如何在 Statsmodels 中获得稳健回归 (RLM) 的 R 平方？

Question

在衡量拟合优度时，R-Squared 似乎是“简单”线性模型的一种普遍理解（和接受）的衡量标准。 但是在statsmodels（以及其他统计软件）的情况下，RLM 不包括 R 平方和回归结果。 有没有办法“手动”计算它，也许类似于Stata 中的计算方式？

或者是否有其他可以使用/根据sm.RLS产生的结果计算的度量？

这是 Statsmodels 正在生产的产品：

import numpy as np
import statsmodels.api as sm

# Sample Data with outliers
nsample = 50
x = np.linspace(0, 20, nsample)
x = sm.add_constant(x)
sig = 0.3
beta = [5, 0.5]
y_true = np.dot(x, beta)
y = y_true + sig * 1. * np.random.normal(size=nsample)
y[[39,41,43,45,48]] -= 5   # add some outliers (10% of nsample)

# Regression with Robust Linear Model
res = sm.RLM(y, x).fit()
print(res.summary())

哪些输出：

                    Robust linear Model Regression Results                    
==============================================================================
Dep. Variable:                      y   No. Observations:                   50
Model:                            RLM   Df Residuals:                       48
Method:                          IRLS   Df Model:                            1
Norm:                          HuberT                                         
Scale Est.:                       mad                                         
Cov Type:                          H1                                         
Date:                 Mo, 27 Jul 2015                                         
Time:                        10:00:00                                         
No. Iterations:                    17                                         
==============================================================================
                 coef    std err          z      P>|z|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
const          5.0254      0.091     55.017      0.000         4.846     5.204
x1             0.4845      0.008     61.555      0.000         0.469     0.500
==============================================================================

Answer 1

由于 OLS 返回 R2，我建议使用简单的线性回归将实际值与拟合值进行回归。无论拟合值来自何处，这种方法都会为您提供相应 R2 的指示。

Answer 2

R2 不能很好地衡量 RLM 模型的拟合优度。问题是异常值对 R2 值有巨大影响，以至于它完全由异常值决定。事后使用加权回归是一种有吸引力的替代方法，但最好查看估计系数的 p 值、标准误差和置信区间。

将 OLS 摘要与 RLM 进行比较（由于随机化不同，结果与您的略有不同）：

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       0.726
Model:                            OLS   Adj. R-squared:                  0.721
Method:                 Least Squares   F-statistic:                     127.4
Date:                Wed, 03 Nov 2021   Prob (F-statistic):           4.15e-15
Time:                        09:33:40   Log-Likelihood:                -87.455
No. Observations:                  50   AIC:                             178.9
Df Residuals:                      48   BIC:                             182.7
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const          5.7071      0.396     14.425      0.000       4.912       6.503
x1             0.3848      0.034     11.288      0.000       0.316       0.453
==============================================================================
Omnibus:                       23.499   Durbin-Watson:                   2.752
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               33.906
Skew:                          -1.649   Prob(JB):                     4.34e-08
Kurtosis:                       5.324   Cond. No.                         23.0
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

                    Robust linear Model Regression Results                    
==============================================================================
Dep. Variable:                      y   No. Observations:                   50
Model:                            RLM   Df Residuals:                       48
Method:                          IRLS   Df Model:                            1
Norm:                          HuberT                                         
Scale Est.:                       mad                                         
Cov Type:                          H1                                         
Date:                Wed, 03 Nov 2021                                         
Time:                        09:34:24                                         
No. Iterations:                    17                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
const          5.1857      0.111     46.590      0.000       4.968       5.404
x1             0.4790      0.010     49.947      0.000       0.460       0.498
==============================================================================

If the model instance has been used for another fit with different fit parameters, then the fit options might not be the correct ones anymore .

您可以看到，从 OLS 到 RLM，截距项和斜率项的标准误和置信区间大小都会减小。这表明估计值更接近实际值。

Answer 3

为什么不使用model.predict 来获取r2？例如：

r2=1. - np.sum(np.abs(model.predict(X) - y) **2) / np.sum(np.abs(y - np.mean(y)) ** 2)