Question

1. FEV (forced expiratory volume) is an index of pulmonary function that measures the volume of air expelled after one second of constant effort. The data fev.csv contains determinations of FEV on 654 children ages 3-19 who were seen in the Childhood Respiratory Disease Study in East Boston, Massachusetts. The variables in the data include:
   
   ID:                       subject ID number
   Age:                    age in years
   FEV:                    FEV in liters
   Height:               height in inches
   Sex:                     Male or Female
   Smoker:              non = nonsmoker, Current = current smoker
   
   1. Make a boxplot of the FEV for children with age 3-8 years, 9-12 years, and 13 years or above. Does it appear that the FEV is the same for children from these three age groups?
      
      
   2. Is FEV the same across the three age groups? Perform a hypothesis test to answer the question. Use α = 0.05.
      
      
   3. Rank the three age groups using the multiple comparison approach.
      
   4. What assumptions are made with regard to the analysis in part b? Check whether these assumptions are violated.
      
      
   5. Is FEV is more strongly related to sex or smoking status? Carry out appropriate statistical analysis to answer the question.

1. The investigator is also interested in how height is associated with age. Construct the scatter plot of height against age. What is the relationship between height and age?
   
2. Regardless of what you observed in f, fit the regression model with height as the response and age as the independent variable. What is the fitted regression equation?
   
   
3. Test whether there is a positive correlation between age and height. Perform the hypothesis test using α = 0.05.
   
4. Is it appropriate to use the above regression model? Why or why not?

Answer 1

SOLUTION

a. Make a boxplot of the FEV for children with age 3-8 years, 9-12 years, and 13 years or above. Does it appear that the FEV is the same for children from these three age groups?

I have used EXCEL to construct box plot of the FEV for children with age 3-8 years, 9-12 years, and 13 years or above. The first step to sort the data from youngest to oldest age group. Divide the data into three groups age 3-8 years, 9-12 years, and 13 years or above. Insert > Box & Whisker

No, it doesn't seem that FEV is same for children from these three age groups.

Findings from boxplot:

• FEV Age 9-12 have few outliers (dots in boxplot)
• FEV Age 3-8 and Age >=13 is slightly skewed towards above.
• All three data is roughly normal as data is almost equally distributed around the median.
• Median of each group seems to be different.

b. Is FEV the same across the three age groups? Perform a hypothesis test to answer the question. Use alpha = 0.05.

To find the difference between the three age groups we performed ANOVA analysis on the data

The null and alternative hypotheses:

Calculate the value of the test statistic:

EXCEL > DATA> Data Analysis (AddIn) > ANOVA Single factor

Anova: Single Factor
SUMMARY
Groups	Count	Sum	Average	Variance
FEV Age 3-8	215	399.519	1.858228	0.176462
FEV Age 9-12	322	903.802	2.806839	0.410088
FEV Age >=13	117	421.133	3.599427	0.633303
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	248.0558	2	124.0279	332.4582	3.2E-100	3.00956
Within Groups	242.8641	651	0.373063
Total	490.9198	653

F(2,651) = 332.4582, p = 3.2E-100

Determine if the results are statistically significant (using rejection region or p-value approaches),

P = 3.2E-100 which is highly significant and F value (332.4582) is way above F critical value of 3.00956

State your conclusion in terms of the problem

Since ANOVA shows that the between-group difference is significant thus we reject the null hypothesis and accept the alternate hypothesis that there is a significant difference in FEV values between the age groups.

c. Rank the three age groups using the multiple comparison approaches.

Do Not function available in EXCEL for post host tests like Tukey etc. So I have perfomed paired t test for each pair

3-8 vs 9-12 p = 8.64E-45

9-12 vs >=13 p = 2.47E-23

3-8 vs >=13 p = 2.57E-50

All three pair are significant

d. What assumptions are made with regard to the analysis in part b? Check whether these assumptions are violated.

There are three main assumptions, listed here:

1. The dependent variable is normally distributed in each group that is being compared in the one-way ANOVA

2. There is the homogeneity of variances. This means that the population variances in each group are equal.

3. Independence of observations. This is mostly a study design issue and, as such, you will need to determine whether you believe it is possible that your observations are not independent based on your study design

We carried out the descriptive analysis of the FEV in three age groups. The result obtained is as follows

