目錄

參數估計

方差比的置信區間

均值差的置信區間

一個正態總體方差的點估計和置信區間

一個正態總體均值的點估計和置信區間

單樣本t檢驗的SciPy實現方式

單樣本t檢驗的statsmodels實現方式

兩樣本t檢驗SciPy的實現方式

兩樣本t檢驗statsmodels的實現方式

配對t檢驗的SciPy實現

方差分析

單因素方差分析的SciPy實現

事后檢驗

非參數方法

SciPy實現有符號秩和檢驗

SciPy實現秩和檢驗

一元線性回歸

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參數估計

方差比的置信區間

import numpy as np from scipy.stats import f # 定義一個實現方差比置信區間的函數 def var_ratio_ci_est(data1,data2,alpha):n1 = len(data1) # 樣本1的樣本容量n2 = len(data2) # 樣本2的樣本容量f_lower_value = f.ppf(alpha/2,n1-1,n2-1) # 左側臨界值f_upper_value = f.ppf(1-alpha/2,n1-1,n2-1) # 右側臨界值var_ratio = np.var(data1) / np.var(data2)return var_ratio / f_upper_value,var_ratio / f_lower_value salary_18 = [1484, 785, 1598, 1366, 1716, 1020, 1716, 785, 3113, 1601] salary_35 = [902, 4508, 3809, 3923, 4276, 2065, 1601, 553, 3345, 2182]print(var_ratio_ci_est(salary_18,salary_35,0.05)) (0.05314530971751432, 0.8614126063098585)

均值差的置信區間

import numpy as np from scipy.stats import t # 兩個正態總體均值差的置信區間 def mean_diff_ci_t_est(data1,data2,alpha,equal=True):n1 = len(data1) # 樣本1的樣本容量n2 = len(data2) # 樣本2的樣本容量mean_diff = np.mean(data1) - np.mean(data2) # 兩個樣本的均值差（兩個總體均值差的無偏點估計）sample1_var = np.var(data1) # 樣本1的方差sample2_var = np.var(data2) # 樣本2的方差if equal: # 方差未知且相等sw = np.sqrt(((n1-1)*sample1_var + (n2-1)*sample2_var) / (n1+n2-2))t_value = np.abs(t.ppf(alpha/2,n1+n2-2)) # t值return mean_diff - sw*np.sqrt(1/n1+1/n2)*t_value, \mean_diff + sw*np.sqrt(1/n1+1/n2)*t_valueelse: # 方差未知且不等df_numerator = (sample1_var/n1+sample2_var/n2)**2 # 自由度分子# 自由度分母df_denominator = (sample1_var/n1)**2/(n1-1) + (sample2_var/n2)**2/(n2-1)df = df_numerator / df_denominator # 自由度t_value = np.abs(t.ppf(alpha/2,df))return mean_diff - np.sqrt(sample1_var/n1 + sample2_var/n2)*t_value, \mean_diff + np.sqrt(sample1_var/n1 + sample2_var/n2)*t_value salary_18 = [1484, 785, 1598, 1366, 1716, 1020, 1716, 785, 3113, 1601] salary_35 = [902, 4508, 3809, 3923, 4276, 2065, 1601, 553, 3345, 2182]print(mean_diff_ci_t_est(salary_18,salary_35,0.05,equal=True)) # 方差相等 print(mean_diff_ci_t_est(salary_18,salary_35,0.05,equal=False)) # 方差不等 (-2196.676574582891, -199.32342541710875) (-2227.5521493017823, -168.4478506982175)

一個正態總體方差的點估計和置信區間

import numpy as np from scipy.stats import chi2 # 定義計算方差的置信區間的函數 def var_ci_est(data,alpha):n = len(data) # 樣本容量s2 = np.var(data) # 樣本方差# chi2.ppf(alpha/2,n-1)的意思是返回左側面積為alpha/2的卡方值chi2_lower_value = chi2.ppf(alpha/2,n-1)# # chi2.ppf(1-alpha/2,n-1)的意思是返回左側面積為1-alpha/2的卡方值chi2_upper_value = chi2.ppf(1-alpha/2,n-1)return (n-1) * s2 / chi2_upper_value,(n-1) * s2 / chi2_lower_value salary_18 = [1484, 785, 1598, 1366, 1716, 1020, 1716, 785, 3113, 1601] salary_35 = [902, 4508, 3809, 3923, 4276, 2065, 1601, 553, 3345, 2182]# 樣本方差是總體方差的無偏點估計,再調用函數計算方差的置信區間 print(np.std(salary_18),np.var(salary_18),var_ci_est(salary_18,0.05)) print(np.std(salary_35),np.var(salary_35),var_ci_est(salary_35,0.05)) 631.0754629994736 398256.24 (188421.9056837747, 1327329.3204650925) 1364.3074580167038 1861334.8399999999 (880629.6611156773, 6203554.546528138)

