Quant Correlation

Correlation Overview

  • Correlation Concept

    • Correlation measures the degree to which two variables (X and Y) are associated.

    • It assesses whether knowledge about one variable can provide information about another variable.

    • Both variables must be QUANTITATIVE (typically interval or ratio level).

    • Correlation is used to evaluate hypotheses of association, rather than differences.

    • Example hypothesis:

    • H1: Participants’ level of verbal aggressiveness will be positively associated with using dominance strategies in conflicts.

Scatterplots

  • Definition: A scatterplot is a graphical representation of the relationship between two variables (X and Y).

    • The Vertical (Y) axis represents the dependent variable (DV).

    • The Horizontal (X) axis represents the independent variable (IV).

    • Each dot on the plot represents one unit (person), indicating the intersection of scores from the two axes.

Steps to Calculate Correlation and Create Scatterplots Using JASP

  1. Navigate to Regression in the software.

  2. Select Correlation from the options.

  3. Move both variables into the designated variables box.

  4. Choose the option for Scatterplot.

  5. Check the box to display sample size (degrees of freedom, df = N - 2).

    • Note: The dependent variable (DV) should be entered first.

Types of Relationships in Correlation

  • There are four types of relationships one may observe between two quantitative variables:

1. No Relationship

  • Definition: Scores on variables X and Y share no systematic relationship (no shared variance).

    • Implication: Knowing the score of variable X does not inform us about variable Y.

2. Positive Linear Relationship

  • Definition: As scores on X increase, scores on Y also increase.

    • This relationship can be represented with a straight line.

    • Example:

    • Variable X = Verbal Aggressiveness (scale: 1-5)

    • Variable Y = Conflict Dominance (scale: 1-5)

3. Negative Linear Relationship

  • Definition: As scores on X increase, scores on Y decrease.

    • These scores maintain a linear relationship.

4. Curvilinear Relationship

  • Definition: The relationship between X and Y is curved, deviating from a straight line.

    • The direction of the relationship between X and Y may change over the range of X.

    • Example: Impact of Grade Point Average (GPA) on the amount of worry about grades can be illustrated by a curvilinear graph.

Pearson Correlation (r)

  • The Pearson correlation coefficient (r) quantifies the direction and strength of the linear relationship between two variables (X and Y).

  • Assumptions for Pearson correlation:

    1. The relationship between X and Y is linear.

    2. Data for each variable should be normally distributed.

    3. Data must be of interval or ratio level.

  • The values of Pearson r range from -1.00 to +1.00.

    • Note: Degrees of freedom (df) for a Pearson correlation is calculated as N-2 due to two groups of scores.

Interpreting the Pearson r Test Statistic

  • Sign of the Correlation:

    • A positive sign indicates a positive relationship; a negative sign indicates a negative relationship.

  • Size of the Correlation (absolute value):

    • Indicates the strength of the relationship between X and Y (effect size).

    • Interpretation of values:

    • r = 0: No relationship; knowing X gives no insight into Y.

    • r = ±1.00: Perfect linear relationship; scores on X are completely predictable from Y.

JASP Output Interpretation Example

  • Example Correlation values:

    • For variables csdomin and verbagg:

    • Pearson's r: 0.377

    • p-value: < .001

    • For variables momconfl and momclose:

    • Pearson's r: -0.342

    • p-value: < .001

Magnitude of the Effect (Effect Size)

  • General rules of thumb for interpreting the strength of the correlation coefficient (r):

    1. r = .10: Small relationship.

    2. r = .24: Medium relationship.

    3. r = .37: Large relationship.

  • Coefficient of Determination (R²): Represents the percentage of variance in one variable that is explained by variance in another variable.

    • Example Calculation: If r = .38, then R² = .14, indicating that 14% of the variance in the second variable can be inferred from the first variable.

Important Note on Correlation and Causation

  • Correlation does not imply causation: Establishing a correlation does not mean X causes Y.

    • Example:

    • Correlation between belief in the supernatural (X) and watching paranormal TV shows (Y). Could (1) X cause Y, (2) Y cause X, or (3) a third variable (Z) influence both?

Writing Up Results of Statistical Analysis

  • Example Results for Hypothesis Testing:

    • Hypothesis: Significant association between verbal aggressiveness and dominating conflict style.

    • Test: Pearson correlation used.

    • Statistical Output:

      • r(911) = .38, p < .001, R² = 11%

    • Interpretation: Results indicate a large positive relationship between the two variables.

    • Write-up requirements:

    1. Restate Research Question/Hypothesis.

    2. State the type of test run.

    3. List any limitations on the sample (if applicable).

    4. Detail the results including test statistic, degrees of freedom, value, and p-value.

    5. Describe the meanings of these results.

Homework Assignment

  • Task: Using the “Kittens2” dataset, test the hypothesis:

    • H1: Mangy kittens will be perceived as less cute than kittens that are less mangy.

  • Instruction: Calculate the Pearson correlation and write up the results. Include the correlation and scatterplot output in the write-up.