Turkey HSD检验法/W法

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python 2.7

# -*- coding: utf-8 -*-
from statsmodels.stats.multicomp import (pairwise_tukeyhsd,
                                         MultiComparison)

# Import standard packages
import numpy as np
from scipy import stats
import pandas as pd
import variance_check

#数据excel名
excel="sample.xlsx"
#读取数据
df=pd.read_excel(excel)
#获取第一组数据，结构为列表
group_mentaln=list(df.StressReduction[(df.Treatment=="mental")])
group_physical=list(df.StressReduction[(df.Treatment=="physical")])
group_medical=list(df.StressReduction[(df.Treatment=="medical")])
list_groups=[group_mentaln,group_physical,group_medical]
list_total=group_mentaln+group_physical+group_medical

print"equal test-----------------------------------------------------"
# #比较组内的样本是否相等，如果不相等，不适合于tukey等方法
equal_lenth=variance_check.Equal_lenth(list_groups)
if equal_lenth==False:
    print("the length of groups are not equal")                               

multiComp = MultiComparison(df['StressReduction'], df['Treatment'])
tukey=multiComp.tukeyhsd()
summary=multiComp.tukeyhsd().summary()
print(summary) 

q=tukey.q_crit
print("q values:",q)
'''
q值
Out[41]: 3.5057698487864877
'''

'''
Multiple Comparison of Means - Tukey HSD,FWER=0.05
===============================================
 group1  group2  meandiff  lower  upper  reject
-----------------------------------------------
medical  mental    1.5     0.3217 2.6783  True
medical physical   1.0    -0.1783 2.1783 False
 mental physical   -0.5   -1.6783 0.6783 False
-----------------------------------------------
'''

print("data details:",summary.data)
'''
[['group1', 'group2', 'meandiff', 'lower', 'upper', 'reject'],
[u'medical', u'mental', 1.5, 0.32169999999999999, 2.6783000000000001, True],
[u'medical', u'physical', 1.0, -0.17829999999999999, 2.1783000000000001, False],
[u'mental', u'physical', -0.5, -1.6782999999999999, 0.67830000000000001, False]]
'''                    

variance_check.py

# -*- coding: utf-8 -*-
'''
用于方差齐性检验
正太性检验
配对相等检验
'''
import scipy,math
from scipy.stats import f
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats
# additional packages
from statsmodels.stats.diagnostic import lillifors
#多重比较
from statsmodels.sandbox.stats.multicomp import multipletests
#用于排列组合
import itertools
'''
#测试数据
group1=[2,3,7,2,6]
group2=[10,8,7,5,10]
group3=[10,13,14,13,15]
list_groups=[group1,group2,group3]
list_total=group1+group2+group3
'''
a=0.05

#正态分布测试
def check_normality(testData):

    #20<样本数<50用normal test算法检验正态分布性
    if 20<len(testData) <50:
       p_value= stats.normaltest(testData)[1]
       if p_value<0.05:
           print"use normaltest"
           print "data are not normal distributed"
           return  False
       else:
           print"use normaltest"
           print "data are normal distributed"
           return True

    #样本数小于50用Shapiro-Wilk算法检验正态分布性
    if len(testData) <50:
       p_value= stats.shapiro(testData)[1]
       if p_value<0.05:
           print "use shapiro:"
           print "data are not normal distributed"
           return  False
       else:
           print "use shapiro:"
           print "data are normal distributed"
           return True

    if 300>=len(testData) >=50:
       p_value= lillifors(testData)[1]
       if p_value<0.05:
           print "use lillifors:"
           print "data are not normal distributed"
           return  False
       else:
           print "use lillifors:"
           print "data are normal distributed"
           return True

    if len(testData) >300:
       p_value= stats.kstest(testData,'norm')[1]
       if p_value<0.05:
           print "use kstest:"
           print "data are not normal distributed"
           return  False
       else:
           print "use kstest:"
           print "data are normal distributed"
           return True

#对所有样本组进行正态性检验
def NormalTest(list_groups):
    for group in list_groups:
        #正态性检验
        status=check_normality(group)
        if status==False :
            return False
    return True

#排列组合函数
def Combination(list_groups):
    combination= []
    for i in range(1,len(list_groups)+1):
        iter = itertools.combinations(list_groups,i)
        combination.append(list(iter))
    #需要排除第一个和最后一个
    return combination[1:-1][0]
'''
Out[57]:
[[([2, 3, 7, 2, 6], [10, 8, 7, 5, 10]),
  ([2, 3, 7, 2, 6], [10, 13, 14, 13, 15]),
  ([10, 8, 7, 5, 10], [10, 13, 14, 13, 15])]]
'''       

