Results

## Calculate F-Fraction
See Keppel, p353

$$
    [A] = \frac{\sum{A^2_j}}{n} = n\sum{\overline{Y}^2_{A}}
$$

$$
    [Y] = \sum{Y^2_{ij}}
$$

$$
    [S] = \frac{\sum{S^2_i}}{a} = a\sum{\overline{Y}^2_S}
$$

$$
    [T] = \frac{T^2}{an} = an\overline{Y}^2_T
$$

    Source   df         SS               MS              F
    ======================================================================
    A        a-1        [A] - [T]        SS_AA / df_A    MS_A / MS_(A x S)
    S        n-1        [S] - [T]        SS_S  / df_S
    A x S    (a-1)(n-1) [Y]-[A]-[S]+[T]  SS_AxS / df_AxS
    ----------------------------------------------------------------------
    Total    an-1       [Y] - [T]

$$
    \text{with  } F_A = \frac{MS_A}{MS_{A \times S}}
$$

n = float(len(df))
a = float(len(df.columns))

A_raw = df.sum().apply(lambda v: v ** 2).sum() / n
S_raw = df.sum(axis=1).apply(lambda v: v ** 2).sum() / a
Y_raw = df.apply(lambda v: v ** 2).sum().sum()
T_raw = df.sum().sum() ** 2 / (a*n)
A = A_raw - T_raw
S = S_raw - T_raw
AxS = Y_raw - A_raw - S_raw + T_raw
Total = Y_raw - T_raw
df_A = a - 1
df_S = n - 1
df_AxS = (a-1)*(n-1)
df_total = (a*n)-1

MS_A = (A / df_A)
MS_AxS = (AxS / df_AxS)
F_emp = MS_A / MS_AxS

from scipy import stats
f_dist = stats.f(df_A, df_AxS, loc=0, scale=1)
F_krit = f_dist.ppf(.95)
p = f_dist.sf(F_emp)
print 'F(%s, %s)= %s, p=%s' % (df_A, df_AxS, F_emp, p)
print F_krit

# Results
    F(2.0, 142.0)= 4.3311275401, p=0.0149332913263
    3.05983068862