Model building in multiple regression analysis is a crucial process in Lean Six Sigma and other data-driven methodologies aimed at improving process efficiency and quality. Multiple regression is a statistical technique used to understand the relationship between one dependent variable and two or more independent variables. The essence of model building in this context is to create a predictive equation that can accurately estimate the outcome (dependent variable) based on the values of the predictors (independent variables). This article delves into the steps and considerations involved in model building in multiple regression.

Understanding the Basics

Before diving into model building, it's essential to grasp the basics of multiple regression. The goal is to fit a model: Y=β0​+β1​X1​+β2​X2​+...+βn​Xn​+ϵ where Y is the dependent variable, X1​,X2​,...,Xn​ are the independent variables, β0​ is the intercept, β1​,β2​,...,βn​ are the coefficients for each independent variable, and ϵ represents the error term.

Step 1: Define the Problem

The first step in model building is clearly defining the problem you're trying to solve. This includes identifying the outcome you wish to predict and the potential predictors. Understanding the domain and the data's context is critical at this stage.

Step 2: Data Collection and Preparation

Gather data that includes the dependent variable and the potential independent variables. Data preparation is a crucial step that involves cleaning the data (handling missing values, outliers), transforming variables if necessary (e.g., log transformation for skewed data), and ensuring that the data meets the assumptions of multiple regression.

Step 3: Selecting Variables

Not all collected data will be relevant for your model. Variable selection is a critical step to ensure the model's simplicity and effectiveness. Techniques such as forward selection, backward elimination, and stepwise regression can help in choosing the most relevant predictors. The goal is to include variables that significantly contribute to the prediction of the dependent variable while avoiding overfitting with too many variables.

Step 4: Model Estimation

Using statistical software, the regression coefficients (β) are estimated. These coefficients indicate the strength and direction of the relationship between each independent variable and the dependent variable.

Step 5: Model Evaluation

After estimating the model, it's essential to evaluate its performance. This involves checking the significance of the coefficients, assessing the model's overall fit (using R-squared and adjusted R-squared), and ensuring that the model meets the assumptions of multiple regression (linearity, independence, homoscedasticity, and normality of residuals).

Step 6: Model Refinement

Based on the evaluation, the model may need refinement. This could involve adding or removing variables, transforming variables, or addressing any violations of the model's assumptions. The goal is to improve the model's predictive accuracy while maintaining simplicity.

Step 7: Validation

Finally, validate the model using a different dataset or through cross-validation techniques. This step is crucial to ensure that the model generalizes well to new data, not just the data on which it was trained.

Conclusion

Model building in multiple regression is a systematic process that requires careful consideration at each step, from defining the problem to validating the model. By following these steps, practitioners of Lean Six Sigma and other methodologies can develop robust models that provide valuable insights and help improve decision-making processes. The key is a balance between model complexity and predictive accuracy, ensuring that the model remains interpretable and applicable to real-world situations.

Real-life scenario

Let's consider a real-life scenario in which a company wants to predict the sales of its product based on advertising expenditure in different media: TV, radio, and newspapers. The company aims to allocate its advertising budget more effectively to maximize sales.

Please note that we will not explore the mathematical complexities in detail, as they are typically handled by software tools. Our focus will instead be on the process of conducting multiple regression analyses.

Step 1: Define the Problem

The company's goal is to predict sales based on advertising spend across TV, radio, and newspapers.

Step 2: Data Collection and Preparation

Imagine we have collected data for 10 periods (e.g., months) on sales (in thousands of units) and advertising spending (in thousands of dollars) across the three media.

Step 3: Selecting Variables

All three media variables (TV, Radio, Newspaper) are initially considered as potential predictors for sales.

Step 4: Model Estimation

We estimate the coefficients of the multiple regression model using the least squares method:

Step 5: Model Evaluation

After estimating the model, we will check the significance of the coefficients, the overall model fit (R-squared), and ensure the assumptions of multiple regression are met.

Step 6: Model Refinement

Based on the initial evaluation, we might decide to remove variables that are not significant or transform variables to meet model assumptions better.

Step 7: Validation

We would ideally split the data into a training and a test set to validate the model's predictive power on unseen data.

Let's proceed with the calculation for Step 4 and some parts of Step 5 using the given data.

Based on the regression analysis, the model for predicting sales based on advertising spend in TV, radio, and newspapers is:

Here's the visualization of the sales prediction model based on the advertising spend in TV and Radio. The blue dots represent the actual sales data points in relation to the TV and Radio advertising spends. The red plane illustrates the predicted sales based on our multiple regression model. Interpretation:

• Intercept (const): When there is no spending on TV, radio, and newspapers, the sales are expected to be approximately 13.5015 thousand units.
  

• TV: For each thousand dollars spent on TV ads, sales are expected to increase by 0.0332 thousand units, holding other media constant.
  

• Radio: The coefficient for radio is -0.0186, indicating that for each thousand dollars spent on radio ads, sales are expected to decrease slightly, which might seem counterintuitive and suggests further investigation is needed.
  

• Newspaper: For each thousand dollars spent on newspaper ads, sales are expected to decrease by 0.0083 thousand units, holding other media constant.

Model Evaluation:

• R-squared: The model's R-squared is 0.292, indicating that approximately 29.2% of the variability in sales is explained by the advertising spend on TV, radio, and newspapers. This is relatively low, suggesting that other factors not included in the model might explain the variation in sales.
  

• Adjusted R-squared: The adjusted R-squared is -0.062, which is negative due to the small sample size and the number of predictors. This indicates that the model might not generalize well beyond the data used to fit it.
  

• Significance of coefficients: The p-values for the coefficients of TV, Radio, and Newspaper advertising are 0.257, 0.923, and 0.946, respectively. This suggests that none of the advertising media types have a statistically significant impact on sales at common significance levels. However, the large p-values could also be a result of the small sample size.

Conclusions:

This example demonstrates the steps of building a multiple regression model to predict sales based on advertising spend. However, the model indicates that, with the given data, spending on TV, radio, and newspapers does not significantly affect sales. This outcome could lead to reconsidering the model, exploring other variables that might impact sales, or gathering more data to refine the analysis.