Policy Research for Front of Package Nutrition Labeling: Developing and Testing a Summary System Algorithm. G: Supplementary Analyses of MAXR Regressions with Whole Grains and Total Sugars

05/01/2011

Supplementary analyses were conducted with the addition of whole grains and the replacement of added sugars with total sugars. There was only slight improvement in R2 with the addition of whole grains (TablesG-1 through G-4). The plateau R2 appeared to occur at eight or nine nutrient/component models. The best nine nutrient/component models had adjusted R2 of 66% and 61%, on per 100 kcal and per RACC bases, respectively.

Additional analyses were performed to examine the distribution of R2 among all 24, 310 eight nutrient/component models. The minimum, 25th, 50th, 75th, and 100th percentiles of R2 were 0.209, 0.435, 0.485, 0.529, and 0.647 for eight nutrient/component models on per 100 kcal basis, and 0.129, 0.351, 0.421, 0.491, and 0.604 for models on per RACC basis.

We also examined the top 10 eight nutrient/component models in terms of R2 to examine differences in nutrients and beta-coefficients (TablesG-5 and G-6). All of the top 10 models included fiber, unsaturated fat, saturated fat, sodium, and added sugar. All but one of the models included protein. Whole grains were not retained in the top two models, but were present in 4 or 5 of the top 10 models for both unit bases. Vitamin D replaced calcium in the second highest R models for both unit bases. Some of the top 10 models had potassium, one of the models included vitamin A, one model included vitamin B12, and none of the models included vitamin E, iron, magnesium, or folic acid.

In analyses with total sugars replacing added sugars (TableG-7), the nine nutrient/component model was the plateau for R2 on a per 100 kcal basis, explaining 64% of the variance in HEI scores. Total sugars did not remain in this model and, in fact, only remained in the 17 nutrient/component model. For the regressions using a per RACC basis (TableG-8), the R2 appears to plateau in the 10 nutrient/component model, explaining 58% of the variance in HEI. Total sugars were retained in models with seven or more nutrient/component variables on a RACC basis. It is unclear why total sugars were apparently explaining the added sugar component of HEI scores on a RACC basis but not on a per 100 kcal basis.

Table G-1. Beta Coefficients from Regression Models Using the Maximum R-Square Option Using Nutrient or Component Values as a Percentage of Recommended Intake Levels per 100 Kcal (Includes Whole Grains)
Number
of Vari-
ables
R-
Square
Adjusted
R-
Square
Protein Fiber Vitamin
E
Vitamin
D
Calcium Iron Potassium Unsat-
urated
Fat
Magnes-
ium
Vitamin
A
Vitamin
C
Folic
Acid
Vitamin
B12
Whole
Grains
Saturated
Fat
Sodium Added
Sugar
— Means no data.
