Install Required Libraries

# Load required packages ---------------------------
library(ggplot2)
library(knitr)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(tidyr)

Overview

Part 2 of the project involves evaluating the effectiveness of the supplement types on tooth growth length vs dose levels. This part of the project utilizes the Tooth Growth data in R.

Loading and Exploring Data

# Load and inspect ToothGrowth dataset -------------
data("ToothGrowth")
tooth_data <- ToothGrowth

Basic EDA

# Structure and unique doses
str(tooth_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 ...

unique(tooth_data$dose)

## [1] 0.5 1.0 2.0

Converting Dose to a Factor for Analyis and Plotting

# Convert dose to factor for analysis/plotting
tooth_data$dose <- factor(tooth_data$dose)

Updated Structure

# Updated structure
str(tooth_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: Factor w/ 3 levels "0.5","1","2": 1 1 1 1 1 1 1 1 1 1 ...

Analysis

Summary Statistics

# Summary statistics table -------------------------
summary_stats <- summary(tooth_data)
kable(summary_stats, caption = "Summary Statistics for ToothGrowth Dataset")

Summary Statistics for ToothGrowth Dataset
len	supp	dose
Min. : 4.20	OJ:30	0.5:20
1st Qu.:13.07	VC:30	1 :20
Median :19.25	NA	2 :20
Mean :18.81	NA	NA
3rd Qu.:25.27	NA	NA
Max. :33.90	NA	NA

Boxplot of Tooth Length by Supplement and Dose

# Boxplot of tooth length by supplement type and dose ---------------
ggplot(tooth_data, aes(x = dose, y = len, fill = supp)) +
  geom_boxplot() +
  scale_fill_manual(values = c("OJ" = "coral", "VC" = "cornsilk")) +
  labs(
    title = "Tooth Length by Supplement Type and Dose",
    x = "Dose (mg)",
    y = "Tooth Length"
  ) +
  theme_classic()

From the plot, we observe that the 0.5 mg/day and 1.0 mg/day dose have larger differences in tooth length compared to the 2.0 mg/day dose. There is an observable trend that as the dosage increases, so does the tooth length.

Additionally, we can also surmise that for supplement OJ, the smaller doses (0.5mg/day, 1.0mg/day) are more effective than supplement VC, while at 2.0 mg/day, the growth trends are similar.

Grouped Mean and Difference Table

# Grouped mean and difference table ---------------------------------
mean_diff <- tooth_data |>
  group_by(supp, dose) |>
  summarise(mean_len = mean(len), .groups = "drop") |>
  pivot_wider(names_from = supp, values_from = mean_len) |>
  mutate(abs_diff = abs(VC - OJ))

kable(mean_diff, caption = "Mean Tooth Length by Supplement and Dose with Absolute Differences")

Mean Tooth Length by Supplement and Dose with Absolute Differences
dose	OJ	VC	abs_diff
0.5	13.23	7.98	5.25
1	22.70	16.77	5.93
2	26.06	26.14	0.08

t-test Hypothesis Testing

The hypotheses are:

Null hypothesis - no difference between OJ and VC.
Alternative hypothesis - observable difference between OJ and VC.
Alpha is set at a standard 0.05

Although the difference between the two supplements at 2.0 mg/day is small, a t-test will be performed on all dose amounts.

Subsetting by Dose Level

dose_half <- filter(tooth_data, dose == 0.5)
dose_one  <- filter(tooth_data, dose == 1)
dose_two  <- filter(tooth_data, dose == 2)

Independent t-tests

# Independent t-tests -----------------------------------------------
t_half <- t.test(len ~ supp, data = dose_half)
t_one  <- t.test(len ~ supp, data = dose_one)
t_two  <- t.test(len ~ supp, data = dose_two)

Summary of t-test Results

t_test_summary <- data.frame(
  dose      = c(0.5, 1.0, 2.0),
  conf_int  = c("1.72, 8.78", "2.80, 9.06", "-3.80, 3.64"),
  p_value   = c(0.0064, 0.0010, 0.9639),
  decision  = c("Reject null", "Reject null", "Do not reject null")
)

kable(t_test_summary, caption = "T-Test Results by Dose Level")

T-Test Results by Dose Level
dose	conf_int	p_value	decision
0.5	1.72, 8.78	0.0064	Reject null
1.0	2.80, 9.06	0.0010	Reject null
2.0	-3.80, 3.64	0.9639	Do not reject null

As expected, the p-values for the 0.5 mg/day and 1.0 mg/day doses are very small due to the larger mean differences between doses.

Thus:

For 0.5 mg/day and 1.0 mg/day, the null hypothesis is rejected.
For 2.0 mg/day, the null hypothesis is not rejected.

Conclusion

The central assumption for the results is that the sample is representative of the population, and the variables are IID random variables.

For the t.test, two assumptions are made,

The data isn’t paired, meaning they’re independent
The variance values are different.

With that, in reviewing the t.test, supplement type OC are more effective than VC for doses less than 1.0. But for dose at 2.0 mg/day, there is no difference between the supplement types.

Basic Inferential Analysis on Tooth Growth Data

Christian Salafia

2025-03-25