Guide: Multiple Regression Analysis

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Welcome to this comprehensive guide on Multiple Regression, an invaluable statistical tool that extends simple linear regression to include multiple independent variables, if you are not familiar with simple linear regression, we suggest you start with that guide first. In today’s data-driven world, understanding the relationships between various factors is essential for continuous improvement across industries. Whether you’re in manufacturing, logistics, or the public sector, mastering multiple regression can provide insights that are both actionable and impactful.

Multiple Regression Analysis Example

This guide aims to provide a deep dive into the subject, breaking it down from its fundamental concepts to its practical applications. We’ll start with the basic assumptions and components of multiple regression, guide you through the steps to perform it, and help you interpret the results for real-world decision-making. By the end of this guide, you’ll not only grasp the mathematical aspects but also learn how to apply these in an industry context, particularly in logistics.

Your journey to becoming proficient in multiple regression starts here. Let’s dive in!

Table of Contents

Basics of Regression Analysis

Simple Linear Regression

Simple linear regression is the starting point for understanding the relationship between a dependent variable and a single independent variable. It provides a linear equation that best fits a set of data points, allowing you to make predictions and understand underlying patterns. The equation takes the form:

y=β0​+β1​x+ϵ

Here, is the dependent variable you are trying to predict, is the independent variable, 0 is the intercept, 1 is the coefficient of , and is the error term.

regression analysis - checking for linearity

If you’re interested in a more comprehensive look at linear regression, please visit our detailed guide on Regression Analysis.

Why Move to Multiple Regression?

While simple linear regression is useful, its limitation lies in the fact that it only considers one independent variable. However, real-world scenarios often involve multiple factors affecting the outcome. For example, predicting delivery times in logistics isn’t just about distance; traffic conditions, weather, and vehicle type can all play a role. Multiple regression allows you to incorporate these multiple independent variables into your model, making your predictions more accurate and actionable. The equation expands as follows:

y=β0​+β1​x1​+β2​x2​+…+βn​xn​+ϵ

Applications in Continuous Improvement

Multiple regression is a powerful tool in the field of Continuous Improvement, especially for those involved in Lean Six Sigma methodologies. Here are some specific applications:

  • Process Optimization: By understanding how multiple variables affect a process, you can fine-tune it for maximum efficiency.

  • Cost Reduction: Multiple regression can help you identify which factors have the most impact on costs, allowing for targeted cost-saving measures.

  • Quality Control: In manufacturing, variables like material quality, temperature, and machine calibration can affect the final product. Multiple regression helps in isolating these factors for better quality control.

  • Forecasting: In logistics and supply chain, multiple regression can be used to forecast demand, delivery times, and inventory levels, helping in better planning and resource allocation.

By integrating multiple regression into your Continuous Improvement initiatives, you can derive insights that are both deeper and more comprehensive, leading to better decision-making and, ultimately, better results.

Multiple Regression

Definition and Basics

What is Multiple Regression?

Think of Multiple Regression as a supercharged version of Simple Linear Regression. In Simple Linear Regression, we use one thing (let’s call it “Factor A”) to guess another thing (let’s call it “Result”). For example, using the speed of a conveyor belt (Factor A) to predict the number of products made in an hour (Result).

But what if more than one thing affects the number of products made? Like, what if the skill level of workers and the quality of raw materials also matter? This is where Multiple Regression helps. It lets you use more than one factor (like conveyor belt speed, worker skill, and material quality) to guess the result (number of products made).

Mathematical Representation

In simple words, the formula for Multiple Regression looks like this:

So, if we’re trying to find out the number of products made, it would look like:

The “Random Stuff” is things we can’t measure or don’t know about.

Components: Multiple Independent Variables

In our example, the multiple independent variables are “Conveyor Belt Speed,” “Worker Skill,” and “Material Quality.” These are the things that we think could affect the number of products made.

Explanation and Examples

Imagine you’re a factory manager and you want to increase the number of products made per hour. You believe the speed of the conveyor belt, the skill level of your workers, and the quality of the raw materials all play a role.

