Overview of Data Collection
Each orange dot represents an individual’s data point.
X value: time spent studying.
Y value: score on the final exam.
Aim is to predict the final score based on study time.
Understanding the Regression Line
The line represents the best fit for the data.
Prediction model includes:
Y-intercept (b0)
Slope (b1) multiplied by x value (study time)
Error term
Example: If study time (x) = 5 hours:
Starting point is the intercept, add the slope times x, yielding the predicted score.
Prediction Error
Error is visualized as green lines from data points to the regression line.
Perfect prediction occurs only if data points lie perfectly on the line, yielding zero error.
Regression Basics Review
Slope (b1): change in Y for each unit change in X.
Y-intercept (b0): expected value of Y when X = 0.
Regression Line Calculation
Best fitting line minimizes the sum of squared distances from the data points to the line.
Notion of minimizing residuals, defined as actual (Y_i) minus predicted (Yhat) values.
Rationale for Squared Residuals
Squaring distances avoids potential cancellation of values (positive and negative).
Provides a method to ensure the total distance is never zero, making calculations numerically stable.
Identifying Best Fitting Line
Different lines yield different prediction errors.
Lesser prediction errors indicate a better fit.
Comparing Residuals
Residuals (green lines) from actual to predicted values illustrate how different models fit the data.
Fitting line must minimize the sum of squared residuals for optimal results.
Slope and Intercept Formulas
The slope is derived using:
Sum of Products: Covariance term between X and Y
Sum of Squares: Variance term for X.
Once slope (b1) is established, the intercept (b0) can be calculated as:
Mean(Y) - b1 * Mean(X).
Interpreting Regression Parameters
Slope indicates how much Y changes per unit increment of X.
The intercept represents the expected Y when X equals zero.
Testing Significance of Slope and Intercept
Statistically, determine if slopes and intercepts significantly differ from zero (indicating predictive power).
Null Hypotheses
Null hypothesis for slope: slope = 0 (no relationship).
T-test formulation for slope and intercept based on these null hypotheses.
Implications of Results
A significant P-value (< 0.05) suggests the slope is unlikely due to chance and may imply a predictive relationship.
Multiple Regression Introduction
Going beyond a single predictor (X) to include multiple predictors to analyze Y.
E.g., predicting exam scores using study hours and sleep hours.
Descriptive Pathways Through Multiple Regression
Explore interactions and moderation by adding third variables and analyzing their impact.
Application of Moderation and Mediation Analyses
Moderation analysis checks if the relationship changes when controlling for a third variable.
Mediation analysis helps identify if the third variable explains the relationship between X and Y.
Final Paper Structure
Individual effort required, leveraging group project inputs as groundwork.
Must implement feedback from prior submissions to enhance quality.
Checklists and Formatting
Adhere to APA style throughout, including specific section requirements and their respective contributions to overall grading.
APA Guidelines
Create a cohesive structure, using feedback to refine each section (Introduction, Methods, Results, Discussion).
Sections Overview
Results section focuses purely on presenting findings with no interpretation.
Discussion section should delve into implications and insights gained from data, highlighting limitations and future directions.
Operational Strategy for Finalizing Papers
Utilize all provided scaffolding (guidelines, checklists, example papers) to shape and refine work for submission.
Use Resources Efficiently
Candidates should attend office hours for additional clarity and assistance in the writing process.