def test_osrt(a, val_col: str = None, group_col: str = None, sort: bool = False):
'''Hayter's one-sided studentised range test (OSRT) against an ordered
alternative for normal data with equal variances [1]_.
Parameters
----------
a : array_like or pandas DataFrame object
An array, any object exposing the array interface or a pandas
DataFrame.
val_col : str, optional
Name of a DataFrame column that contains dependent variable values (test
or response variable). Values should have a non-nominal scale. Must be
specified if `a` is a pandas DataFrame object.
group_col : str, optional
Name of a DataFrame column that contains independent variable values
(grouping or predictor variable). Values should have a nominal scale
(categorical). Must be specified if `a` is a pandas DataFrame object.
sort : bool, optional
If True, sort data by block and group columns.
Returns
-------
P value, statistic.
Notes
-----
P values are computed from the Tukey distribution.
References
----------
.. [1] Hayter, A.J.(1990) A One-Sided Studentised Range Test for Testing
Against a Simple Ordered Alternative, Journal of the American
Statistical Association, 85, 778-785.
Examples
--------
>>> import scikit_posthocs as sp
>>> import pandas as pd
>>> x = pd.DataFrame({"a": [1,2,3,5,1], "b": [12,31,54,62,12], "c": [10,12,6,74,11]})
>>> x = x.melt(var_name='groups', value_name='values')
>>> sp.test_osrt(x, val_col='values', group_col='groups')
'''
x, _val_col, _group_col = __convert_to_df(a, val_col, group_col)
if not sort:
x[_group_col] = Categorical(x[_group_col], categories=x[_group_col].unique(), ordered=True)
x.sort_values(by=[_group_col], ascending=True, inplace=True)
groups = np.unique(x[_group_col])
x_grouped = x.groupby(_group_col)[_val_col]
xi = x_grouped.mean()
ni = x_grouped.count()
k = groups.size
n = len(x.index)
df = n - k
sigma2 = 0
c = -1
for i in range(k):
for j in range(ni.iloc[i]):
c += 1
sigma2 += (x[_val_col].iat[c] - xi[i])**2. / df
sigma = np.sqrt(sigma2)
def compare(i, j):
dif = xi.loc[groups[j]] - xi.loc[groups[i]]
A = sigma / np.sqrt(2.) * np.sqrt(1. / ni[groups[j]] + 1. / ni[groups[i]])
qval = np.abs(dif) / A
return qval
vs = np.zeros((k, k), dtype=float)
combs = it.combinations(range(k), 2)
for i, j in combs:
vs[i, j] = compare(i, j)
stat = np.max(vs)
pval = psturng(stat, k, df)
return pval, stat
def test_mackwolfe(a: Union[List, np.ndarray, DataFrame], val_col: str = None,
group_col: str = None, p: int = None, n_perm: int = 100,
sort: bool = False) -> Tuple[float, float]:
'''Mack-Wolfe Test for Umbrella Alternatives.
In dose-finding studies one may assume an increasing treatment effect with
increasing dose level. However, the test subject may actually succumb to
toxic effects at high doses, which leads to decresing treatment
effects [1]_, [2]_.
The scope of the Mack-Wolfe Test is to test for umbrella alternatives for
either a known or unknown point P (i.e. dose-level), where the peak
(umbrella point) is present.
Parameters
----------
a : array_like or pandas DataFrame object
An array, any object exposing the array interface or a pandas
DataFrame.
val_col : str, optional
Name of a DataFrame column that contains dependent variable values
(test or response variable). Values should have a non-nominal scale.
Must be specified if `a` is a pandas DataFrame object.
group_col : str, optional
Name of a DataFrame column that contains independent variable values
(grouping or predictor variable). Values should have a nominal scale
(categorical). Must be specified if `a` is a pandas DataFrame object.
p : int, optional
The a-priori known peak as an ordinal number of the treatment group
including the zero dose level, i.e. p = {0, ..., k-1}.
Defaults to None.
sort : bool, optional
If True, sort data by block and group columns.
