Regression analysis is a statistical technique used to model the relationship between a dependent variable (often denoted as “Y”) and one or more independent variables (often denoted as “X” or “X1,” “X2,” etc.). In simple linear regression, we focus on a single independent variable, while in multiple linear regression, we consider two or more independent variables.

Here’s a step-by-step guide to fitting a regression line and interpreting the results in the context of simple linear regression:

1. Data Collection:

• Gather a dataset that includes measurements of both the dependent variable (Y) and the independent variable (X). The dataset should have a sufficient number of data points to conduct meaningful analysis.

2. Visual Exploration:

• Start by creating a scatterplot of the data points, with X on the x-axis and Y on the y-axis. This allows you to visually assess the relationship between the variables.

3. Fitting the Regression Line:

• The goal is to find the best-fitting regression line that represents the relationship between X and Y. In simple linear regression, this line is represented as: Y = aX + b
• The parameters “a” and “b” are estimated using statistical techniques. “a” represents the slope of the line (the change in Y for a unit change in X), and “b” represents the intercept (the value of Y when X is zero).

4. Estimating the Coefficients:

• Calculate the values of “a” and “b” using the least squares method. The formulas are as follows: a = Σ[(X – X̄)(Y – Ȳ)] / Σ[(X – X̄)²] b = Ȳ – aX̄
• Where X̄ and Ȳ are the sample means of X and Y, respectively.

5. Fitted Regression Line:

• Once you have estimated “a” and “b,” you can write the equation of the fitted regression line. Ŷ = aX + b
• This equation represents the best linear approximation of the relationship between X and Y.

6. Interpretation of Results:

• Interpretation of the regression results involves understanding the estimated coefficients and their significance:
  • The slope “a” indicates the change in the dependent variable (Y) for a one-unit change in the independent variable (X). A positive “a” suggests a positive relationship, and a negative “a” suggests a negative relationship.
  • The intercept “b” represents the estimated value of Y when X is zero. This interpretation may or may not be meaningful depending on the context.
  • Check the statistical significance of “a” using hypothesis tests (e.g., t-test). A significant “a” suggests that X is a predictor of Y.
  • Assess the goodness of fit using metrics like R-squared (R²), which measures the proportion of variance in Y explained by X. Higher R² values indicate a better fit.

7. Residual Analysis:

• Examine the residuals (the differences between observed Y and predicted Ŷ). A random scatter of residuals around zero indicates that the model assumptions are met. Non-random patterns may indicate problems with the model.

8. Prediction and Inference:

• Use the fitted regression line for prediction. You can predict Y for new values of X using the equation Ŷ = aX + b.
• Make inferences about the population based on the sample data, keeping in mind the limitations and assumptions of the model.

Interpreting the results of regression analysis requires a deep understanding of the context and the variables involved. It’s crucial to consider the assumptions of the regression model and conduct appropriate diagnostic tests to ensure the validity of the results. Additionally, interpreting coefficients should always be done in the context of the specific problem or research question at hand.