PSY 10B Chapter 15: Describing Relationships - Correlation

Describing Relationships - Correlation

• Be able to read and create a scatter plot to identify positive or negative relationships between two variables
• Positive relationship: As X increases, Y increases (e.g., r = 0.8)
• Negative relationship: As X increases, Y decreases (e.g., r = -0.8)
• Correlation coefficient (r) ranges from -1 to 1, with values close to 1 or -1 indicating strong relationships
• Visually, strong relationships have points tightly clustered around a line

Computing a Pearson Correlation Coefficient (r)

• Step 1: Convert raw scores to z-scores (z = X - Ms)
• Step 2: Link responses for each participant to a value reflecting direction and strength (multiply z-scores across)
• Step 3: Average linked values to get a correlation coefficient (r = Σ(zx * zy) / (n - 1))

Correlation vs. Causation

• Correlation does not imply causation
• Direction of causality: Whether X causes Y, Y causes X, or both
• Third variable problem: A third factor may influence both X and Y

Restricted Range and Outliers

• Restricted range: Limits variability and may underestimate the correlation
• Outliers: Extreme values can distort the correlation coefficient

Coefficient of Determination (r2 or R2)

• Measures how well a statistical model predicts an outcome (proportion of variance in Y predictable from X)

Hypothesis Testing for Correlation

• Null hypothesis (H0): No relation between variables (r = 0)
• Alternative hypothesis (H1): There is a relation between variables (r ≠ 0)
• Use t-statistic and p-value to determine significance of correlation

Regression (Chapter 16)

• Builds on correlation to predict Y from X
• Regression line minimizes the sum of squared differences between observed and predicted Y values

Equation for the Regression Line

• Y = a + bX
• a is the intercept (value of Y when X = 0)
• b is the slope of the regression line (direction and strength of relation)
• X is the actual value of the predictor for a particular person

Predicting Y from X

• Plug in the value of X to predict Y (Y-hat) using the regression equation

Analysis of Regression

• Similar to ANOVA in partitioning variance
• SSTotal = SSRegression + SSResidual
• SSResidual is minimized by the best-fitting line

Residuals and the Best-Fitting Line

• Residuals are the differences between observed and predicted values (Y - Y-hat)
• The best-fitting line minimizes SSResidual

Reporting and Interpreting Results of Regression Analysis

• Include the value of r, significance level (p-value), and context of the data
• Report the overall model significance and tests of individual predictors

Multiple Regression

• Incorporates multiple predictors
• Each predictor has its own slope, interpreting the effect of each while controlling for others

Correlation

• Correlation measures the strength and direction of the relationship between two variables
• Positive relationship: As X increases, Y increases
• Negative relationship: As X increases, Y decreases
• Correlation coefficient (r) ranges from -1 to 1:
  • Values close to 1 or -1 indicate strong relationships
  • Values near 0 indicate weak relationships
  • Strong relationships have points tightly clustered around a line

Computing Pearson Correlation Coefficient (r)

• Step 1: Convert raw scores to z-scores: z = (X - M) / s
• Step 2: Link responses for each participant to a value that reflects direction and strength
• Step 3: Average these linked values to get a correlation coefficient that represents strength and direction of relation across all participants: rx,y = (zx * zy) / (n - 1)

Correlation vs. Causation

• Correlation does not imply causation
• Direction of causality: Whether X causes Y, Y causes X, or both
• Third variable problem: A third factor may influence both X and Y

Pearson Correlation Coefficient (r) Limitations

• Only measures linear relationships
• Does not account for non-linear relationships

Coefficient of Determination (r² or R²)

• Measures how well a statistical model predicts an outcome
• Interpreted as the proportion of variation in the dependent variable that is predictable from the independent variable

Hypothesis Testing for Correlation

• Null hypothesis (H0): No relation between variables (no correlation)
• Alternative hypothesis (H1): There is a relation between variables (correlation)
• Test statistic: t = r / sqrt((1 - r²) / (n - 2))
• p-value: If p < α, reject H0 and conclude a significant relationship

Regression

• Builds on correlation and allows you to predict Y from X
• Regression line minimizes the sum of squared differences between observed and predicted Y values
• Equation for the regression line: Y = a + bX

Computing Regression Equation

• b = rx,ys / sy
• a = MY - bMX
• Use regression equation to predict Y: Y-hat = a + bX

Analysis of Regression

• Similar to ANOVA in partitioning variance
• Total variance (SSTotal) = Variance explained by regression (SSRegression) + Variance not explained (SSResidual)
• Residuals: Differences between observed and predicted values (Y - Y-hat)

Testing Significance of Regression Equation

• Null hypothesis (H0): Slope (b) = 0 (no effect)
• Alternative hypothesis (H1): Slope (b) ≠ 0 (effect exists)
• F-ratio: F = MSRegression / MSResidual
• p-value: If p < α, reject H0 and conclude a significant regression equation

Multiple Regression

• Incorporates multiple predictors
• Each predictor has its own slope, interpreting the effect of each while controlling for others