Analysis of the ToothGrowth Dataset

Introduction

This report analyses the ToothGrowth dataset from the R datasets package, using exploratory summaries, confidence intervals, and hypothesis tests. The aim is to determine whether tooth growth in guinea pigs differs by: - Supplement type (VC vs OJ) - Dose level (0.5, 1.0, 2.0 mg/day)

Only methods taught in class—primarily two‑sample t‑tests and confidence intervals—are used.

Load and Explore the Data

data(ToothGrowth)
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 ...
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
# Convert dose to a factor for analysis
ToothGrowth$dose <- factor(ToothGrowth$dose)

Basic Summaries

aggregate(len ~ supp, ToothGrowth, mean)
##   supp      len
## 1   OJ 20.66333
## 2   VC 16.96333
aggregate(len ~ dose, ToothGrowth, mean)
##   dose    len
## 1  0.5 10.605
## 2    1 19.735
## 3    2 26.100
aggregate(len ~ supp + dose, ToothGrowth, mean)
##   supp dose   len
## 1   OJ  0.5 13.23
## 2   VC  0.5  7.98
## 3   OJ    1 22.70
## 4   VC    1 16.77
## 5   OJ    2 26.06
## 6   VC    2 26.14

Exploratory Plots

boxplot(len ~ supp, data = ToothGrowth,
        main = "Tooth Length by Supplement",
        xlab = "Supplement Type", ylab = "Tooth Length")

boxplot(len ~ dose, data = ToothGrowth,
        main = "Tooth Length by Dose",
        xlab = "Dose (mg/day)", ylab = "Tooth Length")

boxplot(len ~ supp:dose, data = ToothGrowth, las = 2,
        main = "Tooth Length by Supplement and Dose",
        xlab = "Supplement : Dose", ylab = "Tooth Length")

Comparison by Supplement Type

We compare tooth length between OJ and VC using a two‑sample Welch t‑test.

t.test(len ~ supp, data = ToothGrowth, var.equal = FALSE)
## 
##  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 between group OJ and group VC 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

Interpretation • The 95% confidence interval for the difference in means (OJ − VC) is positive. • The p‑value is below 0.05.

Conclusion There is statistically significant evidence that OJ produces greater tooth growth than VC, averaged across all doses.

Comparison by dose level

Pairwise comparisons are performed using Welch t‑tests.

0.5mg/day vs 1.0mg/day

df <- subset(ToothGrowth, dose %in% c(0.5, 1.0))
df$dose <- as.numeric(as.character(df$dose))

t.test(len ~ dose, data = df, var.equal = FALSE)
## 
##  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 between group 0.5 and group 1 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

1.0mg vs 2.0mg

df <- subset(ToothGrowth, dose %in% c(1.0, 2.0))
df$dose <- as.numeric(as.character(df$dose))

t.test(len ~ dose, data = df, var.equal = FALSE)
## 
##  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 between group 1 and group 2 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

Interpretation Across all comparisons: • The 95% confidence intervals do not include zero. • The p‑values are very small (typically < 0.001). Conclusion There is strong evidence that higher doses lead to greater tooth growth, with a clear monotonic pattern: 0.5 mg < 1.0 mg < 2.0 mg

Combined Interpretation (Supplement × Dose)

Exploring the mean structure: • At 0.5 mg and 1.0 mg, OJ produces greater tooth growth than VC. • At 2.0 mg, the difference between supplements is small. This suggests that OJ is more effective at lower doses, while at high doses both supplements perform similarly.

Assumptions

The conclusions rely on the following assumptions: 1. Independence Each guinea pig’s measurement is independent. 2. Approximate normality Tooth length within each group is assumed to be approximately normally distributed. (The t‑test is robust with n = 10 per group.) 3. Unequal variances allowed Welch’s t‑test does not assume equal variances. 4. Random assignment Supplement type and dose are treated as if randomly assigned. 5. Validity of confidence intervals The sampling distribution of the mean difference is approximately normal.

Final Summary

  • Tooth length increases substantially with dose.
  • OJ produces greater tooth growth than VC overall, especially at lower doses.
  • All dose comparisons show statistically significant differences.
  • The results are consistent with a monotonic dose–response relationship.

This analysis uses only the statistical inference tools taught in class and provides clear evidence of both supplement and dose effects on tooth growth.