0 0.0417 0.0413
1 0.3657 0.3655 5.3217
2 0.4346 0.4343 4.6677 −1.8039
3 0.5211 0.5209 3.5917 −2.3941 −0.8843
4 0.5526 0.5523 3.5531 1.2383 −2.4270 −0.8187
5 0.5818 0.5815 1.4183 3.5986 −2.3479 −1.3146 −0.7915
6 0.6080 0.6078 1.5862 3.7418 1.7803 −2.7342 −1.4097 −0.6449
7 0.6330 0.6327 1.3749 3.5078 1.1422 2.4089 −3.0957 −1.3670 −0.5563
8 0.6476 0.6473 1.3966 3.1261 1.0008 2.5151 0.3653 −2.9478 −1.3347 −0.5160
9 0.6578 0.6575 1.3963 2.8113 0.8762 2.5319 0.3703 0.3109 −2.9061 −1.2995 −0.5102
10 0.6611 0.6608 1.2993 2.8792 0.5317 0.5864 2.5135 0.3704 0.2980 −2.8630 −1.2406 −0.5140
11 0.6631 0.6628 1.3128 2.8177 0.9412 1.1446 2.6444 −0.9052 0.2937 0.3351 −3.0013 −1.3166 −0.5043
12 0.6648 0.6644 1.2569 2.8697 0.3830 0.7251 0.9703 2.6144 −0.7798 0.3054 0.3223 −2.9576 −1.2720 −0.5084
13 0.6657 0.6654 1.2721 2.8259 0.3238 0.6984 1.0777 2.6792 −0.7482 0.3037 0.2605 0.3051 −2.9207 −1.2952 −0.4923
14 0.6666 0.6662 1.3258 2.8913 0.3458 0.6745 −0.3821 1.0792 2.6935 −0.6733 0.3085 0.4968 0.3130 −2.9243 −1.2759 −0.4813
15 0.6674 0.6669 1.3301 2.8660 0.4226 0.3448 0.6741 −0.3999 1.1062 2.5547 −0.7896 0.2874 0.4745 0.3222 −2.8995 −1.2695 −0.4847
16 0.6676 0.6672 1.3283 2.8485 0.3777 0.3133 0.6531 −0.4387 1.0920 2.5668 −0.7872 0.1833 0.2819 0.4782 0.3208 −2.9183 −1.2711 −0.4895
17 0.6678 0.6674 1.3607 2.8218 0.4038 0.3790 0.6507 −0.3776 1.1092 2.5525 −0.8003 0.1849 0.2807 0.4947 −0.1267 0.3192 −2.8968 −1.2770 −0.4854
Table G-2. P-Values from Regression Models Using the Maximum R-Square Option Using Nutrient or Component Values as a Percentage of Recommended Intake Levels per 100 Kcal (Includes Whole Grains)
Number
of Vari-
ables
R-
Square
Adjusted
R-Square
Protein Fiber Vitamin
E
Vitamin
D
Calcium Iron Potas-
sium
Unsat-
urated
Fat
Magnes-
ium
Vitamin
A
Vitamin
C
Folic
Acid
Vitamin
B12
Whole
Grains
Saturated
Fat
Sodium Added
Sugar
— Means no data.
0 0.0417 0.0413
1 0.3657 0.3655 0.0000
2 0.4346 0.4343 0.0000 0.0000
3 0.5211 0.5209 0.0000 0.0000 0.0000
4 0.5526 0.5523 0.0000 0.0000 0.0000 0.0000
5 0.5818 0.5815 0.0000 0.0000 0.0000 0.0000 0.0000
6 0.6080 0.6078 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
7 0.6330 0.6327 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
8 0.6476 0.6473 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
9 0.6578 0.6575 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
10 0.6611 0.6608 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
11 0.6631 0.6628 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000
12 0.6648 0.6644 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0004 0.0000 0.0000 0.0000 0.0000 0.0000
13 0.6657 0.6654 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0005 0.0000 0.0023 0.0000 0.0000 0.0000 0.0000
14 0.6666 0.6662 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0015 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
15 0.6674 0.6669 0.0000 0.0000 0.0926 0.0000 0.0000 0.0001 0.0000 0.0000 0.0004 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
16 0.6676 0.6672 0.0000 0.0000 0.1380 0.0001 0.0000 0.0000 0.0000 0.0000 0.0004 0.1193 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
17 0.6678 0.6674 0.0000 0.0000 0.1178 0.0000 0.0000 0.0006 0.0000 0.0000 0.0002 0.1191 0.0000 0.0000 0.0345 0.0000 0.0000 0.0000 0.0000
Table G-3. Beta Coefficients from Regression Models Using the Maximum R-square Option Using Nutrient or Component Values as a Percentage of Recommended Intake Levels per RACC (Includes Whole Grains)
Number
of Vari-
ables
R-
Square
Adjusted
R-Square
Protein Fiber Vitamin
E
Vitamin
D
Calcium Iron Potassium Unsaturated
Fat
Mag-
nesium
Vitamin
A
Vitamin
C
Folic
Acid
Vitamin
B12
Whole
Grains
Saturated
Fat
Sodium Added
Sugar
— Means no data.
