Simple Linear Regression is a foundational tool used in Lean Six Sigma and various statistical analysis methodologies aimed at process improvement and optimization. Within this context, building a linear regression model involves a series of steps designed to understand and quantify the relationship between two variables. The objective is to use one variable (the independent variable) to predict the value of another (the dependent variable), assuming that the relationship between the two is linear. This article will guide you through the step-by-step process of model building in the context of Simple Linear Regression.

Step 1: Define the Problem

The first step in the model-building process is to clearly define the problem or the business question that needs answering. This involves identifying the dependent variable (the outcome or the factor you are trying to predict or explain) and the independent variable (the predictor or the factor you believe has an influence on the dependent variable). The clarity of this step is crucial for the direction of the analysis.

Step 2: Collect Data

Once the problem is defined, the next step is to collect data. The data should be relevant to the variables identified in the first step. It’s important to ensure the quality and integrity of the data collected, as this will significantly impact the model's accuracy and reliability. Data can be collected from existing databases, through observations, surveys, or experiments.

Step 3: Explore and Prepare the Data

After collecting the data, it’s essential to explore and prepare it for analysis. This step involves cleaning the data (handling missing values, removing outliers, etc.), visualizing the data to understand its distribution and to identify any apparent relationships or patterns, and preparing the data for modeling. Data preparation may also include transforming variables to better fit the linear model assumptions.

Step 4: Specify the Model

Specify the model by identifying which form of the linear equation will be used. The simplest form of a linear regression model is:

y=β0​+β1​x+ϵ

where: y is the dependent variable x is the independent variable, β0​ is the y-intercept, β1​ is the slope of the line ϵ is the error term.

Step 5: Estimate the Model Parameters

Using statistical software, estimate the model parameters (β0​ and β1​). This is typically done through a method called Ordinary Least Squares (OLS), which minimizes the sum of the squared differences between the observed values and the values predicted by the model.

Step 6: Validate the Model

After estimating the model parameters, it’s important to validate the model to ensure its reliability and accuracy. This involves checking the model assumptions (linearity, normality, homoscedasticity, and independence of errors), analyzing residuals, and potentially using goodness-of-fit measures like R-squared to understand how well the model explains the variation in the dependent variable.

Step 7: Interpret the Results

Interpret the model coefficients (β0​ and β1​) to understand the relationship between the independent and dependent variables. The coefficient β1​ indicates the change in the dependent variable for a one-unit change in the independent variable, holding all other factors constant.

Step 8: Use the Model for Prediction or Decision Making

Finally, use the model to make predictions or to inform decision-making. In Lean Six Sigma projects, this often involves using the model to identify key factors affecting a process and to predict the impact of changes to those factors.

Step 9: Refine the Model

Model building is an iterative process. Based on the results and the validation process, you might need to go back, collect more data, try different transformations, or even redefine the problem. Continual refinement is key to developing a robust model that accurately reflects the relationship between variables.

Simple Linear Regression in Lean Six Sigma is a powerful tool for identifying and quantifying relationships between variables. Following this step-by-step process helps ensure that the model built is valid, reliable, and useful for making informed decisions in process improvement initiatives. Let's dive into a real-life example involving Simple Linear Regression to understand the step-by-step model-building process. Imagine a small business wants to improve its sales strategy by understanding the relationship between its advertising spending (independent variable) and sales revenue (dependent variable).

Scenario: Impact of Advertising on Sales Revenue

The business has collected data over the past 12 months, tracking monthly advertising spending and the corresponding sales revenue for that month.

Step 1: Define the Problem

Objective: Determine how changes in advertising spending affect sales revenue.

• Independent variable (X): Advertising Spending (in thousands of dollars)

• Dependent variable (Y): Sales Revenue (in thousands of dollars)
  

Step 2: Collect Data

Here is a small dataset representing the monthly data:

Step 3: Explore and Prepare the Data

Let's assume the data is clean and ready for analysis. Normally, this step would involve visualizing the data and potentially transforming it, but we'll proceed with the data as-is for simplicity.

Step 4: Specify the Model

We will use the simple linear regression model: Y=β0​+β1​X+ϵ.

Step 5: Estimate the Model Parameters

We need to estimate β0​ (the intercept) and β1​ (the slope) using the Ordinary Least Squares (OLS) method. Slope (β1​):

The formula to calculate the slope (β1​) of the regression line is:

Where:

• n is the number of observations,

• ∑XY is the sum of the product of each pair of X and Y values,

• ∑X and ∑Y are the sums of X and Y values respectively,

• ∑X^2 is the sum of the squares of X values.

Given the data:

• n=12 (since there are 12 months of data),

• X represents the advertising spending,

• Y represents the sales revenue.

• ∑X=54 

• ∑Y=1014

• ∑XY=6260.5

• ∑X^2=301.5

Plugging these values into the formula gives us:

Intercept (β0​):

The formula for the intercept (β0​) is:

Where:

• Yˉ is the mean of the Y values,

• Xˉ is the mean of the X values.

Given the data:

• Yˉ = 54 / 12 = 4.5

Xˉ = 1014 / 12 = 84.5

Plugging the mean values and the calculated slope (β1​) into the formula gives us:

β0​=84.5−8.94×4.5≈42.05

Step 6: Validate the Model

For our example, let's check the linearity through a scatter plot of the observed vs. predicted values and compute the value to assess the goodness-of-fit. We'll also visualize the distribution of residuals to evaluate the assumption of normality.

The validation step yields the following insights:

Observed vs. Predicted Values: The scatter plot shows a strong linear relationship between the observed and predicted values, with the points closely aligned along the line representing perfect predictions. This suggests the model has a good fit for the data.

Residuals vs. Predicted Values: The plot of residuals against predicted values does not show any apparent pattern, indicating homoscedasticity, or equal variance of residuals across the range of predictions. This is a good sign that the model meets the assumption of homoscedasticity.

Distribution of Residuals: The histogram of residuals, with a superimposed kernel density estimate, suggests that the residuals are approximately normally distributed, which is another assumption of linear regression.

The calculated R^2value is approximately 0.982, indicating that the model explains about 98.2% of the variance in sales revenue based on advertising spending. This high R^2 value indicates a very good fit of the model to the data.

These validation steps suggest that the simple linear regression model is reliable and performs well for this particular dataset, making it a useful tool for predicting sales revenue based on advertising spending. However, it's always important to remember that the validation should be thorough, and other diagnostic tests may be necessary depending on the complexity of the data and the specific requirements of the analysis.

Step 7: Interpret the Results

The linear regression equation can be written as:

Y=42.05+8.94X

This means for every thousand dollars spent on advertising, sales revenue is expected to increase by approximately $8,940, assuming the model holds true.

Step 8: Use the Model for Prediction

If the business plans to spend $8,000 (8 in our model's scale) on advertising in a month, we can predict the sales revenue for that month:

Y=42.05+8.94×8

Let's calculate this. ​​

With an advertising spending of $8,000, the model predicts that the sales revenue for that month would be approximately $113,545.

Step 9: Refine the Model

In a real-world scenario, we might refine the model by considering additional variables that could affect sales revenue, checking model assumptions more rigorously, or using a larger dataset for a more robust analysis.

This example provides a practical application of Simple Linear Regression in a business context, demonstrating how it can be used to inform strategic decisions based on data-driven insights. ​