#方差齐性检测
def Levene_test(group1,group2,group3):
    leveneResult=scipy.stats.levene(group1,group2,group3)
    p=leveneResult[1]
    print"levene test:"
    if p<0.05:
        print"variances of groups are not equal"
        return False
    else:
        print"variances of groups are equal"
        return True

'''
H0成立，三组数据方差无显著差异
Out[9]: LeveneResult(statistic=0.24561403508771934, pvalue=0.7860617221429711)
'''

#比较组内的样本是否相等，如果不相等，不适合于tukey等方法
#此函数有问题，无法解决nan排除
def Equal_lenth(list_groups):
    list1=list_groups[0]
    list2=list_groups[1]
    list3=list_groups[2]

    list1_removeNan=[x for x in list1 if str(x) != 'nan' and str(x)!= '-inf']
    list2_removeNan=[x for x in list2 if str(x) != 'nan' and str(x)!= '-inf']
    list3_removeNan=[x for x in list3 if str(x) != 'nan' and str(x)!= '-inf']

    len1=len(list1_removeNan)
    len2=len(list2_removeNan)
    len3=len(list3_removeNan)
    if len1==len2==len3:
        return True
    else:
        return False

'''
#返回True or false
normality=NormalTest(list_groups)
leveneResult=Levene_test(list_groups[0],list_groups[1],list_groups[2])
'''

 

 

数据sample.xlsx

https://en.wikipedia.org/wiki/Tukey's_range_test

• Tukey's range test, also called Tukey method, Tukey's honest significance test, Tukey's HSD (Honestly Significant Difference) test

老鼠试验数据

公式

D_turkey表示平均数差值的关键值，任何大于 D_turkey值的平均数差值都是显著的 

第四组和第五组平均数差值是不显著的，其它组的差值是显著的

Studentized Range q Table

a=0.05

http://www.real-statistics.com/statistics-tables/studentized-range-q-table/

q值

q值是一个残差化范围统计数据表格值；由平均数的数量和组内自由度数量交互决定

a表示分类组数，df表示所有数量自由度 ，a_fw表示0.05犯错概率

MS_S/A 表示 within group的方差

s_m值

s_m是一个标准误

n表示组数

结果

Tukey's range test, also known as the Tukey's test, Tukey method, Tukey's honest significance test, Tukey's HSD (honest significant difference) test,^[1] or the Tukey–Kramer method, is a single-step multiple comparison procedure and statistical test. It can be used on raw data or in conjunction with an ANOVA (post-hoc analysis) to find means that are significantly different from each other. Named after John Tukey,^[2] it compares all possible pairs of means, and is based on a studentized range distribution (q) (this distribution is similar to the distribution of t from the t-test. See below).^[3] The Tukey HSD tests should not be confused with the Tukey Mean Difference tests (also known as the Bland–Altman diagram).

Tukey's test compares the means of every treatment to the means of every other treatment; that is, it applies simultaneously to the set of all pairwise comparisons

μ i − μ j {\displaystyle \mu _{i}-\mu _{j}\,}

and id　　entifies any difference between two means that is greater than the expected standard error. The confidence coefficient for the set, when all sample sizes are equal, is exactly 1 − α. For unequal sample sizes, the confidence coefficient is greater than 1 − α. In other words, the Tukey method is conservative when there are unequal sample sizes.

Contents

• 1 Assumptions of Tukey's test
• 2 The test statistic
• 3 The studentized range (q) distribution
• 4 Confidence limits
• 5 Advantages and disadvantages
• 6 See also
• 7 Notes
• 8 Further reading
• 9 External links

Assumptions of Tukey's test

前提条件：

样本独立性+样本正态分布+所有组方差齐性

1. The observations being tested are independent within and among the groups.
2. The groups associated with each mean in the test are normally distributed.
3. There is equal within-group variance across the groups associated with each mean in the test (homogeneity of variance).

The test statistic

Tukey's test is based on a formula very similar to that of the t-test. In fact, Tukey's test is essentially a t-test, except that it corrects for family-wise error rate (when there are multiple comparisons being made, the probability of making a Type I error within at least one of the comparisons, increases — Tukey's test corrects for that, and is thus more suitable for multiple comparisons than a number of t-tests would be).^[3]

The formula for Tukey's test is:

q s = Y A − Y B S E , {\displaystyle q_{s}={\frac {Y_{A}-Y_{B}}{SE}},}

where Y_A is the larger of the two means being compared, Y_B is the smaller of the two means being compared, and SE is the standard error of the data in question.

