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PRINCIPAL COMPONENTS ANALYSIS (PCA)
Introduction • PCA is considered an exploratory technique that can be used to gain a better
understanding of the interrelationships between variables.
• PCA is performed on a set of data with the hope of simplifying the description of a set of interrelated variables.
• Variables are treated equally and they are not separated into dependent and independent variables.
• In simplest terms, PCA transforms the original interrelated variables into a new set of
uncorrelated variables call Principal Components.
• Each principal component is a linear combination of the original variables.
• The amount of information expressed by each principal component is its variance.
• Principal components often are displayed in rank order of decreasing variance.
• The principal component with the highest variance is termed the “first principal component.”
• An advantage of principal components to researchers is that the complexity in
interpretation that can be caused by having a large number of interrelated variables can be reduced by utilizing only the first few principal components that explain a large proportion of the total variation.
• PCA can be used to test for normality. If the principal components are not normally
distributed, then the original data weren’t either. Basic Concepts
• Suppose we have a random sample of N observations for two variables, X1 and X2.
o To simplify the description of these two variables, we will subtract the mean of each dataset from each observation; thus,
§ 𝑥! = (𝑋! − 𝑋!) and 𝑥! = (𝑋! − 𝑋!)
o The values of x1 and x2 would each have a mean of 0 and the sample variances 𝑆!!and 𝑆!! would be unaffected by using the deviations.
o Our goal through PCA is to create two new variables C1 and C2, called principal components that are uncorrelated.
o The new variables are linear functions of x1 and x2 that can be written as:
§ 𝐶! = 𝑎!!𝑥! + 𝑎!"𝑥! and 𝐶! = 𝑎!"𝑥! + 𝑎!!𝑥!, and
§ Mean C1 =Mean C2 = 0
§ Variance C1 = 𝑎!!! 𝑆!! + 𝑎!"! 𝑆!! + 2𝑎!!𝑎!"𝑟𝑆!𝑆!
§ Variance C2 = 𝑎!"! 𝑆!! + 𝑎!!! 𝑆!! + 2𝑎!"𝑎!!𝑟𝑆!𝑆!
§ The variances for C1 and C2 are referred to the first and second eigenvalues of covariance matrix of X1 and X2
o The coefficients are chosen such that
i. The Variance C1 is maximized and greater than all other variances.
• The Var C1 ≥ Var C2 ≥ . . . ≥ Var CP.
ii. The N values of C1 and C2 are uncorrelated.
iii. 𝑎!!! + 𝑎!"! + 𝑎!"! + 𝑎!!! = 1 (i.e. the sum of the squares of the coefficients is one).
o Hotelling originally derived the mathematical solution for the coefficients.
o PCA can be thought of as a rotation of the original x1 and x2 axes to new axes of
C1 and C2.
o The three items above that are related to how the coefficients are chosen determine the amount of the rotation of the new C1 and C2 axes.
o The values for C1 and C2 are found by drawing perpendicular lines to the new axes
from a given point, x1, x2.
Figure 1. Diagram showing the original x1 and x2 axes and the new C1 and C2 axes.
Figure 2. Plot showing principal components for two variable.
The Number of Components to Retain • An important concept of PCA is to reduce the number of variables or reduce
dimensionality.
• An important decision that the researcher must make when using PCA is to determine the number of principal components to use.
• This decision has no hard-set rules, and the decision may seem subjective at times.
• Common methods to reduce the number of principal components include:
o Determine the minimum amount of variation that you want defined by the
principal components. Some individuals use a cutoff of 80%, or may even go lower to 50%.
o Another option is to eliminate the principal components that explain insufficient variation. A common cutoff is <5%.
o Another method is to eliminate all principal components that explain less than
70/P percent of the variation, where P = the total number variables.
o Scree plots from the SAS analysis also can be used. The place where the plot has an “elbow” can be used as the cutoff. Example of using the scree plots will be discussed in the next section.
Examples of SAS Analyses Using Proc Princomp Example 1: Using PCA to reduce the number of variables. This example starts with 20 variables
X1 through X20.
