library(ggplot2)

Exploring the data

ToothGrowth dataset contains the data about the effect of Vitamin C on Tooth Growth in Guinea Pigs. The response is 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 (OJ) or Ascorbic Acid (VC)).

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 ...
unique(ToothGrowth$dose)
## [1] 0.5 1.0 2.0
as.table(by(ToothGrowth[,1], ToothGrowth[,-1], mean))
##     dose
## supp   0.5     1     2
##   OJ 13.23 22.70 26.06
##   VC  7.98 16.77 26.14

It looks like the bigger dose causes the bigger growth for both suppliments, Orange Juice (OJ) seems to be more effective on lower doses though. Let’s see it on a plot.

qplot(supp, len, data = ToothGrowth, geom = 'boxplot', facets = . ~ dose,
      ylab='Length (microns)', xlab='Dose (mg)', 
      main = 'Effect of supplements on tooth growth')

Confidence intervals and hypothesis tests

Influence of the dose on tooth growth

  • mu0 - the dose doesn’t affect tooth growth
  • mu1 - the bigger dose causes the bigger growth
dose05 <- subset(ToothGrowth, dose == .5)
dose10 <- subset(ToothGrowth, dose == 1)
dose20 <- subset(ToothGrowth, dose == 2)

.5 mg vs 1 mg

t <- t.test(dose05$len, dose10$len)
t$conf.int
## [1] -11.983781  -6.276219
## attr(,"conf.level")
## [1] 0.95
t$p.value
## [1] 1.268301e-07

Confidence interval doesn’t contain zero and p-value is very small, so we can reject mu0.

1 mg vs 2 mg

t <- t.test(dose10$len, dose20$len)
t$conf.int
## [1] -8.996481 -3.733519
## attr(,"conf.level")
## [1] 0.95
t$p.value
## [1] 1.90643e-05

Confidence interval doesn’t contain zero and p-value is very small, so we can reject mu0.

Influence of supplement on tooth growth

  • mu0 - There is no difference of effect of different supplements
  • mu1 - One kind of supplements is more effective
supOJ <- subset(ToothGrowth, supp == 'OJ')
supVC <- subset(ToothGrowth, supp == 'VC')
t <- t.test(supOJ$len, supVC$len)
t$conf.int
## [1] -0.1710156  7.5710156
## attr(,"conf.level")
## [1] 0.95
t$p.value
## [1] 0.06063451

Confidence interval contains zero and p-value is bigger than 5%, so we failed to reject mu0.

Influence of supplement and dose on tooth growth

  • mu0 - There is no difference of effect of different supplements for chosen dose
  • mu1 - One kind of supplements is more effective for chosen dose

Dosage .5 mg

OJ05 <- subset(ToothGrowth, supp == 'OJ' & dose == .5)
VC05 <- subset(ToothGrowth, supp == 'VC' & dose == .5)
t <- t.test(OJ05$len, VC05$len)
t$conf.int
## [1] 1.719057 8.780943
## attr(,"conf.level")
## [1] 0.95
t$p.value
## [1] 0.006358607

Confidence interval doesn’t contain zero and p-value is very small, so we can reject mu0.

Dosage 1 mg

OJ10 <- subset(ToothGrowth, supp == 'OJ' & dose == 1)
VC10 <- subset(ToothGrowth, supp == 'VC' & dose == 1)
t <- t.test(OJ10$len, VC10$len)
t$conf.int
## [1] 2.802148 9.057852
## attr(,"conf.level")
## [1] 0.95
t$p.value
## [1] 0.001038376

Confidence interval doesn’t contain zero and p-value is very small, so we can reject mu0.

Dosage 2 mg

OJ20 <- subset(ToothGrowth, supp == 'OJ' & dose == 2)
VC20 <- subset(ToothGrowth, supp == 'VC' & dose == 2)
t <- t.test(OJ20$len, VC20$len)
t$conf.int
## [1] -3.79807  3.63807
## attr(,"conf.level")
## [1] 0.95
t$p.value
## [1] 0.9638516

Confidence interval contains zero and p-value is bigger than 5%, so we failed to reject mu0.

Conclusions

Based on exploratory analysis and hypothesis tests we proved the positive correlation between dosage and tooth growth. We also proved that Orange Juice (OJ) is more effective for .5 mg and 1 mg dosage than Ascorbic Acid (VC), they are equally effective for 2 mg dosage though.