Synopsis

Data Source

The dataset of ‘ToothGrowth’ is loaded from the R dataset packages.
Data Decription: The response is the length of odontoblasts (cells responsible for tooth growth) in 60 guinea pigs. Each animal received one of three dose levels of vitamin C (0.5,1, and 2 mg/day) by one of two delivery methods, (orange juice or ascorbic acid (a form of vitamin C and coded as VC))

Data Reference

1- NcNeil,D.R.(1977) Interactive Data Analysis. New York:Wiley
2- Crampton, E. W. (1947) The growth of the odontoblast of the incisor teeth as a criterion of vitamin C intake of the guinea pig. The Journal of Nutrition 33(5): 491–504.

Exploratory Data Analysis

library(plyr)
data(ToothGrowth)

As show below, the data frame ‘ToothGrowth’ consists of 60 objects with 3 variables.
1- len: Tooth length
2- supp: Supplement type (VC or OJ)
3- dose: Dose in milligrams/day

head(ToothGrowth)
##    len supp dose
## 1  4.2   VC  0.5
## 2 11.5   VC  0.5
## 3  7.3   VC  0.5
## 4  5.8   VC  0.5
## 5  6.4   VC  0.5
## 6 10.0   VC  0.5
str(ToothGrowth)
## '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 ...

1- The summary of the ToothGrowth Data

summary(ToothGrowth)
##       len        supp         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

2- The Mean and Standard Deviation of Tooth Growth Length by supp.

The Mean of Tooth Growth Length for VC supp is 16.96333.
The Mean of Tooth Growth Length for OJ supp is 20.66333.

 mn_by_supp<-ddply(ToothGrowth,.(supp), function(x) mean(x$len));colnames(mn_by_supp)<-c("supp","mean")
 mn_by_supp
##   supp     mean
## 1   OJ 20.66333
## 2   VC 16.96333

The Standard Deviation of Tooth Growth Length for VC supp is 8.266029.
The Standard Deviation of Tooth Growth Length for OJ supp is 6.605561.

 sd_by_supp<-ddply(ToothGrowth,.(supp), function(x) sd(x$len));colnames(sd_by_supp)<-c("supp","sd")
 sd_by_supp
##   supp       sd
## 1   OJ 6.605561
## 2   VC 8.266029

3- The Histogram of Tooth Growth Length with different supplments are plotted below with the mean value for each group marked.

ToothGrowth$supp<-as.character(ToothGrowth$supp)
par(mfrow=c(2,1))
p1<-hist(ToothGrowth$len[ToothGrowth$supp=="VC"],col=rgb(0,0,1,1/4),xlab="The length of tooth growth with VC group",main="The Histogram of Tooth Growth Length for 'VC' supp group");abline(v=mn_by_supp$mean[2],col="red")

p2<-hist(ToothGrowth$len[ToothGrowth$supp=="OJ"],col=rgb(1,0,0,1/4),xlab="The length of tooth growth with OJ group",main="The Histogram of Tooth Growth Length for 'OJ' supp group");abline(v=mn_by_supp$mean[1],col="red")

4- The Mean and Standard Deviation of Tooth Growth Length by dose.

The Mean of Tooth Growth Length for 0.5 milligrams/day dose is 10.605.
The Mean of Tooth Growth Length for 1.0 milligrams/day dose is 19.735.
The Mean of Tooth Growth Length for 2.0 milligrams/day dose is 26.100.

 mn_by_dose<-ddply(ToothGrowth,.(dose), function(x) mean(x$len));colnames(mn_by_dose)<-c("dose","mean")
 mn_by_dose
##   dose   mean
## 1  0.5 10.605
## 2  1.0 19.735
## 3  2.0 26.100

The Standard Deviation of Tooth Growth Length for 0.5 milligrams/day dose is 4.499763.
The Standard Deviation of Tooth Growth Length for 1.0 milligrams/day dose is 4.415436.
The Standard Deviation of Tooth Growth Length for 2.0 milligrams/day dose is 3.774150.

 sd_by_dose<-ddply(ToothGrowth,.(dose), function(x) sd(x$len));colnames(sd_by_dose)<-c("dose","Sd")
 sd_by_dose
##   dose       Sd
## 1  0.5 4.499763
## 2  1.0 4.415436
## 3  2.0 3.774150

