9 min readMar 20, 2023

Dear friends!

Regression modeling is a versatile and widely used statistical method that can be used to investigate and quantify the relationships between variables and make inferences about causal relationships. Versatile and powerful, it’s perfect for predicting future trends, deciphering interventions, pinpointing influential factors, or sizing up relationships. In this article, I will walk you through the fascinating concept of regression modeling, providing insight into its underlying theory and practical applications. Are you ready? Let’s go! 🚀

Regression analysis

Linear regression models come in two enticing flavors: simple and multiple. Simple linear regression, the more straightforward sibling, involves just one independent variable. It’s the perfect gateway to regression analysis for beginners.

Multiple linear regression is the refined and complex elder sibling, involving two or more independent variables. It’s like upgrading from a tricycle to a mountain bike, providing you with the robustness and versatility to tackle intricate relationships in the data.

Key Concepts in Regression Modeling

Independent and dependent variables

The dependent variable (Y), often referred to as the endogenous variable, is the outcome we aim to explain or predict, relying on the influence of the independent variables (X).

In linear regression, the relationship between the dependent and independent variables can be summarized using three primary parameters: α (constant term), β (coefficient), and ε (noise term). These parameters form the basis of the linear regression equation, Y = α + βX + ε. Here, α represents the y-intercept, the point at which the regression line crosses the y-axis. β, the coefficient, quantifies the slope of the regression line, indicating the strength and direction of the relationship between the independent (X) and dependent (Y) variables. Lastly, ε captures the noise or error term, accounting for the variability in the data that cannot be explained by the linear relationship between X and Y. By altering the values of independent variables, we can observe their impact on the dependent variable.

Model assumptions

Linear regression models are only as good as the assumptions on which they are based, and violations of these assumptions can lead to biased and misleading results. To get a more detailed insight, please refer to this ➡️ article.

Before deploying your linear regression models in real-world applications, you should review the underlying assumptions of the model. There is a series of tools available to support you in the process, ranging from visual residual plots, with which most analysts begin, to statistical tests such as the Shapiro-Wilk test (Normality Tests), Breusch-Pagan test (Homoscedasticity Tests), and Variance Inflation Factor (Multicollinearity Tests). Visualizing residuals remains a potent tool for identifying potential violations of these assumptions or pinpointing potential outliers that you might have missed in the exploratory data analysis (EDA) stage (see below). If these assumptions are violated, addressing them may require engaging in further EDA to better understand the underlying patterns and potential anomalies in the data, which might inform the development of more accurate and effective approaches (e.g., data transformations, application of regularization techniques, or a switch to alternative non-linear modeling techniques).

Residual Plots: Checklist Model Assumptions

We already established that checking for outliers is a necessary step in every data analysis process, as these anomalous data points can significantly skew results and lead to erroneous regression models. To maintain the integrity and reliability of your analysis and model, I suggest reading my articles regarding data cleaning ➡️Intro Data Cleaning, ➡️Detect Outliers, ➡️Deal with Outliers.

Hypothesis testing

Hypothesis testing, the thrilling detective work of regression analysis, enables you to scrutinize the significance of variable relationships. In the context of linear regression, hypothesis testing typically involves the following steps:

State the null hypothesis (H0): There is no relationship between the independent and dependent variables (i.e., the coefficient of the independent variable is equal to zero).
State the alternative hypothesis (H1): There is a relationship between the independent and dependent variables (i.e., the coefficient of the independent variable is not equal to zero).
Calculate the test statistic, such as the t-statistic or F-statistic, which measures the difference between the observed relationship and the null hypothesis.
Determine the p-value, which represents the probability of observing the test statistic (or a more extreme value) if the null hypothesis were true.
Compare the p-value to a predetermined significance level (commonly 0.05 or 0.01). If the p-value is less than the significance level, reject the null hypothesis in favor of the alternative hypothesis, concluding that there is a significant relationship between the variables.

Interpretation of the Coefficients

Regression coefficients denote the average variation in the dependent variable corresponding to a single-unit change in the independent variable, while keeping all other independent variables constant. These coefficients define the slope of the regression line, signifying the strength and direction of the relationship between dependent and independent variables; positive coefficients indicate direct relationships, and negative coefficients suggest inverse relationships. When interpreting regression coefficients, it is essential to take causality into account. This is because correlation does not imply causation, and it is possible that other factors are influencing the relationship between the dependent and independent variables.

👣Let’s consider an example of housing prices and ice cream sales. A regression analysis might reveal a positive relationship between these two variables, with higher ice cream sales associated with higher housing prices. One might be tempted to conclude that increasing ice cream sales will lead to an increase in housing prices or vice versa.

