Question

An endocrinologist was interested in exploring the relationship between the level of a steroid (Y) and age (X) in healthy subjects whose ages ranged from 8 to 25 years. She collected a sample of 27 healthy subject in this age range. The data is located in the file problem01.txt, where the first column represents X = age and the second column represents Y = steroid level. For all R programming, print input and output codes and values.

(a) Read the file problem01.txt into R using the read.table() function. You’ll need to set the working directory to the file location. Make a scatterplot of steroid (Y) versus age (X). Include the plot.

(b) Use R to fit a simple linear regression. Write down the fitted equation and multiple R² from the summary() output. Also comment on the p-value for the ?₁

coefficient

Y_i = ?₀ + ?₁X_i + ?_i

(c) Make a scatterplot of the fitted values versus the standardized residuals for the model in part (b). Are there any violation of assumptions? Include a copy of your plot.

(d) Create a quadratic regression in R. Write down the fitted equation, multiple R², and the p-value for ?₁ from the summary()output. Compare to part (b).

Y_i = ?₀ + ?₁X_i + ?₂X_i² + ?_i

problem01.txt

"age" "steroid"

15 14.1

10 8.5

13 10.8

16 18.4

10 4.7

18 23.3

16 16.4

10 9.4

16 17.7

23 35.8

19 25.4

18 24.9

24 42.1

19 26.5

24 40

12 10.7

13 11.6

10 3.6

23 37.9

17 16.8

19 24

23 37.7

20 29.6

14 13.7

19 23.1

11 8.3

17 19.6

9 7.8

11 7.1

13 13.3

18 20.8

25 44.4

9 9.7

12 12.5

22 34.9

8 4.3

9 5.9

8 6

22 36.2

15 11.7

10 5.3

15 15.6

9 6.6

14 15.7

13 10.5

17 20.7

23 36.8

23 37.2

8 5

16 19.6

16 18.9

15 16.1

10 7.7

14 11.9

12 9

8 4.4

8 2.7

8 5.2

16 19.3

20 27.5

20 27.8

13 12.9

12 12.8

13 9.3

15 16.1

19 25

13 10.5

13 9

18 22.3

22 33.6

9 4.9

19 28.4

15 14

21 30.6

19 24.8

R Outline Sample

########################

####### Part (a) #######

########################

# First save the file 'problem01.txt' on your computer.

# Next, set the working directory to the file location by doing the following:

# 1) Click on 'Session' on the top menu

# 2) Select 'Set Working Directory' > 'Choose Directory'

# 3) Select the folder where 'problem01.txt' is saved

# Read in data using the read.table() function.

dat <-

attach(dat)

# Create a scatterplot of age (X) vs steroid (Y)

# Write code here

########################

####### Part (b) #######

########################

# Fit a simple linear regression, then display the summary

fit <- # Enter code for simple linear regression

summary(fit)

########################

####### Part (c) #######

########################

# Plot the fitted values versus the standardized residuals for the fitted

# equation in part (b). Use the functions: sigma(), resid(), and predict()

y.hat <-

e.std <-

plot(y.hat, e.std, main = "Standardized Residuals vs. Fitted Value")

########################

####### Part (d) #######

########################

Answer 1

rm(list=ls())

# Reading Data

df <- read.table(file.choose(),header=T) #You will put your directory here

head(df)

#Scatter Plot

plot(df$Age,df$Steriod)

# Fitting linear model

model.linear <- lm(Steriod~.,data=df)

#Summary of the model with p-values, Rsquare and p-values

summary(model.linear)

# Adding Age^2 in our data

df$Age2 <- df$Age^2

# Now performing Quadratic Regression

model.quadratic <- lm(Steriod~.,data=df)

# Summary of quadratic model

summary(model.quadratic)

Part(a)

Part(b)

Variable Age is highly correlated with the variable Steriod. Also, the R^2 is very high. Quite a good fit for the data

part(c)

Residual vs fitted

There is a non-linear pattern in data. It means we should try a quadratic model

# Standardized Residuals vs fitted

The plot is useful for heteroscedasticity. But the plot doesn't verify it.

Part(d)

model.quadratic

The quadratic model has high R-squared. Also when we include the squared term in the model. The unsquared do not contribute to our model. Hence there is a quadratic relation between the variables.