import pandas as pd import scipy.stats as stats from statsmodels.stats.anova import AnovaRM import statsmodels.api as sm

Data for the first scenario

data1 = { "Treatment A": [13, 14, 14, 15, 14], "Treatment B": [18, 13, 15, 14, 15], "Treatment C": [22, 16, 20, 18, 19], }

Data for the second scenario (adjusted Treatment C)

data2 = { "Treatment A": [13, 14, 14, 15, 14], "Treatment B": [18, 13, 15, 14, 15], "Treatment C": [19, 13, 17, 15, 16], }

Convert data to DataFrame for first scenario

df1 = pd.DataFrame(data1)

Convert data to DataFrame for second scenario

df2 = pd.DataFrame(data2)

Perform ANOVA for first dataset

df1_melted = df1.melt(var_name="Treatment", value_name="Score") model1 = sm.stats.anova_lm(sm.OLS.from_formula('Score ~ C(Treatment)', data=df1_melted).fit(), typ=2)

Perform ANOVA for second dataset

df2_melted = df2.melt(var_name="Treatment", value_name="Score") model2 = sm.stats.anova_lm(sm.OLS.from_formula('Score ~ C(Treatment)', data=df2_melted).fit(), typ=2)

Extract F-ratio and p-value for both datasets

f_ratio1, p_value1 = model1['F'][0], model1['PR(>F)'][0] f_ratio2, p_value2 = model2['F'][0], model2['PR(>F)'][0]

f_ratio1, p_value1, f_ratio2, p_value2(11.66666666666665, 0.0015345456520061101, 1.666666666666662, 0.22975660989883423)(11.66666666666665, 0.0015345456520061101, 1.666666666666662, 0.22975660989883423)success### First Scenario:

• F-ratio: 11.67
• p-value: 0.0015
• Conclusion: There is a significant difference between the treatments (since $p < \alpha = 0.05$).

Second Scenario:

• F-ratio: 1.67
• p-value: 0.2298
• Conclusion: These data do not provide evidence of a difference between the treatments (since $p > \alpha = 0.05$).

Prediction Validated:

By moving the scores in the third treatment closer to the other two, the difference between treatments decreased. This reduced the $F$-ratio and led to a lack of statistical significance in the second scenario.

Would you like further explanation or a breakdown of the analysis? Here are five questions you might explore:

1. How is the $F$-ratio calculated, and what does it represent?
2. Why does moving the means closer reduce the $F$-ratio?
3. What assumptions are required for ANOVA?
4. How do post-hoc tests help in understanding significant differences?
5. How would the results change with unequal sample sizes or variances?

Tip: Always visualize your data (e.g., using boxplots) to better understand the group differences before conducting statistical tests.