0 0.0417 0.0413
1 0.2074 0.2071 3.9647
2 0.4036 0.4033 4.7469 −2.0714
3 0.4732 0.4730 4.5970 −1.7868 −0.6528
4 0.5218 0.5215 3.1269 3.1410 −2.1354 −0.5857
5 0.5416 0.5412 2.8476 3.1097 1.3403 −2.7630 −0.6113
6 0.5654 0.5651 3.1202 3.3033 1.7656 −2.5134 −1.0465 −0.6170
7 0.5883 0.5880 1.4790 3.6688 1.6051 0.4735 −2.5594 −1.5006 −0.6409
8 0.6042 0.6039 1.1864 3.4482 0.9461 1.9831 0.3893 −2.9860 −1.4492 −0.6370
9 0.6120 0.6116 1.1924 3.1817 0.8237 1.9857 0.3866 0.2925 −2.9430 −1.4111 −0.6318
10 0.6149 0.6145 1.0171 2.8964 0.7073 1.0068 1.9651 0.3142 0.3023 −2.9363 −1.3732 −0.6173
11 0.6158 0.6154 1.0706 2.9918 0.7466 −0.2842 0.9935 1.9732 0.3199 0.3241 −2.9596 −1.3319 −0.6048
12 0.6173 0.6168 1.0876 3.0050 0.6287 −0.3890 0.9161 1.9895 0.4868 0.3113 0.3200 −2.9642 −1.3167 −0.6025
13 0.6177 0.6173 1.1059 3.0101 0.5920 −0.5888 0.9887 2.0044 0.4857 0.3097 0.2496 0.3143 −2.9617 −1.3323 −0.6019
14 0.6186 0.6181 1.1500 3.1133 0.5177 0.6815 −0.3721 1.1414 1.8905 −0.6801 0.4323 0.2883 0.3385 −2.9565 −1.3492 −0.6037
15 0.6189 0.6184 1.1616 3.1103 0.4934 0.6477 −0.5395 1.1884 1.9068 −0.6397 0.4336 0.2881 0.2075 0.3329 −2.9542 −1.3601 −0.6031
16 0.6191 0.6186 1.1545 3.1594 0.5125 0.1516 0.5787 −0.5341 1.0827 1.8949 −0.6280 0.3864 0.2970 0.1878 0.3311 −2.9371 −1.3440 −0.6033
17 0.6191 0.6186 1.1415 3.1719 0.5012 0.1242 0.5824 −0.5624 1.0721 1.9053 −0.6143 0.3833 0.2980 0.1849 0.0534 0.3317 −2.9457 −1.3390 −0.6034
Table G-4. P-Values from Regression Models Using the Maximum R-Square Option Using Nutrient or Component Values as a Percentage of Recommended Intake Levels per RACC (includes Whole Grains)
Number
of Vari-
ables
R-
Square
Adjusted
R-Square
Protein Fiber Vitamin
E
Vitamin
D
Calcium Iron Potas-
sium
Unsat-
urated
Fat
Mag-
nesium
Vitamin
A
Vitamin
C
Folic
Acid
Vitamin
B12
Whole
Grains
Saturated
Fat
Sodium Added
Sugar
— Means no data.
