Examples diet data part I

Comparing Diets A and B

Getting the data

In this exercise, we will replicate the diet data analysis example presented in the lectures slides, and we will compare diets A and B. Our first step is to load the data.

# Import data
import pandas as pd
diet = pd.read_csv("https://raw.githubusercontent.com/ELSTE-Master/Data-Science/main/Data/diet.csv")

We can now take a look at the data.

# Import data
n = 5
print(diet.head(n)) #print the first n rows of the dataset

   id  gender  age  height diet.type  initial.weight  final.weight
0   1  Female   22     159         A              58          54.2
1   2  Female   46     192         A              60          54.0
2   3  Female   55     170         A              64          63.3
3   4  Female   33     171         A              64          61.1
4   5  Female   50     170         A              65          62.2

To compare the effectiveness of the diets, we first compute the weight loss of the participants (i.e. initial weight - final weight), which can be done in Python as follows:

# Compute weight loss
diet["weight.loss"] = diet["initial.weight"] - diet["final.weight"]

For this example, we consider diets A and B so we construct our “variables of interest”, say dietA and dietB, by only selecting these diets. This can be done in Python as follows:

# Variable of interest
dietA = diet["weight.loss"][diet["diet.type"]=="A"]
dietB = diet["weight.loss"][diet["diet.type"]=="B"]

import matplotlib.pyplot as plt
import numpy as np

plt.figure(figsize=(6,4))

# Boxplot without outliers
plt.boxplot([dietA, dietB],
            tick_labels=["Diet A", "Diet B"],
            patch_artist=True,
            boxprops=dict(facecolor="lightblue", color="blue"),
            medianprops=dict(color="red"),
            showfliers=False) # don't show outliers in boxplot

{'whiskers': [<matplotlib.lines.Line2D object at 0x7aee0b7cb490>, <matplotlib.lines.Line2D object at 0x7aee0b7cb790>, <matplotlib.lines.Line2D object at 0x7aee0b61c640>, <matplotlib.lines.Line2D object at 0x7aee0b61c940>], 'caps': [<matplotlib.lines.Line2D object at 0x7aee0b7cba90>, <matplotlib.lines.Line2D object at 0x7aee0b7cbd90>, <matplotlib.lines.Line2D object at 0x7aee0b61cc40>, <matplotlib.lines.Line2D object at 0x7aee0b61cf40>], 'boxes': [<matplotlib.patches.PathPatch object at 0x7aee0b7cb040>, <matplotlib.patches.PathPatch object at 0x7aee0b61c070>], 'medians': [<matplotlib.lines.Line2D object at 0x7aee0b61c0d0>, <matplotlib.lines.Line2D object at 0x7aee0b61d240>], 'fliers': [], 'means': []}

# Overlay all points with horizontal jitter
for i, data in enumerate([dietA, dietB], start=1):
    x = np.random.normal(i, 0.09, size=len(data))  # jitter for visibility
    plt.scatter(x, data, alpha=0.7, color='orange', edgecolor='k', zorder=10)

plt.ylabel("Weight loss (kg)")
plt.title("Comparison of Weight Loss for Diets A and B")
plt.show()

Based on the previous boxplot, Welch’s t-test or Wilcoxon rank sum test are both reasonable choices. For this example we will use the Welch’s t-test. To compare the effectiveness of diets A and B we start by defining the hypotheses:

\[H_{0}: \mu_A=\mu_B \quad \text{and} \quad H_{a}: \mu_A\neq\mu_B,\]

where \(\mu_A\) and \(\mu_B\) denote the mean weight loss for diets A and B, respectively. We consider \(\alpha = 0.05\) and compute the p-value as follows:

from scipy import stats

# Welch's t-test (equal_var=False makes it Welch)
t_stat, p_value = stats.ttest_ind(dietA, dietB, equal_var=False)

print("t-statistic:", round(t_stat, 4))

t-statistic: 0.0476

print("p-value:", round(p_value, 4))

p-value: 0.9622

Since the p-value is equal to \(96.22\%\) which is larger than \(5\%\), we fail to reject the null hypothesis at the \(95\%\) confidence level.