Understanding Linear Regression and Its Applications

Linear Regression

Predicts the dependent variable (y) using independent variables (x).

Purpose of Linear Regression

Estimates one variable based on another with known accuracy.

Regression Equation

Mathematical expression showing the influence of predictor (x) on dependent variable (y).

Assumptions of Linear Regression

Dependent variable (y) is normally distributed.

Assumptions of Linear Regression

A linear relationship exists between x and y.

Assumptions of Linear Regression

Observations are independent.

Simple Linear Regression

Examines the relationship between one predictor (x) and one dependent variable (y).

Graph Representation of Simple Linear Regression

X-axis: Predictor variable; Y-axis: Dependent variable.

Purpose of Regression Line

Develops a line of best fit for predicting y from x.

Method Used in Simple Linear Regression

Least squares method minimizes squared differences between observed and predicted y values.

Perfect Correlation

All data points lie on the regression line if perfectly correlated.

Real-World Application of Simple Linear Regression

Provides the best equation to predict y for any given x.

Regression Equation for Best Fit

y = bx + a

Example Calculation for Regression Line

For a baby born at 28 weeks: y = 20(28) + 500 = 1060 grams.

Residuals

Difference between actual and predicted values, representing prediction error.

Error Minimization

The line of best fit minimizes residuals for the most accurate prediction.

Research Designs for Simple Linear Regression

Used in associational studies where relationships between variables are explored.

Variables in Research Designs

Attributional variables like health status, blood pressure, gender, etc.

Measurement Levels for Variables

Variables must be measured at interval or ratio levels, with correct coding if nominal.

Statistical Assumptions of Linear Regression

Normal Distribution: Dependent variable (y) should be normally distributed.

Statistical Assumptions of Linear Regression

Linear Relationship: Changes in x should correspond to proportional changes in y.

Statistical Assumptions of Linear Regression

Independent Observations: Observations should not be related.

Statistical Assumptions of Linear Regression

No Multicollinearity: Predictor variables should not be highly correlated with each other.

Statistical Assumptions of Linear Regression

Homoscedasticity: Variance of errors should be constant across all x values.