Analysis of the ToothGrowth data in the R datasets package.

The data are about 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).

```
# Basic settings and libraries
knitr::opts_chunk$set(echo = TRUE, tidy.opts=list(width.cutoff=60), tidy=TRUE)
library(pander)
library(ggplot2)
library(dplyr)
library(datasets) # load the data
```

```
# Show the table
pandoc.table(ToothGrowth)
```

len | supp | dose |
---|---|---|

4.2 | VC | 0.5 |

11.5 | VC | 0.5 |

7.3 | VC | 0.5 |

5.8 | VC | 0.5 |

6.4 | VC | 0.5 |

10 | VC | 0.5 |

11.2 | VC | 0.5 |

11.2 | VC | 0.5 |

5.2 | VC | 0.5 |

7 | VC | 0.5 |

16.5 | VC | 1 |

16.5 | VC | 1 |

15.2 | VC | 1 |

17.3 | VC | 1 |

22.5 | VC | 1 |

17.3 | VC | 1 |

13.6 | VC | 1 |

14.5 | VC | 1 |

18.8 | VC | 1 |

15.5 | VC | 1 |

23.6 | VC | 2 |

18.5 | VC | 2 |

33.9 | VC | 2 |

25.5 | VC | 2 |

26.4 | VC | 2 |

32.5 | VC | 2 |

26.7 | VC | 2 |

21.5 | VC | 2 |

23.3 | VC | 2 |

29.5 | VC | 2 |

15.2 | OJ | 0.5 |

21.5 | OJ | 0.5 |

17.6 | OJ | 0.5 |

9.7 | OJ | 0.5 |

14.5 | OJ | 0.5 |

10 | OJ | 0.5 |

8.2 | OJ | 0.5 |

9.4 | OJ | 0.5 |

16.5 | OJ | 0.5 |

9.7 | OJ | 0.5 |

19.7 | OJ | 1 |

23.3 | OJ | 1 |

23.6 | OJ | 1 |

26.4 | OJ | 1 |

20 | OJ | 1 |

25.2 | OJ | 1 |

25.8 | OJ | 1 |

21.2 | OJ | 1 |

14.5 | OJ | 1 |

27.3 | OJ | 1 |

25.5 | OJ | 2 |

26.4 | OJ | 2 |

22.4 | OJ | 2 |

24.5 | OJ | 2 |

24.8 | OJ | 2 |

30.9 | OJ | 2 |

26.4 | OJ | 2 |

27.3 | OJ | 2 |

29.4 | OJ | 2 |

23 | OJ | 2 |

```
ToothGrowth$dose <- as.factor(ToothGrowth$dose) # dose is factor
str(ToothGrowth) # Table variables
```

```
## '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: Factor w/ 3 levels "0.5","1","2": 1 1 1 1 1 1 1 1 1 1 ...
```

The variables are:

- len: the tooth length

- supp: the supplement type of vitamin C (VC is ascorbic acid, OJ is orange juice)

- dose: dose in milligrams/day of vitamin C

```
# Basic summary
summary(ToothGrowth)
```

```
## 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
```

```
# Means for supplement type and dose
TG <- tbl_df(ToothGrowth)
TG2 <- summarize(group_by(TG, supp, dose), mean_of_len = mean(len))
TG2
```

```
## Source: local data frame [6 x 3]
## Groups: supp [?]
##
## supp dose mean_of_len
## <fctr> <fctr> <dbl>
## 1 OJ 0.5 13.23
## 2 OJ 1 22.70
## 3 OJ 2 26.06
## 4 VC 0.5 7.98
## 5 VC 1 16.77
## 6 VC 2 26.14
```

```
# Basic exploratory graph
qplot(supp, data = TG2, geom = "bar", weight = mean_of_len, facets = . ~
dose, fill = supp, main = "Mean of tooth lenght for supplement type and quantity",
xlab = "", ylab = "Tooth length")
```

The lenght seems correlated with dose quantity and, in part, with supplement type.

We perform a first T-test with significance level of 5%, with this Null Hypotesis: *different supplement type hasn’t effect with tooth growth*.

`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
```

We can’t reject null hypotesis because confidence interval contains zero. **We can’t say that supplement types have significant impact on tooth growth**.

Now we perform three tests with significance level of 5%, with this Null Hypotesis: *different dose hasn’t effect with tooth growth*.

```
# dose = 2 vs dose = 0.5
t.test(ToothGrowth$len[ToothGrowth$dose == 2], ToothGrowth$len[ToothGrowth$dose ==
0.5])
```

```
##
## Welch Two Sample t-test
##
## data: ToothGrowth$len[ToothGrowth$dose == 2] and ToothGrowth$len[ToothGrowth$dose == 0.5]
## 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:
## 12.83383 18.15617
## sample estimates:
## mean of x mean of y
## 26.100 10.605
```

```
# dose = 2 vs dose = 1
t.test(ToothGrowth$len[ToothGrowth$dose == 2], ToothGrowth$len[ToothGrowth$dose ==
1])
```

```
##
## Welch Two Sample t-test
##
## data: ToothGrowth$len[ToothGrowth$dose == 2] and ToothGrowth$len[ToothGrowth$dose == 1]
## 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:
## 3.733519 8.996481
## sample estimates:
## mean of x mean of y
## 26.100 19.735
```

```
# dose = 1 vs dose = 0.5
t.test(ToothGrowth$len[ToothGrowth$dose == 1], ToothGrowth$len[ToothGrowth$dose ==
0.5])
```

```
##
## Welch Two Sample t-test
##
## data: ToothGrowth$len[ToothGrowth$dose == 1] and ToothGrowth$len[ToothGrowth$dose == 0.5]
## 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:
## 6.276219 11.983781
## sample estimates:
## mean of x mean of y
## 19.735 10.605
```

All three tests have confidence intervals that don’t contains 0. This means that we can accept the null hypotesis. **Increasing the dose of vitamin C have impact on tooth growth.**

We assume that the experiment was done with random assignment of guinea pigs and the sample is representative of the entire population. The variances are assumed to be different for the groups being compared.

If all the assumptions are true, we can say that increasing the dose of vitamin C have a significant impact on tooth growth of the pig: a higher dose level of vitamin C correspond to longer teeth.

We also can say that supplement types of vitamin C (orange juice or ascorbic acid) don’t have significant impact on tooth growth.