Introduction

The objective of this analysis is to apply Principal Components Analysis to a (large) dataset from the United States Department of Agriculture Food Composition Database. We will walk-through fitting PCA to the dataset and determine how many components would be adequately useful to represent the data in a lower dimension.

Data

The data set we will be using comes from the United States Department of Agriculture Food Composition Database. The USDA National Nutrient Database for Standard Reference (SR) is the major source of food composition data in the United States. It provides the foundation for most food composition databases in the public and private sectors. As information is updated, new versions of the database are released. This version, Release 28 (SR28), contains data on 8,789 food items and up to 150 food components. It replaces SR27, initially issued in August 2014 and revised in May 2015.

There are 8789 different types of food in the data. The columns for the food include various vitamin/mineral content, nutrient profiles, and other food related characteristics.

Exploratory Data Analysis

Let us import the data set from a text file:

setwd('C:/users/crzys/Documents')
SR = read.table("ABBREV.txt", header=F, row.names=1, sep="^", quote="~")

Let us take a look at our data set:

summary(SR)
##                                                             V2      
##  BEEF,CHUCK,UNDER BLADE CNTR STEAK,BNLESS,DENVER CUT,LN,0" FA:   2  
##  OIL,INDUSTRIAL,PALM KERNEL (HYDROGENATED),CONFECTION FAT    :   2  
##  POPCORN,OIL-POPPED,LOFAT                                    :   2  
##  ABALONE,MIXED SPECIES,RAW                                   :   1  
##  ABALONE,MXD SP,CKD,FRIED                                    :   1  
##  ABIYUCH,RAW                                                 :   1  
##  (Other)                                                     :8781  
##        V3               V4              V5              V6        
##  Min.   :  0.00   Min.   :  0.0   Min.   : 0.00   Min.   :  0.00  
##  1st Qu.: 29.70   1st Qu.: 91.0   1st Qu.: 2.38   1st Qu.:  0.95  
##  Median : 63.08   Median :191.0   Median : 8.00   Median :  5.14  
##  Mean   : 54.11   Mean   :226.3   Mean   :11.34   Mean   : 10.55  
##  3rd Qu.: 77.96   3rd Qu.:337.0   3rd Qu.:19.88   3rd Qu.: 13.72  
##  Max.   :100.00   Max.   :902.0   Max.   :88.32   Max.   :100.00  
##  NA's   :1                                                        
##        V7               V8               V9              V10        
##  Min.   : 0.000   Min.   :  0.00   Min.   : 0.000   Min.   : 0.000  
##  1st Qu.: 0.840   1st Qu.:  0.05   1st Qu.: 0.000   1st Qu.: 0.000  
##  Median : 1.200   Median :  9.34   Median : 0.700   Median : 1.840  
##  Mean   : 1.784   Mean   : 22.13   Mean   : 2.187   Mean   : 8.543  
##  3rd Qu.: 2.090   3rd Qu.: 34.91   3rd Qu.: 2.600   3rd Qu.: 9.287  
##  Max.   :99.800   Max.   :100.00   Max.   :79.000   Max.   :99.800  
##  NA's   :325                       NA's   :594      NA's   :1832    
##       V11               V12              V13             V14        
##  Min.   :   0.00   Min.   :  0.00   Min.   :  0.0   Min.   :   0.0  
##  1st Qu.:  10.00   1st Qu.:  0.54   1st Qu.: 12.0   1st Qu.:  51.0  
##  Median :  21.00   Median :  1.38   Median : 21.0   Median : 145.0  
##  Mean   :  76.74   Mean   :  2.70   Mean   : 35.3   Mean   : 165.1  
##  3rd Qu.:  69.00   3rd Qu.:  2.60   3rd Qu.: 30.0   3rd Qu.: 220.0  
##  Max.   :7364.00   Max.   :123.60   Max.   :781.0   Max.   :9918.0  
##  NA's   :348       NA's   :144      NA's   :739     NA's   :579     
##       V15               V16               V17              V18        
##  Min.   :    0.0   Min.   :    0.0   Min.   : 0.000   Min.   : 0.000  
##  1st Qu.:  127.0   1st Qu.:   41.0   1st Qu.: 0.340   1st Qu.: 0.052  
##  Median :  229.5   Median :   88.0   Median : 0.990   Median : 0.090  
##  Mean   :  279.5   Mean   :  312.5   Mean   : 2.117   Mean   : 0.196  
##  3rd Qu.:  336.0   3rd Qu.:  404.5   3rd Qu.: 2.920   3rd Qu.: 0.160  
##  Max.   :16500.0   Max.   :38758.0   Max.   :90.950   Max.   :15.050  
##  NA's   :426       NA's   :83        NA's   :706      NA's   :1257    
##       V19                V20               V21                V22         
##  Min.   :  0.0000   Min.   :   0.00   Min.   :   0.000   Min.   : 0.0000  
##  1st Qu.:  0.0150   1st Qu.:   1.30   1st Qu.:   0.000   1st Qu.: 0.0400  
##  Median :  0.0820   Median :  10.10   Median :   0.000   Median : 0.0840  
##  Mean   :  0.6582   Mean   :  15.59   Mean   :   9.231   Mean   : 0.2231  
##  3rd Qu.:  0.3260   3rd Qu.:  24.30   3rd Qu.:   3.500   3rd Qu.: 0.2500  
##  Max.   :328.0000   Max.   :1917.00   Max.   :2732.000   Max.   :23.3750  
##  NA's   :2160       NA's   :1700      NA's   :818        NA's   :634      
##       V23               V24               V25               V26         
##  Min.   : 0.0000   Min.   :  0.000   Min.   : 0.0000   Min.   : 0.0000  
##  1st Qu.: 0.0630   1st Qu.:  0.552   1st Qu.: 0.1950   1st Qu.: 0.0510  
##  Median : 0.1700   Median :  2.580   Median : 0.4350   Median : 0.1500  
##  Mean   : 0.2522   Mean   :  3.658   Mean   : 0.6510   Mean   : 0.2915  
##  3rd Qu.: 0.2700   3rd Qu.:  5.276   3rd Qu.: 0.7322   3rd Qu.: 0.3900  
##  Max.   :17.5000   Max.   :127.500   Max.   :34.5000   Max.   :12.0000  
##  NA's   :616       NA's   :637       NA's   :2242      NA's   :905      
##       V27              V28               V29               V30         
##  Min.   :   0.0   Min.   :   0.00   Min.   :   0.00   Min.   :   0.00  
##  1st Qu.:   5.0   1st Qu.:   0.00   1st Qu.:   4.00   1st Qu.:   4.00  
##  Median :  12.0   Median :   0.00   Median :   9.00   Median :  11.00  
##  Mean   :  51.6   Mean   :  23.17   Mean   :  24.56   Mean   :  64.28  
##  3rd Qu.:  46.0   3rd Qu.:   0.00   3rd Qu.:  21.00   3rd Qu.:  41.00  
##  Max.   :3786.0   Max.   :2993.00   Max.   :2340.00   Max.   :5881.00  
##  NA's   :1261     NA's   :2039      NA's   :1768      NA's   :2057     
##       V31               V32              V33                V34         
##  Min.   :   0.00   Min.   : 0.000   Min.   :     0.0   Min.   :    0.0  
##  1st Qu.:   9.00   1st Qu.: 0.000   1st Qu.:     0.0   1st Qu.:    0.0  
##  Median :  24.75   Median : 0.240   Median :    32.0   Median :    4.0  
##  Mean   :  43.60   Mean   : 1.393   Mean   :   735.8   Mean   :  111.7  
##  3rd Qu.:  69.67   3rd Qu.: 1.610   3rd Qu.:   252.5   3rd Qu.:   33.0  
##  Max.   :2403.30   Max.   :98.890   Max.   :100000.0   Max.   :30000.0  
##  NA's   :4016      NA's   :1193     NA's   :711        NA's   :1535     
##       V35                V36               V37               V38         
##  Min.   :    0.00   Min.   :    0.0   Min.   :    0.0   Min.   :   0.00  
##  1st Qu.:    0.00   1st Qu.:    0.0   1st Qu.:    0.0   1st Qu.:   0.00  
##  Median :    0.00   Median :    0.0   Median :    0.0   Median :   0.00  
##  Mean   :   89.77   Mean   :   31.5   Mean   :  221.3   Mean   :  10.62  
##  3rd Qu.:   12.00   3rd Qu.:    0.0   3rd Qu.:   21.0   3rd Qu.:   0.00  
##  Max.   :30000.00   Max.   :14251.0   Max.   :42891.0   Max.   :6252.00  
##  NA's   :1806       NA's   :3258      NA's   :3162      NA's   :3270     
##       V39             V40             V41               V42         
##  Min.   :    0   Min.   :    0   Min.   :  0.000   Min.   :  0.000  
##  1st Qu.:    0   1st Qu.:    0   1st Qu.:  0.120   1st Qu.:  0.000  
##  Median :    0   Median :    0   Median :  0.300   Median :  0.000  
##  Mean   :  157   Mean   :  194   Mean   :  1.331   Mean   :  0.579  
##  3rd Qu.:    0   3rd Qu.:   30   3rd Qu.:  0.800   3rd Qu.:  0.200  
##  Max.   :46260   Max.   :19697   Max.   :149.400   Max.   :250.000  
##  NA's   :3292    NA's   :3315    NA's   :2889      NA's   :3262     
##       V43                V44               V45              V46         
##  Min.   :    0.00   Min.   :   0.00   Min.   : 0.000   Min.   : 0.0000  
##  1st Qu.:    0.00   1st Qu.:   0.20   1st Qu.: 0.220   1st Qu.: 0.2358  
##  Median :    0.00   Median :   1.60   Median : 1.592   Median : 1.9385  
##  Mean   :   22.99   Mean   :  15.63   Mean   : 3.576   Mean   : 4.0810  
##  3rd Qu.:    6.00   3rd Qu.:   5.20   3rd Qu.: 4.345   3rd Qu.: 5.1150  
##  Max.   :10000.00   Max.   :1714.50   Max.   :95.600   Max.   :83.6890  
##  NA's   :3211       NA's   :3563      NA's   :349      NA's   :666      
##       V47              V48               V49              V50      
##  Min.   : 0.000   Min.   :   0.00   Min.   :   0.1   3 oz   :1409  
##  1st Qu.: 0.220   1st Qu.:   0.00   1st Qu.:  29.0   1 cup  :1182  
##  Median : 0.696   Median :   4.00   Median :  85.0   1 oz   : 889  
##  Mean   : 2.289   Mean   :  40.61   Mean   :  96.4   1 tbsp : 343  
##  3rd Qu.: 2.081   3rd Qu.:  67.00   3rd Qu.: 130.0          : 299  
##  Max.   :74.623   Max.   :3100.00   Max.   :1184.0   1 fl oz: 268  
##  NA's   :665      NA's   :410       NA's   :300      (Other):4400  
##       V51        
##  Min.   :   0.4  
##  1st Qu.:  42.0  
##  Median : 164.0  
##  Mean   : 250.7  
##  3rd Qu.: 305.0  
##  Max.   :5717.0  
##  NA's   :3965    
##                                                                        V52      
##                                                                          :3965  
##  1 cup                                                                   : 298  
##  1 steak                                                                 : 267  
##  1 lb                                                                    : 260  
##  1 piece, cooked, excluding refuse (yield from 1 lb raw meat with refuse): 201  
##  1 oz                                                                    : 198  
##  (Other)                                                                 :3601  
##       V53        
##  Min.   : 0.000  
##  1st Qu.: 0.000  
##  Median : 0.000  
##  Mean   : 4.964  
##  3rd Qu.: 0.000  
##  Max.   :81.000  
##  NA's   :50

We see that there are a lot of NA’s for many of the variables. In addition, the names for the variables are not very useful.

