Chapter 14: Correlation and Scatterplots

Chapter 14: Correlation and Scatterplots

14.1 Introduction

  • Expands discussion of measures of association in statistics.

  • Focus on methods to measure strength and direction of relationships between numeric variables.

  • Recommendation to load the countries2 data set and import libraries:

    • descr

    • DescTools

    • Hmisc

    • ppcor

14.2 Relationships Between Numeric Variables

Overview of Measurements

  • Traditional methods (crosstabs, chi-square) are inadequate for numeric variables.

  • Numeric variables require specific methods to assess relationships.

Example Variables

  • Dependent Variable: Life expectancy (years) from 190 countries.

  • Independent Variable: Fertility rate (births per woman of childbearing age).

    • Hypothesis: Negative relationship between fertility rate and life expectancy.

Data Characteristics

  • Data characteristics need exploration:

    • Crosstab for life expectancy vs fertility rate would be unmanageable due to size (100+ columns/rows).

Descriptive Statistics

  • Life Expectancy Statistics:

    • Mean: 72.6

    • Median: 73.9

    • Skewness: -0.5 (indicating negative skew).

    • Range: 53 to > 85 years.

  • Fertility Rate Statistics:

    • Mean: 2.74

    • Median: 2.28

    • Skewness: 0.93 (indicating right skew).

    • Range: 1 to > 5 births per woman.

Expectation of the Relationship

  • Expected finding: Low fertility rates correlate with higher life expectancy for reasons:

    • Pregnancy/childbirth remains a cause of maternal mortality.

    • High birth rates often align with higher infant/child mortality.

14.3 Scatterplots

Definition and Utility

  • Scatterplots plot joint outcomes for two variables.

  • Each point illustrates the values for life expectancy and fertility rate for individual countries.

Example Code (R):
par(mfrow = c(1,1))  
plot(countries2$fert1520, countries2$lifexp, xlab="Fertility Rate", ylab="Life Expectancy")
Interpretation of Scatterplot

  • Vertical axis: Life expectancy (dependent).

  • Horizontal axis: Fertility rate (independent).

  • Signals strong correlation:

    • Low fertility rates correlate with high life expectancies.

    • Pattern indicates a negative relationship.

14.4 Pearsons's r

Description and Calculation

  • Pearson's r measures the linear relationship between two continuous variables.

  • For bivariate correlation:

    • Values <0 indicate a negative relationship.

    • Values near 0 indicate no correlation.

    • Values >0 indicate positive correlation.

Formula for Pearson’s r

r=racextcov(X,Y)extstd(X)imesextstd(Y)r = rac{ ext{cov}(X,Y)}{ ext{std}(X) imes ext{std}(Y)}

  • Covariance indicates how changes in one variable correspond to changes in another.

  • Standard deviations normalizes the covariance.

  • r values range from -1 (perfect negative correlation) to +1 (perfect positive correlation).

Example Computation (with 10 countries)

  • R Code for computation of deviation and correlation:

X10countries$y_dev = X10countries$lifexp - mean(X10countries$lifexp)  
X10countries$x_dev = X10countries$fert1520 - mean(X10countries$fert1520)  
X10countries$y_devx_dev = X10countries$y_dev * X10countries$x_dev  
X10countries$yvar = X10countries$y_dev^2  
X10countries$xvar = X10countries$x_dev^2

Pearson’s r Calculation Steps

  1. calculate deviations for y and x.

  2. get cross-products of deviations.

  3. compute variances.

  4. evaluate whether the data fits a positive/negative pattern.

14.5 Partial Correlation

  • Partial correlation assesses the relationship between two variables while controlling for a third variable.

  • Useful to delineate peer impacts, hidden patterns, and co-linearity.

  • Formula:
    rxy.z=rxy−(rxzimesryz)r_{xy.z} = r_{xy} - (r_{xz} imes r_{yz})

14.6 Proportional Reduction in Error (PRE)

  • Indicates how much variance in the dependent variable is explained by the variance in the independent variable.

  • Computed as the square of Pearson’s r.

  • Example:

    • Fertility rate explains 71% of variance in life expectancy.

    • Education explains 59% variance.

    • Population size has minimal impact on variance.

14.7 Correlation and Scatterplot Matrices

  • Highlight relationships among multiple variables.

  • Useful for visual exploration and quick reference to correlation strength.

  • R code: To create a scatterplot matrix

plot(lifexp_corr, cex=.6)

14.8 Overlapping Explanations

  • Warns against overstatement of impact when variables are correlated.

  • Consider overlaps in variance between independent variables that can distort perceived effects on dependent variables.

  • Potential for double-counting errors in influence measurement.

14.9 Next Steps

  • Transition to regression analysis for deeper exploration of relationships among numeric variables.

  • Multivariable regression models will include multiple independent variables to better predict dependent outcomes.

14.10 Exercises

Practical Application

  1. Analyze relationships and identify positive/negative correlations.

  2. Use scatterplots to illustrate correlations and evaluate statistically.

  3. R Functions to derive key statistics for shared variances and relationships.