• SAS commands
ods graphics on; ods rtf file='pca.rtf'; proc princomp; var x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20; run; ods rtf close;
PCA of the Depression Data Set
The PRINCOMP Procedure
Observations 294
Variables 20
Simple Statistics
x1 x2 x3 x4 x5 x6 x7
Mean 0.3639455782 0.5680272109 0.5442176871 0.1938775510 0.5510204082 0.2482993197 0.2448979592
StD 0.7573481288 0.8097938384 0.8916089113 0.5898717961 0.8194202018 0.6261485001 0.6515074328
Simple Statistixs
x8 x9 x10 x11 x12 x13 x14
Mean 0.3503401361 0.5680272109 0.4625850340 0.3605442177 0.5136054422 0.3401360544 0.7210884354
StD 0.7770491263 0.9128200608 0.7990788967 0.6857454351 0.7738154621 0.7058661916 0.9690240890
Simple Statistics
x15 x16 x17 x18 x19 x20
Mean 0.6734693878 0.7482993197 0.6190476190 0.3095238095 0.2551020408 0.2482993197
StD 0.8910879595 0.8924678170 0.8885570105 0.6527268435 0.5720072643 0.5867541003
Correlation Matrix
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13
x1 1.0000 0.6245 0.4284 0.5749 0.5172 0.4061 0.5519 0.2002 0.3615 0.5217 0.4628 0.2099 0.2209
x2 0.6245 1.0000 0.6198 0.4832 0.7303 0.4344 0.5699 0.1328 0.3746 0.5841 0.4597 0.4043 0.1922
x3 0.4284 0.6198 1.0000 0.4411 0.6065 0.4846 0.5688 0.1672 0.2353 0.4646 0.4483 0.2415 0.1496
x4 0.5749 0.4832 0.4411 1.0000 0.5267 0.3774 0.5243 0.2534 0.2068 0.3883 0.3919 0.1923 0.2017
x5 0.5172 0.7303 0.6065 0.5267 1.0000 0.4708 0.4880 0.1353 0.2782 0.5737 0.4105 0.3649 0.2000
x6 0.4061 0.4344 0.4846 0.3774 0.4708 1.0000 0.3942 0.1503 0.1107 0.3154 0.2916 0.3065 0.1326
x7 0.5519 0.5699 0.5688 0.5243 0.4880 0.3942 1.0000 0.2884 0.3564 0.4831 0.4816 0.2574 0.2042
x8 0.2002 0.1328 0.1672 0.2534 0.1353 0.1503 0.2884 1.0000 0.1612 0.1723 0.1593 0.1368 0.1056
x9 0.3615 0.3746 0.2353 0.2068 0.2782 0.1107 0.3564 0.1612 1.0000 0.4808 0.3587 0.1557 0.0328
x10 0.5217 0.5841 0.4646 0.3883 0.5737 0.3154 0.4831 0.1723 0.4808 1.0000 0.5417 0.2934 0.2526
x11 0.4628 0.4597 0.4483 0.3919 0.4105 0.2916 0.4816 0.1593 0.3587 0.5417 1.0000 0.2802 0.1477
x12 0.2099 0.4043 0.2415 0.1923 0.3649 0.3065 0.2574 0.1368 0.1557 0.2934 0.2802 1.0000 0.0790
x13 0.2209 0.1922 0.1496 0.2017 0.2000 0.1326 0.2042 0.1056 0.0328 0.2526 0.1477 0.0790 1.0000
x14 0.2458 0.3200 0.2000 0.1785 0.2716 0.3227 0.2545 0.0940 0.1566 0.3523 0.2803 0.2736 0.2440
x15 0.3638 0.5228 0.3361 0.2247 0.3594 0.2437 0.3969 0.0918 0.1281 0.3279 0.2603 0.3331 0.2911
PCA of the Depression Data Set
The PRINCOMP Procedure
Correlation Matrix
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13
x16 0.2673 0.4110 0.2671 0.1449 0.3303 0.2466 0.3412 0.0439 0.2222 0.3026 0.2547 0.2867 0.2068
x17 0.3893 0.4441 0.2497 0.2195 0.3362 0.3424 0.3563 0.0457 0.1583 0.3404 0.2710 0.3054 0.2236
x18 0.3099 0.2990 0.3019 0.2868 0.2862 0.2706 0.2626 0.1152 0.1908 0.3396 0.2836 0.2248 0.2078
x19 0.1789 0.2682 0.2823 0.1159 0.1724 0.1275 0.2256 0.0132 0.2183 0.1964 0.2346 0.0577 0.0464
x20 0.4257 0.4276 0.4389 0.5113 0.3888 0.2682 0.5368 0.2951 0.2838 0.4020 0.3366 0.2519 0.1662
Correlation Matrix
x14 x15 x16 x17 x18 x19 x20