Let’s pre-process the data set and simplify it a little:

SR = na.omit(SR) # remove rows with missing values
SR = SR[row.names(SR) != "13352",] # remove "duplicate" entry

Re-name the rows:

R = SR[row.names(SR) != "13352",] # remove "duplicate" entry

row.names(SR) = SR[,1] # set more meaningful row names

SR = SR[,-1]

names(SR) = c("Water_(g)", "Energ_Kcal", "Protein_(g)", "Lipid_Tot_(g)", "Ash_(g)", "Carbohydrt_(g)", "Fiber_TD_(g)", "Sugar_Tot_(g)", "Calcium_(mg)", "Iron_(mg)", "Magnesium_(mg)", "Phosphorus_(mg)", "Potassium_(mg)", "Sodium_(mg)", "Zinc_(mg)", "Copper_(mg)", "Manganese_(mg)", "Selenium_(µg)", "Vit_C_(mg)", "Thiamin_(mg)", "Riboflavin_(mg)", "Niacin_(mg)", "Panto_Acid_(mg)", "Vit_B6_(mg)", "Folate_Tot_(µg)", "Folic_Acid_(µg)", "Food_Folate_(µg)", "Folate_DFE_(µg)", "Choline_Tot_(mg)", "Vit_B12_(µg)", "Vit_A_IU", "Vit_A_RAE", "Retinol_(µg)", "Alpha_Carot_(µg)", "Beta_Carot_(µg)", "Beta_Crypt_(µg)", "Lycopene_(µg)", "Lut+Zea_(µg)", "Vit_E_(mg)", "Vit_D_µg", "Vit_D_IU", "Vit_K_(µg)", "FA_Sat_(g)", "FA_Mono_(g)", "FA_Poly_(g)", "Cholestrl_(mg)", "GmWt_1", "GmWt_Desc1", "GmWt_2", "GmWt_Desc2", "Refuse_Pct")

We just want to use the nutrient variables:

SRp = SR[,c(1:46)] # restrict to just the nutrient variables

We will utilize the SRp version of the data for our analysis.

summary(SRp)
##    Water_(g)       Energ_Kcal     Protein_(g)    Lipid_Tot_(g)   
##  Min.   : 0.00   Min.   :  0.0   Min.   : 0.00   Min.   :  0.00  
##  1st Qu.:41.00   1st Qu.: 83.0   1st Qu.: 2.21   1st Qu.:  0.74  
##  Median :64.96   Median :182.0   Median :10.08   Median :  5.34  
##  Mean   :57.57   Mean   :210.9   Mean   :12.85   Mean   : 10.15  
##  3rd Qu.:79.00   3rd Qu.:292.0   3rd Qu.:22.46   3rd Qu.: 13.37  
##  Max.   :99.90   Max.   :902.0   Max.   :85.60   Max.   :100.00  
##     Ash_(g)       Carbohydrt_(g)   Fiber_TD_(g)    Sugar_Tot_(g)   
##  Min.   : 0.000   Min.   : 0.00   Min.   : 0.000   Min.   : 0.000  
##  1st Qu.: 0.840   1st Qu.: 0.00   1st Qu.: 0.000   1st Qu.: 0.000  
##  Median : 1.130   Median : 5.50   Median : 0.000   Median : 1.070  
##  Mean   : 1.868   Mean   :17.55   Mean   : 1.763   Mean   : 7.182  
##  3rd Qu.: 1.900   3rd Qu.:22.70   3rd Qu.: 2.100   3rd Qu.: 6.370  
##  Max.   :99.800   Max.   :99.98   Max.   :53.200   Max.   :99.800  
##   Calcium_(mg)       Iron_(mg)      Magnesium_(mg)   Phosphorus_(mg) 
##  Min.   :   0.00   Min.   : 0.000   Min.   :  0.00   Min.   :   0.0  
##  1st Qu.:  11.00   1st Qu.: 0.445   1st Qu.: 12.00   1st Qu.:  50.0  
##  Median :  19.00   Median : 1.080   Median : 21.00   Median : 152.0  
##  Mean   :  76.16   Mean   : 1.890   Mean   : 33.58   Mean   : 177.0  
##  3rd Qu.:  56.00   3rd Qu.: 2.340   3rd Qu.: 28.00   3rd Qu.: 221.5  
##  Max.   :7364.00   Max.   :89.800   Max.   :711.00   Max.   :9918.0  
##  Potassium_(mg)     Sodium_(mg)      Zinc_(mg)       Copper_(mg)    
##  Min.   :    0.0   Min.   :    0   Min.   : 0.000   Min.   :0.0000  
##  1st Qu.:  135.0   1st Qu.:   41   1st Qu.: 0.350   1st Qu.:0.0490  
##  Median :  241.0   Median :   76   Median : 1.000   Median :0.0810  
##  Mean   :  298.6   Mean   :  339   Mean   : 2.185   Mean   :0.1716  
##  3rd Qu.:  339.0   3rd Qu.:  349   3rd Qu.: 3.100   3rd Qu.:0.1370  
##  Max.   :16500.0   Max.   :38758   Max.   :90.950   Max.   :6.1000  
##  Manganese_(mg)     Selenium_(µg)       Vit_C_(mg)        Thiamin_(mg)    
##  Min.   :  0.0000   Min.   :   0.00   Min.   :   0.000   Min.   : 0.0000  
##  1st Qu.:  0.0130   1st Qu.:   1.30   1st Qu.:   0.000   1st Qu.: 0.0360  
##  Median :  0.0550   Median :  13.00   Median :   0.000   Median : 0.0710  
##  Mean   :  0.4587   Mean   :  17.08   Mean   :   6.356   Mean   : 0.1753  
##  3rd Qu.:  0.2800   3rd Qu.:  26.85   3rd Qu.:   1.800   3rd Qu.: 0.1725  
##  Max.   :125.0000   Max.   :1917.00   Max.   :1900.000   Max.   :10.9900  
##  Riboflavin_(mg)  Niacin_(mg)     Panto_Acid_(mg)    Vit_B6_(mg)    
##  Min.   :0.000   Min.   : 0.000   Min.   : 0.0000   Min.   :0.0000  
##  1st Qu.:0.057   1st Qu.: 0.438   1st Qu.: 0.1970   1st Qu.:0.0500  
##  Median :0.175   Median : 2.486   Median : 0.4330   Median :0.1520  
##  Mean   :0.212   Mean   : 3.317   Mean   : 0.5864   Mean   :0.2565  
##  3rd Qu.:0.265   3rd Qu.: 5.279   3rd Qu.: 0.7320   3rd Qu.:0.4035  
##  Max.   :4.000   Max.   :45.249   Max.   :22.6240   Max.   :5.8990  
##  Folate_Tot_(µg)   Folic_Acid_(µg)    Food_Folate_(µg)  Folate_DFE_(µg) 
##  Min.   :   0.00   Min.   :   0.000   Min.   :   0.00   Min.   :   0.0  
##  1st Qu.:   3.00   1st Qu.:   0.000   1st Qu.:   3.00   1st Qu.:   3.0  
##  Median :   9.00   Median :   0.000   Median :   8.00   Median :   9.0  
##  Mean   :  31.08   Mean   :   9.052   Mean   :  22.01   Mean   :  37.4  
##  3rd Qu.:  28.00   3rd Qu.:   0.000   3rd Qu.:  20.00   3rd Qu.:  31.0  
##  Max.   :2340.00   Max.   :1286.000   Max.   :2340.00   Max.   :2340.0  
##  Choline_Tot_(mg)   Vit_B12_(µg)       Vit_A_IU         Vit_A_RAE      
##  Min.   :   0.00   Min.   : 0.000   Min.   :    0.0   Min.   :   0.00  
##  1st Qu.:  10.60   1st Qu.: 0.000   1st Qu.:    1.0   1st Qu.:   0.00  
##  Median :  29.10   Median : 0.200   Median :   16.0   Median :   3.00  
##  Mean   :  46.63   Mean   : 0.961   Mean   :  499.5   Mean   :  48.21  
##  3rd Qu.:  70.45   3rd Qu.: 1.390   3rd Qu.:  142.5   3rd Qu.:  22.00  
##  Max.   :2403.30   Max.   :20.090   Max.   :49254.0   Max.   :4220.00  
##   Retinol_(µg)     Alpha_Carot_(µg) Beta_Carot_(µg)   Beta_Crypt_(µg)  
##  Min.   :   0.00   Min.   :   0.0   Min.   :    0.0   Min.   :   0.00  
##  1st Qu.:   0.00   1st Qu.:   0.0   1st Qu.:    0.0   1st Qu.:   0.00  
##  Median :   0.00   Median :   0.0   Median :    0.0   Median :   0.00  
##  Mean   :  27.89   Mean   :  26.8   Mean   :  225.2   Mean   :  10.92  
##  3rd Qu.:   8.00   3rd Qu.:   0.0   3rd Qu.:   12.0   3rd Qu.:   0.00  
##  Max.   :4220.00   Max.   :4342.0   Max.   :26162.0   Max.   :6186.00  
##  Lycopene_(µg)      Lut+Zea_(µg)       Vit_E_(mg)        Vit_D_µg      
##  Min.   :    0.0   Min.   :    0.0   Min.   : 0.000   Min.   : 0.0000  
##  1st Qu.:    0.0   1st Qu.:    0.0   1st Qu.: 0.110   1st Qu.: 0.0000  
##  Median :    0.0   Median :    0.0   Median : 0.250   Median : 0.0000  
##  Mean   :  162.4   Mean   :  173.9   Mean   : 0.999   Mean   : 0.5171  
##  3rd Qu.:    0.0   3rd Qu.:   26.5   3rd Qu.: 0.630   3rd Qu.: 0.2000  
##  Max.   :45902.0   Max.   :19697.0   Max.   :41.080   Max.   :31.9000  
##     Vit_D_IU         Vit_K_(µg)        FA_Sat_(g)      FA_Mono_(g)     
##  Min.   :   0.00   Min.   :   0.00   Min.   : 0.000   Min.   : 0.0000  
##  1st Qu.:   0.00   1st Qu.:   0.10   1st Qu.: 0.136   1st Qu.: 0.1215  
##  Median :   0.00   Median :   1.50   Median : 1.617   Median : 1.8820  
##  Mean   :  20.68   Mean   :  13.27   Mean   : 3.396   Mean   : 3.9440  
##  3rd Qu.:   7.00   3rd Qu.:   4.25   3rd Qu.: 4.249   3rd Qu.: 5.2865  
##  Max.   :1276.00   Max.   :1714.50   Max.   :86.503   Max.   :83.6890  
##   FA_Poly_(g)     Cholestrl_(mg)  
##  Min.   : 0.000   Min.   :   0.0  
##  1st Qu.: 0.164   1st Qu.:   0.0  
##  Median : 0.559   Median :   8.0  
##  Mean   : 1.910   Mean   :  42.2  
##  3rd Qu.: 1.619   3rd Qu.:  73.0  
##  Max.   :57.740   Max.   :3100.0

We now have useful variable names and there are no NA’s.

