3: Chapter 3: Relationships: Scatterplots and Correlation

Biostatistics & Statistical Analysis - Chapter 3

• Chapter 3 focuses on relationships using scatterplots and correlation.
• Previous Learning Objectives covered describing distributions using numbers:
  • Measures of center (mean and median)
  • Measures of spread (quartiles and standard deviation)
  • Five-number summary and boxplots
  • Interquartile range (IQR) and outliers
  • Dealing with outliers
  • Choosing among summary statistics
  • Organizing statistical problems

Learning Objectives

• Demonstrate relationships using scatterplots and correlation
• Understand bivariate data
• Create scatterplots
• Interpret scatterplots
• Add categorical variables to scatterplots
• Define the correlation coefficient (r)
• Understand facts about correlation

Bivariate Data

• For each individual, data is recorded on two variables.
• Examine relationships between variables.
• Changes in one variable often correlate with changes in another.
• Example: Number of beers consumed and resulting blood alcohol content (BAC) for 16 students.

Scatterplots

• Used to display quantitative bivariate data.
• Each variable maps to an axis.
• Each individual is represented as a point on the plot.

Explanatory and Response Variables

• A response (dependent) variable measures the outcome of a study.
• An explanatory (independent) variable influences the response variable
• In a scatterplot, explanatory variable is typically plotted on the x-axis.
• Example: Number of beers is the explanatory variable (independent variable) and BAC is the response variable (dependent variable)

Scaling a Scatterplot

• Data is often displayed similarly across different plots.
• Both variables should occupy a similar amount of space.
• The plot should be roughly square, points ideally filling the plot space without blank areas.

Interpreting Scatterplots

• Describe the overall relationship pattern.
• Look for:
  • Form (linear, curved, clusters, no pattern)
  • Direction (positive, negative, no direction)
  • Strength (how closely points fit the form and deviations from it)
  • Outliers (points that deviate significantly from the pattern). 

Types of Relationships

• Linear Relationship: The points show a linear pattern.
• No Relationship: The points show little or no connection.
• Nonlinear Relationship: A pattern not represented by a straight line.
• Positive Association: Higher values of one variable tend to occur with higher values in the other variable.
• Negative Association: Higher values of one variable tend to occur with lower values in the other variable.

Outliers in Scatterplots

• An outlier is unusual or unexpected data point, with a low probability of occurrence.
• Outliers in a scatterplot appear outside the overall pattern of the relationship.

Adding Categorical Variables to Scatterplots

• Compare two or more relationships on a single plot by using different symbols for groups of points.
• Example: compare thorax length and longevity for male fruit flies that either reproduce or do not reproduce.

The Correlation Coefficient (r)

• A measure of the direction and strength of the relationship between two variables.
• Calculated using the means and standard deviations of both variables
• r = 1 or r =−1 represents a perfect linear relationship
• r = 0 represents no linear relationship.

The Roles of the Variables in r

• r treats both variables symmetrically (x and y)
• One variable can be explanatory for the other, typically put on the x-axis.
• The correlation (r) is the same regardless.

r has no units

• r isn't influenced by the units of measurement of the variables.
• It's a standardized measure.

Correlation and Strength/Direction

• Strength is indicated by the absolute value of r.
• Direction is indicated by the sign of r (positive or negative).
• Range of r: -1 to 1

r is Not Resistant to Outliers

• Correlations are based on means & standard deviations, making them sensitive to outliers.
• Outliers can influence the correlation value.
• In this case, moving a point can reduce the correlation strength.

Software: SPSS

• Software for statistical analysis
  • Student discount and regular pricing information available online for the appropriate software.

Variance

• Another measure of data spread.
• Mean sum of squares.
• Population vs sample variance formulas.
• Standard deviation is the positive square root of variance.

Application of SPSS

• A variety of tools, analyses and reporting options.
• Data entry, manipulation, and descriptive analyses.