This q_s value can then be compared to a q value from the studentized range distribution. If the q_s value is larger than the q_critical value obtained from the distribution, the two means are said to be significantly different.^[3]

Since the null hypothesis for Tukey's test states that all means being compared are from the same population (i.e. μ₁ = μ₂ = μ₃ = ... = μ_k), the means should be normally distributed (according to the central limit theorem). This gives rise to the normality assumption of Tukey's test.

The studentized range (q) distribution

The Tukey method uses the studentized range distribution. Suppose that we take a sample of size n from each of k populations with the same normal distribution N(μ, σ) and suppose that y ¯ {\displaystyle {\bar {y}}} _min is the smallest of these sample means and y ¯ {\displaystyle {\bar {y}}} _max is the largest of these sample means, and suppose S² is the pooled sample variance from these samples. Then the following random variable has a Studentized range distribution.

q = ( y ¯ max − y ¯ min ) S 2 / n {\displaystyle q={\frac {({\overline {y}}_{\max }-{\overline {y}}_{\min })}{S{\sqrt {2/n}}}}}

This value of q is the basis of the critical value of q, based on three factors:

1. α (the Type I error rate, or the probability of rejecting a true null hypothesis)
2. k (the number of populations)
3. df (the number of degrees of freedom (N-k) where N is the total number of observations)

The distribution of q has been tabulated and appears in many textbooks on statistics. In some tables the distribution of q has been tabulated without the 2 {\displaystyle {\sqrt {2}}} factor. To understand which table it is, we can compute the result for k=2 and compare it to the result of the Student's t-distribution with the same degrees of freedom and the same α. In addition, R offers a cumulative distribution function (ptukey) and a quantile function (qtukey) for q.

Confidence limits

The Tukey confidence limits for all pairwise comparisons with confidence coefficient of at least 1 − α are

y ¯ i ∙ − y ¯ j ∙ ± q α ; k ; N − k 2 σ ^ ε 2 n i , j = 1 , … , k i ≠ j . {\displaystyle {\bar {y}}_{i\bullet }-{\bar {y}}_{j\bullet }\pm {\frac {q_{\alpha ;k;N-k}}{\sqrt {2}}}{\widehat {\sigma }}_{\varepsilon }{\sqrt {\frac {2}{n}}}\qquad i,j=1,\ldots ,k\quad i\neq j.}

Notice that the point estimator and the estimated variance are the same as those for a single pairwise comparison. The only difference between the confidence limits for simultaneous comparisons and those for a single comparison is the multiple of the estimated standard deviation.

Also note that the sample sizes must be equal when using the studentized range approach. σ ^ ε {\displaystyle {\widehat {\sigma }}_{\varepsilon }} is the standard deviation of the entire design, not just that of the two groups being compared. It is possible to work with unequal sample sizes. In this case, one has to calculate the estimated standard deviation for each pairwise comparison as formalized by Clyde Kramer in 1956, so the procedure for unequal sample sizes is sometimes referred to as the Tukey–Kramer method which is as follows:

y ¯ i ∙ − y ¯ j ∙ ± q α ; k ; N − k 2 σ ^ ε 1 n i + 1 n j {\displaystyle {\bar {y}}_{i\bullet }-{\bar {y}}_{j\bullet }\pm {\frac {q_{\alpha ;k;N-k}}{\sqrt {2}}}{\widehat {\sigma }}_{\varepsilon }{\sqrt {{\frac {1}{n}}_{i}+{\frac {1}{n}}_{j}}}\qquad }

where n _i and n _j are the sizes of groups i and j respectively. The degrees of freedom for the whole design is also applied.

Advantages and disadvantages

When doing all pairwise comparisons, this method is considered the best available when confidence intervals are needed or sample sizes are not equal. When samples sizes are equal and confidence intervals are not needed Tukey’s test is slightly less powerful than the stepdown procedures, but if they are not available Tukey’s is the next-best choice, and unless the number of groups is large, the loss in power will be slight. In the general case when many or all contrasts might be of interest, Scheffé's method tends to give narrower confidence limits and is therefore the preferred method.

https://github.com/thomas-haslwanter/statsintro_python/tree/master/ISP/Code_Quantlets/08_TestsMeanValues/multipleTesting

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