Let us see the means of the columns:

apply(SRp, 2, mean)
##        Water_(g)       Energ_Kcal      Protein_(g)    Lipid_Tot_(g) 
##       57.5733648      210.8547009       12.8536302       10.1525641 
##          Ash_(g)   Carbohydrt_(g)     Fiber_TD_(g)    Sugar_Tot_(g) 
##        1.8684480       17.5526361        1.7632029        7.1816374 
##     Calcium_(mg)        Iron_(mg)   Magnesium_(mg)  Phosphorus_(mg) 
##       76.1605938        1.8895322       33.5802969      176.9505173 
##   Potassium_(mg)      Sodium_(mg)        Zinc_(mg)      Copper_(mg) 
##      298.5600540      338.9874044        2.1851192        0.1715866 
##   Manganese_(mg)    Selenium_(µg)       Vit_C_(mg)     Thiamin_(mg) 
##        0.4586914       17.0785425        6.3556905        0.1753297 
##  Riboflavin_(mg)      Niacin_(mg)  Panto_Acid_(mg)      Vit_B6_(mg) 
##        0.2120067        3.3172353        0.5863590        0.2565119 
##  Folate_Tot_(µg)  Folic_Acid_(µg) Food_Folate_(µg)  Folate_DFE_(µg) 
##       31.0800720        9.0517319       22.0107962       37.3958614 
## Choline_Tot_(mg)     Vit_B12_(µg)         Vit_A_IU        Vit_A_RAE 
##       46.6255511        0.9609717      499.4898785       48.2069276 
##     Retinol_(µg) Alpha_Carot_(µg)  Beta_Carot_(µg)  Beta_Crypt_(µg) 
##       27.8884390       26.7953216      225.1785875       10.9221772 
##    Lycopene_(µg)     Lut+Zea_(µg)       Vit_E_(mg)         Vit_D_µg 
##      162.4291498      173.9185785        0.9990463        0.5170490 
##         Vit_D_IU       Vit_K_(µg)       FA_Sat_(g)      FA_Mono_(g) 
##       20.6842105       13.2708052        3.3955789        3.9440193 
##      FA_Poly_(g)   Cholestrl_(mg) 
##        1.9098399       42.1983806

The mean varies a lot between the variables and they are in different units, like mg and g.

How about the variance of the columns?

apply(SRp, 2, var)
##        Water_(g)       Energ_Kcal      Protein_(g)    Lipid_Tot_(g) 
##     8.349523e+02     2.516972e+04     1.284037e+02     2.060145e+02 
##          Ash_(g)   Carbohydrt_(g)     Fiber_TD_(g)    Sugar_Tot_(g) 
##     1.596343e+01     6.076003e+02     1.622151e+01     2.164031e+02 
##     Calcium_(mg)        Iron_(mg)   Magnesium_(mg)  Phosphorus_(mg) 
##     7.473869e+04     1.505961e+01     3.197176e+03     9.443834e+04 
##   Potassium_(mg)      Sodium_(mg)        Zinc_(mg)      Copper_(mg) 
##     2.602524e+05     1.918025e+06     1.438623e+01     1.584133e-01 
##   Manganese_(mg)    Selenium_(µg)       Vit_C_(mg)     Thiamin_(mg) 
##     1.076104e+01     1.998693e+03     2.032894e+03     1.268544e-01 
##  Riboflavin_(mg)      Niacin_(mg)  Panto_Acid_(mg)      Vit_B6_(mg) 
##     8.725530e-02     1.292544e+01     9.630418e-01     9.814600e-02 
##  Folate_Tot_(µg)  Folic_Acid_(µg) Food_Folate_(µg)  Folate_DFE_(µg) 
##     6.855954e+03     2.064629e+03     4.682490e+03     1.083398e+04 
## Choline_Tot_(mg)     Vit_B12_(µg)         Vit_A_IU        Vit_A_RAE 
##     5.790941e+03     3.266657e+00     5.098720e+06     3.547239e+04 
##     Retinol_(µg) Alpha_Carot_(µg)  Beta_Carot_(µg)  Beta_Crypt_(µg) 
##     2.411153e+04     5.225111e+04     1.490032e+06     2.447177e+04 
##    Lycopene_(µg)     Lut+Zea_(µg)       Vit_E_(mg)         Vit_D_µg 
##     2.469441e+06     1.226786e+06     1.050709e+01     4.662972e+00 
##         Vit_D_IU       Vit_K_(µg)       FA_Sat_(g)      FA_Mono_(g) 
##     7.462127e+03     5.920390e+03     3.401565e+01     4.299196e+01 
##      FA_Poly_(g)   Cholestrl_(mg) 
##     1.995847e+01     1.160956e+04

Not as spread compared to the mean of the variables but there is still some difference among the variance.

We will need to scale the variables to mean zero and standard deviation one.

If we do not scale the variables before performing PCA, then most of the principal components that we observed would be driven by the variable with the largest mean and variance. Thus, it is important to standardize the variables to have mean zero and standard deviation one before performing PCA.

What is the new structure of our data?

str(SRp)
## 'data.frame':    2223 obs. of  46 variables:
##  $ Water_(g)       : num  15.87 16.72 0.24 42.41 41.11 ...
##  $ Energ_Kcal      : int  717 718 876 353 371 334 300 404 394 98 ...
##  $ Protein_(g)     : num  0.85 0.49 0.28 21.4 23.24 ...
##  $ Lipid_Tot_(g)   : num  81.1 78.3 99.5 28.7 29.7 ...
##  $ Ash_(g)         : num  2.11 1.62 0 5.11 3.18 2.7 3.68 3.71 3.36 1.41 ...
##  $ Carbohydrt_(g)  : num  0.06 2.87 0 2.34 2.79 0.45 0.46 3.09 2.57 3.38 ...
##  $ Fiber_TD_(g)    : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ Sugar_Tot_(g)   : num  0.06 0.06 0 0.5 0.51 0.45 0.46 0.48 0.52 2.67 ...
##  $ Calcium_(mg)    : int  24 23 4 528 674 184 388 710 685 83 ...
##  $ Iron_(mg)       : num  0.02 0.05 0 0.31 0.43 0.5 0.33 0.14 0.76 0.07 ...
##  $ Magnesium_(mg)  : int  2 1 0 23 24 20 20 27 26 8 ...
##  $ Phosphorus_(mg) : int  24 24 3 387 451 188 347 455 457 159 ...
##  $ Potassium_(mg)  : int  24 41 5 256 136 152 187 76 127 104 ...
##  $ Sodium_(mg)     : int  643 583 2 1146 560 629 842 653 604 364 ...
##  $ Zinc_(mg)       : num  0.09 0.05 0.01 2.66 2.6 2.38 2.38 3.64 3.07 0.4 ...
##  $ Copper_(mg)     : num  0 0.01 0.001 0.04 0.024 0.019 0.021 0.03 0.042 0.029 ...
##  $ Manganese_(mg)  : num  0 0.001 0 0.009 0.012 0.034 0.038 0.027 0.012 0.002 ...
##  $ Selenium_(µg)   : num  1 0 0 14.5 14.5 14.5 14.5 28.5 14.5 9.7 ...
##  $ Vit_C_(mg)      : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ Thiamin_(mg)    : num  0.005 0.007 0.001 0.029 0.014 0.07 0.028 0.029 0.015 0.027 ...
##  $ Riboflavin_(mg) : num  0.034 0.064 0.005 0.382 0.351 0.52 0.488 0.428 0.375 0.163 ...
##  $ Niacin_(mg)     : num  0.042 0.022 0.003 1.016 0.118 ...
##  $ Panto_Acid_(mg) : num  0.11 0.097 0.01 1.729 0.288 ...
##  $ Vit_B6_(mg)     : num  0.003 0.008 0.001 0.166 0.065 0.235 0.227 0.066 0.079 0.046 ...
##  $ Folate_Tot_(µg) : int  3 4 0 36 20 65 62 27 18 12 ...
##  $ Folic_Acid_(µg) : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ Food_Folate_(µg): int  3 4 0 36 20 65 62 27 18 12 ...
##  $ Folate_DFE_(µg) : int  3 4 0 36 20 65 62 27 18 12 ...
##  $ Choline_Tot_(mg): num  18.8 18.8 22.3 15.4 15.4 15.4 15.4 16.5 15.4 18.4 ...
##  $ Vit_B12_(µg)    : num  0.17 0.07 0.01 1.22 1.26 1.65 1.3 1.1 0.83 0.43 ...
##  $ Vit_A_IU        : int  2499 2468 3069 721 1080 592 820 1242 994 140 ...
##  $ Vit_A_RAE       : int  684 683 840 198 292 174 241 330 264 37 ...
##  $ Retinol_(µg)    : int  671 671 824 192 286 173 240 330 257 36 ...
##  $ Alpha_Carot_(µg): int  0 1 0 0 0 0 0 0 0 0 ...
##  $ Beta_Carot_(µg) : int  158 135 193 74 76 9 12 85 82 12 ...
##  $ Beta_Crypt_(µg) : int  0 6 0 0 0 0 0 0 0 0 ...
##  $ Lycopene_(µg)   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ Lut+Zea_(µg)    : int  0 13 0 0 0 0 0 0 0 0 ...
##  $ Vit_E_(mg)      : num  2.32 1.37 2.8 0.25 0.26 0.24 0.21 0.71 0.28 0.08 ...
##  $ Vit_D_µg        : num  0 0 0 0.5 0.5 0.5 0.4 0.6 0.6 0.1 ...
##  $ Vit_D_IU        : int  0 0 0 21 22 20 18 24 24 3 ...
##  $ Vit_K_(µg)      : num  7 4.6 8.6 2.4 2.5 2.3 2 2.4 2.7 0 ...
##  $ FA_Sat_(g)      : num  51.4 45.4 61.9 18.7 18.8 ...
##  $ FA_Mono_(g)     : num  21.02 19.87 28.73 7.78 8.6 ...
##  $ FA_Poly_(g)     : num  3.043 3.331 3.694 0.8 0.784 ...
##  $ Cholestrl_(mg)  : int  215 225 256 75 94 100 72 99 95 17 ...

We have 2223 observations with 46 variables.

We now perform principal components analysis using the prcomp() function, which is one of several functions in R that perform PCA.

pca = prcomp(SRp, scale = TRUE)

By default, the prcomp function already sets the mean to zero for the variables, to set the standard deviation to one we let scale be true.

