The Effect of Vitamin C on Tooth Growth in Guinea Pigs

Goal

We’re going to analyze the ToothGrowth data in the R datasets package.

According to the description of the help article of the dataset, this contains:
- The length of odontoblasts (teeth) in each of 10 guinea pigs at each of three dose levels of Vitamin C (0.5, 1, and 2 mg) with each of two delivery methods (orange juice or ascorbic acid).

data("ToothGrowth")

Exploratory Data Analysis

Let’s have a look to 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
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 ...

Summary of the data

We have a data frame with 60 observations on 3 variables:

1. len: Contains the tooth length with numeric values  
    - Mean: 18.81  
    - Standard deviation: 7.65    
2. supp: Two types of supplements with class factor  
    - VC (ascorbic acid) with 30 observations  
    - OJ (orange juice) with 30 observations  
3. dose: Dose of Vitamin C (.5, 1 and 2 miligrams) with 20 observations of each dose 
table(ToothGrowth$dose, ToothGrowth$supp)
##      
##       OJ VC
##   0.5 10 10
##   1   10 10
##   2   10 10

There are 6 groups of 10 observations each depending on the dose and the supplement type.

library(ggplot2)
ggplot(ToothGrowth, aes(x=dose, y=len)) + 
        ggtitle("Length vs Dose per supplement") + 
        xlab("Dose") + ylab("Length") +
        geom_boxplot(aes(fill=factor(dose))) + 
        geom_jitter() +
        facet_grid(.~supp)

Hypotheses

Based on the previous graph, it seems that:
1. The supplement orange juice (OJ) is more effetive than ascorbic acid (VC) for .5 and 1 dose
2. The lenght for dose 2 is independent of the supplement

1.Hypothesis:
- Ho (null hypothesis): for .5 dose, mean(OJ) = mean(VC)
- Ha (alternativve hypothesis): for .5 dose, mean(OJ) != mean(VC)

Null model:

t.test(len ~ supp, paired=FALSE, data=ToothGrowth[ToothGrowth$dose=="0.5",])
## 
##  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

Ho has to be rejected as the p-value (.006) is realy low and t>3!!. This tells me that the type of supplement affects clearly to the length of the tooth.

2.Hypothesis:
- Ho (null hypothesis): for 1 dose, mean(OJ) = mean(VC)
- Ha (alternativve hypothesis): for 1 dose, mean(OJ) != mean(VC)

Null model:

t.test(len ~ supp, paired=FALSE, data=ToothGrowth[ToothGrowth$dose=="1",])
## 
##  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

Ho has to be rejected as the p-value (.001) is realy low and t>4!!. This tells me that the type of supplement affects clearly to the length of the tooth.

3.Hypothesis:
- Ho (null hypothesis): for 2 dose, mean(OJ) = mean(VC)
- Ha (alternativve hypothesis): for 2 dose, mean(OJ) != mean(VC)

Null model:

t.test(len ~ supp, paired=FALSE, data=ToothGrowth[ToothGrowth$dose=="2",])
## 
##  Welch Two Sample t-test
## 
## data:  len by supp
## t = -0.046136, 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

Ho can not be rejected as the 95% confidence interval includes 0 and the p-value is very high. This means that the null hypothesis stays valid. For dose = 2, length is independent of the type of supplement.

Conclusions

  • The higher the dose, the longer the tooth.
  • The supplement orange juice (OJ) is more effetive than ascorbic acid (VC) for .5 and 1 dose. But the effect on the length for dose 2 is independent of the supplement.

Assumptions

  • Dose and type of supplement are independient variables
  • The Guinea pigs where choosen on a randomized way
  • The samples are independent, so we performed a unpaired analysis
  • The samples follow a normal distribution