Module 10-1 (2023)

Introduction to Bivariate Data

• Observing the relationship between two numerical variables (e.g., height and weight).

• Aim is to understand how one numerical variable responds to changes in another.

Variables Definitions

Response Variable (Dependent Variable)

• A variable that changes in response to the independent variable.

• Denoted as y.

Independent Variable (Explanatory Variable)

• A variable used to explain changes in the dependent variable.

• Denoted as x.

• Acts independently to cause differences in the response variable y.

Examples of Variables

Example 1:

• Effect of rainfall on crop yield.

  • X = amount of rainfall (independent variable).

  • Y = crop yield (dependent variable).

Example 2:

• Effect of midterm score on final grade.

  • X = midterm score (independent variable).

  • Y = final grade (dependent variable).

Data Recording and Visualization

• Data for two numerical variables should be recorded as pairs (X, Y).

• Use scatter plots to visualize these bivariate observations:

  • X-axis: Independent variable (x).

  • Y-axis: Dependent variable (y).

  • Plot data points based on bivariate observations (e.g., (x1, y1), (x2, y2)).

Evaluating Relationships in Scatter Plots

• Example: Does schooling affect salary?

  • X = years of schooling.

  • Y = salary.

• Scatter plots show relationships and can indicate differences among age groups by using different symbols for data points.

Examining Scatterplots

1. Direction of Relationship

• Positive Association: as X increases, Y also increases.

• Negative Association: as X increases, Y decreases.

2. Form of Relationship

• Linear: points follow a straight line.

• Curvilinear: points follow a curved line.

• Clustered data: points are loose and hard to identify a trend.

3. Strength of Relationship

• Strong linear relationship: data points closely align with a linear trend.

• Moderate linear relationship: data points are somewhat clustered around a trend line.

• Weak relationship: data points are scattered with no clear trend.

Outliers

• Observations that deviate significantly from overall pattern.

• Could mislead interpretations of the relationship.

Correlation Coefficient

• Measures strength and direction of linear relationships between two numerical variables.

• Denoted by r (or R).

• Ranges from -1 to 1:

  • r = 1: Perfect positive linear correlation.

  • r = -1: Perfect negative linear correlation.

  • r = 0: No linear correlation.

• Calculated using means and standard deviations: sensitive to outliers.

Interpreting Correlation Coefficient

Positive Correlation (r > 0)

• Indicates positive association, where increases in X lead to increases in Y.

  • Example: Years of schooling (X) and salary (Y) have r = 0.9941, indicating strong positive linear relationship.

Negative Correlation (r < 0)

• Indicates negative association, where increases in X lead to decreases in Y.

Correlation Coefficient Characteristics

• Symmetrical: Switching X and Y does not change the r value.

• Dimensionless: Has no units, purely a numerical signifier.

• Assess strength using the absolute value of r:

  • Close to 1 indicates strong relation, close to 0 indicates weak relation.

• Correlation does not imply causation:

  • Correlation can exist due to lurking variables.

Lurking Variables and Causation

• Lurking Variables: Hidden influences that affect both x and y.

• Example: Association between years of schooling and salary does not imply causation (factors like experience, company size affect salary).

• Correct conclusions require controlling for all lurking variables.

Conclusion

• Need scatter plot to confirm linearity before using correlation coefficient.

• Strong positive correlation noted between years of schooling and salary.

• Important to refrain from assuming causation based solely on correlation.