一個正態總體均值的點估計和置信區間

import numpy as np from scipy.stats import norm, t # 計算一個正態總體均值的置信區間的函數 def mean_ci_est(data,alpha,sigma=None):n = len(data) # 樣本容量sample_mean = np.mean(data) # 樣本均值if sigma is None: # 方差未知的情況s = np.std(data)se = s / np.sqrt(n)# t.ppf(alpha / 2,n-1)返回的是左側面積為alpha / 2對應的t值t_value = np.abs(t.ppf(alpha / 2,n-1))# 根據公式返回置信區間return sample_mean - se * t_value,sample_mean + se * t_valueelse: # 方差已知的情況se = sigma / np.sqrt(n) # 標準誤# norm.ppf(alpha / 2)返回的是左側面積為alpha / 2對應的z值z_value = np.abs(norm.ppf(alpha / 2))# 根據公式返回置信區間return sample_mean - se * z_value,sample_mean + se * z_value salary_18 = [1484, 785, 1598, 1366, 1716, 1020, 1716, 785, 3113, 1601] salary_35 = [902, 4508, 3809, 3923, 4276, 2065, 1601, 553, 3345, 2182]# 樣本均值是總體均值的無偏點估計,并調用函數計算均值的置信區間 print(np.mean(salary_18),mean_ci_est(salary_18,0.05)) print(np.mean(salary_35),mean_ci_est(salary_35,0.05)) 1518.4 (1066.9558093661096, 1969.8441906338905) 2716.4 (1740.433238065152, 3692.366761934848)

假設檢驗

初始化設置

import pandas as pd import scipy.stats as ss import matplotlib # 解決繪圖的兼容問題 %matplotlib inline matplotlib.rcParams['font.sans-serif'] = ['SimHei']

單樣本t檢驗的SciPy實現方式

ccss = pd.read_excel("CCSS_sample.xlsx",sheet_name="CCSS") # 讀取Excel文件中名稱為CCSS的表單 ccss.head() # 廣州基期的消費信心指數的描述統計 ccss.query("s0 == '廣州' & time == 203004" ).index1.describe()from scipy import stats as ss # 單樣本t檢驗 ss.ttest_1samp(ccss.query("s0 == '廣州' & time == 203004" ).index1,100)

Ttest_1sampResult(statistic=-1.3625667518512996, pvalue=0.17611075148299993)

我們可以看出P值大于0.05，接受H0,沒有顯著性差異

單樣本t檢驗的statsmodels實現方式

from statsmodels.stats import weightstats as wsdes = ws.DescrStatsW(ccss.query("s0 == '廣州' & time == 203004").index1) des.mean 97.16472701710536 # 置信水平為95%的置信區間（可信區間） des.tconfint_mean() (93.03590418536487, 101.29354984884586) # 單樣本t檢驗 des.ttest_mean(100) (-1.3625667518512996, 0.17611075148299993, 99.0)

我們可以看出P值大于0.05，接受H0,沒有顯著性差異

兩樣本t檢驗SciPy的實現方式

from scipy import stats as ss # 消費的信心指數分布的對稱性考察 ccss.index1.plot.hist()

# 不同婚姻狀況的消費信心指數分組描述 ccss.groupby('s7').index1.describe()

# 方差齊性檢驗 ss.levene(ccss.index1[ccss.s7 == '未婚'],ccss.index1[ccss.s7 == '已婚']) LeveneResult(statistic=0.6178738290825966, pvalue=0.4320031337363959) # 兩樣本t檢驗(方差齊性) ss.ttest_ind(ccss.index1[ccss.s7 == '未婚'],ccss.index1[ccss.s7 == '已婚']) Ttest_indResult(statistic=2.4052614244262576, pvalue=0.01632071963816213) # 兩樣本t檢驗(方差不齊性) ss.ttest_ind(ccss.index1[ccss.s7 == '未婚'],ccss.index1[ccss.s7 == '已婚'],equal_var=False) Ttest_indResult(statistic=2.466907208318663, pvalue=0.013870358702526698)