0 0.0417 0.0413
1 0.2074 0.2071 0.0000
2 0.4036 0.4033 0.0000 0.0000
3 0.4732 0.4730 0.0000 0.0000 0.0000
4 0.5218 0.5215 0.0000 0.0000 0.0000 0.0000
5 0.5416 0.5412 0.0000 0.0000 0.0000 0.0000 0.0000
6 0.5654 0.5651 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
7 0.5883 0.5880 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
8 0.6042 0.6039 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
9 0.6120 0.6116 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
10 0.6149 0.6145 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
11 0.6158 0.6154 0.0000 0.0000 0.0000 0.0139 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
12 0.6173 0.6168 0.0000 0.0000 0.0000 0.0006 0.0000 0.0000 0.0050 0.0000 0.0000 0.0000 0.0000 0.0000
13 0.6177 0.6173 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0047 0.0000 0.0121 0.0000 0.0000 0.0000 0.0000
14 0.6186 0.6181 0.0000 0.0000 0.0302 0.0000 0.0002 0.0000 0.0000 0.1105 0.0077 0.0000 0.0000 0.0000 0.0000 0.0000
15 0.6189 0.6184 0.0000 0.0000 0.0349 0.0000 0.0000 0.0000 0.0000 0.1219 0.0073 0.0000 0.0233 0.0000 0.0000 0.0000 0.0000
16 0.6191 0.6186 0.0000 0.0000 0.0275 0.0363 0.0000 0.0000 0.0000 0.0000 0.1243 0.0160 0.0000 0.0350 0.0000 0.0000 0.0000 0.0000
17 0.6191 0.6186 0.0000 0.0000 0.0294 0.0986 0.0000 0.0000 0.0000 0.0000 0.1331 0.0171 0.0000 0.0401 0.3695 0.0000 0.0000 0.0000 0.0000
Table G-5. Beta Coefficients from the Top 10 8-Nutrient/Component Regression Models Using the Maximum R-Square Option Using Nutrient or Component Values as a Percentage of Recommended Intake Levels per 100 Kcal (Includes Whole Grains)
Number
of Vari-
ables
R-
Square
Adjusted
R-Square
Protein Fiber Vitamin
E
Vitamin
D
Calcium Iron Potas-
sium
Unsat-
urated
Fat
Mag-
nesium
Vitamin
A
Vitamin
C
Folic
Acid
Vitamin
B12
Whole
Grains
Saturated
Fat
Sodium Added
Sugar
— Means no data.
8 0.6476 0.6473 1.3966 3.1261 1.0008 2.5151 0.3653 −2.9478 −1.3347 −0.5160
8 0.6462 0.6459 1.3006 3.2871 0.9852 2.2474 0.3959 −2.7125 −1.2367 −0.5536
8 0.6438 0.6435 1.5531 2.8613 2.0946 0.4276 0.3714 −2.6125 −1.3188 −0.5672
8 0.6428 0.6425 1.3743 3.2043 1.0221 2.4240 0.3047 −3.0568 −1.3329 −0.5512
8 0.6398 0.6395 1.2087 3.1517 0.9945 1.1475 2.5123 −3.0261 −1.3517 −0.5086
8 0.6393 0.6390 1.2902 3.3917 0.9555 2.1242 0.3165 −2.8207 −1.2414 −0.5953
8 0.6377 0.6374 1.2853 2.7882 1.6148 2.1417 0.3807 −2.7009 −1.3323 −0.5436
8 0.6372 0.6369 1.2668 3.5694 0.5930 0.8133 2.3890 −3.0458 −1.2997 −0.5603
8 0.6368 0.6365 1.7522 3.2094 1.0236 3.0447 0.4119 0.3182 −2.7854 −1.1477
8 0.6365 0.6362 1.3273 3.3569 0.9702 2.4179 0.6240 −3.1058 −1.3640 −0.5683
Table G-6. Beta Coefficients from the Top 10 8-Nutrient/Component Regression Models Using the Maximum R-Square Option Using Nutrient or Component Values as a Percentage of Recommended Intake Levels per RACC (Includes Whole Grains)
Number
of Vari-
ables
R-
Square
Adjusted
R-Square
Protein Fiber Vitamin
E
Vitamin
D
Calcium Iron Potas-
sium
Unsat-
urated
Fat
Magnes-
ium
Vitamin
A
Vitamin
C
Folic
Acid
Vitamin
B12
WholevGrains SaturatedvFat Sodium Added
Sugar
— Means no data.
