Untitled

Chapter 6 – Scatterplots, Associations, & Correlation

Overview of Bivariate Analysis

  • Bivariate analysis involves two variables with objectives:

    • Plotting/Graphing for visualization.

    • Characteristics Description of associations.

    • Measurement of these characteristics.

    • Inference Method to draw conclusions.

Visualizing and Describing Relationships

  • Scatterplot characteristics:

    • Form: Linear, curved, etc.

    • Direction:

    • Positive: both variables increase together.

    • Negative: one variable increases while the other decreases.

    • Strength: Clustering of points around a line.

    • Presence of Outliers: Points that deviate from the pattern.

Measuring the Strength of Relationships

  • Use of:

    • Standardized (z) values for analysis.

    • Correlation Coefficient (r):

    • Ranges from -1 to +1.

    • 1: perfect positive correlation.

    • -1: perfect negative correlation.

    • 0: no correlation.

Distinction Between Correlation and Causation

  • Correlation does not imply causation; correlating variables do not necessarily cause changes in each other.

Lurking Variables

  • Lurking variables affect both studied variables but are not analyzed.

Correlation Tables

  • Provide a summary of relationships across variables (e.g., Amazon books correlation).

Chapter 7 – Intro to Linear Regression

Key Elements of Linear Regression

  • Necessary components:

    • Scatterplot visualization.

    • Measures of relationship characteristics.

    • Statistical methods for inference.

Bivariate Analysis Characteristics

  • Form, Direction, Strength using correlation coefficient (r), Outliers.

Model Development

  • Independent variable (x): Manipulated or categorized.

  • Dependent variable (y): Observed and measured.

Regression Analysis Steps

  1. Create a scatterplot.

  2. Analyze relationship strength (correlation coefficient).

  3. Determine the line of best fit.

  4. Assess model adequacy (R-squared).

  5. Predict using the model.

Interpretation of Regression Coefficients

  • Slope (b1): Indicates change in y per unit change in x.

  • Intercept (b0): Expected value of y when x=0.

Regression to the Mean

  • Extreme values in x predict closer values to the mean in y.

Conditions for Valid Linear Models

  1. Variables must be quantitative.

  2. The relationship should be linear.

  3. Check for outliers.

  4. Residuals must have equal spread.

Residual Analysis

  • Residuals must not show patterns; check through residual plots.

Adequacy of the Model and R-squared

  • R-squared indicates variance explained by the model.

Nonlinear Relationships

  • Some data may require transformations to achieve linearity.

Extrapolation Considerations

  • Predictions outside observed data may be inaccurate.

Outliers and Influential Points

  • Outliers: Significantly deviate from patterns.

  • Influential Points: Affect slope/intercept of the regression line.