兩樣本t檢驗statsmodels的實現方式

from statsmodels.stats import weightstats as ws d1 = ws.DescrStatsW(ccss.index1[ccss.s7 == '未婚']) # 未婚的消費信心指數數據 d2 = ws.DescrStatsW(ccss.index1[ccss.s7 == '已婚']) # 已婚的消費信心指數數據 comp = ws.CompareMeans(d1,d2) # 創建CompareMeans對象 comp.ttest_ind() # 兩組獨立樣本t檢驗(方差齊性) (2.4052614244262576, 0.01632071963816213, 1131.0) comp.ttest_ind(usevar='unequal') # 兩組獨立樣本t檢驗(方差不齊性) (2.4669072083186636, 0.013870358702526675, 690.0875773844764)

配對t檢驗的SciPy實現

import scipy.stats as ss import pandas as pd ccss_p.loc[:,['index1','index1n']].describe() # 取出4月和12月信心指數，并描述

# 用相關分析確認配對信息是否的確存在 ss.pearsonr(ccss_p.index1,ccss_p.index1n) (0.2638011798615908, 0.01301162367951006) # 配對t檢驗 ss.ttest_rel(ccss_p.index1,ccss_p.index1n) Ttest_relResult(statistic=1.1616334792419984, pvalue=0.24856144386191056)

方差分析

單因素方差分析的SciPy實現

導包

import pandas as pd import scipy.stats as ss import matplotlib # 解決繪圖的兼容問題 %matplotlib inline matplotlib.rcParams['font.sans-serif'] = ['SimHei'] ccss = pd.read_excel("CCSS_sample.xlsx",sheet_name="CCSS") # 讀取Excel文件中名稱為CCSS的表單 # 分別提取出北京四個時間段的消費信心數據 a = ccss.query("s0 == '北京' & time == 203004 ").index1 b = ccss.query("s0 == '北京' & time == 203012 ").index1 c = ccss.query("s0 == '北京' & time == 203112 ").index1 d = ccss.query("s0 == '北京' & time == 203212 ").index1 ss.levene(a,b,c,d) # 方差齊性檢驗 LeveneResult(statistic=0.44332330387152036, pvalue=0.7221678627997157) ss.f_oneway(a,b,c,d) # 單因素方差分析 F_onewayResult(statistic=5.630155391280303, pvalue=0.0008777240313291846)

事后檢驗

# 需要先在環境中安裝scikit_posthocs包：pip install scikit_posthocs import scikit_posthocs as sp# 創建對象，該對象接收事后檢驗的數據，并且設置p值校正的方法（控制兩兩比較的α值）為'bonferroni' pc = sp.posthoc_conover(ccss,val_col='index1',group_col='time',p_adjust='bonferroni') # 使用熱力圖顯示比較結果 heatmap_args = {'linewidths': 0.25, 'linecolor': '0.5', 'clip_on': False, 'square': True, 'cbar_ax_bbox': [0.80, 0.35, 0.04, 0.3]} sp.sign_plot(pc,**heatmap_args) # 繪制熱力圖

非參數方法

SciPy實現有符號秩和檢驗

import numpy as np from scipy import stats data = [36, 32, 31, 25, 28, 36, 40, 32, 41, 26, 35, 35, 32, 87, 33, 35] data = np.array(data) # 轉換為numpy數組，方便運算# 檢驗總體均值是否為37 k = min(len(data[data>37]),len(data[data<37])) pvalue = 2*stats.binom.cdf(k,len(data),0.5) print(pvalue) 0.021270751953125

SciPy實現秩和檢驗

import scipy.stats as stats weight_high=[134,146,104,119,124,161,107,83,113,129] # 高蛋白人群的體重的樣本 weight_low=[70,118,101,85,112,132,94,100,68,86] # 低蛋白人群的體重的樣本# mannwhitneyu方法針對樣本容量相等或不等都是適用的 stats.mannwhitneyu(weight_high,weight_low,alternative='two-sided') MannwhitneyuResult(statistic=81.0, pvalue=0.0211339281291611)

一元線性回歸

import pandas as pd import scipy.stats as ss ccss = pd.read_excel("CCSS_sample.xlsx",sheet_name="CCSS") # 讀取Excel文件中名稱為CCSS的表單 # 建立年齡和總消費信心指數的回歸方程 ss.linregress(ccss.s3,ccss.index1) LinregressResult(slope=-0.3576848241677336, intercept=108.89832302796889, rvalue=-0.2190793138600294, pvalue=6.243013355018415e-14, stderr=0.047077765734542185, intercept_stderr=1.815504369310354) # 以k*2形式的二維數組提供數據 ss.linregress(ccss.loc[:,['s3','index1']]) LinregressResult(slope=-0.3576848241677336, intercept=108.89832302796889, rvalue=-0.2190793138600294, pvalue=6.243013355018415e-14, stderr=0.047077765734542185, intercept_stderr=1.815504369310354)

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