The underlying economic relationship between housing prices and ice cream sales, however, is not direct. Instead, a lurking variable, such as temperature or seasonality, might be influencing both variables. During summer months, for instance, warmer weather might lead to an increase in both ice cream sales and housing market activity. In this case, it would be incorrect to infer that ice cream sales directly impact housing prices, as the true causal relationship lies with the lurking variable (temperature or seasonality).

Goodness-of-fit measures

Performance evaluation metrics, or goodness-of-fit measures, are essential tools for assessing the effectiveness and predictive accuracy of regression models. The most prevalent goodness-of-fit measures include:

These goodness-of-fit measures help analysts evaluate the performance of regression models and make informed decisions about model selection, improvement, and interpretation.

Building a Linear Regression Model

Now that we have our variables in place, it’s time to obtain the best-fit line. We employ the Ordinary Least Squares (OLS) method, which is the most commonly used method for estimating the best fit line in linear regression modeling. OLS works by minimizing the sum of the squared differences between the actual data points and the predicted values based on the regression line. Let’s dive a little deeper into how OLS works, with the help of this graph.

Imagine you have a scatter plot of data points representing the relationship between two variables, say ‘X’ and ‘Y’. In linear regression, we assume there is a linear relationship between these variables. The goal is to find a straight line (blue) that best represents this relationship.

Here’s a step-by-step explanation of how OLS works:

Define the regression line: The equation for the regression line is Y = a + bX + e, where ‘Y’ is the dependent variable, ‘X’ is the independent variable, ‘a’ is the intercept, ‘b’ is the slope, and ‘e’ is the error term.
Calculate the differences: For each data point, calculate the vertical difference between the actual ‘Y’ value and the predicted ‘Y’ value based on the regression line. These differences are called residuals or errors.
Square the differences: Next, square each residual. Squaring ensures that positive and negative differences do not cancel each other out, and it emphasizes larger deviations.
Minimize the sum of squared differences: OLS seeks to find the values of ‘a’ (intercept) and ‘b’ (slope) that minimize the sum of the squared residuals. This is called the “least squares” criterion.
Identify the best fit line: Once the optimal values of ‘a’ and ‘b’ are found, the best fit line can be plotted on the graph, representing the linear relationship between ‘X’ and ‘Y’ with the least amount of error.

In the analysis of our car dealer dataset (LINK), the Ordinary Least Squares (OLS) method was employed to estimate the optimal multiple regression model, yielding an adjusted R-squared value of 73.69%. This result implies that an increase in mileage or age negatively affects car sales prices. Conversely, vehicles with greater horsepower and higher fuel efficiency (MPG) contribute to increased sales prices.

✅Advantages and Disadvantages❌

A standout advantage of linear regression models is their simplicity and interpretability. These models offer a clear view of the relationships between independent and dependent variables. When the relationships in your data are essentially linear, these models can provide excellent predictive performance with minimal computational resources. The coefficients of the model provide direct insights, allowing users to see which variables have the greatest impact on the predicted outcome, and in which direction that impact lies. Additionally, linear regression is versatile, with variants like ridge and lasso regression that can handle multicollinearity and prevent overfitting.

However, linear regression models come with their set of challenges. They inherently assume a linear relationship between the dependent and independent variables, which may not always hold true in real-world scenarios. This can lead to underfitting, where the model fails to capture the underlying patterns in the data. Multicollinearity, the presence of high correlations among independent variables, can inflate the variance of coefficient estimates, making them difficult to interpret and potentially unreliable. Outliers in the data can disproportionately impact the regression line, leading to inaccurate predictions. Moreover, linear regression models are sensitive to heteroscedasticity, where the variance of errors differs across data points, potentially leading to inefficient and biased estimates.

👣Practical Applications of Linear Regression

Linear regression’s ability to model and interpret relationships between variables has found its footing in many real-world applications. This modeling technique is invaluable in scenarios demanding prediction or understanding of the weight of driving factors. Here are a few noteworthy applications:

🏘️ Real Estate Valuation: Estimating the price of a property can be complex, given the numerous influencing factors such as size, location, age, and amenities. Real estate professionals employ linear regression to predict property prices based on these features, ensuring both sellers and buyers get a fair deal.
📊 Sales Prediction: For businesses, predicting future sales can aid in inventory management, staffing, and marketing efforts. Linear regression can model the relationship between sales and factors like advertising spend, seasonality, and competitor activity, providing businesses with forecasts to strategize effectively.
📈 Stock Market Predictions: In the financial sector, analysts employ linear regression to predict stock prices. By considering factors like historical returns, trading volumes, and economic indicators, they aim to provide forecasts that can guide investment strategies. Although the stock market is influenced by myriad factors and can be inherently volatile, linear regression provides a foundational tool for making informed predictions.

Regression modeling is the gateway to advanced machine learning techniques like deep learning and neural networks. As data volumes continue to grow, mastering regression modeling becomes an invaluable skill for those seeking to excel in the field of data analytics.