Chapter 14: Correlation and Regression – Understanding Correlation

Overview of Chapter 14: Correlation and Regression

• Chapter 14 encompasses both correlation and regression, though this lecture focuses strictly on correlation, specifically Pearson correlation.
• The material covered corresponds to pages $446$ through $466$ in the textbook.
• While the textbook mentions various types of correlations, the Pearson correlation is the most central and significant type for this course and is the primary focus of the discussion.
• Pearson correlation is directly related to regression, which will be the subject of a subsequent lecture.

Experimental vs. Non-Experimental Research Designs

• Every hypothesis test previously learned in this class—including the $z$-test ($8$), two $t$-tests (chapters $9$, $10$, and $11$), and two ANOVAs (chapters $12$ and $13$)—has been designed for experimental or quasi-experimental studies.
• Experimental and quasi-experimental studies involve an independent variable (IV) and a dependent variable (DV).
  • The goal is to determine if the IV has an effect on, causes a change in, or affects the DV.
  • The independent variable is manipulated by the researcher, who decides the group structures and assignments.
  • The dependent variable is simply measured; the researcher does not attempt to manipulate it, as doing so would invalidate the experiment.
• In a typical experiment:
  • The Independent Variable (IV) is discrete (a grouping variable with a fixed number of levels or groups).
  • The Dependent Variable (DV) is continuous (scores can be any value within a range).
• Examples of experimental setups:
  • Does the dose of a pill affect pain levels? (Pill dose is the discrete IV; pain is the continuous DV).
  • Does wording (e.g., "bumped," "hit," "smashed") affect speed estimates of a witnessed car accident? (Wordings are discrete IVs; speed estimates are the continuous DV).
  • Does alcohol consumption affect reaction deficits? (Alcohol doses are discrete IVs; reaction time is the continuous DV).

Characteristics of Correlation in Non-Experimental Research

• Correlation and regression are used for non-experiments. They do not involve traditional independent variables.
• In a correlational study, there are two variables that are both "dependent-variable-like."
  • Both variables are measured, not manipulated.
  • Both variables are continuous.
• Correlation looks for a relationship rather than causation.
• The relationship is often represented by an arrow pointing both ways, suggesting that neither variable is necessarily causing the other; they are simply associated.
• Example: Investigating the relationship between height and weight. The researcher does not tell someone how tall to be or how much to weigh; they simply measure both continuous values to see if a relationship exists.

Understanding Directionality: Positive vs. Negative Correlations

• Positive Correlations:
  • Defined as two variables moving in the same direction.
  • If one variable increases ($\uparrow$), the other also increases ($\uparrow$).
  • If one variable decreases ($\downarrow$), the other also decreases ($\downarrow$).
  • Example: Height and weight. Taller people tend to be heavier. Conversely, shorter people tend to be lighter. Because they move together, the correlation is positive.
  • Example: Education and income. Higher education levels are generally associated with higher income levels. Lower education is associated with lower income. Since they move in the same direction, it is a positive correlation.
  • Example: Violent video games and aggression. If more game play is associated with more aggression, and less game play is associated with less aggression, it is a positive correlation.
• Negative Correlations:
  • Defined as two variables moving in opposite directions.
  • If one variable increases ($\uparrow$), the other decreases ($\downarrow$).
  • Example: Violent video games and school grades. As the amount of time playing games increases, a student's grades might decrease.
  • Example: Income and financial stress. As income increases ($\uparrow$), stress about meeting daily survival needs might decrease ($\downarrow$).

Strength of Relationships and Scatter Plots

• The strength of a relationship can range from weak to strong.
• A scatter plot is the primary visual tool for analyzing correlation.
  • The horizontal axis is the $x$-axis; the vertical axis is the $y$-axis.
  • Each point on the plot represents one individual with two scores ($x, y$).
• Visualizing the Slope:
  • If a line drawn through the middle of the points slopes upward from left to right, the correlation is positive.
  • If a line drawn through the middle of the points slopes downward from left to right, the correlation is negative.
• Determining Strength:
  • Strong Relationship: The points are clustered very tightly around a straight line.
  • Weak Relationship: The points are widely scattered, though a general trend may still be visible.
  • Perfect Relationship: Every single point falls exactly on a straight line. This is studied via algebra/geometry rather than statistics, which accounts for variability.
  • No Correlation: Points appear as a random cloud; it is impossible to determine if a positive or negative slope fits better.
• Pearson correlation specifically measures linear (straight-line) relationships. It is not appropriate for non-linear or curved relationships.

The Pearson Correlation Coefficient ($r$)

• Symbols:
  • Sample Correlation: $r$ (Latin/English letter).
  • Population Correlation: $\rho$ (Greek letter rho, which corresponds to the letter $r$ but looks somewhat like a lowercase $p$).
• Range and Interpretation of $r$:
  • The value of $r$ always falls between $-1.00$ and $+1.00$.
  • The sign (+ or -) indicates the type of relationship (positive or negative).
  • The numerical value indicates the strength.
  • $1.00$: Perfect positive correlation.
  • $-1.00$: Perfect negative correlation.
  • $0.00$: No correlation.
  • Values closer to $1$ or $-1$ (e.g., $0.90$ or $-0.85$) indicate strong relationships.
  • Values closer to $0$ (e.g., $0.10$ or $-0.20$) indicate weak relationships.
  • Note: The sign does not dictate strength. $-0.90$ is a much stronger correlation than $+0.10$.

Case Study: Violent Video Games and Aggression

• Researchers wanted to see if more hours of violent video games per week ($x$) were associated with more fights per week ($y$) among children.
• Sample size: $n = 5$ children.
• Data points:
  1. Child 1: $x = 0$ hours, $y = 2$ fights.
  2. Child 2: $x = 10$ hours, $y = 6$ fights.
  3. Child 3: $x = 4$ hours, $y = 2$ fights.
  4. Child 4: $x = 8$ hours, $y = 4$ fights.
  5. Child 5: $x = 8$ hours, $y = 6$ fights.
• Analysis of Scatter Plot:
  • A positive slope fits the data best.
  • The relationship appears to be strong or moderate-to-strong because points are fairly close to a potential line.
  • This indicates a positive trend: more games relate to more fights; fewer games relate to fewer fights.
• Descriptive statistics for the data:
  • Sum of $x$ ($\sum x$): $0 + 10 + 4 + 8 + 8 = 30$.
  • Mean of $x$ ($M_x$): $\frac{30}{5} = 6$ hours per week.
  • Sum of $y$ ($\sum y$): $2 + 6 + 2 + 4 + 6 = 20$.
  • Mean of $y$ ($M_y$): $\frac{20}{5} = 4$ fights per week.

Mathematical Formulas for Calculation

• Sum of Squares for $x$ ($SS_x$): Measures the squared deviations for the $x$ variable.
  • $SS_x = \sum(x - M_x)^2$
• Sum of Squares for $y$ ($SS_y$): Measures the squared deviations for the $y$ variable.
  • $SS_y = \sum(y - M_y)^2$
• Sum of Products ($SP$): A new concept required for correlation that measures how the variables move together.
  • Instead of squaring a single deviation, you multiply the deviation of $x$ by the deviation of $y$.
  • $SP = \sum(x - M_x)(y - M_y)$
• Pearson Correlation Coefficient Calculation Formula:
  • $r = \frac{SP}{\sqrt{SS_x \times SS_y}}$
• The process involves creating several columns of data: deviations for $x$, deviations for $y$, squared deviations for both, and the products of the deviations for both variables, eventually summing these columns to plug into the final equation.