1. STATS 324 - Multiple Regression Project
We have a dataset with various cereals and their
nutritional breakdowns including: Calories, Total Fat,
Saturated Fat, Carbohydrates, Fiber, Sugar, Protein,
and Trans-Fat.
Our goal is to construct a model based on a subset of
these variables to predict the number of calories a
box of cereal will have.

2. Simple Regression

3. Simple Regression
Ho:Carbs have no significant relationship with Calories.
Ha:Carbs have a significant relationship with Calories.
Calories = -2.01654 + 4.61713 Carbs

4. Results
The R-Sq is 85.76% with an S of 14.6795. From the
scatterplot we can see a strong, positive linear
association between carbs and calories.
From the regression we find that Carbs do have a
statistically significant (t(df=50)=17.3535, p~0)
relationship with Calories. This matches what we would
expect because carbohydrates always contain calories,
so any increase in carbs would lead to an increase in
calories.

5. Model Formulation:
0 = Transfat
1= No Transfat
Model1:
General Regression Analysis: Calories versus Carbs, Sugar, Trans
Regression Equation: Trans
0 Calories = -5.64709 + 4.49934 Carbs + 0.59055 Sugar
1 Calories = -3.58016 + 4.49934 Carbs + 0.59055 Sugar
Model 2:
General Regression Analysis: Calories versus Carbs, ln(sugar), Trans
Regression Equation: Trans
0 Calories = -4.32483 + 4.36409 Carbs + 3.83766 ln(sugar)
1 Calories = -1.87922 + 4.36409 Carbs + 3.83766 ln(sugar)
Multicollinearity:
-VIF’s of 1st model:
Trans 0 =1.21029
Carbs = 1.14868
Sugar = 1.31693
-VIF of 2nd model:
Trans 0 =1.19284
Carbs = 1.13480
Sugar = 1.29197

6. Transformations:
Calories as response, carbs, sugar, and trans fat as our first model:
Calories and carbs show to have a moderately strong and positive
association. Calories and sugar show to be not as strong but positive
association. Thus the first thought is to transform sugar, since calories
and carbs are fine. We ln(sugar) since ln is our go to transformation.

7. Comparing Models
Model 1, F= 2.451 and Pvalue = 0.076850, R- sq =86.18%, R-sq adjusted =85.31%
thus at a 10% level, evidence of lack of fitting a line through data is not doing well.
L:Seems ok, but may want to make it better.
I: OK.
N: OK.
E: OK
Model 2, F= 1.549 and Pvalue = 0.249241 , R-Sq = 89.28%, R-sq adjusted 89.28%
The P-value increased thus less evidence against the null that the linear model is appropriate.
L:Seems ok, but safe to use model one.
I: OK.
N: OK.
E: OK
The VIF’s in model 2 are smaller than model 1, yet model
one did not have high VIF’s values. Thus it is better to
use model 1 with the untransformed data.

9. Case Influence Diagnostics

12. -the model remain statistically significant after adding
the interaction between carbs and trans-fat
-adjustment carbs*trans-fat is unnecessary
-did not improve the association between the
interaction and response.

13. Best Subsets Regression: Calories versus Carbs, Sugar
Response is Calories
C S
a u
r g
Mallows b a
Vars R-Sq R-Sq(adj) Cp S s r
1 85.8 85.5 2.3 14.680 x
1 11.0 9.3 266.4 36.691 x
2 86.1 85.6 3.0 14.631 x x
-Carbs alone has the lowest Mallows Cp
-Carbs and sugar together have a better R2 and S value as they are larger
-We used Carbs as our best predictor.

14. Conclusion
Analyzing the general regression model with
Calories, Trans Fat, Carbs and Sugar as our variables,
we believe the overall model to be statistically significant since our analysis of variance show - -
an extremely small p-value (≈ 0)
- R^2 is fairly large at 86.18% with the adjusted at 85.31% and the predicted at 82.80%.
- VIFs also look to be reasonably small (however it makes intuitive sense for there to be
multicollinearity between sugar and carbs since carbs contain sugar.)
Ho: β1=β2=0
Ha: β1orβ2≠0
Based on our hypotheses, we have evidence that there is an association between sugar, carb, and
trans-fat to calories since β1and β2 are not equal to 0 (since trans-fat is categorical binary,
the equations just need to add either 0 or 1 for that value).