  • Conveyor Belt Speed: You notice that a faster conveyor belt usually means more products are made.
  • Worker Skill: Skilled workers seem to make products more quickly.
  • Material Quality: Good-quality materials are easier and faster to work with, leading to more products.

You gather information on these three factors for different hours of operation and use Multiple Regression to see how each one affects the number of products made. After crunching the numbers, you find out that all three factors are important, but the skill level of workers has the biggest impact.

Multiple Regression Analysis Example

What the Graph Shows

  • The X-axis represents the speed of the conveyor belt in meters per second (m/s). You’ve noticed that a faster conveyor belt usually leads to more products being made.

  • The Y-axis shows the skill level of the workers on a scale of 1 to 10. Skilled workers seem to produce items more quickly.

  • The Z-axis indicates the quality of raw materials in percentage terms. Good-quality materials appear to be easier and faster to work with, contributing to higher production rates.

  • The color of each point represents the number of products made per hour. Darker colors indicate higher production.

Interpretation

By looking at this graph, you can get an idea of how each of these factors interacts with each other to affect the number of products made per hour. For instance, points that are darker (indicating higher production) tend to have:

  • Faster conveyor belt speeds
  • Higher worker skill levels
  • Better material quality

You can use such graphical representations to make informed decisions. Maybe you’ll decide to invest in training for your workers or opt for higher-quality materials to boost production.

Assumptions

Before you start using Multiple Regression, there are a few important things—called assumptions—that you need to check. Making sure these assumptions hold true can help you get results that you can trust. Let’s go through them one by one.

Multicollinearity

Definition

Multicollinearity happens when two or more of the factors you’re looking at are very closely related to each other. Imagine you’re a factory manager and you’re using both “Worker Skill Level” and “Years of Experience” as factors. These two might be so closely linked that it’s hard to tell which one is actually affecting the output.

Multicollinearity Example

In the scatter plot above, you can see that “Worker Skill Level” and “Years of Experience” are closely related to each other. The dots form a pattern that slopes upwards, indicating that as one variable increases, the other tends to increase as well. This close relationship is an example of multicollinearity.

In a real-world scenario like a factory setting, this could be problematic when using Multiple Regression. It becomes difficult to determine which factor—Worker Skill Level or Years of Experience—is actually influencing the output, such as the number of products made.

When you encounter multicollinearity, you might need to make adjustments like removing one of the factors or combining them into a new single factor to get more reliable results.

How to Detect

You can spot multicollinearity by looking at the data carefully. Some tools and software can also give you a number called the “Variance Inflation Factor” (VIF). A VIF greater than 5 usually means that multicollinearity might be a problem.

Solutions

If you find multicollinearity, you can try a few things:

  • Remove one of the closely related factors.
  • Combine the related factors into a single new factor.

Independence

Definition

Independence means that the factors you’re studying should not depend on each other. For example, the speed of a conveyor belt in a factory should not be affected by the quality of materials used.

Indipendence Example

The graphs above show the relationship between two factors—”Conveyor Speed” and “Material Quality”—and the number of “Products Made” in a factory.

Conveyor Speed vs Products Made

The first graph shows a pattern: as the “Conveyor Speed” increases, the number of “Products Made” also generally goes up. This suggests that the speed of the conveyor belt has an effect on production.

Material Quality vs Products Made

The second graph, on the other hand, shows no clear pattern between “Material Quality” and the number of “Products Made.” This suggests that in this example, the quality of the material doesn’t really affect the number of products made.

What Does This Mean?

In terms of the independence assumption, these graphs demonstrate that “Conveyor Speed” and “Material Quality” are independent factors affecting “Products Made.” That is, changing the material quality doesn’t seem to directly impact how many products are made, and vice versa. This is good because it means we can use both factors in a multiple regression model without worrying about them messing up our results.

By ensuring factors are independent, you make your multiple regression model more reliable and easier to interpret, which is crucial for making informed decisions in a manufacturing setting.

Importance

If the factors are not independent, your Multiple Regression results might be misleading. You might think one factor is really important when it’s actually not.

