Correlation and Regression txtbook

Chapter 14: Correlation and Regression

14-1 Introduction

• Learning Objective 1: Describe the sign and the numerical value of a correlation.

  • The correlational design involves measuring two variables for each individual in one group.

  • Correlation is a statistical technique used to measure and describe the relationship between two variables.

  • Example: Correlating high school grades with family income without manipulating either variable.

• Scatter Plot:

  • Displays relationships between two variables (e.g., X and Y).

  • Each point represents an individual observation. E.g., higher family income correlating with higher grades.

Characteristics of a Correlation

1. Direction of Relationship:

   • Positive correlation: Both variables increase together.

   • Negative correlation: One variable increases while the other decreases.

2. Form of the Relationship:

   • Most common: Linear relationships.

3. Strength of Relationship:

   • Measured numerically; ranges from -1.00 to +1.00.

   • A correlation of 1.00 indicates a perfect direct relationship; 0 means no relationship.

   • Example Values:

     • Perfect Positive: 1.00

     • Strong Positive: ~(0.80-0.99)

     • Moderate: ~(0.4-0.7)

Examples of Positive and Negative Relationships

• Positive: Time spent studying vs. course grades.

• Negative: Smartphone usage vs. student performance.

14-2 The Pearson Correlation

• Learning Objectives:

1. Calculate SP (Sum of products of deviations) using definitional and computational formulas.

2. Calculate the Pearson correlation and explain its significance.

3. Describe effects of linear transformations on Pearson correlation.

• Pearson Correlation:

  • Represents linear relationship; denoted as r for samples, ρ (rho) for populations.

  • Formula:
    $r = \frac{SP}{\sqrt{SS<em>X SS</em>Y}}$

  • SP calculated as both the definitional and computational formulas:

  1. Definitional: $SP = \sum (X - M<em>X)(Y - M</em>Y)$

  2. Computational: $SP = \sum XY - \frac{S<em>X S</em>Y}{n}$

Changes in Correlation Values

• Adding a constant to scores: does not change the correlation.

• Multiplying by constant: affects the magnitude but not the direction unless the constant is negative, which flips the sign of the correlation.

Example Calculation of Pearson Correlation

• Performance data calculated with both formulas will yield the same value for SP.

14-4 Hypothesis Tests with the Pearson Correlation

• Learning Objective: Conduct hypothesis tests using correlation.

• The null hypothesis states: H0: ρ = 0.

• To test hypotheses, we use a t-test or an F-ratio to determine significance of correlation.

Example of Hypothesis Tests

• Calculate the t-statistic using
  $t = \frac{r \sqrt{n - 2}}{\sqrt{1 - r^2}}$

• Degrees of freedom: df = n - 2.

Practical Example:

• Sample size n = 30, with a correlation r = 0.35 yielding t = 1.97.

• Critical value comparison to determine if null can be rejected.

Summary of Key Points

1. Correlation characterizes relationships among two numerical variables.

2. Pearson correlation r is a key statistic with defined calculations for describing linear relationships.

3. Results from hypothesis tests affirm relationships beyond mere correlation, recommending further exploration into causation.

14-6 Introduction to Linear Equations and Regression

• Regression allows us to model relationships and make predictions based on the established correlation.

• Linear Equation Model:
  $Y = bX + a$

• Regression Calculation: The slope (b) and intercept (a) computed using SP and SS.

Example Regression Calculation

• Coefficients yield regression equations for predicting future values based on known data, derived using statistical formulas, leading to predictive capabilities across various domains.