library(dplyr)
library(tidyr)
library(ggplot2)

Objective

The purpose of this paper is to perform a simple interential analysis of the tooth growth data set that comes with the base R installation. The dataset measures the impact of Vitamin C on tooth growth in guinea pigs. includes observations for different combinations of 2 supplements and 3 dosages. There are 10 observations for each combination of dosage and supplement for a total of 60 observations. The supplments are orange juice (OJ) or ascorbic acid (VC).

Inferential Data Analysis

Figure 1 below shows the range of tooth length for each combination of supplements and dosages. Each box includes a horizontal range indicated the mean value for the values. At first visual inspection, we note that there is an increasing trend in tooth length as the supplement dosage increases. We can also see that the mean value for ‘VC’ supplment is less than ‘OJ’ for smaller doses but the difference seems to diminish as the dose size increases. This is confirmed by the values in Table 1 below.

We can perform simple T-test to determine in tooth growth length is statistically different between the two supplments for each dosage level We’ll do seperate test for each of dosage level: 0.5 mL, 1 mL, and 2 mL. Our test will be to accept or reject the hypothesis that supplment does have an impact on the tooth growth. After another look at Figure 1, it looks like there is difference in the two supplements at the 0.5 mL and 1 mL level but there may not be a difference at 2 mL. If the resulting p-valuefrom these tests are less than 0.05, then we can state our conclusions with 95% confidence. Before continuing, we have to note one primary assumption is this test. That is that the data is paired. This is appears to be a safe assumption because for each combination, there are exacly 10 observations, but none the less, it is an assumption

The p-values for both tests were very small as expected. I say the test values are as expected because we can see clearly from figure 1 that the mean tooth length between orange juice and ascorbic acid are not the same for the 0.5 mL and 1 mL dose. On the other hand, this is not true for the 2 mL dosage since the mean values are close together. Overall, we can conclude from these tests that teeth grow longer with orange juice over ascorbic acid when the dose is 1 mL or less. Greater than 1 mL, there does not appear to be a difference in tooth growth.

Dosage T Test p-value Decision
0.5 mL 0.006359 Do not reject
1 mL 0.001038 Do not reject
2 mL 0.9639 Reject

R Code

Data<-ToothGrowth
Data$dose<- factor(Data$dose)
ggplot(Data) + 
  aes(x=dose,y=len, fill= supp) +
  geom_boxplot() +
  labs(title= "Fig 1: Tooth growth by Supplement and Dose size")

#   The output shows the summary statistics for each of the variables. It
#   does not provide information about the combination of values. Most
#   importantly, it shows the quartile distribution of the 'len' or lenght
#   variable.
summary(Data)
##       len        supp     dose   
##  Min.   : 4.20   OJ:30   0.5:20  
##  1st Qu.:13.07   VC:30   1  :20  
##  Median :19.25           2  :20  
##  Mean   :18.81                   
##  3rd Qu.:25.27                   
##  Max.   :33.90

Tbl 1: Mean Tooth length by combination

# Summary of means
#   The table shows the mean tooth length for each combination of 
#   suppment and dose size.
Data %>%
  group_by(supp,dose) %>%
  summarise(Mean= mean(len)) %>%
  spread(dose,Mean)
## Source: local data frame [2 x 4]
## 
##   supp   0.5     1     2
## 1   OJ 13.23 22.70 26.06
## 2   VC  7.98 16.77 26.14

T- Test for significance of impact of supplment

t.test(len~supp,Data[Data$dose=="0.5",])
t.test(len~supp,Data[Data$dose=="1",])
t.test(len~supp,Data[Data$dose=="2",])
## 
##  Welch Two Sample t-test
## 
## data:  len by supp
## t = 3.1697, df = 14.969, p-value = 0.006359
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  1.719057 8.780943
## sample estimates:
## mean in group OJ mean in group VC 
##            13.23             7.98 
## 
## 
##  Welch Two Sample t-test
## 
## data:  len by supp
## t = 4.0328, df = 15.358, p-value = 0.001038
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  2.802148 9.057852
## sample estimates:
## mean in group OJ mean in group VC 
##            22.70            16.77 
## 
## 
##  Welch Two Sample t-test
## 
## data:  len by supp
## t = -0.0461, df = 14.04, p-value = 0.9639
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -3.79807  3.63807
## sample estimates:
## mean in group OJ mean in group VC 
##            26.06            26.14