Variable | Standardized Coefficient | Real Coefficient | t-stat | VIF | Correlation |
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R2 | MAPE | AIC |
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Variable | Residual Correlation |
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Our feature-rich Linear Regression App serves as an indispensable tool for conducting various linear regression techniques on your dataset. With applications ranging from linear regression with regularization methods for handling multicollinearity to exploratory data analysis (EDA) prior to fitting Bayesian Marketing-Mix models, this comprehensive guide delves into the software's features, usage, input CSV data requirements, and outputs in detail

Before utilizing the Linear Regression App, ensure that your data is provided in CSV format. The CSV file should include rows representing data points and columns signifying variables. The first column must contain the dates, the second column the dependent variable, followed by columns for each independent variable. The header row should list the names of the variables.

Verify that your data is clean and free of missing values or inconsistencies, as such issues could compromise the accuracy of your results. If necessary, preprocess your data using appropriate imputation or data cleaning techniques before uploading the CSV file to the app. Performing these steps will ensure that your dataset is in the best possible condition for analysis.

Once you have uploaded your CSV file, you can choose from an array of linear regression methods to best suit your analysis:

- OLS (Ordinary Least Squares) – This classic linear regression method minimizes the sum of squared residuals, finding the best-fitting line for the given data points. OLS is a popular choice for EDA as it provides a quick and straightforward way to understand the relationships between variables.
- Positive OLS – A variation of OLS that enforces positivity constraints on the coefficients, ensuring that they remain non-negative during the estimation process. This method can be useful in situations where negative coefficients are not meaningful or could lead to incorrect interpretations.
- Lasso – This regularization method adds an L1 penalty to the OLS objective function, helping to address multicollinearity and perform feature selection by shrinking some coefficients to zero. Lasso can be especially useful for high-dimensional datasets or when performing EDA to identify the most relevant predictors.
- Ridge – Another regularization method that adds an L2 penalty to the OLS objective function, reducing the impact of multicollinearity by preventing the coefficients from becoming too large. Ridge regression is often employed when dealing with correlated predictors or when aiming to improve model stability.
- LARS (Least Angle Regression) – An algorithm for fitting linear regression models, particularly useful when the number of predictors is much larger than the number of observations. LARS selects a subset of relevant predictors during the estimation process, which can be helpful for EDA and feature selection.
- Elastic Net – A combination of Lasso and Ridge techniques, Elastic Net balances feature selection with multicollinearity reduction by applying both L1 and L2 penalties to the OLS objective function. This method can be advantageous when there are groups of correlated variables, as it tends to select all variables within the group or none at all, making it easier to interpret the results.

For regularized regression methods like Lasso, Ridge, and Elastic Net, you can also adjust the "Alpha" value, which serves as a tuning parameter that controls the strength of the regularization. Selecting the appropriate alpha value is crucial for striking the right balance between model complexity and predictive accuracy.

After you have chosen your desired method and submitted your data, the Linear Regression App will furnish the following outputs:

- A summary of the results, complete with R2 (coefficient of determination) to evaluate the proportion of the variance in the dependent variable explained by the model, MAPE (Mean Absolute Percentage Error) to gauge the model's accuracy by contrasting actual and predicted values, and AIC (Akaike Information Criterion) for comparing models with varying numbers of parameters while penalizing those with more complex structures.
- A table of selected variables, encompassing their standardized and real coefficients, t-statistics for hypothesis testing (determining whether a variable has a statistically significant impact on the dependent variable), VIF (Variance Inflation Factor) to diagnose multicollinearity by quantifying the extent to which the variance of the estimated regression coefficients is increased due to multicollinearity, and correlations with the dependent variable to measure the strength and direction of the linear relationship between each independent variable and the dependent variable.
- A table of non-selected variables and their residual correlations, enabling you to identify potentially relevant variables that were not included in the model. This information can prove invaluable for further analysis and model refinement.
- AVM (Actual vs. Model) Plot – A visual comparison of the actual values of the dependent variable and the values predicted by the model, allowing you to assess the overall fit of the model and identify potential patterns or discrepancies in the residuals, which could suggest areas for improvement.
- Individual Variables Plot – A plot that visualizes the impact of individual variables on the dependent variable, facilitating the interpretation of the relationship between each independent variable and the dependent variable. You can choose to display the raw variable values, the impact of the variable, and/or the residual values, and select between single or dual scale axes for easier interpretation.

The Linear Regression App is an excellent tool for EDA, enabling you to uncover patterns, trends, and relationships in your data that can inform subsequent analyses. By examining the outputs provided, you can gain a deeper understanding of the relationships between variables, identify potential multicollinearity issues, and discover the most relevant predictors for your dependent variable.

This exploratory process can be particularly useful when preparing to fit a Bayesian MMM. By first conducting EDA with the Linear Regression App, you can make informed decisions regarding which variables to include in your Bayesian analysis, the appropriate priors to use, and how to structure your model. This ultimately leads to a more accurate and efficient Bayesian model, allowing you to extract even more value from your data.

With its versatile functionality, the Linear Regression App is well-suited for various real-world applications, such as marketing mix modeling. By analyzing the coefficients and the strength of the relationships between independent variables (e.g., marketing channels, promotional activities, pricing strategies) and the dependent variable (e.g., sales, conversions, revenue), you can gain insights into the effectiveness of your marketing mix and make data-driven decisions to optimize your marketing strategy.

Similarly, the app's regularization techniques, such as Lasso and Ridge, can address multicollinearity in your data, ensuring a more stable and interpretable model. This is particularly useful in situations where you have a large number of correlated predictors, which can lead to unstable estimates and hinder the interpretability of your model.