Section 38 Simple Linear Regression Model: Issues
38.0.1 Regression Model
\[ \Huge y_{i} = \beta_0 + \beta_1 x_{i} + \epsilon_{i}, \space i=1,...,n \]
38.1 Response & Predictor
How to identify response and predictor variable(s)?
Example: Preparing a Standard Curve in laboratory works, e.g. RT-PCR study.
Example: Chemical composition and spectral data
38.2 Regressing X on Y & Regressing Y on X
Regressing X on Y is not the same as Y on X
Regression coefficient X on Y
Regression coefficient Y on X
38.3 Regression & Correlation
Correlation quantifies the degree to which two variables are related.
Correlation does not fit a line through the data points. It provides information regarding how much one variable tends to change when the other one does.
Correlation assumes linear relationhsip.
The estimate of correlation ranges between -1 to +1
Correlation is not causation.
In regression, we predict one variable (reponse) based on the other variable (predictor).
The strength of the relationship is explicit in the regression analysis, and uncertainty can be seen clearly from confidence intervals or prediction intervals.
Regression can capture polynomial or non-linear relationship.
However, interpretation of regression coefficient as the causal relationship should be avoided.
38.4 Zero intercept
The meaning of zero intercept.
The fitting of zero intercept should be considered cautiously.
The \(R^2\) for models with and without intercept cannot be compared.
38.5 Interpreting intercept
Check if the interpretation of intercept is meaningful
Centering of data
38.6 Interpretation of regression coefficient
Standard interpretation
Effects and association
Regression coefficient does not answer causality
38.10 Regression toward the mean
Regression towards the mean is a mathematical phenomenon; it is not a causal phenomenon.
If a variable is extreme on its first measurement, it will tend to be closer to the average on its second measurement.
Consider carefully the regression toward the mean while planning an experimental design and interpreting the data.