What is the best way to undertake a multiple linear regression?

Multiple linear regression is a standard technique implemented by any serious statistics package. The legacy way of rigorous inference is using hypothesies tests, e.g. t-tests. The now recommended procedure of interpretation is by using coefficient tables and draw conclusions on the magnitude of effect by looking at the center estimate and on the level of certainty by looking at the confidence or credibility intervals. https://schmettow.github.io/New_Stats/mpm.html#mrm

Since you ask about multiple independent variables, there is a pitfall especially with the legacy way. If you have many potential predictors there is a risk of contamination by researchers degree of freedom. That means you are trying all predictors, but only report those that are "significant". This is a questionnable research practice.

If your goal is to explore and find the model with the best predictive accuracy (rather than testing a prior hypotheses!), model selection by information criteria is useful to find the best combination of predictors.

https://schmettow.github.io/New_Stats/wwm.html#model-comp

Munazza Kanwal

Understand the Problem: Clearly define your research question or problem. Identify the dependent variable (the one you want to predict) and the independent variables (the ones you think might influence the dependent variable).

Data Collection: Gather the necessary data for the dependent and independent variables. Ensure your data is accurate, complete, and representative of the population you're studying.

Data Preprocessing:Missing Data: Deal with missing values through imputation methods or removing incomplete cases. Outliers: Identify and handle outliers, which can significantly influence regression results. Variable Scaling: If your independent variables are on different scales, consider standardizing or normalizing them to ensure fairness in their impact.

Assumptions Checking: Before conducting the regression analysis, make sure to check the assumptions of linear regression:Linearity: The relationship between the dependent and independent variables should be linear. Independence: The residuals (errors) should be independent of each other. Homoscedasticity: The residuals should have constant variance across all levels of the independent variables. Normality: The residuals should follow a normal distribution.

Model Building:Model Specification: Decide on the form of the model by selecting the independent variables to include and their functional relationships (linear, quadratic, etc.). Dummy Variables: If you have categorical independent variables, create dummy variables to represent them in the regression equation.

Model Estimation:Estimation Method: Use a suitable method to estimate the model coefficients, such as the least squares method. Interpretation: Interpret the coefficients of the independent variables. A coefficient represents the change in the dependent variable for a unit change in the corresponding independent variable, holding other variables constant.

Model Assessment:R-squared: Evaluate the goodness of fit using the R-squared value, which indicates the proportion of variance in the dependent variable explained by the independent variables. Adjusted R-squared: Consider the adjusted R-squared value, which penalizes adding unnecessary variables to the model. Residual Analysis: Examine the residual plots to ensure they meet the assumptions.

Model Refinement:Feature Selection: Use techniques like forward selection, backward elimination, or stepwise regression to refine the model by selecting the most relevant variables. Interaction Terms: Consider adding interaction terms if you suspect that the relationship between variables is not constant across different levels.

Interpretation and Reporting: Interpret the final model's coefficients in the context of your problem. Explain how each independent variable impacts the dependent variable while considering their statistical significance.

Validation: If possible, validate the model's performance on new data to ensure its generalizability.

Remember that multiple linear regression is just one tool in a statistician's toolkit. Depending on your research question and the nature of your data, other regression techniques or machine learning algorithms might be more appropriate.

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