Interpreting the regression line constants, explanatory and response variables

Understanding Regression Line Constants, Explanatory and Response Variables

A simple linear regression establishes a relationship between two variables, known as the explanatory variable and the response variable.
The explanatory variable (also known as the predictor variable) is one that might explain changes in another variable.
The response variable (also known as the dependent variable) is the one that might be predicted or explained by one or more variables.
The relationship is typically presented in a formulaic manner: y = a + bX, where ‘y’ is the response variable and ‘X’ is the explanatory variable.
‘a’ and ‘b’ are the constants of regression. ‘a’ is the y-intercept (where the line crosses the y-axis when X=0) and ‘b’ is the slope (the change in ‘y’ with each unit change in ‘X’).

The constant ‘a’, or the y-intercept, provides the value of the dependent variable when the independent variable is zero. If values of a are far from zero, and X=0 is within the range of data, then this may not be plausible in reality.
The constant ‘b’, or the slope, indicates the amount by which the dependent variable changes, on average, per unit increase in the independent variable. If ‘b’ is negative, it suggests an inverse relationship: as X increases, y decreases.
These constants are obtained by minimising the sum of the squared differences between the actual and predicted values of the response variable - a method known as least squares.

It’s essential to correctly identify the explanatory and response variables to construct a meaningful regression model. Misidentification can lead to incorrect interpretations.
The role of the variables depends on the specific question being asked. For example, if we’re investigating “Does studying time (explanatory variable) affect test scores (response variable)?”
Graphs are helpful for visualising the relationship. If a straight line seems to fit the data, linear regression may be appropriate. If the relationship appears curved, a different type of model, perhaps quadratic or exponential, might be more suitable.

Once a regression model is established, it can be used to make predictions but with caution. The model is reliable within the range of data used to develop it. Predicting outside of this range is known as extrapolation and can lead to misleading results.
Remember that the predictive accuracy of the regression model depends on its appropriateness for the data and the reliability of the data itself.
Most importantly, correlation in your data does not imply causation. Even if a significant relationship exists, it does not mean one variable causes changes in the other.

Assumptions in regression analysis include: linearity and additivity between variables, statistical independence of errors, homoscedasticity (constant variance of errors), and normality of error distribution.
Violating these assumptions may lead to inaccurate estimations and weaken the analysis results.
Graphical analysis and statistical tests (e.g., residuals plots, formal normality tests) can be conducted to check these assumptions.
Remember, the quality of your regression analysis findings is tied to these assumptions, so comprehending them is crucial.