Unit 7-Measures of Association: Correlation and Regression

Measures of Association

Correlation & Regression

• Focus on understanding the relationships between variables through correlation and regression analyses.

Correlational Study Key Points

• Criteria Variable (Dependent): The outcome or variable you are trying to predict.
• Predictor Variable (Independent): The variable you manipulate or measure for prediction.
• Scatterplot: Visual representation to examine the relationship between the two variables.
• Intra-ocular Test: Evaluate your subjective impression of the scatterplot; does the relationship make sense psychologically, physiologically, or sociologically?

Correlation Analysis Overview

• Describes:
  • Direction and strength of the relationship between two variables.
  • Consistency in paired (X, Y) scores.
  • Variability of Y scores with respect to X scores.
• Regression Analysis:
  • Fit of scatterplot to the regression line (line of best fit).
  • Predictive accuracy from the analysis.

Relationship Types

Positive Relationship

• Example: Grade Point Average (Y) vs. Lectures Attended (X)
• Attending more lectures positively correlates with higher GPAs.

Negative Relationship

• Example: Time Spent in Gym (X) vs. Body Mass (Y)
• More time in the gym does not necessarily reduce body mass but indicates an inverse relationship.

Correlation Coefficients

• r (Correlation Coefficient): Measures the strength and direction of linear relationships.
• Ranges from -1 (perfect negative) to +1 (perfect positive).
• Values Interpretation:
  • 0.00 - 0.25: Very weak
  • 0.25 - 0.50: Weak
  • 0.50 - 0.75: Moderate
  • >0.75: Strong
• A high r-value suggests a stronger predictive relationship but does not imply causation.

Pearson Product Moment Correlation (PPMC)

• Defines strength of a linear relationship: High r indicates strong predictability.
• Relies on the notion of co-variance between variables.

Coefficient of Determination

• Measure of the proportion of variability in one variable explained by another (expressed as (r_{XY})^2).
• Evaluates how much variation can be ascribed to the predictor variable.

Linear Regression Equation

• General form: Y = a + bX
  • b is the slope of the line, corresponding to the covariance ratio of correlated variables.
  • a is the y-intercept, representing the value of Y when X = 0.

Regression Analysis Steps

1. Calculate descriptive statistics (mean, standard deviation) for X and Y.
2. Calculate Pearson's r to find the correlation coefficient.
3. Determine line of best fit (regression equation): Calculate slope b and intercept a.
4. Calculate the Standard Error of Estimate (SEE): SEE = s_Y imes ext{sqr}(1 - r^2)
5. With a hypothetical X value, predict Y using the regression equation; express predicted Y with the SEE.
6. Calculate the range of predicted values considering error.

Example: Correlation in Student Grades

• Correlation between first-year (X) and second-year (Y) psychology course grades.
• Based on sample scores: (e.g., X means: 50,56,52… Y means: 80,85,83…)
• Derived regression equation can help predict second-year grades based on first-year grades.
  • Example linear regression output: Y = 36.08 + 0.88X

Limitations of PPMC

1. Data range influence: Only captures relationships in the studied range.
2. Nonlinear relationships may go undetected.
3. Extreme data points can skew results significantly.

Hypothesis Testing in Correlation

• Statistical null hypothesis (H0): Presume no correlation between X and Y.
• Alternative hypothesis (H1): Posit a significant relationship exists.
• Degrees of freedom: Typically N - 2 for correlation significance tests.

Covid-19 Case Study Example

• Examining correlation between country vaccination rates (predictor variable X) and 28-day new case counts (criterion variable Y).
• Found statistically significant positive correlation (r = 0.214) suggesting higher vaccination rates are associated with increased case counts in studied countries.