Statistical inference - Course Project: Part 2
Overview
This project aims to analyze the Tooth Growth database using confidence intervals and/or tests. This dataset has 60 observations and 3 variables, and a summary was provided with a brief of exploratory analysis. As a results of this project, supplement type has no effect on tooth growth and increasing the dose level leads to increased tooth growth.
Requeriments and Settings
Requirements to reproduce this exercise
# Loading libraries
library(ggplot2)
library(dplyr)
library(datasets)
# Force results to be in English
Sys.setlocale("LC_ALL","English")## [1] "LC_COLLATE=English_United States.1252;LC_CTYPE=English_United States.1252;LC_MONETARY=English_United States.1252;LC_NUMERIC=C;LC_TIME=English_United States.1252"Load Data
Make a copy of the original dataset and converting into a dplyr table.
dataset_tg <- ToothGrowth
dataset_tg <- tbl_df(dataset_tg)A brief exploratory analysis of the dataset
How is the data structure:
str(dataset_tg)## Classes 'tbl_df', 'tbl' and '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 ...How is the data loaded:
head(dataset_tg)## Source: local data frame [6 x 3]
##
## len supp dose
## 1 4.2 VC 0.5
## 2 11.5 VC 0.5
## 3 7.3 VC 0.5
## 4 5.8 VC 0.5
## 5 6.4 VC 0.5
## 6 10.0 VC 0.5How many observation and variable are this dataset
dim(dataset_tg)## [1] 60 3A data frame with 60 observations on 3 variables.
How the data are organized:
table(dataset_tg$supp,dataset_tg$dose)##
## 0.5 1 2
## OJ 10 10 10
## VC 10 10 10Note: The experiment counts with two kinds of supplements (VC or JC) and there are 3 dose variation (0.5, 1.0, and 2.0 mg/day).
To synthesize all information we will use two graph.
boxplot(len ~ supp * dose, dataset_tg, col=5, ylab="Tooth Length (mm)", xlab="Supplement type & Dose",main = "Tooth Growth by supplement and Dose") 
ggplot(aes(x=supp, y=len), data=dataset_tg) + geom_boxplot(aes(fill=supp)) + xlab("Supplement type") +ylab("Tooth length (mm)") 
Provide a basic summary of the data
Summary of this dataset:
summary(dataset_tg)## 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. :2.000For futher more information about each variable, please read this document.
Compare tooth growth by supplement and dose
There are more than one comparison of tooth growth by supplement (OJ and VC) and dose. Thus, to turn this study much clearly we divided this section into 3 parts: Comparison between supplements, 1mg and 0.5 mg dose, and 2 and 1 mg dose.
Test 1: Growth Tooth Differences between supplements OJ and VC
We are testing if exist some differences between those supplements. It means, we are looking for a p value greater then 0.05. Thus, we need to assume two hypotesis: Ho equals means and H1 means are differents.
First of all, we need to check the len variance between OJ and VC supplement.
# Subsetting the ToothGrowth to acquire only OJ observations
dataset_tg_OJ <- filter(dataset_tg,supp == "OJ")
# Len Variance when used OJ supplement
var_len_OJ <- var(dataset_tg_OJ$len)
# Subsetting the ToothGrowth to acquire only VC observations
dataset_tg_VC <- filter(dataset_tg,supp == "VC")
# Len Variance when used VC supplement
var_len_VC <- var(dataset_tg_VC$len)Those variance are far different so the var.equal should be set to FALSE (Len Variance using OJ =43.6334368 and Len Variance using VC = 68.3272299). Now, we can use de t.test to compare the supplements performances are the same.
t.test(dataset_tg_OJ$len, dataset_tg_VC$len, paired = FALSE, var.equal = FALSE)##
## Welch Two Sample t-test
##
## data: dataset_tg_OJ$len and dataset_tg_VC$len
## 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 of x mean of y
## 20.66333 16.96333The p-value of the test is 0.06. It means we do not have evidence to reject the null hypothesis. Supplement types seems to have no impact on Tooth growth.
Test 2: Growth Tooth Differences by dosages 2 and 1 mg/day
For this test we define the Ho as the null hypotheses of equal means between the two groups, versus the alternative hypothesis (H1) that the two means are different.
t.test(filter(dataset_tg,dose==2)$len, filter(dataset_tg,dose==1)$len, paired = FALSE, var.equal = TRUE)##
## Two Sample t-test
##
## data: filter(dataset_tg, dose == 2)$len and filter(dataset_tg, dose == 1)$len
## t = 4.9005, df = 38, p-value = 1.811e-05
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 3.735613 8.994387
## sample estimates:
## mean of x mean of y
## 26.100 19.735As results of the t.test we have enough evidence to reject the null hypothesis (Ho). It means if I increase the dosage from 1mg to 2mg creates an positive effect on tooth length.
Test 3: Growth Tooth Differences between supplements OJ and VC
For this test we define the Ho as the null hypotheses of equal means between the two groups, versus the alternative hypothesis (H1) that the two means are different.
t.test(filter(dataset_tg,dose==1)$len, filter(dataset_tg,dose==0.5)$len, paired = FALSE, var.equal = TRUE)##
## Two Sample t-test
##
## data: filter(dataset_tg, dose == 1)$len and filter(dataset_tg, dose == 0.5)$len
## t = 6.4766, df = 38, p-value = 1.266e-07
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 6.276252 11.983748
## sample estimates:
## mean of x mean of y
## 19.735 10.605As results of the t.test we have enough evidence to reject the null hypothesis (Ho). It means if I increase the dosage from 0.5mg to 1mg creates an positive effect on tooth length.
Conclusions
By the Test 1 we can conclude that supplement has no effect on tooth growth.
By the Test 2 and Test 3 we can conclude that increasing the dose level leads to increased tooth growth.
Assumptions
- For t-tests regarding tooth length per dosage level, the variances are assumed to be equal for the three combinations of the two groups being compared.
References and Sources
Source
C. I. Bliss (1952) The Statistics of Bioassay. Academic Press.
References
McNeil, D. R. (1977) Interactive Data Analysis. New York: Wiley.
Crampton, E. W. (1947) The growth of the odontoblast of the incisor teeth as a criterion of vitamin C intake of the guinea pig. The Journal of Nutrition 33(5): 491-504. http://jn.nutrition.org/content/33/5/491.full.pdf
R Documention: https://stat.ethz.ch/R-manual/R-devel/library/datasets/html/ToothGrowth.html