Loadings for each variable(Warning: long output)

pca$rotation
##                          PC1          PC2          PC3           PC4
## Water_(g)         0.25460314 -0.030649880  0.275735871 -0.0244049168
## Energ_Kcal       -0.23217104  0.123951819 -0.304485403 -0.0885494474
## Protein_(g)      -0.13832714  0.263225869  0.188174696 -0.0688227059
## Lipid_Tot_(g)    -0.16063707  0.185105935 -0.278239852 -0.2414205566
## Ash_(g)          -0.10962435 -0.025576868 -0.047623246  0.0002234890
## Carbohydrt_(g)   -0.12539693 -0.185198377 -0.236180806  0.1988645570
## Fiber_TD_(g)     -0.18792946 -0.216355955 -0.107442527  0.0292432574
## Sugar_Tot_(g)    -0.02067393 -0.115511278 -0.207879778  0.1159356556
## Calcium_(mg)     -0.11297690 -0.033148530 -0.033195333 -0.0155171005
## Iron_(mg)        -0.22064970 -0.084132847  0.049783200  0.0421019451
## Magnesium_(mg)   -0.24623296 -0.049370783 -0.092055908 -0.0114746967
## Phosphorus_(mg)  -0.15250627  0.068673518 -0.001799448 -0.0249868167
## Potassium_(mg)   -0.13673839 -0.050692745  0.013868072 -0.0075262814
## Sodium_(mg)      -0.03433893 -0.005676561 -0.048013167  0.0069608166
## Zinc_(mg)        -0.10883508  0.154934364  0.117896372 -0.0534759842
## Copper_(mg)      -0.19511597 -0.016186944 -0.053962392 -0.0056339073
## Manganese_(mg)   -0.13726796 -0.061552421  0.016416080  0.0224209614
## Selenium_(µg)    -0.07660596  0.120797800  0.041985399 -0.0519902736
## Vit_C_(mg)       -0.03601389 -0.103087627  0.047192615  0.0280217798
## Thiamin_(mg)     -0.21526854 -0.033963895  0.098325139  0.1969802063
## Riboflavin_(mg)  -0.23710138  0.020026061  0.160042492  0.0437181249
## Niacin_(mg)      -0.19045488  0.120644279  0.212208264  0.0617048684
## Panto_Acid_(mg)  -0.21396980  0.062012980  0.167206020  0.0222886178
## Vit_B6_(mg)      -0.18119480  0.058106066  0.234629534  0.0002396016
## Folate_Tot_(µg)  -0.24681845 -0.137512630  0.089586301  0.2337041384
## Folic_Acid_(µg)  -0.15198931 -0.080650638  0.060582038  0.1908837991
## Food_Folate_(µg) -0.19771823 -0.112855664  0.068209754  0.1560383817
## Folate_DFE_(µg)  -0.24280039 -0.134038070  0.089819786  0.2442552465
## Choline_Tot_(mg) -0.10566587  0.206805861  0.173680905 -0.1362816065
## Vit_B12_(µg)     -0.07176037  0.196802588  0.234479124 -0.1080010479
## Vit_A_IU         -0.03631815 -0.322392453  0.110969469 -0.3368634324
## Vit_A_RAE        -0.08498682 -0.191037744  0.132309805 -0.3034870112
## Retinol_(µg)     -0.09205135  0.003266532  0.095721606 -0.1478413038
## Alpha_Carot_(µg)  0.01449421 -0.177530371  0.054075487 -0.1738005689
## Beta_Carot_(µg)  -0.01530264 -0.328913585  0.090635342 -0.3058131949
## Beta_Crypt_(µg)  -0.04679916 -0.202713609  0.050628744 -0.2235796300
## Lycopene_(µg)     0.01163187 -0.038522773 -0.004412801  0.0216547697
## Lut+Zea_(µg)     -0.01611010 -0.248993851  0.073083451 -0.1947269929
## Vit_E_(mg)       -0.18235994 -0.057388829 -0.111262091 -0.1617291642
## Vit_D_µg         -0.05017610  0.097758135  0.162345669 -0.0632476186
## Vit_D_IU         -0.05018942  0.097730767  0.162415829 -0.0632852690
## Vit_K_(µg)       -0.04979110 -0.198271839  0.026651249 -0.1069872891
## FA_Sat_(g)       -0.08590427  0.168427030 -0.201919177 -0.1963835342
## FA_Mono_(g)      -0.14316517  0.177481192 -0.237823128 -0.2148891831
## FA_Poly_(g)      -0.15657290  0.068684400 -0.233389253 -0.1488748831
## Cholestrl_(mg)   -0.05175551  0.204613549  0.146429411 -0.1502284746
##                            PC5          PC6          PC7          PC8
## Water_(g)         0.0158023856 -0.120826592  0.019556963 -0.188041437
## Energ_Kcal        0.1153632829  0.084026633 -0.033839902  0.094826029
## Protein_(g)      -0.0599307489 -0.166649221 -0.077501332  0.099163557
## Lipid_Tot_(g)     0.1490077278  0.003200098 -0.068918565 -0.133808821
## Ash_(g)          -0.4797746490  0.150294202 -0.231443440 -0.126778140
## Carbohydrt_(g)    0.0037956474  0.186974324  0.093410743  0.277529828
## Fiber_TD_(g)     -0.0645653415 -0.191893953  0.174540153 -0.037501846
## Sugar_Tot_(g)     0.0127277010  0.231098735  0.093329068  0.333074498
## Calcium_(mg)     -0.4203592608  0.150206085 -0.064522928 -0.045263605
## Iron_(mg)        -0.0862671324 -0.052696468  0.147985789  0.153930130
## Magnesium_(mg)   -0.1279497614 -0.213266482  0.236869228  0.005979502
## Phosphorus_(mg)  -0.3960676060  0.065427256 -0.084827002 -0.049705718
## Potassium_(mg)   -0.2620469026 -0.105839281  0.089523557  0.049787146
## Sodium_(mg)      -0.3074885157  0.176692400 -0.281711039 -0.141470036
## Zinc_(mg)        -0.0433215869 -0.223237487  0.040332111  0.283692209
## Copper_(mg)      -0.0811970985 -0.250650506  0.227721692  0.100023075
## Manganese_(mg)   -0.0320443698  0.003108003  0.217548733  0.069224890
## Selenium_(µg)    -0.0233959546 -0.075044785  0.016307486  0.010799916
## Vit_C_(mg)        0.0055538464 -0.041692899  0.033554447 -0.017684716
## Thiamin_(mg)      0.1382478058 -0.004238438 -0.164966338 -0.156911684
## Riboflavin_(mg)   0.0095021371  0.058798903 -0.082826103  0.113693276
## Niacin_(mg)       0.0769717076 -0.081927018 -0.064888754  0.069995554
## Panto_Acid_(mg)   0.0329036979 -0.013886704 -0.032339594 -0.054851392
## Vit_B6_(mg)       0.0435808550 -0.108120615 -0.049960399  0.124949363
## Folate_Tot_(µg)   0.1661587221  0.043521210 -0.130747035 -0.189180916
## Folic_Acid_(µg)   0.1592278322  0.224988654 -0.043810805  0.134768433
## Food_Folate_(µg)  0.0952917063 -0.096737870 -0.129079844 -0.318357611
## Folate_DFE_(µg)   0.1807939426  0.103367276 -0.117329342 -0.109277022
## Choline_Tot_(mg) -0.0243953027  0.023315617 -0.053253452  0.006045273
## Vit_B12_(µg)     -0.0180520902  0.043491017  0.044323209  0.244383220
## Vit_A_IU          0.0470248632  0.038041692 -0.081781438  0.061358481
## Vit_A_RAE         0.0924278434  0.267118314 -0.065176768  0.106797528
## Retinol_(µg)      0.0935006194  0.355581905 -0.023556532  0.101818682
## Alpha_Carot_(µg)  0.0298560864  0.014030115 -0.086576462  0.065165407
## Beta_Carot_(µg)   0.0255779625 -0.046174330 -0.074560281  0.029274493
## Beta_Crypt_(µg)   0.0001772272 -0.050575164 -0.033558355  0.105065154
## Lycopene_(µg)    -0.0374455782 -0.026343502  0.045732406 -0.005256978
## Lut+Zea_(µg)      0.0004372259 -0.092633419 -0.006729845 -0.113209759
## Vit_E_(mg)        0.0655053144 -0.085050497  0.098222784 -0.146429308
## Vit_D_µg         -0.0169969413  0.321928761  0.464343942 -0.250409842
## Vit_D_IU         -0.0171054741  0.321893637  0.464372200 -0.250261014
## Vit_K_(µg)       -0.0565340864 -0.130384417  0.094080817 -0.148185282
## FA_Sat_(g)        0.1098178894  0.104458834 -0.129057274  0.005947157
## FA_Mono_(g)       0.1399168101 -0.035228529 -0.048256010 -0.137525373
## FA_Poly_(g)       0.0968297259 -0.078703050  0.035306458 -0.213172156
## Cholestrl_(mg)    0.0124605083  0.070142451 -0.115936118  0.017223496
##                          PC9         PC10         PC11         PC12
## Water_(g)         0.17236567 -0.039852757  0.067173005 -0.061890731
## Energ_Kcal       -0.09933829  0.033389101 -0.085357687 -0.006384096
## Protein_(g)      -0.08643103  0.107018461 -0.068776879 -0.050227434
## Lipid_Tot_(g)     0.06838099  0.025514755 -0.053005388 -0.056267693
## Ash_(g)           0.04223017  0.113591609  0.016049397  0.195523833
## Carbohydrt_(g)   -0.21184993 -0.036362063 -0.021504053  0.083692379
## Fiber_TD_(g)     -0.04350149 -0.068674241  0.113833827  0.078387284
## Sugar_Tot_(g)    -0.30290957 -0.115466167 -0.025202001 -0.017465496
## Calcium_(mg)      0.09993492 -0.084479287 -0.082180083 -0.316075907
## Iron_(mg)         0.18460401  0.002675676 -0.227730320  0.135161876
## Magnesium_(mg)    0.04898012 -0.055344871  0.080183968  0.015328729
## Phosphorus_(mg)   0.01713432 -0.011109140  0.033302305 -0.318900682
## Potassium_(mg)   -0.15287683 -0.024702344  0.049819277 -0.262766247
## Sodium_(mg)       0.06252956  0.180385392  0.022405981  0.493906939
## Zinc_(mg)         0.19909002 -0.062410680  0.039481713  0.202211756
## Copper_(mg)       0.12437795 -0.117001827  0.242680966  0.173763449
## Manganese_(mg)    0.26702081 -0.025807871  0.102160768  0.031166189
## Selenium_(µg)    -0.04481206  0.051075879  0.056887015  0.129256608
## Vit_C_(mg)       -0.09371019 -0.032572148 -0.121416548 -0.270493052
## Thiamin_(mg)     -0.06747351  0.075171895  0.057991752 -0.057763367
## Riboflavin_(mg)  -0.08967849 -0.018842614 -0.064039642 -0.128509660
## Niacin_(mg)      -0.04034297  0.337016900 -0.116813184 -0.081731914
## Panto_Acid_(mg)  -0.14292437 -0.094708289  0.189017873 -0.080516143
## Vit_B6_(mg)      -0.06614108  0.284295010 -0.133327533 -0.150923515
## Folate_Tot_(µg)   0.03205372 -0.107424589  0.037768076  0.046536858
## Folic_Acid_(µg)   0.22577989  0.202681737 -0.199326425  0.120821058
## Food_Folate_(µg) -0.11114297 -0.264667629  0.178091587 -0.023912779
## Folate_DFE_(µg)   0.09447718 -0.023559953 -0.030870811  0.073936073
## Choline_Tot_(mg) -0.30646811 -0.310671990  0.007179353  0.210503728
## Vit_B12_(µg)      0.14983465 -0.072431775 -0.017243592  0.095379758
## Vit_A_IU         -0.05995189  0.072308530  0.069628089  0.010251970
## Vit_A_RAE         0.20401111 -0.123807448  0.152391651 -0.090079520
## Retinol_(µg)      0.34926190 -0.243365734  0.161069002 -0.140067440
## Alpha_Carot_(µg) -0.07785001  0.162082861  0.322904551  0.024193442
## Beta_Carot_(µg)  -0.13543230  0.111915322  0.003785514  0.041611068
## Beta_Crypt_(µg)  -0.20319280  0.228353757  0.036579167  0.053501574
## Lycopene_(µg)    -0.00814251 -0.032596544  0.131866313 -0.033742856
## Lut+Zea_(µg)     -0.06114091 -0.135595446 -0.427435611  0.069540764
## Vit_E_(mg)        0.04164395  0.160700287  0.077706684  0.009363860
## Vit_D_µg         -0.13017467  0.135604477 -0.020924112  0.055179310
## Vit_D_IU         -0.13002261  0.135452978 -0.020767104  0.055357366
## Vit_K_(µg)        0.11472382 -0.240491408 -0.515465434  0.061942205
## FA_Sat_(g)        0.04401531 -0.081389742 -0.130289947 -0.117966908
## FA_Mono_(g)       0.07150091  0.070119438 -0.044387294 -0.038924257
## FA_Poly_(g)       