What is the best way to interpret regression model coefficients?

One intuitive explanation of a regression coefficient is that it is a average sensitivity coefficient, how sensitive is the dependent variable Y to input X. For example, is X is GDP and has linear relationship to Sales Revenue Y, Y = 1.2*GDP, then for every 1% increase in GDP, sales revenue is expected on average to increase by 1.2%.

Interpreting regression model coefficients requires a sound blend of statistical rigor and domain knowledge. Sometimes simply asking yourself if the changes in the dependent variable associated with the independent variable are meaningful in your specific context might just be the moment when the lightbulb switches on

We can have one example of linear regression so that we can see the coefficient interpretation with variables. Let us consider the regression equation with one independent variable as Result = Smartwork + 2* Hardwork Where Intercept= Smartwork X = Hardwork Y = Result From here we can understand that for one unit change in Hardwork we will have 2 times change in Result. And when we have the coefficient of an independent variable as zero then Result = Smartwork. So think accordingly!!

Interpreting regression model coefficients is crucial to understand the relationships between variables and making predictions. It is essential to consider the context of your specific problem and the domain knowledge when interpreting these coefficients. Generally, both p-value (Significance of Coefficients) and R-squared (Coefficient of Determination) are 2 indices to assess and evaluate the performance and significance of regression models.

One thing I would like to emphasize is the necessity of taking into account the units of the variable when assessing its impact on the target. A commonly employed technique by many data scientists involves standardizing the dependent variables.

There is so much p-hacking nowadays so that the most knowledgeable folks are moving to looking at the magnitude of effects rather than p-values.

the p-value is a statistical estimate. researchers often believe that p == P(H|D)—the probability of the hypothesis given the data—when p == P(D|H)—the *long-term* probability of the data given the hypothesis. it is also binary; one either reaches statistical significance or the estimate is statistically non-distinct from zero. there is neither such thing as "marginal significance," nor interpreting the magnitude of p-values in any fashion other than the above.

I agree with fellow collaborators regarding the discussion on the manipulation of p-values, and I would like to provide a more detailed explanation for the rationale behind this practice. The primary motivation stems from the fact that much of the data encountered in the real world does not adhere to linear relationships. P-values are effective tools for assessing linearity, but they may prove less reliable when dealing with non-linear yet crucial variables. Consequently, employing p-value analysis can serve as a means to identify significant variables, but it is imperative not to place undue reliance solely on p-values.

It is possible to assess the significance of coefficients thanks to the p-value. This value, also known as alpha, can vary with sample size. The standard value for a p-value is generally 0.05, but when dealing with a small sample size, you may consider shifting it to 0.1. Conversely, with a large sample size, you can consider lowering it to 0.01 It's important to understand that this value is a convention, so while it's useful to consider the p-value, its robustness isn't absolute. You shouldn't be overly concerned if you get a p-value of 0.051, just as you shouldn't overly celebrate a p-value of 0.049. It serves as a good indicator of the significance of your coefficients but should be used judiciously.

The significance of the intercept tells us that (if using the arbitrarily agreed-upon p-value of 0.05) that there is less than a 5% chance that the intercept is zero. A non-zero intercept should be expected if we predict that when the X-value is 0, there is already an existing Y-value. This is generally the case when there is a starting amount from which to build upon or deplete. The significance of the slope tells us that there is less than a 5% chance that the slope is 0. This is important, since a 0 slope indicates that there is no relationship between the X and Y values. That being said, there are many ways to influence the data and experiment to achieve the p-value of 0.05 and inappropriately find such trends.

Coefficients, especially their magnitudes, are the heartbeats of regression models, revealing the nuances of relationships within the data. However, interpretation requires caution. The raw magnitude might be misleading due to differing scales. Always consider the context and the unit of measurement. Standardized coefficients are invaluable tools in this regard, offering a clearer lens to discern the true impact of predictors. As data science progresses, ensuring the interpretability and comparability of model outputs remains paramount, making concepts like coefficient magnitudes pivotal for insightful decision-making.

It is common for starters in statistics to think that coefficients of a linear model must be naturally static. However, there are actually better, statistically more sound methods to treat them as random variables and having intervals of coefficients with statistical guarantees. This might suggest for instance, the magnitude of coefficient B1 for variable X1 is between 0.4 and 0.6 for 95% of the time. This type of interpretation shall be vital for certain sensitive use cases.

Like many concepts of statistical modeling even for a relatively simple aspect like linear regression, there is actually quite a vast literature. This is particularly due to quest to develop robust methods to infer its coefficients. So, one needs to know different kinds of linear regression algorithms and understand in what type of cases which one should be chosen : OLS, Theil Sen, RANSAC, Stochastic Gradient Based, etc etc

1) Understand the Model Type 2) Coefficient Sign 3) Magnitude 4) Control for Other Variables 5) Unit of Measurement 6) Statistical Significance 7) Confidence Intervals 8) Interaction Effects 9) Coefficient of Determination (R-squared) 10) Interpretation Based on Domain Knowledge 11) Visualize Relationships 12) Check for Multicollinearity 13) Use Standardized Coefficients 14) Avoid Overinterpretation 15) Validate Your Model