Returns
-------
p : float
P value.
stat : float
Statistic.
References
----------
.. [1] Chen, I.Y. (1991) Notes on the Mack-Wolfe and Chen-Wolfe Tests for
Umbrella Alternatives. Biom. J., 33, 281-290.
.. [2] Mack, G.A., Wolfe, D. A. (1981) K-sample rank tests for umbrella
alternatives. J. Amer. Statist. Assoc., 76, 175-181.
Examples
--------
>>> x = [[22, 23, 35], [60, 59, 54], [98, 78, 50], [60, 82, 59], [22, 44, 33], [23, 21, 25]]
>>> sp.posthoc_mackwolfe(x)
'''
x, _val_col, _group_col = __convert_to_df(a, val_col, group_col)
if not sort:
x[_group_col] = Categorical(x[_group_col], categories=x[_group_col].unique(), ordered=True)
x.sort_values(by=[_group_col], ascending=True, inplace=True)
k = x[_group_col].unique().size
if p and p > k:
print("Selected 'p' > number of groups:", str(p), " > ", str(k))
return False
elif p is not None and p < 1:
print("Selected 'p' < 1: ", str(p))
return False
Rij = x[_val_col].rank()
n = x.groupby(_group_col)[_val_col].count()
def _fn(Ri, Rj):
return np.sum(Ri.apply(lambda x: Rj[Rj > x].size))
def _ustat(Rij, g, k):
levels = np.unique(g)
U = np.identity(k)
for i in range(k):
for j in range(i):
U[i, j] = _fn(Rij[x[_group_col] == levels[i]], Rij[x[_group_col] == levels[j]])
U[j, i] = _fn(Rij[x[_group_col] == levels[j]], Rij[x[_group_col] == levels[i]])
return U
def _ap(p, U):
tmp1 = 0.
if p > 0:
for i in range(p):
for j in range(i+1, p+1):
tmp1 += U[i, j]
tmp2 = 0.
if p < k:
for i in range(p, k):
for j in range(i+1, k):
tmp2 += U[j, i]
return tmp1 + tmp2
def _n1(p, n):
return np.sum(n[:p+1])
def _n2(p, n):
return np.sum(n[p:k])
def _mean_at(p, n):
N1 = _n1(p, n)
N2 = _n2(p, n)
return (N1**2. + N2**2. - np.sum(n**2.) - n.iloc[p]**2.)/4.
def _var_at(p, n):
N1 = _n1(p, n)
N2 = _n2(p, n)
N = np.sum(n)
var = (2. * (N1**3 + N2**3) + 3. * (N1**2 + N2**2) -
np.sum(n**2 * (2*n + 3.)) - n.iloc[p]**2. * (2. * n.iloc[p] + 3.) +
12. * n.iloc[p] * N1 * N2 - 12. * n.iloc[p] ** 2. * N) / 72.
return var
if p:
# if (x.groupby(_val_col).count() > 1).any().any():
# print("Ties are present")
U = _ustat(Rij, x[_group_col], k)
est = _ap(p, U)
mean = _mean_at(p, n)
sd = np.sqrt(_var_at(p, n))
stat = (est - mean)/sd
p_value = ss.norm.sf(stat)
else:
U = _ustat(Rij, x[_group_col], k)
Ap = np.array([_ap(i, U) for i in range(k)]).ravel()
mean = np.array([_mean_at(i, n) for i in range(k)]).ravel()
var = np.array([_var_at(i, n) for i in range(k)]).ravel()
A = (Ap - mean) / np.sqrt(var)
stat = np.max(A)
mt = []
for _ in range(n_perm):
ix = Series(np.random.permutation(Rij))
uix = _ustat(ix, x[_group_col], k)
apix = np.array([_ap(i, uix) for i in range(k)])
astarix = (apix - mean) / np.sqrt(var)
mt.append(np.max(astarix))
mt = np.array(mt)
p_value = mt[mt > stat] / n_perm
return p_value, stat