如何比较两条回归线的斜率？

Question

假设您有两个 numpy 数组：blue_y 和 light_blue_y 例如：

import numpy as np

x=np.array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
blue_y = np.array([0.94732871, 0.85729212, 0.86039587, 0.89169027, 0.90817473, 0.93606619, 0.93890423, 1., 0.97783521, 0.93035495])
light_blue_y = np.array([0.81346023, 0.72248919, 0.72406021, 0.74823437, 0.77759055, 0.81167983,  0.84050726, 0.90357904, 0.97354455, 1. ])

blue_m, blue_b = np.polyfit(x, blue_y, 1)
light_blue_m, light_blue_b = np.polyfit(x, light_blue_y, 1)

将线性回归线拟合到这两个 numpy 数组后，我得到以下斜率：

>>> blue_m
0.009446010787878795
>>> light_blue_m
0.028149985151515147

如何比较这两个斜率并显示它们在统计上是否彼此不同？

Answer 1

import numpy as np
import statsmodels.api as sm

x=np.array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
blue_y = np.array([0.94732871, 0.85729212, 0.86039587, 0.89169027, 0.90817473, 0.93606619, 0.93890423, 1., 0.97783521, 0.93035495])
light_blue_y = np.array([0.81346023, 0.72248919, 0.72406021, 0.74823437, 0.77759055, 0.81167983,  0.84050726, 0.90357904, 0.97354455, 1. ])

为简单起见，我们可以尝试查看差异。如果它们具有相同的常数和相同的斜率，则应该在线性回归中可见。

y = blue_y-light_blue_y
# Let create a linear regression
mod = sm.OLS(y, sm.add_constant(x))
res = mod.fit()

有输出

print(res.summary())

输出

==============================================================================
Dep. Variable:                      y   R-squared:                       0.647
Model:                            OLS   Adj. R-squared:                  0.603
Method:                 Least Squares   F-statistic:                     14.67
Date:                Mon, 08 Feb 2021   Prob (F-statistic):            0.00502
Time:                        23:12:25   Log-Likelihood:                 18.081
No. Observations:                  10   AIC:                            -32.16
Df Residuals:                       8   BIC:                            -31.56
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const          0.1775      0.026      6.807      0.000       0.117       0.238
x1            -0.0187      0.005     -3.830      0.005      -0.030      -0.007
==============================================================================
Omnibus:                        0.981   Durbin-Watson:                   0.662
Prob(Omnibus):                  0.612   Jarque-Bera (JB):                0.780

解读: 常数和斜率看起来不同，因此 blue_y 和 light_blue_y 具有不同的斜率和常数项。

备选方案：更传统的是，您可以运行两种情况的线性回归和运行您自己的 F 检验。

如此处所示：

如何比较两条回归线的斜率？

How to compare the slopes of two regression lines?

statistics

linear-regression