8 0.6042 0.6039 1.1864 3.4482 0.9461 1.9831 0.3893 −2.9860 −1.4492 −0.6370
8 0.6010 0.6007 1.1780 3.6412 0.8370 1.7915 0.4293 −2.7085 −1.3792 −0.6360
8 0.6003 0.6000 1.4400 3.3083 1.6680 0.4569 0.3575 −2.5742 −1.4459 −0.6340
8 0.5973 0.5970 0.9098 3.2602 0.9075 1.6320 1.9643 −3.0688 −1.3947 −0.6113
8 0.5968 0.5965 1.0217 2.9211 2.2568 1.6805 0.3736 −2.7045 −1.3609 −0.5986
8 0.5968 0.5964 1.3557 3.4904 1.7490 0.9986 0.4141 −2.6813 −1.4756 −0.6418
8 0.5962 0.5959 1.2036 3.5476 1.0261 2.0067 0.2966 −3.0721 −1.4175 −0.6283
8 0.5948 0.5945 1.1752 3.2315 1.4549 1.6401 0.3544 −2.6250 −1.4389 −0.6195
8 0.5932 0.5929 2.7850 0.9033 2.4461 2.1462 0.3154 −2.9290 −1.0388 −0.6155
8 0.5919 0.5915 1.2708 3.6709 1.7102 0.4580 0.3907 −2.6426 −1.4692 −0.6498
Table G-7. Beta Coefficients from Regression Models Using the Maximum R-Square Option Using Nutrient or Component Values as a Percentage of Recommended Intake Levels per 100 Kcal (Total Sugar Replacing Added Sugar)
Number
of Vari-
ables
R-
Square
Adjusted
R-Square
Protein Fiber Vitamin
E
Vitamin
D
Calcium Iron Potas-
sium
Unsat-
urated
Fat
Mag-
nesium
Vitamin
A
Vitamin
C
Folic
Acid
Vitamin
B12
Whole
Grains
Saturated
Fat
Sodium Total
Sugar
— Means no data.
0 0.0417 0.0413
1 0.3657 0.3655 5.3217
2 0.4346 0.4343 4.6677 −1.8039
3 0.5020 0.5017 1.6699 4.2784 −2.0087
4 0.5406 0.5403 1.6797 4.2618 2.0661 −2.5611
5 0.5761 0.5758 1.3562 3.8857 1.3456 2.7242 −3.0194
6 0.6079 0.6076 1.7640 4.0018 1.3252 2.9597 −2.9824 −1.2048
7 0.6262 0.6259 1.7567 3.5363 1.1527 3.0335 0.4073 −2.8267 −1.1819
8 0.6368 0.6365 1.7522 3.2094 1.0236 3.0447 0.4119 0.3182 −2.7854 −1.1477
9 0.6421 0.6418 1.5703 2.9171 0.9005 1.0527 3.0833 0.3379 0.3293 −2.7573 −1.1494
10 0.6456 0.6453 1.5701 2.8548 0.7978 1.2268 3.1744 0.3345 0.4866 0.2946 −2.6953 −1.1903
11 0.6476 0.6472 1.6495 2.9614 0.7946 −0.5683 1.2759 3.1893 0.3374 0.8353 0.3091 −2.7220 −1.1721
12 0.6487 0.6483 1.6138 3.0438 0.3193 0.6447 −0.5952 1.1589 3.1670 0.3451 0.7948 0.3051 −2.7067 −1.1302
13 0.6492 0.6488 1.6546 3.0032 0.4143 0.6382 −0.5060 1.1725 3.1516 0.3454 0.8176 −0.1786 0.3018 −2.6790 −1.1404
14 0.6497 0.6492 1.6366 3.0980 0.3536 0.2837 0.7259 −0.5769 1.3451 3.0852 −0.4657 0.3169 0.7499 0.3199 −2.7346 −1.1357