x1 0.2458 0.3638 0.2673 0.3893 0.3099 0.1789 0.4257
x2 0.3200 0.5228 0.4110 0.4441 0.2990 0.2682 0.4276
x3 0.2000 0.3361 0.2671 0.2497 0.3019 0.2823 0.4389
x4 0.1785 0.2247 0.1449 0.2195 0.2868 0.1159 0.5113
x5 0.2716 0.3594 0.3303 0.3362 0.2862 0.1724 0.3888
x6 0.3227 0.2437 0.2466 0.3424 0.2706 0.1275 0.2682
x7 0.2545 0.3969 0.3412 0.3563 0.2626 0.2256 0.5368
x8 0.0940 0.0918 0.0439 0.0457 0.1152 0.0132 0.2951
x9 0.1566 0.1281 0.2222 0.1583 0.1908 0.2183 0.2838
x10 0.3523 0.3279 0.3026 0.3404 0.3396 0.1964 0.4020
x11 0.2803 0.2603 0.2547 0.2710 0.2836 0.2346 0.3366
x12 0.2736 0.3331 0.2867 0.3054 0.2248 0.0577 0.2519
x13 0.2440 0.2911 0.2068 0.2236 0.2078 0.0464 0.1662
x14 1.0000 0.2459 0.3921 0.4232 0.1316 0.1842 0.2603
x15 0.2459 1.0000 0.3469 0.2303 0.1802 0.0568 0.2274
x16 0.3921 0.3469 1.0000 0.4253 0.1401 0.2532 0.2371
x17 0.4232 0.2303 0.4253 1.0000 0.3511 0.2792 0.2213
x18 0.1316 0.1802 0.1401 0.3511 1.0000 0.0620 0.2086
x19 0.1842 0.0568 0.2532 0.2792 0.0620 1.0000 0.3699
x20 0.2603 0.2274 0.2371 0.2213 0.2086 0.3699 1.0000
PCA of the Depression Data Set
The PRINCOMP Procedure
Eigenvalues of the Correlation Matrix
Eigenvalue Difference Proportion Cumulative
1 7.05541769 5.56984843 0.3528 0.3528
2 1.48556925 0.25405950 0.0743 0.4270
3 1.23150975 0.16582473 0.0616 0.4886
4 1.06568502 0.05305242 0.0533 0.5419
5 1.01263260 0.04517187 0.0506 0.5925
6 0.96746073 0.02064438 0.0484 0.6409
7 0.94681635 0.17757881 0.0473 0.6883
8 0.76923754 0.07460055 0.0385 0.7267
9 0.69463699 0.03451014 0.0347 0.7614
10 0.66012685 0.05249140 0.0330 0.7945
11 0.60763545 0.05829812 0.0304 0.8248
12 0.54933734 0.01270376 0.0275 0.8523
13 0.53663358 0.02791603 0.0268 0.8791
14 0.50871755 0.05781498 0.0254 0.9046
15 0.45090258 0.07572163 0.0225 0.9271
16 0.37518095 0.05399659 0.0188 0.9459
17 0.32118435 0.02662157 0.0161 0.9619
18 0.29456278 0.02618829 0.0147 0.9767
19 0.26837449 0.06999633 0.0134 0.9901
20 0.19837816 0.0099 1.0000
PCA of the Depression Data Set
The PRINCOMP Procedure
Eigenvectors
Prin1 Prin2 Prin3 Prin4 Prin5 Prin6 Prin7 Prin8 Prin9 Prin10 Prin11
x1 0.277438 -.144979 -.057702 0.002724 -.088268 -.118951 -.120184 -.180910 -.388430 -.197252 -.104234
x2 0.313183 0.027136 -.031630 -.247811 -.024397 0.100961 -.130714 0.042606 -.114186 -.008324 0.159686
x3 0.267798 -.154720 -.034590 -.247247 0.218305 -.042497 -.109251 0.111386 0.246253 0.392895 -.000589
x4 0.243554 -.319404 -.176943 0.071551 0.172925 -.141394 -.060163 -.152562 -.127640 -.410196 0.005257
x5 0.286784 -.049717 -.138388 -.279353 0.041111 0.013724 -.073581 -.097235 0.087690 0.008652 0.400198
x6 0.220570 0.053395 -.224213 -.182286 0.339874 -.150884 0.255087 -.273927 0.043571 0.339431 0.053202
x7 0.284370 -.164359 0.018960 0.076061 0.086999 0.096669 -.088212 0.036050 -.196078 0.080462 -.284635
x8 0.108096 -.304519 -.110325 0.556703 0.097606 0.320860 0.381442 0.066608 -.115406 0.374184 0.046032
x9 0.175781 -.168999 0.396225 0.014631 -.535478 0.126622 0.083942 -.070534 -.136258 0.120102 0.309623
x10 0.276625 -.045423 0.083456 -.008415 -.365054 -.017871 -.025569 -.163073 0.232840 0.022423 0.147770
