Executive Summary

This inferential analysis explores the ToothGrowth data set using the following steps:

  1. Basic exploratory data analysis
  2. Summarize the data
  3. Examine the effect of supp and dose on tooth growth using confidence intervals and hypothesis tests
  4. State conclusions and assumptions

Step 1: Exploratory data analysis

library(ggplot2)

data(ToothGrowth)

# Examine data set stucture
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 ...
# Determine number of observations
dim(ToothGrowth)
[1] 60  3
# Coerce dose variable to doses
ToothGrowth$dose <- as.factor(ToothGrowth$dose)

# Examine data structure after conversion
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: Factor w/ 3 levels "0.5","1","2": 1 1 1 1 1 1 1 1 1 1 ...

Step 2: Summarize the data

# Summary statistics for all variables
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                   
# Table of dose levels and delivery methods
table(ToothGrowth$dose, ToothGrowth$supp)
     
      OJ VC
  0.5 10 10
  1   10 10
  2   10 10
# Exploratory plots
ggplot(ToothGrowth,aes(x = dose, y = len))+ geom_boxplot(aes(fill = dose))

ggplot(ToothGrowth,aes(x = supp, y = len)) + geom_boxplot(aes(fill = supp))

Step 3: Examine the effect of supp and dose on tooth growth using confidence intervals and hypothesis tests

A T-test was performed to check for group differences explained by supplement type.

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

As shown above, the p value is 0.06 and the confidence interval contains 0. Thus, we can not reject the null hypothesis that different supplement types have no effect on tooth length.

T-tests were then performed to determine if group differences are due to dose levels.

# Subgroup does by their level pairs
ToothGrowth_Dose_0.5_1.0<-subset(ToothGrowth, dose %in% c(0.5, 1.0))
ToothGrowth_Dose_0.5_2.0<-subset(ToothGrowth, dose %in% c(0.5, 2.0))
ToothGrowth_Dose_1.0_2.0<-subset(ToothGrowth, dose %in% c(1.0, 2.0))

# Group Differences due to 0.5 to 1.0 dose levels
t.test(len ~ dose, data = ToothGrowth_Dose_0.5_1.0)

    Welch Two Sample t-test

data:  len by dose
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:
 -11.983781  -6.276219
sample estimates:
mean in group 0.5   mean in group 1 
           10.605            19.735 
# Group Differences due to 0.5 to 2.0 dose levels
t.test(len ~ dose, data = ToothGrowth_Dose_0.5_2.0)

    Welch Two Sample t-test

data:  len by dose
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:
 -18.15617 -12.83383
sample estimates:
mean in group 0.5   mean in group 2 
           10.605            26.100 
# Group Differences due to 1.0 to 2.0 dose levels
t.test(len ~ dose, data = ToothGrowth_Dose_1.0_2.0)

    Welch Two Sample t-test

data:  len by dose
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:
 -8.996481 -3.733519
sample estimates:
mean in group 1 mean in group 2 
         19.735          26.100

For all three tests, the p value is less than 0.05 and the confidence intervals does not contain zero. Thus, we can reject the null hypothesis and conclude that increasing dose levels does in fact lead to an increase in tooth length.

Step 4: State conclusions and assumptions

Based on my analysis above my conclusions are as follows:
1. Supplement type does not effect tooth growth
2. Increasing dose levels does lead to increased tooth growth.

This conclusion is based on the following assumptions:
- The experiment was conducted using random assignment to control for confounding factors
- The sample population is representative of the entire population
- Variance for the t-tests are different for the two groups being compared