Structure: Exploratory data analyses, Summary of the data, Hypothesis testing

Exploratory data analyses

Loading in the data.
set.seed(12345)
library(datasets)
data <- ToothGrowth
Exploratory analyses.
str(data)
## 'data.frame':    60 obs. of  3 variables:
##  $ len : num  4.2 11.5 7.3 5.8 6.4 10 11.2 11.2 5.2 7 ...
##  $ supp: Factor w/ 2 levels "OJ","VC": 2 2 2 2 2 2 2 2 2 2 ...
##  $ dose: num  0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 ...
library(ggplot2)
library(gridExtra)
## Loading required package: grid
data$dose <- factor(data$dose)
gplot1 <- ggplot(data, aes(x=dose, y=len))+ geom_violin(alpha=0.5, color="darkorange3") +
    geom_jitter(alpha=0.5, aes(color=supp), position = position_jitter(width = 0.1)) +
    labs(x="Dose",y="") + facet_grid(.~supp) + theme(legend.position="none") +      
    scale_colour_brewer(palette="Set2")

data <- ToothGrowth

gplot2<- ggplot(data, aes(x=dose, y=len, colour=supp)) +
    geom_point(alpha=.5, size=5) + facet_grid(.~supp) + scale_colour_brewer(palette="Set2") +
    stat_smooth(method="auto") + theme(legend.position=c(1.04,0), legend.justification=c(1,0)) + 
    labs(x="",y="") + 
    guides(colour = guide_legend(title="Supplement",override.aes = list(size=4)))

grid.arrange(gplot1, gplot2, ncol=2,main="Tooth length by dose based on the supplement type",respect=F,just="center",left="Tooth Length")

Summary of the data

names(data) <- c("Tooth length","Supplement","Dose")
summary(data)
##   Tooth length   Supplement      Dose      
##  Min.   : 4.20   OJ:30      Min.   :0.500  
##  1st Qu.:13.07   VC:30      1st Qu.:0.500  
##  Median :19.25              Median :1.000  
##  Mean   :18.81              Mean   :1.167  
##  3rd Qu.:25.27              3rd Qu.:2.000  
##  Max.   :33.90              Max.   :2.000
library(dplyr)
data <- ToothGrowth
dataSum <- group_by(data,supp,dose)
dataSum %>% summarise(Mean=mean(len),SD=sd(len))
## Source: local data frame [6 x 4]
## Groups: supp
## 
##   supp dose  Mean       SD
## 1   OJ  0.5 13.23 4.459709
## 2   OJ  1.0 22.70 3.910953
## 3   OJ  2.0 26.06 2.655058
## 4   VC  0.5  7.98 2.746634
## 5   VC  1.0 16.77 2.515309
## 6   VC  2.0 26.14 4.797731

Hypothesis testing

Setting the confidence level to 95%.

Find confidence intervals for tooth lengths based on the supplement type.

\(# Ho: The supplement type does not affect tooth growth.\) \(#Ha: The supplement type does affect tooth growth.\)

t.test(len~supp,data=data,conf.level=.95,var.equal=T,paired=F,alternative ="two.sided")
## 
##  Two Sample t-test
## 
## data:  len by supp
## t = 1.9153, df = 58, p-value = 0.06039
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.1670064  7.5670064
## sample estimates:
## mean in group OJ mean in group VC 
##         20.66333         16.96333
Conclusion: Based on the confidence interval we fail to reject the null hypothesis as the interval includes 0. P-value is around 6%, which is higher than 5% significance level we set for the test.

Find whether dosage has a significant impact on toothgrowth.

\(# Ho: The dosage does not affect tooth growth.\) \(#Ha: The dosage does affect tooth growth.\)

t.test(len~dose,dose %in% c(0.5,1.0),data=data,conf.level=.95,var.equal=T,paired=F,alternative ="two.sided")
## 
##  Two Sample t-test
## 
## data:  len by dose
## t = -6.4766, df = 38, p-value = 1.266e-07
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -11.983748  -6.276252
## sample estimates:
## mean in group 0.5   mean in group 1 
##            10.605            19.735
t.test(len~dose,dose %in% c(1.0,2.0),data=data,conf.level=.95,var.equal=T,paired=F,alternative ="two.sided")
## 
##  Two Sample t-test
## 
## data:  len by dose
## t = -4.9005, df = 38, p-value = 1.811e-05
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -8.994387 -3.735613
## sample estimates:
## mean in group 1 mean in group 2 
##          19.735          26.100
t.test(len~dose,dose %in% c(0.5,2.0),data=data,conf.level=.95,var.equal=T,paired=F,alternative ="two.sided")
## 
##  Two Sample t-test
## 
## data:  len by dose
## t = -11.799, df = 38, p-value = 2.838e-14
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -18.15352 -12.83648
## sample estimates:
## mean in group 0.5   mean in group 2 
##            10.605            26.100
Conclusion: We have to reject Ho hypothesis based on the t-statistics for all three tests, and conclude that the dosage level does affect tooth growth.
We can also conclude that the impact is higher for low ~ high dosage level, and is less significant for medium ~ high dosage level.

Our findings in this project indicate that dosage level has high impact on tooth growth whereas the supplement type is irrelevant.