x11 0.243270 -.104819 0.131395 -.041382 -.241930 -.030600 0.052013 -.123516 0.363904 0.008163 -.668991
x12 0.179019 0.229984 -.163432 -.145063 -.036843 0.367000 0.460876 0.303827 0.254277 -.390629 0.096363
x13 0.125906 0.212631 -.264529 0.540020 -.095286 -.190807 -.452218 0.040916 0.301849 0.039170 0.234516
x14 0.180253 0.401483 0.101409 0.246105 0.084712 0.045743 0.171322 -.507914 0.228825 -.092695 -.053963
x15 0.200363 0.209780 -.270323 -.031213 -.083408 0.390090 -.355778 0.287622 -.092704 0.050316 -.263350
x16 0.192430 0.417446 0.185012 0.046739 0.039931 0.202204 -.073058 0.033934 -.304982 0.199647 0.012610
x17 0.209684 0.390480 0.086023 0.068401 0.049913 -.318303 0.203555 0.027570 -.382192 -.080368 -.061428
x18 0.171712 0.015325 -.201930 0.062862 -.275225 -.549661 0.292328 0.474241 0.002129 0.088240 -.035984
x19 0.131489 0.056866 0.632607 0.023179 0.334922 -.157717 -.116624 0.316230 0.162291 0.040590 0.030883
x20 0.235700 -.228260 0.193254 0.240427 0.290938 0.082201 -.008931 0.176681 0.102174 -.374999 0.097475
Eigenvectors
Prin12 Prin13 Prin14 Prin15 Prin16 Prin17 Prin18 Prin19 Prin20
x1 0.286574 -.012444 0.125514 0.224443 -.224906 -.549164 -.265853 0.026949 0.238605
x2 0.182666 -.077524 -.189498 0.017131 0.132969 -.182041 0.091523 -.158884 -.781315
x3 -.133670 0.017281 -.142106 -.228717 0.188980 -.366600 0.032081 0.513338 0.166970
x4 -.192715 -.087906 0.044586 0.263833 0.360904 0.393507 -.014676 0.365430 -.074480
x5 -.028886 -.275472 -.241924 0.060494 0.086076 0.128577 0.084836 -.500952 0.460898
x6 0.102265 0.257438 0.506858 0.120893 -.240812 0.176439 0.113868 -.071734 -.105756
x7 -.179040 0.094940 0.117573 -.559414 0.102489 0.181793 -.468761 -.319636 -.027376
x8 0.210580 -.231186 -.193353 0.139832 0.016659 0.002992 0.020863 0.003605 -.012852
x9 -.002924 0.269038 0.335431 -.045605 0.312700 0.015411 0.207553 0.051066 0.094840
x10 0.053062 -.042353 -.242867 -.088813 -.572512 0.337189 -.255973 0.302199 -.084866
x11 -.030848 -.323119 0.110391 0.126584 0.028622 -.027464 0.300615 -.165863 -.019095
PCA of the Depression Data Set
The PRINCOMP Procedure
Eigenvectors
Prin12 Prin13 Prin14 Prin15 Prin16 Prin17 Prin18 Prin19 Prin20
x12 0.055942 -.150989 0.295300 -.079415 0.058212 -.108218 -.223006 0.116261 0.047158
x13 0.041837 -.176416 0.336937 -.109570 0.074537 -.097993 0.015767 -.033185 -.055836
x14 -.007998 0.370796 -.331885 0.061372 0.282669 -.135965 -.148451 -.050682 0.023338
x15 0.259095 0.380741 -.094768 0.176289 0.007444 0.259252 0.175924 0.076887 0.191330
x16 -.622118 -.235488 0.079951 0.300664 -.151649 -.037694 -.038036 0.069687 -.017238
x17 0.251275 -.230142 -.099636 -.417317 0.012245 0.136290 0.355917 0.174599 0.114463
x18 -.208372 0.262276 -.177396 0.213171 0.049702 -.034009 -.111069 -.149200 -.029813
x19 0.343680 -.052032 0.074344 0.268064 0.082175 0.162671 -.255453 -.034123 0.036257
x20 -.232928 0.293801 -.051536 -.125060 -.376759 -.161253 0.414382 -.124051 -.002511
Initial PCA Analysis of Malt Data to Determine the Number of Principal Components to Retain
• How many principal components should be retained?
