How do you interpret the coefficients of a regression model?

In a regression model, coefficients show how the dependent variable changes when independent variables change. Positive coefficients mean the dependent variable increases with the independent variable, while negative coefficients mean it decreases. The coefficient's size indicates the strength of the relationship, and statistical significance shows if the relationship is likely real. In more complex models, interactions and higher-order terms add nuance to interpretation.

Interpreting regression coefficients is like understanding how much and in what direction each predictor affects the target. If a coefficient is positive, it means increasing that predictor tends to increase the target, and vice versa for negative coefficients. The size of the coefficient shows how much the target is expected to change for each unit increase in the predictor, holding other variables constant. Statistically significant coefficients are reliable, indicating a genuine relationship. If there are interaction terms, the interpretation becomes more complex, considering how different predictors interact to affect the target.

Intercept (b0): The intercept (b0) represents the expected value of the dependent variable (y) when all independent variables are zero. Coefficient of the Independent Variable (b1): The coefficient (b1) indicates the change in the dependent variable (y) for a one-unit change in the independent variable (x). Direction of Effect: The sign of the coefficient (positive or negative) indicates the direction of the effect of the independent variable on the dependent variable. Magnitude of the Coefficient: The magnitude of the coefficient represents the strength of the relationship between the independent and dependent variables.

Actually, the AI-given answer in this context of scale is incorrect. A large coefficient has nothing to do with a strong relationship regardless of scale. Period. A strong relationship (or lack thereof) is indicated by the t-statistic or the p-value in a regression. Think absolute value of t-score > 1.96 or p-value < 0.05 for "significance" of the relationship. Size of coefficient can be changed by adjusting the scale but its significance CANNOT be impacted by adjusting scale... However, transforming the variable (in logarithmic or exponential form) can impact both the size and significance of the variable because the measured relationship changes.

It would help to show what an interaction model would look like. If X1 and X2 were two variables, the Regression model with their interaction included would be: Y = b0 + b1X1 + b2X2 + b3X1*X2 Here X1*X2 would need to be created as a third variable X3 = X1*X2. Thus, when considering the effect of say, X1 on Y, we would need to be mindful of both the b1 and the b3 values.

One major limitation is the assumption of linearity. Regression models assume that the relationship between the independent and dependent variables is linear. This means that for every one-unit increase in the independent variable, the dependent variable changes by a constant amount, represented by the coefficient. However, in reality, many relationships may not be strictly linear, which can lead to misinterpretation of coefficients. Another limitation is multicollinearity, which occurs when independent variables in the regression model are highly correlated with each other. In such cases, it can be difficult to discern the unique contribution of each predictor to the outcome, leading to unstable and unreliable coefficient estimates.