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多元回归的置信区间

news 来源：原创 2025/5/24 6:43:38

本文是实验设计与分析（第6版，Montgomery著傅珏生译)第10章拟合回归模型第10.5节的python解决方案。本文尽量避免重复书中的理论，着于提供python解决方案，并与原书的运算结果进行对比。您可以从Detail 下载实验设计与分析（第6版，Montgomery著傅珏生译)电子版。本文假定您已具备python基础，如果您还没有python的基础，可以从Detail 下载相关资料进行学习。

我们常常要对回归系数{Bj}以及回归模型中其他感兴趣的量构造置信区间估计。这些置信区间的推导过程需要假定误差{εi}服从均值为零、方差为σ2的独立的正态分布，与10.4节的假设检验中的假定相同。

10.5.1单个回归系数的置信区间

因为最小二乘估计量是观测的线性组合，所以，服从均值为、协方差矩阵为的正态分布。故统计量

服从自由度为n-p的t分布，这里，是矩阵的第（jj）个元素，是由（10.17）式得到的误差方差的估计。因此，回归系数的100（1-α）％置信区间是

因为，这个置信区间也可写成

例10.7求例10.1中β1的95％置信区间。现在，由，C11=1.429184×10-3，可得

故β1的95％置信区间是

>>> print(model.summary2())

C:\Users\Administrator\AppData\Local\Programs\Python\Python311\Lib\site-packages\scipy\stats\_stats_py.py:1736: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=16

warnings.warn("kurtosistest only valid for n>=20 ... continuing "

Results: Ordinary least squares

====================================================================

Model: OLS Adj. R-squared: 0.916

Dependent Variable: df.Viscosity AIC: 137.5159

Date: 2024-03-14 10:31 BIC: 139.8337

No. Observations: 16 Log-Likelihood: -65.758

Df Model: 2 F-statistic: 82.50

Df Residuals: 13 Prob (F-statistic): 4.10e-08

R-squared: 0.927 Scale: 267.60

--------------------------------------------------------------------

Coef. Std.Err. t P>|t| [0.025 0.975]

--------------------------------------------------------------------

Intercept 1566.0778 61.5918 25.4267 0.0000 1433.0167 1699.1388

df.Temperature 7.6213 0.6184 12.3236 0.0000 6.2853 8.9573

df.Catalyst 8.5848 2.4387 3.5203 0.0038 3.3164 13.8533

--------------------------------------------------------------------

Omnibus: 1.215 Durbin-Watson: 2.607

Prob(Omnibus): 0.545 Jarque-Bera (JB): 0.779

Skew: -0.004 Prob(JB): 0.677

Kurtosis: 1.919 Condition No.: 1385

====================================================================

Notes:

[1] Standard Errors assume that the covariance matrix of the errors

is correctly specified.

[2] The condition number is large, 1.38e+03. This might indicate

that there are strong multicollinearity or other numerical

problems.

>>> print(model.params)

Intercept 1566.077771

df.Temperature 7.621290

df.Catalyst 8.584846

dtype: float64

>>> anovatable=sm.stats.anova_lm(model)

>>> anovatable

df sum_sq mean_sq F PR(>F)

df.Temperature 1.0 40840.842466 40840.842466 152.616757 1.473645e-08

df.Catalyst 1.0 3316.244074 3316.244074 12.392360 3.764806e-03

Residual 13.0 3478.850960 267.603920 NaN NaN

>>> print(model.summary2())

C:\Users\Administrator\AppData\Local\Programs\Python\Python311\Lib\site-packages\scipy\stats\_stats_py.py:1736: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=16

warnings.warn("kurtosistest only valid for n>=20 ... continuing "

Results: Ordinary least squares

====================================================================

Model: OLS Adj. R-squared: 0.916

Dependent Variable: df.Viscosity AIC: 137.5159

Date: 2024-03-14 10:31 BIC: 139.8337

No. Observations: 16 Log-Likelihood: -65.758

Df Model: 2 F-statistic: 82.50

Df Residuals: 13 Prob (F-statistic): 4.10e-08

R-squared: 0.927 Scale: 267.60

--------------------------------------------------------------------

Coef. Std.Err. t P>|t| [0.025 0.975]