1. Eleven principal components should be retained based on the rule of maintaining the total variation >80%.
2. Five principal components should be retained based on the rule of eliminating all principal components that explain less than 5% of the total variation.
3. Nine principal components should be retained based on the rule of eliminating all
principal components that explain <70/P% of the variation (70/20 = 3.5%).
4. Around three principal components should be retained based on the scree plots.
• So what is the correct answer?
o Decision should be based on your knowledge of the subject area.
o I would select a number between 5-7. Example 2: Using PCA to determine the interrelationships between variables related to malt
quality, particularly malt extract.
• Malt quality of barley lines is determined using a large number of correlated traits.
• PCA will be used to:
o Reduce dimensionality between the 10 variables that define malt quality.
o Determine which of the 10 variables contribute to explaining the “most” variability in each principal component based on the load.
Initial PCA Analysis of Malt Data to Determine the Number of Principal Components to Retain (Abbreviated Output)
Eigenvalues of the Correlation Matrix
Eigenvalue Difference Proportion Cumulative
1 3.25651383 1.32360184 0.3257 0.3257
2 1.93291199 0.29774838 0.1933 0.5189
3 1.63516362 0.45047335 0.1635 0.6825
4 1.18469026 0.23755389 0.1185 0.8009
5 0.94713637 0.52141083 0.0947 0.8956
6 0.42572554 0.12523121 0.0426 0.9382
7 0.30049433 0.12518352 0.0300 0.9683
8 0.17531082 0.05986813 0.0175 0.9858
9 0.11544268 0.08883213 0.0115 0.9973
10 0.02661056 0.0027 1.0000
Eigenvectors
Prin1 Prin2 Prin3 Prin4 Prin5 Prin6 Prin7 Prin8 Prin9 Prin10
kwt 0.460908 0.098994 -.191527 0.024931 -.208176 0.436128 0.416468 -.553296 -.001091 0.165322
plump 0.504392 0.093371 -.006892 -.054249 -.118362 0.021649 0.372624 0.726388 0.149609 -.174729
barcolor -.221913 -.226217 0.621107 -.046569 0.136818 0.175687 0.349279 -.099545 0.574247 -.019714
wrtcolor 0.107532 0.546630 -.186266 0.098715 0.493803 0.246585 -.334806 -.007926 0.473256 -.066470
protein 0.420932 -.168517 0.252873 0.408924 -.035322 -.235921 -.301670 0.038411 0.148166 0.626032
wrtprt 0.128291 0.440959 0.401698 0.424598 0.016312 -.388182 0.172970 -.235822 -.221777 -.402225
kolbach -.331075 0.526391 0.139328 -.107046 0.079364 0.056819 0.318532 0.201344 -.254311 0.605167
dp 0.308405 -.156832 0.400847 -.225186 0.484449 0.368787 -.173206 0.040258 -.512496 -.062732
alpha 0.033714 0.313990 0.370631 -.304447 -.640944 0.226079 -.446256 0.004503 0.064486 -.070536
bglucan -.269890 -.091582 -.006101 0.694381 -.163648 0.566538 -.045656 0.238714 -.145816 -.092158
Initial PCA Analysis of Malt Data to Determine the Number of Principal Components to Retain (Abbreviated Output)
• Based on the different methods of determining how many principal components to retain, I would keep five.
• The next step is to redo the analysis keeping in only five principal components.
• The different plots to interpret the results should be requested.