15 0.6503 0.6498 1.6859 3.0507 0.3985 0.3939 0.7214 −0.4727 1.3704 3.0539 −0.4920 0.3147 0.7734 −0.2098 0.3173 −2.7011 −1.1473
16 0.6503 0.6498 1.6865 3.0458 0.3843 0.3842 0.7150 −0.4851 1.3668 3.0592 −0.4904 0.0574 0.3130 0.7755 −0.2105 0.3169 −2.7064 −1.1474
17 0.6503 0.6498 1.6815 3.0453 0.3850 0.3870 0.7165 −0.4842 1.3707 3.0516 −0.4988 0.0590 0.3144 0.7722 −0.2096 0.3174 −2.7093 −1.1504 −0.0082
Table G-8. Beta Coefficients from Regression Models Using the Maximum R-Square Option Using Nutrient or Component Values as a Percentage of Recommended Intake Levels per RACC (Total Sugar Replacing Added Sugar)
Number
of Vari-
ables
R-Square Adjusted R-Square Protein Fiber Vitamin E Vitamin D Calcium Iron Potas-
sium
Unsat-
urated Fat
Mag-
nesium
Vitamin A Vitamin C Folic Acid Vitamin B12 Whole Grains Saturated Fat Sodium Total Sugar
— Means no data.
0 0.0417 0.0413
1 0.2074 0.2071 3.9647
2 0.4036 0.4033 4.7469 −2.0714
3 0.4666 0.4663 3.0678 3.5502 −2.4323
4 0.4822 0.4819 2.6922 3.4790 0.4040 −2.3962
5 0.5046 0.5043 1.2483 3.5089 2.4916 −2.3969 −1.1935
6 0.5251 0.5247 1.1241 3.2833 2.6485 1.4255 −2.9348 −1.4166
7 0.5473 0.5469 3.1408 1.2390 3.3263 2.0305 −3.1199 −1.1449 −0.4433
8 0.5646 0.5642 1.3083 3.6994 1.2320 1.8897 0.5343 −3.1118 −1.5330 −0.4623
9 0.5746 0.5742 1.3130 3.3989 1.0989 1.8970 0.5347 0.3315 −3.0578 −1.4912 −0.4726
10 0.5847 0.5843 0.9766 2.8680 0.8991 1.8844 1.8754 0.4108 0.3504 −3.0248 −1.4245 −0.5068
11 0.5865 0.5861 1.0547 3.0040 0.9466 −0.4029 1.8360 1.8881 0.4146 0.3796 −3.0570 −1.3639 −0.4865
12 0.5883 0.5878 1.0732 3.0183 0.8171 −0.5167 1.7482 1.9064 0.5324 0.4050 0.3749 −3.0614 −1.3472 −0.4850
13 0.5890 0.5885 1.0956 3.0243 0.7716 −0.7666 1.8403 1.9257 0.5308 0.4033 0.3138 0.3677 −3.0575 −1.3672 −0.4866
14 0.5899 0.5894 1.1405 3.1314 0.6054 0.8759 −0.4981 1.9989 1.7867 −0.7423 0.4665 0.3800 0.3960 −3.0474 −1.3827 −0.4900
15 0.5906 0.5901 1.1285 3.2207 0.6355 0.2768 0.7465 −0.5144 1.8195 1.7693 −0.7180 0.3804 0.3974 0.3920 −3.0147 −1.3564 −0.4951
16 0.5910 0.5905 1.1424 3.2093 0.6050 0.2517 0.7204 −0.7016 1.8895 1.7898 −0.6754 0.3896 0.3958 0.2348 0.3859 −3.0146 −1.3714 −0.4954
17 0.5910 0.5904 1.1361 3.2153 0.5995 0.2385 0.7221 −0.7153 1.8843 1.7948 −0.6686 0.3881 0.3963 0.2335 0.0258 0.3862 −3.0188 −1.3690 −0.4953