0.05584598  0.085554496  0.082894463  0.037146487
## Cholestrl_(mg)   -0.26914748 -0.351740500 -0.037430180  0.214225207
##                          PC13         PC14         PC15         PC16
## Water_(g)         0.027361032  0.028802314  0.006208485 -0.088613649
## Energ_Kcal       -0.043787747 -0.038262059  0.025148445  0.055653210
## Protein_(g)      -0.069534187  0.006895857 -0.016664066  0.190771439
## Lipid_Tot_(g)    -0.009381487 -0.061332277  0.056295576 -0.027002179
## Ash_(g)           0.128891639 -0.092110010 -0.015412166  0.021100622
## Carbohydrt_(g)   -0.021811517  0.017792681 -0.025991875  0.028746327
## Fiber_TD_(g)     -0.025900200  0.081637453 -0.012390222 -0.035394691
## Sugar_Tot_(g)     0.062024094 -0.135866155 -0.079554620  0.032031805
## Calcium_(mg)     -0.237735307  0.075704094  0.054667518 -0.163068425
## Iron_(mg)        -0.071767369  0.131090855  0.151275014 -0.146312351
## Magnesium_(mg)    0.017854772  0.094384387 -0.065849854  0.120146640
## Phosphorus_(mg)  -0.232939011  0.110591836  0.027640396 -0.110651179
## Potassium_(mg)    0.095172710 -0.028064142  0.117492287  0.137535489
## Sodium_(mg)       0.262318057 -0.167748973 -0.113699683  0.069443610
## Zinc_(mg)        -0.174597550 -0.374115879  0.057442302 -0.149955619
## Copper_(mg)      -0.027960688 -0.236784327 -0.023041212 -0.000343847
## Manganese_(mg)    0.177046619  0.351186433 -0.168630332  0.001097793
## Selenium_(µg)    -0.101572117  0.217433915 -0.116539522  0.638078162
## Vit_C_(mg)        0.457109978 -0.310092442 -0.217249841 -0.181448214
## Thiamin_(mg)     -0.080779370 -0.084521796  0.014152100  0.137109341
## Riboflavin_(mg)   0.074665249 -0.060610411 -0.038107144  0.067808606
## Niacin_(mg)       0.059112969  0.059050869  0.024129686  0.092706413
## Panto_Acid_(mg)   0.184216238 -0.002925710 -0.144827261 -0.070922418
## Vit_B6_(mg)       0.187062644  0.037754050 -0.062858801 -0.050332142
## Folate_Tot_(µg)  -0.117393404 -0.058884226  0.058476916 -0.035851293
## Folic_Acid_(µg)  -0.040510114  0.193553188  0.137566852 -0.147812946
## Food_Folate_(µg) -0.115160951 -0.199860437 -0.020530943  0.054785711
## Folate_DFE_(µg)  -0.105747339  0.012251118  0.088530057 -0.073698731
## Choline_Tot_(mg)  0.070341930  0.248418270  0.035649008 -0.158943377
## Vit_B12_(µg)     -0.037805115 -0.296760934  0.029167607 -0.081990999
## Vit_A_IU         -0.088557201 -0.026446378  0.025195395  0.032831092
## Vit_A_RAE         0.075511900  0.001963557 -0.054276417  0.143498405
## Retinol_(µg)      0.187067497  0.026001522 -0.101025820  0.180158508
## Alpha_Carot_(µg) -0.259521903  0.024520173  0.096260516 -0.100116431
## Beta_Carot_(µg)  -0.122354201 -0.039848228  0.044561969  0.008104839
## Beta_Crypt_(µg)   0.013931228  0.029851259  0.001632166 -0.130582918
## Lycopene_(µg)     0.386812787 -0.016597969  0.851282833  0.157887445
## Lut+Zea_(µg)      0.014992897 -0.084659562 -0.014117973  0.140460484
## Vit_E_(mg)        0.208988452  0.114937819 -0.058085625 -0.285432147
## Vit_D_µg         -0.093061916 -0.131244182  0.034824206  0.011929956
## Vit_D_IU         -0.093146239 -0.131369734  0.034878440  0.011790423
## Vit_K_(µg)        0.017034223  0.002464280  0.006798459  0.125601737
## FA_Sat_(g)       -0.107597006 -0.177188614  0.110419355  0.061050382
## FA_Mono_(g)       0.015946368 -0.036380468  0.049840761 -0.068454638
## FA_Poly_(g)       0.089737922  0.094887572 -0.052106038 -0.063737664
## Cholestrl_(mg)    0.044180021  0.276273250  0.068454688 -0.219929796
##                          PC17          PC18         PC19          PC20
## Water_(g)        -0.018421336  0.0707406056  0.020474959  3.082089e-02
## Energ_Kcal       -0.009877687 -0.0648216959 -0.018597424  7.692856e-03
## Protein_(g)       0.068205697 -0.1710762132 -0.035763783 -1.835528e-02
## Lipid_Tot_(g)    -0.048701738 -0.0433346589  0.023178192  6.382243e-02
## Ash_(g)          -0.009516064 -0.0613404601  0.030049504 -7.028874e-03
## Carbohydrt_(g)    0.019851409  0.0327886745 -0.026713133 -6.462344e-02
## Fiber_TD_(g)      0.062587684 -0.1422753641  0.304389874 -4.121189e-02
## Sugar_Tot_(g)     0.037676307  0.1090898851 -0.249132483  9.858038e-03
## Calcium_(mg)     -0.025776453  0.1786913985 -0.061772035  6.859202e-02
## Iron_(mg)        -0.057632950 -0.1013515555  0.213254178 -2.237166e-01
## Magnesium_(mg)   -0.049276155 -0.0911532442  0.010037110 -6.757503e-02
## Phosphorus_(mg)  -0.003900714  0.2187619648 -0.078308547  9.203901e-02
## Potassium_(mg)    0.017771819 -0.2248870390  0.223993986 -1.577811e-01
## Sodium_(mg)      -0.006627813 -0.1157855725 -0.025080519  1.208283e-03
## Zinc_(mg)        -0.014829978  0.1297289510 -0.056513399  1.661495e-01
## Copper_(mg)      -0.036059526  0.0940218658 -0.060616713  4.536897e-02
## Manganese_(mg)   -0.040009288 -0.2941189289 -0.182325725  5.780262e-01
## Selenium_(µg)    -0.324224406  0.4828713391  0.165870930  8.029713e-02
## Vit_C_(mg)       -0.562553483  0.1096983193  0.274044150  1.028654e-01
## Thiamin_(mg)      0.022160676 -0.0293588302 -0.009604800  5.091345e-02
## Riboflavin_(mg)   0.048684698  0.0259139826 -0.158482390 -9.721392e-02
## Niacin_(mg)       0.077236198 -0.1484782705 -0.076586972 -8.759815e-02
## Panto_Acid_(mg)   0.091536408  0.0606047804 -0.374407786  1.659316e-01
## Vit_B6_(mg)       0.011857413 -0.0519799252 -0.020955636 -4.740665e-02
## Folate_Tot_(µg)  -0.009102698  0.0351667684  0.069714282  3.598119e-02
## Folic_Acid_(µg)  -0.182354237  0.1574296057  0.043328009  8.270567e-02
## Food_Folate_(µg)  0.110049328 -0.0619921938  0.055597030 -1.129484e-02
## Folate_DFE_(µg)  -0.062943470  0.0760676682  0.068676025  5.395109e-02
## Choline_Tot_(mg) -0.079024007 -0.0236910499  0.035225330 -1.839716e-02
## Vit_B12_(µg)      0.041112106  0.1273264803  0.001968244 -1.244105e-01
## Vit_A_IU         -0.035510470 -0.0141535239 -0.038740182  3.636816e-02
## Vit_A_RAE         0.070155387 -0.0198106051  0.061959003 -1.168460e-01
## Retinol_(µg)      0.133020071 -0.0164916082  0.123955355 -2.017934e-01
## Alpha_Carot_(µg) -0.512663598 -0.2534210731 -0.236857882 -1.736631e-01
## Beta_Carot_(µg)  -0.049613267  0.0002274929 -0.076559457  9.199495e-02
## Beta_Crypt_(µg)   0.378730863  0.1875833705  0.379470779  2.495721e-01
## Lycopene_(µg)    -0.013112521  0.1133018339 -0.116516019  1.037795e-01
## Lut+Zea_(µg)      0.096987379  0.1251181696 -0.157300880  1.679650e-01
## Vit_E_(mg)        0.129723141  0.2805225263 -0.103008046 -1.689613e-01
## Vit_D_µg         -0.011446829 -0.0329153395  0.015281448  3.180696e-02
## Vit_D_IU         -0.011534658 -0.0327811232  0.015357874  3.184933e-02
## Vit_K_(µg)       -0.058574720 -0.0988729463 -0.247747317 -1.749407e-01
## FA_Sat_(g)       -0.064303032 -0.2390481955  0.171621340  3.672485e-01
## FA_Mono_(g)      -0.031000597 -0.0072890557 -0.001425355 -5.178278e-02
## FA_Poly_(g)      -0.021680279  0.2095506870 -0.151448255 -2.103806e-01
## Cholestrl_(mg)   -0.110737412 -0.0081167080  0.121278113  8.798673e-05
##                          PC21         PC22          PC23         PC24
## Water_(g)        -0.041147753  0.124612117  0.0424817013  0.038401022
## Energ_Kcal        0.040624797 -0.073844957 -0.0234985060 -0.031380051
## Protein_(g)      -0.039156404 -0.158539448 -0.0529366357 -0.181747261
## Lipid_Tot_(g)     0.006372160  0.041477091  0.0260705535  0.014764674
## Ash_(g)           0.063154623  0.036470194 -0.0153718343  0.023896854
## Carbohydrt_(g)    0.054426118 -0.107747848 -0.0400787268  0.027412333
## Fiber_TD_(g)     -0.222245943 -0.125865043 -0.1035252658  0.095064837
## Sugar_Tot_(g)    -0.051888381 -0.003922464 -0.1712683801  0.057544718
## Calcium_(mg)     -0.210115828 -0.012930917 -0.0178374705 -0.019855119
## Iron_(mg)        -0.297630606  0.109124074  0.0159856680  0.041474781
## Magnesium_(mg)   -0.071514964 -0.107083564  0.1149221092  0.002285444
## Phosphorus_(mg)   0.085886032 -0.256223016  0.0009629337  0.047863762
## Potassium_(mg)    0.623669395  0.403579873 -0.0042470479 -0.020242428
## Sodium_(mg)      -0.072577125  0.008713030 -0.0040800306  0.019223041
## Zinc_(mg)         0.142573837 -0.050260167 -0.0821387149 -0.059954574
## Copper_(mg)      -0.001690185 -0.005000995  0.4926311636  0.086623486
## Manganese_(mg)    0.092556088 -0.020711484 -0.2750239820 -0.246730758
## Selenium_(µg)    -0.070759574  0.168216745 -0.1725681599  0.105793024
## Vit_C_(mg)       -0.049836924 -0.166101300 -0.0005721713 -0.102081168
## Thiamin_(mg)     -0.112553630 -0.122011046  0.1031010141 -0.347473163
## Riboflavin_(mg)  -0.302291603  0.412063286  0.2042648934 -0.477802656
## Niacin_(mg)       0.032143636 -0.248290355  0.0237929687  0.213080775
## Panto_Acid_(mg)  -0.061019942  0.271154901  0.1553238340  0.393343829
## Vit_B6_(mg)      -0.060706003 -0.133122993 -0.0763433182  0.311541263
## Folate_Tot_(µg)   0.104653731  0.009746240 -0.1138679681  0.054145312
## Folic_Acid_(µg)   0.235324307  0.114859323  0.2182060564  0.027650107
## Food_Folate_(µg) -0.029686093 -0.064378842 -0.2827653547  0.047131519
## Folate_DFE_(µg)   0.155079777  0.042905794 -0.0239721806  0.051534375
## Choline_Tot_(mg)  0.064648695 -0.071658943  0.0593938807 -0.035512125
## Vit_B12_(µg)      0.075536766  0.013316453 -0.4154283798 -0.042073208
## Vit_A_IU          0.037317698 -0.045688708  0.0349589736 -0.013941447
## Vit_A_RAE         0.015593070 -0.110787194  0.0582546391  0.022692353
## Retinol_(µg)     -0.009900807 -0.121347971  0.0543813303  0.045197356
## Alpha_Carot_(µg) -0.069777037  0.077730169 -0.1155078059  0.014730576
## Beta_Carot_(µg)   0.057065757 -0.031815311  0.0348657401 -0.022991836
## Beta_Crypt_(µg)  -0.101965124  0.072117379  0.0168380751 -0.082765950
## Lycopene_(µg)    -0.118323343 -0.114629002 -0.0645405262 -0.018314347
## Lut+Zea_(µg)      0.194781045 -0.124319764  0.0854956387 -0.049646831
## Vit_E_(mg)       -0.041874924  0.256051072 -0.3242966397  0.035340061
## Vit_D_µg         -0.008368863 -0.015089219  0.0181429567 -0.008039319
## Vit_D_IU         -0.008404713 -0.015342226  0.0181521343 -0.007845153
## Vit_K_(µg)       -0.071317834  