Step-by-Step 1-Way ANOVA from scratch notebook

1. Data Creation

In [0]:
import pandas as pd
In [0]:
A = [12.6, 12, 11.8, 11.9, 13, 12.5, 14]
B = [10, 10.2, 10, 12, 14, 13]
C = [10.1, 13, 13.4, 12.9, 8.9, 10.7, 13.6, 12]
In [0]:
all_scores = A + B + C
company_names = (['A'] * len(A)) +  (['B'] * len(B)) +  (['C'] * len(C))
In [0]:
data = pd.DataFrame({'company': company_names, 'score': all_scores})
In [25]:
data
Out[25]:
company score
0 A 12.6
1 A 12.0
2 A 11.8
3 A 11.9
4 A 13.0
5 A 12.5
6 A 14.0
7 B 10.0
8 B 10.2
9 B 10.0
10 B 12.0
11 B 14.0
12 B 13.0
13 C 10.1
14 C 13.0
15 C 13.4
16 C 12.9
17 C 8.9
18 C 10.7
19 C 13.6
20 C 12.0

2. Descriptives

In [26]:
data.groupby('company').mean()
Out[26]:
score
company
A 12.542857
B 11.533333
C 11.825000

2. A 1-Way ANOVA Using StatsModels

In [0]:
import statsmodels.api as sm
from statsmodels.formula.api import ols
In [28]:
lm = ols('score ~ company',data=data).fit()
table = sm.stats.anova_lm(lm)
print(table)

            df     sum_sq   mean_sq         F    PR(>F)
company    2.0   3.606905  1.803452  0.821297  0.455683
Residual  18.0  39.525476  2.195860       NaN       NaN

3. 1-Way ANOVA by hand (from scratch)

In [29]:
# compute overall mean
overall_mean = data['score'].mean()
overall_mean
Out[29]:
11.980952380952381
In [30]:
# compute Sum of Squares Total
data['overall_mean'] = overall_mean
ss_total = sum((data['score'] - data['overall_mean'])**2)
ss_total
Out[30]:
43.132380952380956
In [31]:
# compute group means
group_means = data.groupby('company').mean()
group_means = group_means.rename(columns = {'score': 'group_mean'})
group_means
Out[31]:
group_mean overall_mean
company
A 12.542857 11.980952
B 11.533333 11.980952
C 11.825000 11.980952
In [0]:
# add group means and overall mean to the original data frame
data = data.merge(group_means, left_on = 'company', right_index = True)
In [33]:
# compute Sum of Squares Residual
ss_residual = sum((data['score'] - data['group_mean'])**2)
ss_residual
Out[33]:
39.52547619047619
In [14]:
# compute Sum of Squares Model
ss_explained = sum((data['overall_mean'] - data['group_mean'])**2)
ss_explained
Out[14]:
3.6069047619047776
In [14]:
# compute Mean Square Residual
n_groups = len(set(data['company']))
n_obs = data.shape[0]
df_residual = n_obs - n_groups
ms_residual = ss_residual / df_residual
ms_residual
Out[14]:
2.1958597883597886
In [15]:
# compute Mean Square Explained
df_explained = n_groups - 1
ms_explained = ss_explained / df_explained
ms_explained
Out[15]:
1.8034523809523888
In [16]:
# compute F-Value
f = ms_explained / ms_residual
f
Out[16]:
0.8212966923081592
In [17]:
# compute p-value
import scipy.stats
p_value = 1 - scipy.stats.f.cdf(f, df_explained, df_residual)
p_value
Out[17]:
0.4556832940515221