 

Homoscedasticity

Definition

This fancy word basically means that as you go along your data, the spread of the results should be about the same. In simpler terms, if you’re predicting the number of products made in an hour, the errors in your predictions should be about the same whether it’s the first hour of the workday or the last.

Homoscedasticity Example

In the graphs above, we have two scenarios that show the number of products made over time in a factory. The black dashed line in both graphs represents the “ideal” number of products that should be made each hour according to our Multiple Regression model.

Homoscedastic Example (Left Graph)

In the left graph labeled “Homoscedastic Example,” you’ll notice that the green dots (actual number of products made) are scattered around the black dashed line. Importantly, the “spread” or “distance” of these dots from the line is about the same, whether it’s the start of the workday (left side) or the end (right side). This is a good example of Homoscedasticity, where the variation in our predictions remains consistent over time.

Heteroscedastic Example (Right Graph)

In contrast, the right graph labeled “Heteroscedastic Example” shows a different pattern. Here, the spread of the green dots increases as time goes on. This indicates that the errors in our predictions are not consistent; they get worse as the day progresses. This is what we call Heteroscedasticity, and it’s something we’d like to avoid in a good Multiple Regression model.

Understanding Homoscedasticity is crucial because it ensures that the insights you derive from your model are reliable throughout the range of your data.

How to Validate

You can check for homoscedasticity by looking at a plot of your data. If the plot looks like a random scatter without any pattern, you’re probably good to go. If not, you might need to adjust your model.

Normality

What it Means

Normality means that if you look at the errors in your predictions, they should roughly form a bell curve. This helps make sure that your results are reliable.

Test for normality Example

The graph above is an example of what normality in prediction errors would look like. The curve you see is commonly called a “bell curve” or a “normal distribution.” In this idealized example, the errors in the predictions are spread out in a way that most of them are close to zero, and fewer errors are found as you move further away from zero.

In a real-world scenario, if the errors in your predictions form a curve similar to this, you can be more confident that your Multiple Regression model is reliable.

Tests for Normality

There are statistical tests like the Shapiro-Wilk test that can help you check for normality. But a simpler way is to look at a plot of your errors. If it looks like a bell curve, you’re likely in the clear.

Steps to Perform Multiple Regression

Doing Multiple Regression involves a few key steps. Each step is important to make sure you get reliable and useful results. Let’s dig into each one.

Data Collection

Before you can run any kind of analysis, you need data—lots of it!

Types of Data Needed

  1. Dependent Variable: This is what you’re trying to predict or understand. For example, the number of products manufactured in an hour.

  2. Independent Variables: These are the factors you think might affect the dependent variable. Like conveyor belt speed, worker skill level, and quality of materials.

  3. Control Variables: Sometimes there are other factors that you know affect the dependent variable, but you’re not studying them right now. You’ll want to collect data on these too so you can account for them in your analysis.

Sources

Where you get your data from is crucial. Here are some common sources:

  1. Internal Records: If you’re a factory manager, you might already have records on production levels, worker performance, and so on.

  2. Surveys and Questionnaires: Sometimes, you need to gather data yourself. Surveys can help capture things like worker satisfaction.

  3. Public Databases: For some projects, you might find useful data in public or industry-specific databases.

Here is an example dataset that you could use for Multiple Regression analysis in a manufacturing context:

Products_MadeConveyor_SpeedWorker_SkillMaterial_QualityMaintenance_Level
942.28777
972.45588
502.16687
531.51768
532.12587
892.17879
592.47699
692.38698
712.01497
861.56577

  • Products_Made: This is our dependent variable. We’re trying to understand how many products are made in an hour.

  • Conveyor_Speed: This represents the speed of the conveyor belt in meters per second (m/s).

  • Worker_Skill: This is a rating of worker skill level on a scale from 1 to 10.

  • Material_Quality: This is a rating of the quality of the materials used, also on a scale from 1 to 10.

  • Maintenance_Level: This is our control variable. It’s a rating of how well the machines are maintained, on a scale from 1 to 10.

In this example, the data for the independent variables and the control variable could be collected from internal records, sensors on the factory floor, and worker assessments. The dependent variable data could come from production logs.