5- The Histogram of Tooth Growth Length with different doses are plotted below with the mean value for each group marked.

par(mfrow=c(3,1))
p1<-hist(ToothGrowth$len[ToothGrowth$dose==0.5],col=rgb(0,0,1,1/4),xlab="The length of tooth growth with 0.5 dose group",main="The Histogram of Tooth Growth Length for 0.5 dose group");abline(v=mn_by_dose$mean[1],col="red")
p2<-hist(ToothGrowth$len[ToothGrowth$dose==1.0],col=rgb(1,0,0,1/4),xlab="The length of tooth growth with 1.0 dose group",xlim=c(10,30),main="The Histogram of Tooth Growth Length for 1.0 dose group");abline(v=mn_by_dose$mean[2],col="red")
p3<-hist(ToothGrowth$len[ToothGrowth$dose==1.0],col=rgb(0,1,0,1/4),xlab="The length of tooth growth with 2.0 dose group",xlim=c(10,30),main="The Histogram of Tooth Growth Length for 2.0 dose group");abline(v=mn_by_dose$mean[3],col="red")

The Statistical Inference of Tooth Growth Data

The two-sample T test on the Tooth Growth Length for OJ supp and VC supp group

t.test(len~supp,data=ToothGrowth)
## 
##  Welch Two Sample t-test
## 
## data:  len by supp
## t = 1.9153, df = 55.309, p-value = 0.06063
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.1710156  7.5710156
## sample estimates:
## mean in group OJ mean in group VC 
##         20.66333         16.96333

The Null Hypothesis: Assume the mean of tooth growth length is the same. According to the result of t-test, the p-value is 0.061, which is higher than the significant value 0.05. So the Null Hypothesis is accepted and the conclusion that there is no difference between mean of tooth growth length is arrived.

The two-sample T test on the Tooth Growth Length for 0.5 and 1.0 dose group

len_dose1<-ToothGrowth$len[ToothGrowth$dose==0.5];len_dose2<-ToothGrowth$len[ToothGrowth$dose==1.0];len_dose3<-ToothGrowth$len[ToothGrowth$dose==2.0]
t.test(len_dose1,len_dose2)
## 
##  Welch Two Sample t-test
## 
## data:  len_dose1 and len_dose2
## t = -6.4766, df = 37.986, p-value = 1.268e-07
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -11.983781  -6.276219
## sample estimates:
## mean of x mean of y 
##    10.605    19.735

The Null Hypothesis: Assume the mean of tooth growth length is the same. According to the result of t-test, the p-value is 1.268e-07, which is less than the significant value 0.05. So the Null Hypothesis is rejected and the conclusion that there is difference between mean of tooth growth length is arrived.

The two-sample T test on the Tooth Growth Length for 0.5 and 2.0 dose group

t.test(len_dose1,len_dose3)
## 
##  Welch Two Sample t-test
## 
## data:  len_dose1 and len_dose3
## t = -11.799, df = 36.883, p-value = 4.398e-14
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -18.15617 -12.83383
## sample estimates:
## mean of x mean of y 
##    10.605    26.100

The Null Hypothesis: Assume the mean of tooth growth length is the same. According to the result of t-test, the p-value is 4.398e-14, which is less than the significant value 0.05. So the Null Hypothesis is rejected and the conclusion that there is difference between mean of tooth growth length is arrived.

The two-sample T test on the Tooth Growth Length for 1.0 and 2.0 dose group

t.test(len_dose2,len_dose3)
## 
##  Welch Two Sample t-test
## 
## data:  len_dose2 and len_dose3
## t = -4.9005, df = 37.101, p-value = 1.906e-05
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -8.996481 -3.733519
## sample estimates:
## mean of x mean of y 
##    19.735    26.100

The Null Hypothesis: Assume the mean of tooth growth length is the same. According to the result of t-test, the p-value is 1.906e-05, which is less than the significant value 0.05. So the Null Hypothesis is rejected and the conclusion that there is difference between mean of tooth growth length is arrived.

Conclusion and Assumption

To do the two-sample t test for different groups, we have to assume that the data of two populations must be independently obtained, the variance of the two groups is the same and the two populations should follow a normal distribution.
1- There is no significant effect on the tooth growth length for different supplments.
2- The increasing dose levels of vitamin C has positive effect on the tooth growth length.