FEV Age 3-8		FEV Age 9-12		FEV Age >=13
Mean	1.858228	Mean	2.806839	Mean	3.59942735
Standard Error	0.028649	Standard Error	0.035687	Standard Error	0.07357204
Median	1.79	Median	2.756	Median	3.519
Mode	1.624	Mode	2.352	Mode	3.297
Standard Deviation	0.420073	Standard Deviation	0.640381	Standard Deviation	0.795803284
Sample Variance	0.176462	Sample Variance	0.410088	Sample Variance	0.633302868
Kurtosis	-0.0875	Kurtosis	0.804245	Kurtosis	-0.227353716
Skewness	0.242282	Skewness	0.693698	Skewness	0.411303102
Range	2.202	Range	3.766	Range	3.595
Minimum	0.791	Minimum	1.458	Minimum	2.198
Maximum	2.993	Maximum	5.224	Maximum	5.793
Sum	399.519	Sum	903.802	Sum	421.133
Count	215	Count	322	Count	117
Confidence Level(95.0%)	0.05647	Confidence Level(95.0%)	0.07021	Confidence Level(95.0%)	0.145718695

Shapiro-Wilk Test
	FEV Age 3-8	FEV Age 9-12	FEV Age >=13
W-stat	0.988854391	0.97202266	0.977644226
p-value	0.093440645	6.72808E-06	0.047848341
alpha	0.05	0.05	0.05
normal	yes	no	no

We also performed Shapiro Wilks test RealStats(AddIn) in EXCEL. Which shows that two of the age groups is not normal.

Sample variance of three groups are different in numerical value.

e. Is FEV is more strongly related to sex or smoking status? Carry out appropriate statistical analysis to answer the question.

Using EXCEL > AddIn > RealStats > Data Analysis > Corr > Correlation test

Carried correlation test between FEV and SEX and FEV and SMOKING

RESULT of correlation test on FEV and SMOKING

Correlation Coefficients
Pearson	-0.245424571
Spearman	-0.258349236
Kendall	-0.211145277
Pearson's coeff (t test)			Pearson's coeff (Fisher)
Alpha	0.05		Rho	0
Tails	2		Alpha	0.05
			Tails	2
corr	-0.245424571
std err	0.037965248		corr	-0.245424571
t	-6.464453173		std err	0.039133024
p-value	1.99285E-10		z	-6.392409034
lower	-0.319973478		p-value	1.63292E-10
upper	-0.170875664		lower	-0.316142406
			upper	-0.171994479

RESULT of correlation test on FEV and SEX

Correlation Coefficients
Pearson	-0.20841
Spearman	-0.14364
Kendall	-0.11739
Pearson's coeff (t test)			Pearson's coeff (Fisher)
Alpha	0.05		Rho	0
Tails	2		Alpha	0.05
			Tails	2
corr	-0.20841
std err	0.038303		corr	-0.208414959
t	-5.44121		std err	0.039133024
p-value	7.5E-08		z	-5.39671039
lower	-0.28363		p-value	6.78738E-08
upper	-0.1332		lower	-0.28059776
			upper	-0.13388797

Conclusion:

• Both sex and smoking have a significant correlation with FEV and have small negative correlation (-0.1 to -0.3).
• Smoking is more strongly correlated with spearmen correlation of -0.25 while has a correlation of -0.14

f. The investigator is also interested in how height is associated with age. Construct the scatter plot of height against age. What is the relationship between height and age?

The height and Age have a linear relationship with R square valued of 0.6272

g. Regardless of what you observed in f, fit the regression model with height as the response and age as the independent variable. What is the fitted regression equation?

Regression Analysis of height as response and Age as the independent variable

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.791943602
R Square	0.627174669
Adjusted R Square	0.626602851
Standard Error	3.485201717
Observations	654
ANOVA
	df	SS	MS	F	Significance F
Regression	1	13322.52461	13322.52461	1096.808	8.0598E-142
Residual	652	7919.603418	12.14663101
Total	653	21242.12803
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	45.95779737	0.478358112	96.07404205	0	45.01848904	46.89710571
Age	1.529099387	0.046171116	33.1180949	8.1E-142	1.438437365	1.619761409

The regression equation will be

height = 45.95 + 1.52*Age

Thus for one every unit increase in Age, height will increase with 1.52. The regression analysis is significant with p <0.00000001. The coefficient of determination, R square = 0.62 which implies 62% variance in height is explained by Age.

h.Test whether there is a positive correlation between age and height. Perform the hypothesis test using alpha = 0.05.

Correlation test on height and age

Correlation Coefficients
Pearson	0.791973295
Spearman	0.818796185
Kendall	0.660161965
Pearson's coeff (t test)			Pearson's coeff (Fisher)
Alpha	0.05		Rho	0
Tails	2		Alpha	0.05
			Tails	2
corr	0.791973295
std err	0.023929566		corr	0.791973295
t	33.09601623		std err	0.039163022
p-value	0		z	27.450651
lower	0.744984849		p-value	6.8244E-166
upper	0.838961742		lower	0.761521493
			upper	0.818936333

The Pearson correlation coefficient is 0.79 which depicts a strong positive correlation.