0.082907709 -0.1077920057  0.092071213
## FA_Sat_(g)       -0.175052073  0.215241181  0.0498196822  0.339214504
## FA_Mono_(g)       0.038065977  0.072196880 -0.0895153045 -0.224179482
## FA_Poly_(g)       0.186170721 -0.229510440  0.1357096240 -0.072167752
## Cholestrl_(mg)    0.025290491 -0.056033588  0.0571044289 -0.022197227
##                           PC25         PC26          PC27          PC28
## Water_(g)        -0.0722959427 -0.047356728 -0.1811259579  0.0988556394
## Energ_Kcal        0.0251053116  0.057434957  0.1180857089 -0.0737833387
## Protein_(g)       0.3586080420  0.144502409 -0.0400287272 -0.3533479070
## Lipid_Tot_(g)    -0.0723794787  0.037146275 -0.0575521348  0.0704319895
## Ash_(g)          -0.0421300993 -0.032158243 -0.0133009555  0.0097466155
## Carbohydrt_(g)   -0.0313301718 -0.028449664  0.2652700620  0.0001423372
## Fiber_TD_(g)      0.1263340911  0.077197363  0.2883222431  0.2701125922
## Sugar_Tot_(g)    -0.0798481490 -0.122597019 -0.4521353991 -0.0088998511
## Calcium_(mg)     -0.0352777576  0.007265212  0.0088853758 -0.0093397867
## Iron_(mg)        -0.2857985114 -0.047950069  0.0729240278 -0.1781200890
## Magnesium_(mg)    0.2867581107  0.070398273 -0.3157808861  0.2217416716
## Phosphorus_(mg)   0.0605882873 -0.053648595 -0.0030488569  0.0239328661
## Potassium_(mg)   -0.1527556134 -0.034366078 -0.0112664561 -0.0014236876
## Sodium_(mg)       0.0432522619  0.009510301 -0.0546052987  0.0270699215
## Zinc_(mg)        -0.1430796663 -0.144916574 -0.0371028184 -0.2246645899
## Copper_(mg)       0.0375286120 -0.177666103 -0.0821584199  0.0031199796
## Manganese_(mg)   -0.1296783593  0.036985463 -0.0153598437 -0.0126518450
## Selenium_(µg)    -0.0779772751 -0.049010393  0.0233104360 -0.0147103546
## Vit_C_(mg)        0.0518543622  0.039595882  0.0849790329 -0.1624839099
## Thiamin_(mg)     -0.4857682663 -0.319794606  0.1214894820  0.1613781275
## Riboflavin_(mg)   0.1255954404  0.209670643 -0.0375602746  0.1193375056
## Niacin_(mg)       0.0026807817 -0.155304485  0.1375507250 -0.1874259883
## Panto_Acid_(mg)  -0.0418079060  0.099459296  0.3503647937 -0.0575300573
## Vit_B6_(mg)      -0.1553749061 -0.063866723 -0.3447391822  0.3983370433
## Folate_Tot_(µg)   0.1130006893  0.071347439 -0.1305665969 -0.0585808601
## Folic_Acid_(µg)   0.1842484500  0.059755539 -0.0346532354  0.0586228724
## Food_Folate_(µg)  0.0145345175  0.046381750 -0.1350598961 -0.1098107868
## Folate_DFE_(µg)   0.1461802724  0.074877754 -0.1144645735 -0.0286924372
## Choline_Tot_(mg) -0.0136039850 -0.053479116 -0.0692239153 -0.0351861590
## Vit_B12_(µg)     -0.0207433011  0.297306900  0.1987159835  0.3984841453
## Vit_A_IU          0.0246282411  0.003188483  0.0392612054  0.0117942927
## Vit_A_RAE         0.0125271592 -0.066636264 -0.0003629327 -0.0574688282
## Retinol_(µg)     -0.0032630284 -0.099725719 -0.0347402578 -0.0939479949
## Alpha_Carot_(µg) -0.0267733681  0.017658688 -0.0543366004 -0.0343235443
## Beta_Carot_(µg)   0.0386873822  0.016984592  0.0701303415  0.0531494989
## Beta_Crypt_(µg)  -0.1268229618  0.156389529 -0.2000149163 -0.2008002235
## Lycopene_(µg)     0.0293489847  0.038198112  0.0055658820 -0.0153638912
## Lut+Zea_(µg)      0.1190071843 -0.101321292  0.1414111727  0.2086978955
## Vit_E_(mg)        0.1701401710 -0.292742861  0.0603690928 -0.1545459772
## Vit_D_µg          0.0006401993 -0.009981052 -0.0148394206 -0.0225152284
## Vit_D_IU          0.0003637848 -0.010829608 -0.0150512340 -0.0237247933
## Vit_K_(µg)       -0.1495683318  0.042419569 -0.1007815107 -0.2087959491
## FA_Sat_(g)       -0.0433860237  0.086354650 -0.0503114226  0.0336258421
## FA_Mono_(g)       0.1564966844 -0.395190763 -0.0242122008  0.1613004675
## FA_Poly_(g)      -0.3650160691  0.524073113 -0.0829648753 -0.0473967337
## Cholestrl_(mg)   -0.0462176540 -0.104080864 -0.0202126379  0.0823442241
##                           PC29          PC30         PC31          PC32
## Water_(g)         0.0793717447  0.0009116682 -0.022793211 -0.0436202011
## Energ_Kcal       -0.0306043388 -0.0182306681  0.033782103  0.0524427227
## Protein_(g)      -0.1024470253  0.0036687644 -0.094139049  0.1104117282
## Lipid_Tot_(g)     0.0519261480 -0.0295121131  0.052699155  0.0020991550
## Ash_(g)           0.0109884816 -0.0175078266  0.011270547 -0.0083240273
## Carbohydrt_(g)   -0.0798070341  0.0160590145  0.038205627 -0.0006872018
## Fiber_TD_(g)     -0.3407455827 -0.0626724746  0.165410856 -0.1873980066
## Sugar_Tot_(g)     0.1448561734 -0.0196498483 -0.058958608 -0.0267515560
## Calcium_(mg)     -0.0257291160 -0.0921836312  0.084977383 -0.0412868036
## Iron_(mg)         0.3203812380 -0.0563930313 -0.068376790  0.4743789799
## Magnesium_(mg)    0.3262704486  0.1278683216 -0.112627847 -0.2278806966
## Phosphorus_(mg)   0.0144636761  0.0949487320 -0.055840930  0.0021947629
## Potassium_(mg)   -0.0260847719  0.0061729283 -0.025765938 -0.0085587401
## Sodium_(mg)      -0.0035877496  0.0031088782 -0.011020310  0.0017402928
## Zinc_(mg)        -0.3018677803 -0.1173601474  0.141495843 -0.1009392214
## Copper_(mg)      -0.0018600723  0.0535987913 -0.096221920  0.0660711734
## Manganese_(mg)    0.0264651716 -0.0239420604 -0.023578127  0.0714716242
## Selenium_(µg)    -0.0071616520 -0.0432684751  0.046264942  0.0270531566
## Vit_C_(mg)        0.0723073771  0.0060305600 -0.002488762 -0.0216935807
## Thiamin_(mg)     -0.0009771528  0.1488914841 -0.198503287 -0.2179549556
## Riboflavin_(mg)  -0.1788731751 -0.0428787729 -0.008573513  0.0193746311
## Niacin_(mg)       0.2520044914  0.0866054005 -0.050462055 -0.2482749347
## Panto_Acid_(mg)   0.1919621292 -0.0334587982  0.252686682 -0.0125245411
## Vit_B6_(mg)      -0.3381129811 -0.1445878099  0.114252470  0.2215483093
## Folate_Tot_(µg)   0.0296042983 -0.0332756659  0.018798915  0.0566595613
## Folic_Acid_(µg)  -0.0521968318  0.0189164884  0.027205380 -0.1636762902
## Food_Folate_(µg)  0.0705650936 -0.0529106706  0.004722086  0.1772200088
## Folate_DFE_(µg)   0.0076826193 -0.0207761859  0.023260731 -0.0048260111
## Choline_Tot_(mg) -0.0566314822 -0.0318980365  0.097655766  0.0852442755
## Vit_B12_(µg)      0.2860408805  0.1332499650 -0.099594041 -0.1115199158
## Vit_A_IU          0.0730957946 -0.2927236847 -0.129451996 -0.0193780887
## Vit_A_RAE        -0.0299169925 -0.0687122805 -0.006582915 -0.0057286546
## Retinol_(µg)     -0.1073921271  0.1553991728  0.103330389  0.0086647311
## Alpha_Carot_(µg) -0.0824162065  0.4514435734  0.189545153  0.0413719859
## Beta_Carot_(µg)   0.1080498508 -0.4262444390 -0.205953755 -0.0173237076
## Beta_Crypt_(µg)   0.1231566823  0.3034454972  0.289217474 -0.1583705213
## Lycopene_(µg)    -0.0200389921 -0.0021125850  0.028275151  0.0019542113
## Lut+Zea_(µg)     -0.0437019818  0.4400698225  0.034646902  0.3843022305
## Vit_E_(mg)       -0.2037808968  0.1474753269 -0.426087970 -0.0158366393
## Vit_D_µg         -0.0481728541 -0.0238251092  0.016931981  0.0042290457
## Vit_D_IU         -0.0489896460 -0.0241755762  0.017081446  0.0046015919
## Vit_K_(µg)       -0.0984028937 -0.0177329480  0.152244287 -0.4362757397
## FA_Sat_(g)       -0.1017561089  0.1417839212 -0.316907191 -0.0447197479
## FA_Mono_(g)       0.2522043712 -0.1751123423  0.468248036  0.0101396210
## FA_Poly_(g)      -0.0650369042 -0.0188795428 -0.058074817  0.0566368899
## Cholestrl_(mg)    0.0067031491  0.0235662230 -0.184208095 -0.1911368908
##                          PC33         PC34         PC35          PC36
## Water_(g)        -0.088509881 -0.083894784  0.121350782  0.0988253826
## Energ_Kcal        0.056242044  0.061545872 -0.100896393 -0.0858855885
## Protein_(g)       0.084550636  0.257054654  0.379467397 -0.2041151502
## Lipid_Tot_(g)    -0.015088541 -0.044009033  0.052637743  0.0684308451
## Ash_(g)           0.020974669 -0.058559909  0.048742934  0.0278444124
## Carbohydrt_(g)    0.068952938  0.012011055 -0.340617759 -0.0586821027
## Fiber_TD_(g)     -0.089442737 -0.092941444  0.387394861  0.1675135696
## Sugar_Tot_(g)    -0.080884590 -0.072055700  0.451979704  0.1261633631
## Calcium_(mg)      0.000679246 -0.193406831  0.178357547 -0.1062444672
## Iron_(mg)        -0.072191380  0.013802884  0.049966833 -0.0600915907
## Magnesium_(mg)   -0.366610303 -0.036602036 -0.237129827 -0.3232168061
## Phosphorus_(mg)   0.054248525  0.212050681 -0.196177738  0.1841760850
## Potassium_(mg)   -0.015455229 -0.027477225  0.085958346 -0.0253782806
## Sodium_(mg)      -0.039560201  0.018979554 -0.010812262 -0.0187895185
## Zinc_(mg)        -0.319824107 -0.251802996 -0.106402666 -0.1723599530
## Copper_(mg)       0.461045862  0.135109277  0.065316985  0.2470581499
## Manganese_(mg)    0.133969769 -0.041968749 -0.025011010  0.0971346725
## Selenium_(µg)     0.024680842 -0.057657831 -0.002277248  0.0326031358
## Vit_C_(mg)       -0.003934352  0.022642231 -0.002904127 -0.0004927224
## Thiamin_(mg)     -0.133973179  0.271678893  0.138361277 -0.1415182383
## Riboflavin_(mg)   0.018274892 -0.253084658 -0.193363598  0.2087953481
## Niacin_(mg)       0.006348039 -0.456547326 -0.021735042  0.3818870828
## Panto_Acid_(mg)  -0.109954292  0.104541061  0.112544497 -0.2511118513
## Vit_B6_(mg)       0.142965131  0.098862056 -0.138320183 -0.1381865115
## Folate_Tot_(µg)   0.051442631 -0.027460840 -0.015982450  0.0202115970
## Folic_Acid_(µg)  -0.060843923  0.116743414  0.177111016 -0.0190927288
## Food_Folate_(µg)  0.102554003 -0.110656818 -0.136829798  0.0369388959
## Folate_DFE_(µg)   0.022230634  0.013758611  0.041414843  0.0100204482
## Choline_Tot_(mg) -0.410083433  0.279471167 -0.081456565  0.4388212456
## Vit_B12_(µg)      0.124828778  0.164847397  0.044559321  0.1093567044
## Vit_A_IU         -0.017247159  0.026304133 -0.012799739 -0.0005303386
## Vit_A_RAE        -0.031187749  0.004727798  0.003700649 -0.0246913238
## Retinol_(µg)     -0.030422612 -0.016277277  0.016643066 -0.0357264957
## Alpha_Carot_(µg)  0.039429248 -0.029347057  0.011048423 -0.0096963519
## Beta_Carot_(µg)  -0.023043514  0.030954243 -0.014958424  0.0111694590
## Beta_Crypt_(µg)   0.112239639  0.088422538 -0.074537366 -0.0168449148
