Using R
ProteinHeight.txt:
|
Height |
Age |
Protein—Rich |
|
54 |
0.2 |
yes |
|
54.3 |
0.5 |
yes |
|
63 |
0.8 |
yes |
|
66 |
1 |
yes |
|
69 |
1 |
yes |
|
73 |
1.4 |
yes |
|
82 |
1.8 |
yes |
|
83 |
2 |
yes |
|
80.3 |
2 |
yes |
|
91 |
2.5 |
yes |
|
93.2 |
2.5 |
yes |
|
94 |
3 |
yes |
|
94 |
2.7 |
yes |
|
52 |
0.4 |
no |
|
55 |
0.7 |
no |
|
61 |
1 |
no |
|
63.4 |
1 |
no |
|
66 |
1.5 |
no |
|
68.5 |
2 |
no |
|
67.9 |
2 |
no |
|
72 |
2.4 |
no |
|
76 |
2.8 |
no |
|
74 |
3 |
no |
|
65 |
1.3 |
no |
|
69 |
1.8 |
no |
|
51 |
0.2 |
no |
|
77 |
3 |
no |
Exercise 1: Protein content in diet
A team of anthropologists and nutrition experts investigated the influence of protein content in diet on the relationship between age and height among children. The data ProteinHeight.txt gives height (in cm) and age (in years) for a sample of children with protein—rich and protein—poor diets.
(a) Provide numerical and graphical summaries for height across the two diet types. Describe and compare the distributions.
(b) Perform a two—sample t—test assessing the mean difference in height between protein—rich and protein—poor diets. What do you conclude?
(c) Consider a simple linear regression model of height versus age
(1) Draw a scatterplot of height versus age and describe the relationship.
(2) Write the equation of the regression line.
(3) Check that the model assumptions are satisfied.
(4) Interpret the regression coefficient of age in context.
(5) Interpret the R2 in context.
(d) Consider a regression model with age and diet type as covariates
(1) Fit a regression model with age and diet type as covariates. Write the regression equation.
(2) Check that the model assumptions are satisfied.
(3) Interpret the regression coefficients of age and diet type.
(4) Interpret the R2 in context.
(e) Consider a regression model with interaction between age and diet type
(1) Draw a scatterplot of height versus age across diet types. Describe the relationships.
(2) Fit a regression model with interaction between age and diet types. Write the regression equation.
(3) Check that the model assumptions are satisfied.
(4) Test the significance of the main effect for diet type (i.e., the regression parameter of diet type). State your conclusion in context.
(5) Is there evidence that the effect of age on height varies across diet types? State the null and alternative hypothesis, perform the appropriate test and state your conclusion at a = 0.05.
(6) Interpret the regression coefficients of age, diet type and the interaction term.
(7) Construct a 95% confidence interval for the interaction effect (i.e., the regression parameter of the interaction term) and interpret it in context.
(8) Interpret the R2 in context.
(f) Based on the adjusted-R2, which model would you choose between the simple linear regression model in (c), the additive model in (d), and the model with interaction term in (e). Why?
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