Multivariate analysis

The PRINCOMP Procedure

n=30, # x's=10

Observations 30

Variables (x) used in PCA

Variables 10

Simple Statistics

TASTE ODOR PH ACID1 ACID2

Mean 0.2633333333 0.0733333333 4.202000000 1.542666667 0.8236666667

StD 0.3978245440 0.4298623142 0.165995430 0.153912947 0.2627898509

Mean & Std Dev reported for each x

Simple Statistics

SAKE SUGARDIR SUGARTOT ALCOHOL FNITRO

Mean -1.406666667 3.488333333 4.404000000 15.97433333 121.7333333

StD 2.240371480 0.364805884 0.509269930 0.59266600 39.2566474

Pairwise correlations for x's

Correlation Matrix

TASTE ODOR PH ACID1 ACID2 SAKE SUGARDIR SUGARTOT ALCOHOL FNITRO

TASTE 1.0000 0.5627 0.2163 0.1041 0.2006 -.0444 0.1295 0.0327 -.0702 0.0922

ODOR 0.5627 1.0000 -.0809 0.1330 0.2048 -.1713 0.1794 0.2413 0.1519 0.0645

PH 0.2163 -.0809 1.0000 0.1581 0.7036 -.2906 -.4529 -.3445 -.1143 0.6825

ACID1 0.1041 0.1330 0.1581 1.0000 0.4865 -.0346 -.1646 0.0048 0.4185 0.3693

ACID2 0.2006 0.2048 0.7036 0.4865 1.0000 -.3103 -.3369 -.1904 0.2999 0.8749

SAKE -.0444 -.1713 -.2906 -.0346 -.3103 1.0000 -.4281 -.5873 -.0911 -.2628

SUGARDIR 0.1295 0.1794 -.4529 -.1646 -.3369 -.4281 1.0000 0.8230 0.2340 -.2977

SUGARTOT 0.0327 0.2413 -.3445 0.0048 -.1904 -.5873 0.8230 1.0000 0.4453 -.1734

ALCOHOL -.0702 0.1519 -.1143 0.4185 0.2999 -.0911 0.2340 0.4453 1.0000 0.2892

FNITRO 0.0922 0.0645 0.6825 0.3693 0.8749 -.2628 -.2977 -.1734 0.2892 1.0000

With R matrix, keep PCs with λ's greater than 1

Eigenvalue (λ) represents variance of extracted PC

Eigenvalues of the Correlation Matrix

Eigenvalue Difference Proportion Cumulative

1 3.17516850 0.60555170 0.3175 0.3175

Cumulative is sum of all Proportions on that line and above

2 2.56961680 1.13405256 0.2570 0.5745

1st PC has largest variance (λ), 2nd PC has next largest variance, etc.

3 1.43556424 0.16292084 0.1436 0.7180

4 1.27264340 0.73255863 0.1273 0.8453

5 0.54008477 0.07085030 0.0540 0.8993

6 0.46923447 0.21869127 0.0469 0.9462

7 0.25054320 0.13179249 0.0251 0.9713

8 0.11875071 0.01412773 0.0119 0.9832

9 0.10462299 0.04085206 0.0105 0.9936

10 0.06377093 0.0064 1.0000

The PRINCOMP Procedure

Proportion is λ divided by sum of all 10 λ's

Difference between successive variances (Eigenvalue (λ) represents variance of extracted PC

10 PCs extracted since # variables = 10

Scree Plot shows decreasing values of variances (λ's)

These PCs are "new" variables that are uncorrelated

The PRINCOMP Procedure

Values shown in each column are "weights" assigned to each variable (x) in the PC linear combination

Eigenvectors

PC1 PC2 PC3 PC4 PC5

TASTE 0.121117 0.186763 0.687027 0.119507 -.123337

ODOR 0.058578 0.321794 0.536887 0.269035 0.253609

PH 0.463397 -.058975 0.076576 -.376941 -.050550

ACID1 0.286904 0.163316 -.185837 0.485428 -.762026

ACID2 0.523825 0.141770 -.038462 -.015263 0.115793

SAKE -.089950 -.423482 0.064458 0.556492 0.240452

SUGARDIR -.312368 0.450915 -.002795 -.138754 -.032312

SUGARTOT -.224605 0.535005 -.139801 -.105888 -.040059

ALCOHOL 0.093332 0.361365 -.395331 0.432719 0.422154

FNITRO 0.495613 0.108829 -.133692 -.089368 0.291793

Eigenvectors

PC6 PC7 PC8 PC9 PC10

TASTE 0.552644 -.071707 -.234965 -.286170 -.035933

ODOR -.603234 -.041160 0.244279 0.183576 -.095468

PH 0.237662 -.446215 0.459149 0.404325 -.037341

ACID1 -.075112 0.073284 0.140631 0.044888 -.085736

ACID2 -.081558 0.188919 -.454547 0.257048 0.612666

SAKE 0.311685 0.163319 0.397015 0.158032 0.366744

SUGARDIR 0.317048 0.493828 0.068749 0.559588 -.128635

SUGARTOT 0.044087 -.190745 0.381853 -.359814 0.563344

ALCOHOL 0.237993 -.408005 -.189331 0.076490 -.272310

FNITRO 0.075269 0.528777 0.322346 -.423568 -.250864

PC1 = .12*TASTE + .06*ODOR + .46*PH + .29*ACID1 + .52*ACID2 -.09*SAKE - .31*SUGARDIR - .22*SUGARTOT + .09*ALCOHOL + .50*FNITRO