## Lycopene_(µg)     0.010522475  0.013118164 -0.023634398 -0.0261730080
## Lut+Zea_(µg)     -0.099003125 -0.195906127  0.111969406 -0.0389574722
## Vit_E_(mg)       -0.049742900  0.050983154 -0.026094306 -0.0045908754
## Vit_D_µg          0.004734483 -0.019632054 -0.014201172 -0.0294162771
## Vit_D_IU          0.003448209 -0.020423872 -0.014378643 -0.0308557653
## Vit_K_(µg)        0.175976328  0.236005633 -0.093780211  0.0517627073
## FA_Sat_(g)       -0.102988614  0.005161271 -0.030802568  0.0988381010
## FA_Mono_(g)       0.094908418  0.010696796  0.058034563  0.0146025859
## FA_Poly_(g)      -0.042011377 -0.146234427  0.132569163  0.0228600108
## Cholestrl_(mg)    0.387936664 -0.320298911  0.045825125 -0.3388158099
##                          PC37          PC38          PC39          PC40
## Water_(g)        -0.087166129 -0.0358333483  1.397059e-01  6.468417e-02
## Energ_Kcal        0.059846855  0.0009920859 -6.340626e-01  2.829830e-01
## Protein_(g)       0.013743576 -0.0361712480  2.550920e-01 -5.992446e-02
## Lipid_Tot_(g)    -0.029181465 -0.0080465795 -5.065693e-02 -8.253504e-01
## Ash_(g)           0.092782731 -0.7433551002 -1.112564e-02  1.037330e-02
## Carbohydrt_(g)    0.100645176 -0.0156570225  5.449289e-01 -1.240138e-01
## Fiber_TD_(g)     -0.191624915 -0.0013198263 -4.211971e-02  1.483538e-02
## Sugar_Tot_(g)    -0.159019661 -0.0171682097 -2.696323e-02  5.171371e-03
## Calcium_(mg)      0.554750662  0.2625381323  9.505016e-03  1.759757e-03
## Iron_(mg)        -0.183464384  0.0006575262  1.605699e-03 -2.685370e-03
## Magnesium_(mg)    0.061927475 -0.0349417923 -3.996860e-03  1.706885e-05
## Phosphorus_(mg)  -0.580067979  0.0598526125 -8.080612e-03  8.772446e-04
## Potassium_(mg)   -0.007449825  0.1900995239  3.910642e-03  9.979479e-05
## Sodium_(mg)      -0.121842737  0.5711873055  2.448749e-02 -5.679336e-04
## Zinc_(mg)        -0.121478371 -0.0179300536  3.814612e-03 -4.255791e-03
## Copper_(mg)       0.177734120  0.0361691039 -1.352802e-02  6.859803e-03
## Manganese_(mg)    0.024109445  0.0098068394  1.525791e-03  2.334403e-03
## Selenium_(µg)     0.001105878  0.0070822376 -3.196597e-03 -2.786812e-03
## Vit_C_(mg)       -0.006289094  0.0018945489  6.356357e-04 -1.326943e-05
## Thiamin_(mg)      0.038567664  0.0015399205  4.638714e-04  8.003433e-04
## Riboflavin_(mg)  -0.151699503 -0.0362174020 -5.265855e-03  3.391033e-03
## Niacin_(mg)       0.074771173  0.0526598394 -2.772841e-03  1.107285e-02
## Panto_Acid_(mg)  -0.100093953 -0.0251483884  3.254324e-04 -7.932855e-03
## Vit_B6_(mg)       0.083645068 -0.0046475014 -1.452616e-03 -1.739975e-03
## Folate_Tot_(µg)   0.019597412  0.0049694373  1.123253e-03 -3.891060e-04
## Folic_Acid_(µg)  -0.014336268 -0.0121319651 -4.481112e-03 -1.811124e-03
## Food_Folate_(µg)  0.033027076  0.0140967449  4.206595e-03  7.013434e-05
## Folate_DFE_(µg)   0.011000525  0.0003050978 -5.977357e-04 -1.270149e-03
## Choline_Tot_(mg)  0.258505539  0.0184859678  3.275491e-03  8.714117e-03
## Vit_B12_(µg)      0.109269928  0.0098092198  1.429700e-03  3.904636e-03
## Vit_A_IU         -0.003335435  0.0016424125  2.959317e-03  8.520154e-04
## Vit_A_RAE        -0.012153418 -0.0039500462  1.502767e-03 -3.028698e-04
## Retinol_(µg)     -0.014795983 -0.0069886482 -6.082496e-04 -7.826665e-04
## Alpha_Carot_(µg)  0.005950496 -0.0026471428 -3.475066e-03 -1.547120e-03
## Beta_Carot_(µg)  -0.001912158  0.0030990872  4.393444e-03  2.471362e-03
## Beta_Crypt_(µg)   0.031008443  0.0074029975 -3.433818e-03  1.459487e-04
## Lycopene_(µg)    -0.005110033  0.0127588689  4.505721e-04  5.286756e-04
## Lut+Zea_(µg)      0.013338729  0.0017314122 -3.547164e-03 -6.701708e-04
## Vit_E_(mg)        0.073637257 -0.0044567680 -1.155880e-03 -8.360449e-04
## Vit_D_µg         -0.021466963 -0.0073697711  8.236550e-04 -1.400859e-03
## Vit_D_IU         -0.021107165 -0.0058453039  2.124602e-03  7.579625e-04
## Vit_K_(µg)       -0.038844771 -0.0075388048 -9.749889e-06  1.122532e-03
## FA_Sat_(g)       -0.009448123 -0.0161175335  2.791647e-01  2.807126e-01
## FA_Mono_(g)      -0.059908466  0.0043698229  2.926651e-01  2.950248e-01
## FA_Poly_(g)      -0.024909845  0.0040279753  2.155936e-01  2.212853e-01
## Cholestrl_(mg)   -0.148499348  0.0025819626  2.572089e-03 -4.449529e-04
##                           PC41          PC42          PC43          PC44
## Water_(g)        -7.745814e-01 -9.699277e-04 -3.524119e-04 -6.458704e-04
## Energ_Kcal       -4.714079e-01 -3.921920e-04 -2.614109e-04 -4.245799e-04
## Protein_(g)      -1.480405e-01 -3.937682e-04 -7.037654e-05 -3.353801e-04
## Lipid_Tot_(g)    -1.150911e-01  1.261177e-03  5.813583e-04 -7.782514e-04
## Ash_(g)          -2.493854e-02  7.163873e-04  9.274405e-05  1.556101e-04
## Carbohydrt_(g)   -3.568793e-01 -7.792409e-04 -2.730772e-04 -2.712571e-04
## Fiber_TD_(g)     -2.534843e-02  2.132576e-04  4.144647e-05 -6.132843e-06
## Sugar_Tot_(g)    -9.318598e-03  4.105216e-05  1.217832e-04 -3.683451e-05
## Calcium_(mg)     -2.722189e-02 -7.237008e-04 -1.913924e-04 -7.262487e-06
## Iron_(mg)         1.336901e-03 -2.969678e-04  3.894966e-05 -7.090426e-05
## Magnesium_(mg)   -1.250745e-03 -1.997935e-04  3.521309e-05  1.831093e-05
## Phosphorus_(mg)  -1.200880e-02  3.532959e-04  1.153143e-04 -7.710118e-05
## Potassium_(mg)   -2.197899e-02 -2.088687e-04 -3.102231e-05 -9.726836e-05
## Sodium_(mg)      -7.723914e-02 -7.299998e-04 -1.199787e-04 -1.792839e-04
## Zinc_(mg)         6.994345e-03 -1.236684e-03 -1.350672e-04 -4.128135e-06
## Copper_(mg)      -1.560111e-02  4.067509e-04  1.243772e-05  6.324060e-06
## Manganese_(mg)   -2.152726e-03  1.804961e-04  3.654048e-05 -3.038462e-05
## Selenium_(µg)    -5.516817e-04 -6.120392e-05  8.802852e-07 -7.901456e-06
## Vit_C_(mg)        7.813579e-05 -7.642109e-05  2.694772e-07 -1.669589e-05
## Thiamin_(mg)     -1.618032e-04 -6.011257e-05  1.408565e-05  5.007039e-05
## Riboflavin_(mg)  -7.353199e-04  4.295351e-04  6.114166e-05 -6.146424e-05
## Niacin_(mg)      -6.627113e-03  8.805325e-05 -9.015555e-05 -1.285892e-04
## Panto_Acid_(mg)   5.673421e-03 -8.326293e-05  7.466745e-05 -1.700635e-05
## Vit_B6_(mg)       1.084770e-03 -6.260468e-05 -2.440647e-05  8.432161e-05
## Folate_Tot_(µg)  -1.922773e-04 -1.607429e-01  7.927117e-01  2.988696e-03
## Folic_Acid_(µg)   9.429182e-04  2.958401e-02 -1.517611e-01 -3.917737e-04
## Food_Folate_(µg) -5.363091e-04  8.066287e-02 -4.042053e-01 -1.403525e-03
## Folate_DFE_(µg)   2.822753e-04  7.927977e-02 -3.816778e-01 -1.635723e-03
## Choline_Tot_(mg) -1.602472e-03 -7.075633e-05 -1.005698e-04  1.039399e-03
## Vit_B12_(µg)     -2.180999e-03  9.467855e-04  1.421093e-04 -1.071556e-04
## Vit_A_IU          6.158621e-04 -1.193712e-02 -5.747796e-03  6.830073e-01
## Vit_A_RAE         1.439305e-03 -1.036761e-04  1.113731e-03  1.091212e-01
## Retinol_(µg)      1.290852e-03  2.354823e-03  3.206590e-04 -2.464471e-01
## Alpha_Carot_(µg) -3.724210e-04  9.930793e-04  3.945817e-04 -6.316159e-02
## Beta_Carot_(µg)   2.130547e-03  1.070840e-02  4.604957e-03 -6.743079e-01
## Beta_Crypt_(µg)  -4.172804e-03  8.224213e-04  2.781002e-04 -4.320130e-02
## Lycopene_(µg)    -1.109269e-03 -7.188599e-05  1.339671e-05 -2.144607e-05
## Lut+Zea_(µg)     -1.187794e-03 -6.536927e-05 -6.087041e-06 -1.312597e-05
## Vit_E_(mg)       -2.156968e-04 -2.842977e-04 -3.505751e-05  3.729263e-06
## Vit_D_µg          2.285977e-03 -6.929751e-01 -1.404538e-01 -1.210590e-02
## Vit_D_IU          6.209264e-04  6.928253e-01  1.404735e-01  1.213845e-02
## Vit_K_(µg)        4.585577e-04  2.393979e-04  1.549846e-05  1.264921e-05
## FA_Sat_(g)        4.960000e-02 -4.794724e-04 -2.508621e-04  3.656333e-04
## FA_Mono_(g)       4.909922e-02 -4.674991e-04 -1.474063e-04  3.826164e-04
## FA_Poly_(g)       3.598844e-02 -3.725550e-04 -3.617560e-04  2.136160e-04
## Cholestrl_(mg)    2.588582e-04 -1.143872e-04  2.883091e-05 -1.582452e-04
##                           PC45          PC46
## Water_(g)        -2.655622e-05 -4.825874e-05
## Energ_Kcal       -1.080079e-04 -1.273608e-04
## Protein_(g)       9.539227e-06  5.356825e-05
## Lipid_Tot_(g)     1.703748e-04  1.257370e-04
## Ash_(g)           3.745134e-05  4.854239e-05
## Carbohydrt_(g)    8.439780e-05  2.883045e-05
## Fiber_TD_(g)     -3.391581e-05  3.750611e-06
## Sugar_Tot_(g)     2.365790e-05  2.267522e-05
## Calcium_(mg)     -8.411550e-05 -5.456168e-05
## Iron_(mg)         3.001109e-05 -9.926471e-05
## Magnesium_(mg)   -2.237052e-05 -2.305642e-05
## Phosphorus_(mg)   2.716133e-05  1.166685e-05
## Potassium_(mg)   -5.487332e-06  1.503099e-06
## Sodium_(mg)      -1.725164e-05 -4.297549e-05
## Zinc_(mg)        -2.143926e-05 -5.423469e-05
## Copper_(mg)      -7.210187e-07  6.286773e-05
## Manganese_(mg)    3.143399e-05 -1.410344e-05
## Selenium_(µg)    -5.337063e-06  2.619346e-07
## Vit_C_(mg)        1.414778e-05  5.464298e-06
## Thiamin_(mg)     -1.172043e-05  1.105011e-05
## Riboflavin_(mg)   1.030832e-05 -2.612334e-05
## Niacin_(mg)       5.128329e-05  1.042351e-05
## Panto_Acid_(mg)  -1.039745e-05 -6.733073e-05
## Vit_B6_(mg)       3.379042e-05 -2.867595e-05
## Folate_Tot_(µg)   5.949628e-05  2.867534e-03
## Folic_Acid_(µg)   2.861150e-01 -4.430456e-01
## Food_Folate_(µg)  2.534715e-01 -3.935202e-01
## Folate_DFE_(µg)  -3.856616e-01  5.949767e-01
## Choline_Tot_(mg) -7.747696e-05  1.568174e-05
## Vit_B12_(µg)      6.499915e-05  6.466321e-05
## Vit_A_IU          3.177742e-01  2.057591e-01
## Vit_A_RAE        -6.323786e-01 -4.088654e-01
## Retinol_(µg)      4.485162e-01  2.899405e-01
## Alpha_Carot_(µg)  5.129183e-03  3.325947e-03
## Beta_Carot_(µg)   5.524289e-02  3.545293e-02
## Beta_Crypt_(µg)   3.574520e-03  2.316464e-03
## Lycopene_(µg)     2.129339e-05  4.863814e-05
## Lut+Zea_(µg)     -2.217173e-05 -5.557383e-05
## Vit_E_(mg)       -3.331782e-06  2.767053e-06
## Vit_D_µg         -2.692475e-03  3.548610e-04
## Vit_D_IU          2.713812e-03 -3.862952e-04
## Vit_K_(µg)        2.072978e-05  6.647431e-05
## FA_Sat_(g)       -7.474717e-05 -3.168058e-05
## FA_Mono_(g)      -6.235300e-05  1.536868e-05
## FA_Poly_(g)      -5.661687e-05 -9.407731e-05
## Cholestrl_(mg)   -4.857557e-05 -5.822621e-05