Data Visualization

Once you have your data, the next step is to take a good look at it. This is where data visualization comes in.

Ways to Visualize the Data

  • Scatter Plots: As shown above, scatter plots can help you see how each factor might be affecting the number of products made. For example, you can see if higher conveyor speeds generally lead to more products made.

  • Correlation Heatmaps: This would help you quickly understand the strength and direction of the relationship between variables.

Correlation Heatmaps Example

  • Box Plots: These could be useful for understanding the distribution of each variable, helping you spot outliers or errors in the data.

Box Plots Example

  • Pair Plots: These plots show pairwise relationships in your dataset. This can help you understand how variables interact with each other, not just the dependent variable.

Pairplots Example

  • Histograms: These can be useful for understanding the distribution of each independent variable and the dependent variable. It helps in checking the assumption of normality.

Histogram examples

Importance

Visualizing your data helps you in several ways:

  1. Spotting Patterns: Sometimes, just looking at a bunch of numbers doesn’t tell you much. Charts and graphs can help you see patterns you might miss otherwise.

  2. Finding Errors: If some of your data looks “off” in a chart, you can catch it before you run your analysis.

  3. Understanding Relationships: Visuals can help you get an early idea of how your independent variables might be related to your dependent variable.

Tools and Techniques

  1. Spreadsheet Software: Programs like Excel have basic charting functions. These are easy to use and good for simple data sets.

  2. Statistical Software: More advanced tools like R or Python’s libraries (like Matplotlib or Seaborn) offer more types of graphs and are better for complex data.

  3. Business Intelligence Tools: If you’re working in a business environment, BI tools like Tableau can be very helpful for data visualization.

Model Fitting

Once you’ve collected and visualized your data, the next crucial step is fitting the model. This is where the magic happens: the software takes all your data and calculates how each independent variable affects the dependent variable.

Software Options

  1. Excel: Good for simple models and great for quick-and-dirty analyses. However, it lacks the more advanced statistical tests and diagnostic plots that other software offers.

  2. R: A powerful statistical software that’s highly extensible. Perfect for complex models but has a steeper learning curve.

  3. Python (with libraries like scikit-learn or Statsmodels): Similar to R in its capabilities but perhaps more versatile because Python is a general-purpose programming language.

  4. SPSS, SAS, Stata: These are more specialized statistical software options commonly used in academia and certain industries. They are user-friendly but come with a cost.

  5. Business Intelligence Tools: Software like Tableau now offers some regression capabilities, though they are not as flexible or detailed as dedicated statistical software.

Step By Step Guide to Multiple Regression Analysis in Excel

 Step 1: Choose Your Software

For this guide, we’re using Excel. Make sure you have it installed and open it.

Step 2: Input Data

  1. Open a new Excel worksheet.
  2. In the first row, enter the column headers: Products_Made, Conveyor_Speed, Worker_Skill, Material_Quality, and Maintenance_Level.
  3. Paste your data under each corresponding column header.

Example data in Excel

If you would like to practice with our example data can do so

Example Data Download

Step 3: Specify Model

  1. Go to the Data tab on the Excel Ribbon.
  2. In the Analysis group, click on Data Analysis. If you don’t see Data Analysis, you’ll need to install the Data Analysis Toolpak.

Data Analysis in Excel

  1. From the dialog box, select Regression and then click OK.

Select Regression

Step 4: Run Model

  1. A new dialog box for Regression will appear.
  2. In the Input Y Range field, select the range of cells containing your dependent variable (Products_Made).
  3. In the Input X Range field, select the range of cells containing your independent variables (Conveyor_Speed, Worker_Skill, Material_Quality, etc.).
  4. Under Output Range, choose where you want the output to be displayed.
  5. Make sure to tick the box that says Labels if you have included column headers.
  6. You can also tick the Confidence Level box and set it to 95% (or your desired level).
Regression Settings
  1. Click OK to run the model.
Example Regression Output

Regression Statistics:

  1. Multiple R: The correlation coefficient is 0.861, which indicates a strong correlation between the predictors and the dependent variable.

  2. R Square: The R-squared value is approximately 0.74, indicating that around 74% of the variability in the dependent variable is explained by the model. This is a good fit.

