 12.85 80 36 11 27.553

13 Cornell 12.80 83 33 13 21.860

14 Northwestern 12.60 85 39 11 28.052

15 Columbia 13.10 76 24 12 31.510

16 NotreDame 12.55 81 42 13 15.122

17 UVir 12.25 77 44 14 13.349

18 Georgetown 12.55 74 24 12 20.126

19 CarnegieMellon 12.60 62 59 9 25.026

20 Umichigan 11.80 65 68 16 15.470

21 UCBerkeley 12.40 95 40 17 15.140

22 Uwisconsin 10.85 40 69 15 11.857

23 PennState 10.81 38 54 18 10.180

24 Purdue 10.05 28 90 19 9.066

25 TexasA&M 10.75 49 67 25 8.704

Universities – Cluster Analysis-Table 12-9Cluster Analysis of Observations: SAT, Top10, Accept, SFRatio, Expenses, Grad

Euclidean Distance, Single Linkage

Amalgamation Steps

Number

Number of obs.

of Similarity Distance Clusters New in new

Step clusters level level joined cluster cluster

1 24 95.2869 5.3135 16 17 16 2 Notre Dame UVirginia

2 23 94.7291 5.9423 12 14 12 2 UPenn Northwestern

3 22 94.6465 6.0355 4 8 4 2 Stanford Dartmouth

4 21 93.9052 6.8712 5 6 5 2 MIT Duke

5 20 93.4580 7.3753 4 5 4 4 (Stanford, Dartmouth) (MIT, Duke)

6 19 93.4570 7.3765 12 13 12 3

7 18 93.2462 7.6141 1 3 1 2

8 17 92.9253 7.9759 1 4 1 6

9 16 91.4509 9.6381 1 2 1 7

10 15 91.1059 10.0272 1 9 1 8

11 14 89.2653 12.1022 12 16 12 5

12 13 89.1401 12.2433 15 18 15 2

13 12 88.9290 12.4814 1 12 1 13

14 11 88.4325 13.0411 1 15 1 15

15 10 87.1170 14.5242 22 25 22 2

16 9 84.0878 17.9392 22 23 22 3

17 8 83.7468 18.3237 1 11 1 16

18 7 82.2972 19.9579 1 21 1 17

19 6 82.2608 19.9989 19 20 19 2

20 5 80.4746 22.0127 1 10 1 18

21 4 78.0311 24.7675 22 24 22 4

22 3 77.0504 25.8731 1 19 1 20

23 2 76.3836 26.6248 1 22 1 24

24 1 76.3051 26.7134 1 7 1 25


Observations

Similarity

CalTech

Purdue

PennState

TexasA&M

Uwisconsin

Umichigan

CarnegieMellon

JohnsHopkins

UCBerkeley

Uchicago

Georgetown

Columbia

Uvir

NotreDame

Cornell

Northwestern

Upenn

Brown

Princeton

Duke

MIT

Dartmouth

Stanford

Yale

Harvard

76.31

84.20

92.10

100.00

Dendrogram with Single Linkage and Euclidean Distance


Cluster Analysis of Variables: SAT, Top10, Accept, SFRatio, Expenses, Grad

Correlation Coefficient Distance, Single Linkage- Amalgamation Steps

Number

Number of obs.

of Similarity Distance Clusters New in new

Step clusters level level joined cluster cluster

1 5 96.1261 0.07748 1 2 1 2

2 4 88.9491 0.22102 1 5 1 3

3 3 87.3856 0.25229 1 6 1 4

4 2 81.5832 0.36834 3 4 3 2

5 1 22.0783 1.55843 1 3 1 6

Correlations: SAT, Top10, Accept, SFRatio, Expenses, Grad

SAT Top10 Accept SFRatio Expenses

Top10 0.923

Accept -0.886 -0.859

SFRatio -0.813 -0.643 0.632

Expenses 0.779 0.611 -0.558 -0.782

Grad 0.748 0.746 -0.820 -0.561 0.394

V a r ia b le s

Similarity

S F R a tioA c c e p tG r a dE xp e n s e sT o p 1 0S A T

2 2 .0 8

4 8 .0 5

7 4 .0 3

1 0 0 .0 0

D e nd r o g r am w i th S i n g l e L in k a g e a n d C o r r e l a t i o n C o e f f i c i e n t D i s ta n c e

Universities – Cluster Analysis-Table 12-9Cluster Analysis of Observations: SAT, Top10, Accept, SFRatio, Expenses, Grad

Euclidean Distance, Single Linkage

Amalgamation Steps

K-means Cluster Analysis: SAT, Top10, Accept, SFRatio, Expenses, Grad

Number of clusters: 2 - Cluster Centroids

Variable Cluster1 Cluster2 centroid

SAT 13.1447 11.1433 12.6644

Top10 85.7895 47.0000 76.4800

Accept 30.1579 67.8333 39.2000

SFRatio 11.3684 17.0000 12.7200

Expenses 31.8099 13.3838 27.3876

Grad 90.7368 74.0000 86.7200

Number of clusters: 3 - Cluster Centroids

Grand

Variable Cluster1 Cluster2 Cluster3 centroid

SAT 13.1031 11.1433 13.3667 12.6644

Top10 86.2500 47.0000 83.3333 76.4800

Accept 28.3750 67.8333 39.6667 39.2000

SFRatio 11.8750 17.0000 8.6667 12.7200

Expenses 27.7339 13.3838 53.5487 27.3876

Grad 91.8125 74.0000 85.0000 86.7200

Documents

Multivariate Analysis -Overviefuchs/Introduction And Summary.pdf · •z4 =Diastolic blood pressure (DIAP) •z5 =QRS wave measurements (QRS) ... 860 722 1 1750 135 90 79 500 384