Question

An experiment was conducted for better understanding of the effectiveness of a particular type of drug for reducing bad cholesterol (LDL) level. The purpose of the experiment was to determine whether different dosages used have significant different outcomes in average LDL reduction. Twenty subjects with LDL at around 250 to 300 mg/dL had participated in the study and were randomly divided into four groups. Each group was given a specific level of dosage of the drug each day for one month, with a control group that only provided with placebo. The reduction in LDL was recorded and showed in the following table. Positive number indicates reduction and negative numbers indicates increasing in DLD. Use statistical software to analyze the data and answer the following question.

Control	Light Dosage Level	Medium Dosage Level	Heavy Dosage Level
7	25	73	81
-3	17	60	71
6	22	55	79
5	21	41	60
15	12	36	85

1. Use statistical software to determine whether the average reduction in LDL differ among people who used different dosages of the drug. Perform a one-way ANOVA F-test and report p-value of the test and draw a conclusion using the level of significance at 5%.

Null hypothesis:

Alternative hypothesis:

Report the value of the F-test statistic =

Report p-value from the F-test and the conclusion:

[Place your software output here.]

1. Verify the assumptions behind the ANOVA F-test using statistical software and explain your result.

[Place your software output here.]

1. Use Tukey’s multiple comparison methods to compare all the treatment groups to see where the differences are and also identify homogeneous subsets from these treatment groups.

[Place your software output here.]

Answer 1

Null hypothesis: H0: The mean reduction in LDL is same for control group and all dosage levels.

Alternative hypothesis: Ha: At least one of the dosage levels or control group have different mean reduction in LDL.

Report the value of the F-test statistic = 51.44

Report p-value from the F-test and the conclusion: p-value = 1.94e-08

Since, p-value is less than 0.05 significance level, we reject null hypothesis H0 and conclude that there is strong evidence that at least one of the dosage levels or control group have different mean reduction in LDL.

By Tukey’s multiple comparison method output, the significant mean differences of reduction in LDL is found in,

Control and Medium Dosage Level ,  Control and Heavy Dosage Level , Light and Medium Dosage Level, Light and Heavy Dosage Level,  Medium and Heavy Dosage Level

The homogeneous subsets are,

Subset 1 - {Control , Light Dosage Level}

Subset 2 - {Medium Dosage Level }

Subset 3 - {Heavy Dosage Level}

On running the below R code, we get the following output.

R code -

# Load the data
LDL <- c(7,-3,6,5,15,
25,17,22,21,12,
73,60,55,41,36,
81,71,79,60,85)
# Create vectors of factors (4 levels) for Dosage levels
Dosage=factor(c(rep(1,5),rep(2,5),rep(3,5),rep(4,5)))

# Fit a regression model on scale for different factors of Dosage levels and run the anova test
summary(model <- aov(LDL ~ Dosage))

# Run the TukeyHSD test
TukeyHSD(model, "Dosage", ordered = TRUE)

> summary(model <- aov(LDL ~ Dosage))
Df Sum Sq Mean Sq F value Pr(>F)
Dosage 3 14891 4964 51.44 1.94e-08 ***
Residuals 16 1544 96   
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> # Run the TukeyHSD test
> TukeyHSD(model, "Dosage", ordered = TRUE)
Tukey multiple comparisons of means
95% family-wise confidence level
factor levels have been ordered

Fit: aov(formula = LDL ~ Dosage)

$Dosage
diff lwr upr p adj
2-1 13.4 -4.375201 31.1752 0.1777685
3-1 47.0 29.224799 64.7752 0.0000062
4-1 69.2 51.424799 86.9752 0.0000000
3-2 33.6 15.824799 51.3752 0.0003061
4-2 55.8 38.024799 73.5752 0.0000007
4-3 22.2 4.424799 39.9752 0.0121973