  3. Adjusted R Square: The adjusted R-squared value is 0.732, which is also good. It’s slightly less than the R-squared value, indicating that the model isn’t overly complex.

  4. Standard Error: A value of 5.28 suggests that the model’s predictions are relatively close to the actual data points.

  5. Observations: You have 99 observations, which is generally a good sample size for regression analysis.

Dont worry if none of this makes sense yet. We will explain it in more detail in the following section. 
 

ANOVA:

  1. Significance F: The F-statistic has an extremely low p-value (almost zero), indicating that the model is statistically significant. This means that at least one predictor variable has a significant relationship with the dependent variable.

Coefficients:

  1. Intercept: The intercept is 21.5, which is the estimated number of products made when all the independent variables are zero.

  2. Coefficients for Independent Variables: All the p-values for the coefficients are less than 0.05, indicating that they are all statistically significant. The coefficients themselves tell you about the direction and strength of the relationship between each independent variable and the dependent variable.

  3. Confidence Intervals: The lower and upper 95% intervals give you a range in which you can expect the true population parameter to lie.

Step 5: Check Assumptions

  1. Normality: Excel does not provide direct tests for normality. However, you can create a histogram or a Q-Q plot of the residuals to check for normality visually.

  2. Multicollinearity: Excel doesn’t directly provide the Variance Inflation Factor (VIF). You could either calculate it manually or use more advanced software for this check.

  3. Homoscedasticity and Independence: These assumptions can be checked by plotting the residuals against the predicted values. In Excel, you can plot the RESIDUAL output column against the Predicted Value output column.

After running the model, you’ll get an output with coefficients, intercept, R-squared value, and p-values, among other statistics. Use these to interpret your model, as discussed in the earlier section on interpretation.

Interpretation of Results

Understanding the output of a multiple regression analysis is crucial for making informed decisions. Let’s dive into the key components of the output:

Coefficients and Intercept

What They Mean:

  1. Intercept: This is the value of the dependent variable when all independent variables are zero. In a manufacturing context, it could represent the base number of products made when all influencing factors are at their minimum level.

  2. Coefficients: These numbers tell you how much the dependent variable changes when you change each independent variable by one unit, keeping all other variables constant.

How to Use Them:

  1. Predicting Outcomes: You can use the coefficients and intercept to predict the dependent variable. The general formula is:

 

Dependent Variable=Intercept+(Coefficient 1×Variable 1)+(Coefficient 2×Variable 2)+…

  1. Decision Making: Understanding the size and sign of each coefficient can help you focus on which variables need attention for optimization.

R-squared and Adjusted R-squared

Definition:

  1. R-squared: This is the proportion of the variance in the dependent variable that is explained by the independent variables. It ranges from 0 to 1.

  2. Adjusted R-squared: This is a modified version of R-squared that has been adjusted for the number of predictors in the model. It penalizes you for adding variables that don’t improve the model.

Importance:

  1. Goodness of Fit: Both these metrics tell you how well your model fits the data. Higher values generally mean a better fit.

  2. Model Comparison: Adjusted R-squared is useful for comparing models with different numbers of predictors.

Significance Testing

p-values:

  1. Definition: The p-value for each coefficient tests the null hypothesis that the coefficient is equal to zero. A low p-value indicates that you can reject the null hypothesis and conclude the variable is significant.

  2. Usage: In general, a p-value below 0.05 is considered significant. This can guide you on which variables are most important.

Confidence Intervals:

  1. Definition: These intervals provide an estimated range the coefficient is likely to fall within.

  2. Usage: If the confidence interval for a coefficient doesn’t cross zero, it’s a good sign that the variable is a significant predictor.

Limitations and Caveats

Understanding the limitations and potential pitfalls of multiple regression is crucial for making accurate and meaningful interpretations. Let’s dive into some of these aspects:

When to Use and When Not to Use

When to Use:

  1. Multiple Influencing Factors: Use multiple regression when you have more than one independent variable affecting the dependent variable. This is common in sectors like manufacturing, where factors such as worker skill and material quality both impact the production rate.