• SAS Commands
ods graphics on; ods rtf file='maltpca.rtf'; proc princomp n=5 plots (ncomp=3)=pattern; id variety; var kwt plump barcolor wrtcolor protein wrtprt kolbach dp alpha bglucan; *title 'PCA of Malt Quality Using All Variables'; title 'PCA of Malt Quality Analyses Using 5 Principal Components'; run; ods rtf close;
• In the Proc Princomp statement, I use the option n=5 to have the PCA only calculate only the first five principal components. Additionally, the statement “plots=pattern” will provide the graphical plots of the PCA. The option “(ncomp=3)” requests for the graphical output comparing only the first three principal components.
PCA of Malt Quality Analyses Using 5 Principal Components
The PRINCOMP Procedure
Observations 20
Variables 10
Simple Statistics
kwt plump barcolor wrtcolor protein wrtprt kolbach
Mean 36.00000000 97.09000000 38.50000000 1.810000000 13.67500000 5.947000000 45.55000000
StD 1.47362782 1.73384271 2.68524232 0.158612404 0.54567679 0.220838116 2.22911358
Simple Statistics
dp alpha bglucan
Mean 181.0000000 69.06000000 264.5000000
StD 17.7378572 2.68139791 73.1570049
Correlation Matrix
kwt plump barcolor wrtcolor protein wrtprt kolbach dp alpha bglucan
kwt 1.0000 0.7782 -.5134 0.2342 0.4569 0.1312 -.4251 0.2479 0.0974 -.2920
plump 0.7782 1.0000 -.3866 0.1879 0.6044 0.2405 -.3978 0.4145 0.1534 -.4576
barcolor -.5134 -.3866 1.0000 -.4325 -.0413 0.0799 0.1838 0.3028 0.1228 0.1932
wrtcolor 0.2342 0.1879 -.4325 1.0000 -.0639 0.3765 0.3811 0.0486 -.0324 -.1288
protein 0.4569 0.6044 -.0413 -.0639 1.0000 0.4147 -.6497 0.4834 -.0112 -.0560
wrtprt 0.1312 0.2405 0.0799 0.3765 0.4147 1.0000 0.3481 0.0949 0.3005 0.0508
kolbach -.4251 -.3978 0.1838 0.3811 -.6497 0.3481 1.0000 -.3280 0.3179 0.1166
dp 0.2479 0.4145 0.3028 0.0486 0.4834 0.0949 -.3280 1.0000 0.0238 -.4058
alpha 0.0974 0.1534 0.1228 -.0324 -.0112 0.3005 0.3179 0.0238 1.0000 -.1801
bglucan -.2920 -.4576 0.1932 -.1288 -.0560 0.0508 0.1166 -.4058 -.1801 1.0000
Eigenvalues of the Correlation Matrix
Eigenvalue Difference Proportion Cumulative
1 3.25651383 1.32360184 0.3257 0.3257
2 1.93291199 0.29774838 0.1933 0.5189
3 1.63516362 0.45047335 0.1635 0.6825
4 1.18469026 0.23755389 0.1185 0.8009
5 0.94713637 0.0947 0.8956
PCA of Malt Quality Analyses Using 5 Principal Components
The PRINCOMP Procedure
Eigenvectors
Prin1 Prin2 Prin3 Prin4 Prin5
kwt 0.460908 0.098994 -.191527 0.024931 -.208176
plump 0.504392 0.093371 -.006892 -.054249 -.118362
barcolor -.221913 -.226217 0.621107 -.046569 0.136818
wrtcolor 0.107532 0.546630 -.186266 0.098715 0.493803
protein 0.420932 -.168517 0.252873 0.408924 -.035322
wrtprt 0.128291 0.440959 0.401698 0.424598 0.016312
kolbach -.331075 0.526391 0.139328 -.107046 0.079364
dp 0.308405 -.156832 0.400847 -.225186 0.484449
alpha 0.033714 0.313990 0.370631 -.304447 -.640944
bglucan -.269890 -.091582 -.006101 0.694381 -.163648
PCA of Malt Quality Analyses Using 5 Principal Components
The PRINCOMP Procedure
PCA of Malt Quality Analyses Using 5 Principal Components
The PRINCOMP Procedure
PCA of Malt Quality Analyses Using 5 Principal Components
The PRINCOMP Procedure