Interpreting Regression Coefficients in Simple Regression

How are the regression coefficients interpreted in simple regression?

The simple regression model is

The formula for Regression Coefficients in Simple Regression Models is:

$$b = \frac{n\Sigma XY – \Sigma X \Sigma Y}{n \Sigma X^2 – (\Sigma X)^2}$$

$$a = \bar{Y} – b \bar{X}$$

The basic or unstandardized regression coefficient is interpreted as the predicted change in $Y$ (i.e., the dependent variable abbreviated as DV) given a one-unit change in $X$ (i.e., the independent variable abbreviated as IV). It is in the same units as the dependent variable.

Interpreting Regression Coefficients

Interpreting regression coefficients involves understanding the relationship between the IV(s) and the DV in a regression model.

Magnitude: For simple linear regression models, the coefficient (slope) tells about the change in the DV associated with a one-unit change in the IV. For example, if the regression coefficient for IV (regressor) is 0.5, then it means that for every one-unit increase in that predictor, the DV is expected to increase by 0.5 units while keeping all else equal.
Direction: The sign of the regression coefficient (+ or -) indicates the direction of the relationship between the IV and DV. A positive coefficient means that as the IV increases, the DV is expected to increase as well. A negative coefficient means that as the IV increases, the DV is expected to decrease.
Statistical Significance: The statistical significance of the coefficient is important to consider. The significance of a regression coefficient tells whether the relationship between the IV and the DV is likely to be due to chance or if it’s statistically meaningful. Generally, if the p-value of a regression coefficient is less than a chosen significance level (say 0.05), then that coefficient will be considered to be statistically significant.
Interaction Effects: The relationship between an IV and the DV may depend on the value of another variable. In such cases, the interpretation of regression coefficients may involve the interaction effects, where the effect of one variable on the DV varies depending on the value of another variable.
Context: Always interpret coefficients in the context of the specific problem being investigated. It is quite possible that a coefficient might not make practical sense without considering the nature of the data and the underlying phenomenon being studied.

Therefore, the interpretation of regression coefficients should be done carefully. The assumptions of the regression model, and the limitations of the data, should be considered. On the other hand, interpretation may differ based on the type of regression model being used (e.g., linear regression, logistic regression) and the specific research question being addressed.

Note that there is another form of the regression coefficient that is important: the standardized regression coefficient. The standardized coefficient varies from –1.00 to +1.00 just like a simple correlation coefficient;
If the regression coefficient is in standardized units, then in simple regression the regression coefficient is the same thing as the correlation coefficient.