Analysis of Tooth Growth by Supplement and Dosage

Project Overview

In this project we analyze the ToothGrowth data set. The data set gives the results of an experiment to determine the effect of two supplements (Vitamin C and Orange Juice), each at three different doses (0.5, 1 or 2 mg) on tooth length in guinea pigs. The len variable gives the tooth growth, the supp variable gives the supplement type and the dose variable gives the supplement dose. We will perform an analysis to compare tooth growth by supp and dose.

Loading Necessary Libraries

For our analysis the following libraries need to be loaded.

library(datasets)
library(ggplot2)
library(dplyr)
library(knitr)
library(printr)

Looking at Data

We first look at the structure of the ToothGrowth data set:

data <- ToothGrowth
str(data)

## '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 ...

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Its also useful to look at the first few records of the data:

knitr::kable(head(ToothGrowth),align = 'c')

Basic Descriptive Statistics and Exploratory Data Analysis

In this section we provide a basic summary of the data as well as exploratory data analysis to better observe the supplements and dosage effects on tooth growth.

We start off with a basic summary of the three variables in the data set:

summary(data)

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Now we find a summary statistics of tooth growth by supplement:

group_supp <- group_by(data, supp)
summary <- summarise(group_supp, count= n(), mean=mean(len), median=median(len), 
                     "standard deviation" = sd(len))
summaryData <- as.data.frame(summary)
kable (summaryData,digits = 3,align = 'c')

The summary indicates that the average of tooth growth for orange juice is more than vitamin C. The Box plot of tooth length by supplement below also confirms a higher tooth growth for orange juice.

ggplot(data=data, aes(x=supp, y=len))+
        geom_boxplot(aes(fill=supp))+
        xlab("Supplement type") +
        ylab("Tooth length")+ 
        ggtitle(" Boxplot of tooth lenth by supplement type ")

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Now we find a summary statistics of tooth growth by supplement and dosage :

group_supp <- group_by(data, supp, dose)
summary <- summarise(group_supp, count= n(), mean=mean(len), median=median(len), 
                     "standard deviation" = sd(len))
summaryData <- as.data.frame(summary)
kable (summaryData,digits = 3,align = 'c')

According to this result, the average tooth growth in both OJ and VC categories increase as dosage increases. Also the average of growth in orange juice category is higher for all dosage comparing to the vitamin C category. The Box plot below of the tooth length by supplement type and dosage also confirm this statement.

ggplot(data=data, aes(x=factor(dose), y=len)) +
        geom_boxplot(aes(fill=factor(dose)))+
        facet_grid(.~supp) +
        xlab("Dose") +
        ylab("Tooth length")+
        ggtitle(" Boxplot of tooth lenth by supplement type and does")

Hypothesis Test for Tooth Growth Comparison

As mentioned in the project overview, the goal of this analysis is to analyze the effect of two supplements (Vitamin C and Orange Juice), each at three different doses (0.5, 1 or 2 mg) on tooth length in guinea pigs. In the previous section we speculated that the overall effect of orange juice (OJ) on tooth growth is more than vitamin C (VC). In this section we try to verify this statement by performing several hypothesis test.

test <- t.test(len ~ supp, data= data, var.equal = FALSE, paired=FALSE ,conf.level = .95)

result1 <- data.frame( "t-statistic"  = test$statistic, 
                       "df" = test$parameter,
                        "p-value"  = test$p.value,
                        "lower CL" = test$conf.int[1],
                        "upper CL" = test$conf.int[2],
                        "OJ mean" = test$estimate[1],
                        "VC mean" = test$estimate[2],
                         row.names = "OJ vs VC ")

kable(x = round(result1,3),align = 'c' ,
      caption = "Summary of two sample t-test for tooth growth by supplement")

data_does0.5 <- filter(data, dose==0.5) 
data_does1 <- filter(data, dose==1) 
data_does2 <- filter(data, dose==2) 

test0.5 <- t.test(len ~ supp, data= data_does0.5, var.equal = FALSE, paired=FALSE ,conf.level = .95)
test1 <- t.test(len ~ supp, data= data_does1, var.equal = FALSE, paired=FALSE ,conf.level = .95)
test2 <- t.test(len ~ supp, data= data_does2, var.equal = FALSE, paired=FALSE ,conf.level = .95)


result2 <- data.frame( "t-statistic"  = c(test0.5$statistic,test1$statistic,test2$statistic), 
                       "df" = c(test0.5$parameter,test1$parameter,test2$parameter),
                        "p-value"  = c(test0.5$p.value,test1$p.value,test2$p.value),
                        "lower CL" = c(test0.5$conf.int[1],test1$conf.int[1],test2$conf.int[1]),
                        "upper CL" = c(test0.5 $conf.int[2],test1$conf.int[2],test2$conf.int[2]),
                        "OJ mean" = c(test0.5 $estimate[1],test1 $estimate[1],test2 $estimate[1]),
                        "VC mean" = c(test0.5 $estimate[2],test1 $estimate[2],test2 $estimate[2]),
                         row.names = c("OJ vs VC at dose = 0.5","OJ vs VC at dose = 1","OJ vs VC at dose = 2" ))

kable(round (x = result2, 3),align = 'c', 
      caption = "Summary of two sample t-test for tooth growth by supplement and dosage")

Final Conclusion

Based on the analysis performed the previous section, we can conclude that low levels of dosage (0.5 & 1.0) of orange juice are effective in tooth growth comparing to vitamin C. However the result from higher dosage (2.0) is uncertain whether there will be a greater effect from either OJ or VC.