Overview

This reseach gives basic undestanding of Vitamin C effect on teeth growth for guinea pigs based on data collected in the Tooth Growth dataset from the standard R dataset package.

Set up for the graphical library in R

options("scipen"=10)
if (!require(ggplot2)) {
  install.packages("ggplot2")
  library(ggplot2)
}
if (!require(datasets)) {
  install.packages("datasets")
  library(datasets)
}

Dataset Overview

The structure of Tooth Growth dataset is:

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 ...

The dataset contains data for a length of odontoblasts (teeth) for each of 10 guinea pigs broken down by three dose levels of Vitamin C (0.5, 1, and 2 mg) using two feeding approaches: green juice or ascorbic acid.

There are 60 observations with 3 variables.

  1. [,1] len numeric: Tooth length
  2. [,2] supp factor: Supplement type (VC or OJ).
  3. [,3] dose numeric: Dose in milligrams.

Data graphical overview

g <- ggplot(aes(x=dose, y = len), data = ToothGrowth)  
g <- g + geom_point(aes(color = supp))
g <- g + theme_bw()
g <- g + labs(x = "Dose", y="Teeth length", title="Guinea teeth length dependecy from Vitamin C ")
g

The shape of distribution density for both types and doses:

g <- ggplot(ToothGrowth, aes(x = len))  
g <- g + geom_histogram(aes(y=..density..), colour = "black", fill= "green")
g <- g + geom_density()
g <- g + facet_grid(supp ~ dose)
g <- g + theme_bw()
g <- g + labs(x = "Teeth length", y="Density", title="Teeth length distribution by suppliment type and dose")
g

There are some dependecies visible on the graph:

g <- ggplot(ToothGrowth, aes(x = supp, y = len))
g <- g + geom_boxplot(aes(fill = supp)) + facet_wrap(~ dose)
g <- g + theme_bw()
g <- g + labs(x = "Dose", y="Teeth length", title="Teeth length distribution by suppliment type and dose")
g

Data Summary

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

Each combination of the dose and type of vitamin C should have the same number of samples:

with(ToothGrowth, table(supp, dose))
##     dose
## supp 0.5  1  2
##   OJ  10 10 10
##   VC  10 10 10

Average values of the dose and type of vitamin C are the following:

tapply(ToothGrowth$len, ToothGrowth$dose, mean)
##    0.5      1      2 
## 10.605 19.735 26.100
tapply(ToothGrowth$len, ToothGrowth$supp, mean)
##       OJ       VC 
## 20.66333 16.96333

Hypothesis

The graphes and summary data analysis could be used to suggest the following hypothesis:

  1. Vitamin C type does not have a significant effect on teeth length.
  2. Vitamin C dose increases teeth length for guinea pigs.

Hypothesis analysis

Assumptions

  1. The variation is different for different cross-classifyng factors in the tests.
  2. Populations for all cross-classifyng factors are different, in other words there were 60 guinea pigs in the research and each pig was fed with its own dose and Vitamin C type.
  3. Statistical regulations have been abided: each subject has almost same weight, initial teeth length and etc.
  4. Sample lengths are t-distributed. This assumption is based on small number of degree of freedom.

Tests

Hypothehsis 1

Null hypothesis: Vitamin C type does not have a significant effect on teeth length

t.test(len ~ supp, paired = F, var.equal = F, 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

95% confidence interval includes 0. Even though there are differerent means of differenet Vitamin C type methods on 95% confidence level it does not provide absolute confidence that true difference is not equal to zero. The overall interval is located in area with positive numbers which indicates possible hidden dependecies requiring additional data to be confirmed.

Result: Hypothesis could be true under some conditions.

Hypothehsis 2

Null hypothesis: VVitamin C dose increases teeth length for guinea pigs

Reseach the effect of dose for each pair: 0.5 - 1 mg, 1 - 2 mg and 0.5 - 2 mg:

t.test(len ~ dose, paired = F, var.equal = F, data = ToothGrowth[ToothGrowth$dose %in% c(1.0,0.5),])
## 
##  Welch Two Sample t-test
## 
## data:  len by dose
## t = -6.4766, df = 37.986, p-value = 0.0000001268
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -11.983781  -6.276219
## sample estimates:
## mean in group 0.5   mean in group 1 
##            10.605            19.735

The means difference does not include 0. Thus,increasing the dose from 0.5 to 1.0 leads to the changes in teeth length.

t.test(len ~ dose, paired = F, var.equal = F, data = ToothGrowth[ToothGrowth$dose %in% c(2.0,1.0),])
## 
##  Welch Two Sample t-test
## 
## data:  len by dose
## t = -4.9005, df = 37.101, p-value = 0.00001906
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -8.996481 -3.733519
## sample estimates:
## mean in group 1 mean in group 2 
##          19.735          26.100

The means difference does not include 0. Thus,increasing the dose from 1.0 to 2.0 leads to the changes in teeth length.

t.test(len ~ dose, paired = F, var.equal = F, data = ToothGrowth[ToothGrowth$dose %in% c(2.0,0.5),])
## 
##  Welch Two Sample t-test
## 
## data:  len by dose
## t = -11.799, df = 36.883, p-value = 0.00000000000004398
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -18.15617 -12.83383
## sample estimates:
## mean in group 0.5   mean in group 2 
##            10.605            26.100

The means difference does not include 0. Thus,increasing the dose from 0.5 to 2.0 leads to the changes in teeth length.

Result: More high dose of Vitamin C leads in extensive teeth growth. Hypothesis confirmed.

Conclusion

Provided data in the dataset introduces some links between guinea pigs teeth growth and Vitamin C consumption. During the work with the dataset, two hypothesis were made.

First, Vitamin C type does not have a significant effect on teeth length. Second, Vitamin C dose increases teeth length for guinea pigs. Both hypothesis were confirmed.