Some rows were omitted because there are so many variables and components.

Here is a plot of the components:

biplot(pca, scale = 0)

Seems like there are so many loadings and scores that we are not able to get a clear plot.

Let us see how much variation is described by each component:

pr.var=pca$sdev^2
pve=pr.var/sum(pr.var)
pve
##  [1] 1.585931e-01 9.422046e-02 9.050765e-02 7.542896e-02 5.717441e-02
##  [6] 4.990966e-02 3.889550e-02 3.585142e-02 3.204046e-02 3.187225e-02
## [11] 2.846698e-02 2.492714e-02 2.412063e-02 2.259442e-02 2.146365e-02
## [16] 2.065716e-02 1.957926e-02 1.901520e-02 1.699657e-02 1.586561e-02
## [21] 1.441477e-02 1.240928e-02 1.144438e-02 1.010251e-02 9.391225e-03
## [26] 8.285616e-03 7.223179e-03 6.888471e-03 6.410823e-03 6.015062e-03
## [31] 5.657776e-03 4.906323e-03 4.444971e-03 4.035838e-03 3.589334e-03
## [36] 3.077027e-03 2.691105e-03 5.981964e-04 1.417999e-04 5.544217e-05
## [41] 3.420745e-05 1.106810e-06 8.623930e-07 1.218490e-07 3.829398e-08
## [46] 3.732215e-08

We can also figure out the propotion of variance and cumulative variance using the summary function:

summary(pca)
## Importance of components:
##                           PC1     PC2     PC3     PC4     PC5     PC6
## Standard deviation     2.7010 2.08186 2.04043 1.86272 1.62173 1.51520
## Proportion of Variance 0.1586 0.09422 0.09051 0.07543 0.05717 0.04991
## Cumulative Proportion  0.1586 0.25281 0.34332 0.41875 0.47592 0.52583
##                           PC7     PC8     PC9    PC10    PC11    PC12
## Standard deviation     1.3376 1.28420 1.21403 1.21084 1.14433 1.07082
## Proportion of Variance 0.0389 0.03585 0.03204 0.03187 0.02847 0.02493
## Cumulative Proportion  0.5647 0.60058 0.63262 0.66449 0.69296 0.71789
##                           PC13    PC14    PC15    PC16    PC17    PC18
## Standard deviation     1.05335 1.01948 0.99364 0.97480 0.94902 0.93525
## Proportion of Variance 0.02412 0.02259 0.02146 0.02066 0.01958 0.01902
## Cumulative Proportion  0.74201 0.76460 0.78607 0.80672 0.82630 0.84532
##                          PC19    PC20    PC21    PC22    PC23   PC24
## Standard deviation     0.8842 0.85429 0.81430 0.75553 0.72556 0.6817
## Proportion of Variance 0.0170 0.01587 0.01441 0.01241 0.01144 0.0101
## Cumulative Proportion  0.8623 0.87818 0.89260 0.90500 0.91645 0.9265
##                           PC25    PC26    PC27    PC28    PC29    PC30
## Standard deviation     0.65726 0.61736 0.57643 0.56291 0.54304 0.52602
## Proportion of Variance 0.00939 0.00829 0.00722 0.00689 0.00641 0.00602
## Cumulative Proportion  0.93594 0.94423 0.95145 0.95834 0.96475 0.97077
##                           PC31    PC32    PC33    PC34    PC35    PC36
## Standard deviation     0.51015 0.47507 0.45218 0.43087 0.40634 0.37622
## Proportion of Variance 0.00566 0.00491 0.00444 0.00404 0.00359 0.00308
## Cumulative Proportion  0.97642 0.98133 0.98577 0.98981 0.99340 0.99648
##                           PC37   PC38    PC39    PC40    PC41     PC42
## Standard deviation     0.35184 0.1659 0.08076 0.05050 0.03967 0.007135
## Proportion of Variance 0.00269 0.0006 0.00014 0.00006 0.00003 0.000000
## Cumulative Proportion  0.99917 0.9998 0.99991 0.99996 1.00000 1.000000
##                            PC43     PC44     PC45    PC46
## Standard deviation     0.006298 0.002367 0.001327 0.00131
## Proportion of Variance 0.000000 0.000000 0.000000 0.00000
## Cumulative Proportion  1.000000 1.000000 1.000000 1.00000

We see that the proportion of variance expained by each component is very small and it looks like we will need a large number of components to explain a reasonable amount of variance.

Let us try to find the ‘elbow’ on the plot of the propotion of variance explained:

plot(pve, xlab="Principal Component", ylab="Proportion of Variance Explained", ylim=c(0,1),type='b')

plot(cumsum(pve), xlab="Principal Component", ylab="Cumulative Proportion of Variance Explained", ylim=c(0,1),type='b')

We see that there really is not much improvement when you add an additional component. There is not discernible ‘elbow’ but if you look very carefully at the plots, you can see that there is a slight bend near the sixth component. From the previous summary output, we see that the sixth component has a cumulative proportion of variance explained of 52.583%, which is not very great. This number of components does not adequately explain the variation in the data. This goes to show that the first few components are not able to capture any interpretable combinations of nutrients.

Conclusion

How many components would we need to explain a good amount of the variance? It depends on what we can consider a ‘good’ amount. For 75% of the variance explained we will need 14 components which explains 76.46% of it. For 90%+ we need at least 22 components which would explain 90.5% of the variance. Therefore, we see that PCA is able to represent the data in a lower dimension - although using a lot of components - compared to the original list of 46 variables. Lowering dimensions from 46 to 22 is a great change.