Math Problem Statement

расчет стандартных коэффицентов регрессии

x1 = (df4['x1'] - df4['x1'].mean()) / (df4['x1'].var() ** 0.5)

x2 = (df4['x2'] - df4['x2'].mean()) / (df4['x2'].var() ** 0.5)

y = (df4['y'] - df4['y'].mean()) / (df4['y'].var() ** 0.5)

x = [x1, x2]

def reg_m(y, x):

ones = np.ones(len(x[0]))

X = sm.add_constant(np.column_stack((x[0], ones)))

for ele in x[1:]:

    X = sm.add_constant(np.column_stack((ele, X)))

results = sm.OLS(y, X).fit()

return results

print(reg_m(y, x).summary())

Solution

To calculate the standardized coefficients of regression (often referred to as "beta coefficients"), you are normalizing your independent variables (x1,x2x_1, x_2) and dependent variable (yy) so that they have a mean of 0 and a standard deviation of 1. This normalization allows for direct comparisons of the relative strength of predictors in the regression model.

Here’s an explanation of the code provided:

  1. Standardizing the Variables:
    The independent variables (x1,x2x_1, x_2) and the dependent variable (yy) are standardized by subtracting their means and dividing by the square root of their variances:

    python
    x1 = (df4['x1'] - df4['x1'].mean()) / (df4['x1'].var() ** 0.5) x2 = (df4['x2'] - df4['x2'].mean()) / (df4['x2'].var() ** 0.5) y = (df4['y'] - df4['y'].mean()) / (df4['y'].var() ** 0.5)
  2. Setting up the Regression:
    The reg_m function:

    • Adds a constant (intercept) to the regression matrix.
    • Constructs the design matrix XX by stacking x1x_1, x2x_2, and a column of ones (for the intercept).
    • Uses statsmodels' Ordinary Least Squares (OLS) method to fit the regression.
  3. Output Summary:
    The summary() method provides details such as coefficients, their statistical significance (p-values), confidence intervals, and goodness-of-fit measures like R2R^2.

Here’s how the core function works:

python
def reg_m(y, x): ones = np.ones(len(x[0])) # Column of ones for the intercept X = sm.add_constant(np.column_stack((x[0], ones))) # Add first variable + intercept for ele in x[1:]: # Add remaining variables X = sm.add_constant(np.column_stack((ele, X))) results = sm.OLS(y, X).fit() # Fit the model using OLS return results

Outputs in Regression Summary

  • Coefficients: Represent the standardized beta values for each predictor.
  • Standard Error: Indicates the precision of the coefficients.
  • t-values and p-values: Help test the null hypothesis that a coefficient is zero.
  • R2R^2: Shows the proportion of variance explained by the model.

Implementation Tip:

Make sure df4 is a pandas DataFrame, and libraries like numpy (np) and statsmodels.api (sm) are imported. For example:

python
import numpy as np import statsmodels.api as sm import pandas as pd

If you need assistance running or interpreting the results, feel free to share your data or results.


5 Related Questions:

  1. What are the main advantages of standardizing variables before regression?
  2. How does the presence of multicollinearity affect the interpretation of standardized coefficients?
  3. What alternative methods exist for comparing the importance of predictors in a regression model?
  4. How can you assess the assumptions of linear regression for this model?
  5. How would you extend this approach to handle interaction terms or polynomial features?

Tip: Always examine the residuals to ensure the model assumptions (linearity, independence, homoscedasticity, and normality) are met!

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Math Problem Analysis

Mathematical Concepts

Regression Analysis
Standardization of Variables
Ordinary Least Squares (OLS)

Formulas

z = (x - mean) / standard deviation
Regression equation: y = β0 + β1x1 + β2x2 + ε

Theorems

Gauss-Markov Theorem
Least Squares Method

Suitable Grade Level

Undergraduate (Statistics or Data Science courses)