  2. Predictive Modeling: When you need to forecast a dependent variable based on various known factors, multiple regression is a good choice.

When Not to Use:

  1. Not Enough Data: If you don’t have enough data points, your model might not be reliable. A general rule of thumb is to have at least 20 observations per independent variable.

  2. Non-linear Relationships: Multiple regression assumes a linear relationship between variables. If the relationship is curved, consider using other techniques like polynomial regression.

Common Mistakes

  1. P-Value Fishing: Adding variables just to get a low p-value can make your model misleading. Always have a theoretical basis for including variables.

  2. Ignoring Multicollinearity: If two or more variables are highly correlated, it can make it difficult to determine the effect of each one individually.

Overfitting

What It Is: Overfitting occurs when your model is too complex and starts to capture the random noise in the data, rather than the actual relationship.

How to Avoid:

  1. Variable Selection: Use techniques like backward elimination or forward selection to include only significant variables.

  2. Cross-Validation: Split your data into training and testing sets to validate the model’s predictive power.

Ignoring Assumptions

What It Means: Multiple regression has several underlying assumptions like normality, linearity, and homoscedasticity. Ignoring these can make your results invalid.

How to Check:

  1. Normality: Use plots or tests to check if the residuals are normally distributed.

  2. Homoscedasticity: Plot residuals against predicted values to check for constant variance.

  3. Independence: Make sure observations are independent of each other, especially in time-series data where this might not be the case.

Being aware of these limitations and caveats ensures that you apply multiple regression appropriately and interpret its results with caution. This is particularly relevant for sectors like manufacturing or logistics where data-driven decisions are crucial.

Conclusion

In this guide, we’ve navigated the complexities of Multiple Regression, a powerful statistical tool crucial for understanding the interplay among various factors in sectors like manufacturing, logistics, and beyond. From setting up your data to interpreting results, each step has been detailed to empower you to make informed, data-driven decisions. However, it’s vital to acknowledge the limitations and assumptions that come with this tool. Ignoring them can lead to misleading results and poor decisions.

By being meticulous in data collection, cautious in interpretation, and mindful of assumptions, you can leverage Multiple Regression as a reliable instrument in your toolkit for continuous improvement. This guide aims to be a comprehensive resource for both beginners and seasoned professionals who are keen on harnessing the power of data to drive efficiency and innovation.

References

A: Multiple Regression is a statistical technique that allows you to understand the relationship between one dependent variable and two or more independent variables. It’s essential for making informed decisions in various fields like manufacturing, logistics, and healthcare by helping to predict outcomes and identify key factors affecting performance.

A: Generally, it’s recommended to have at least 20 observations per independent variable for a reliable model. A small dataset may lead to unreliable estimates and overfitting, where the model performs well on the existing data but poorly on new or unseen data.

A: Look at the R-squared and Adjusted R-squared values. Higher values (closer to 1) indicate a better fit. Also, check the p-values of the coefficients; values less than 0.05 generally indicate significance. Don’t forget to validate the model’s assumptions like normality, multicollinearity, and homoscedasticity.

A: Some common pitfalls include overfitting the model by adding too many variables, ignoring the assumptions of the model, and not validating the model on a separate dataset. Also, beware of ‘p-value fishing,’ where variables are included just to achieve statistical significance.

A: Coefficients indicate the change in the dependent variable for a one-unit change in the independent variable while holding other variables constant. For example, a coefficient of 3 for material quality means that for each unit increase in material quality, the dependent variable (e.g., products made) increases by 3 units, assuming other factors remain constant.

Author

Daniel Croft

Daniel Croft

Daniel Croft is a seasoned continuous improvement manager with a Black Belt in Lean Six Sigma. With over 10 years of real-world application experience across diverse sectors, Daniel has a passion for optimizing processes and fostering a culture of efficiency. He's not just a practitioner but also an avid learner, constantly seeking to expand his knowledge. Outside of his professional life, Daniel has a keen Investing, statistics and knowledge-sharing, which led him to create the website learnleansigma.com, a platform dedicated to Lean Six Sigma and process improvement insights.

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