notes on correlation.

Lecture Supplement on Correlation

  • Overview of Correlation

    • This lecture serves as a replacement for the missed class focused on correlation.

    • Assumes familiarity among psychology students with correlation concepts.

  • Recap of Previous Topics

    • Recent classes covered analysis of experience.

    • Discussed logic behind ANOVA and calculations involved.

    • Reviewed post hoc tests following significant F tests to identify which groups differ.

  • Correlation as a New Concept

    • Correlation distinguishes itself from previous analyses, focusing on the strength of association between two quantitative variables which can be on an interval or ratio scale.

    • Key Characteristics:

    • Deviates from t-tests and ANOVA which analyze categorical predictor variables against quantitative outcome variables.

Understanding Correlation

  • Definition of Correlation

    • The term "correlation" combines Latin roots:

    • Cor or Co: Meaning together.

    • Relatio: Meaning relation.

    • A correlation quantifies the degree of relationship between two variables (bivariate correlation).

    • Focused on two quantitative variables at a time.

  • Types of Correlation

    • Bivariate Correlations: Relationships between two variables.

    • Multiple Correlations: Involving more than two variables, not discussed in detail here.

  • Example Correlations

    • Children’s weight and height.

    • Generally, taller children are expected to weigh more, but it doesn't guarantee it.

    • SAT scores and college freshman GPA.

    • SAT scores predict freshman GPA, helping forecast college retention.

Correlation vs. Causation

  • The Importance of Distinction

    • Causation is not implied by correlation.

    • Concurrent relationships do not establish causal links.

    • Experimental designs and testing are necessary for causal conclusions.

    • Reasons for Misconceptions:

    • Spurious Correlations: Illusory correlations that occur by chance. Example: Ice cream sales correlate with crime rates, both driven by temperature.

    • Causality Limitations:

    • Many variables in psychology cannot be manipulated ethically or practically.

  • Examples of Spurious Correlations:

    • Age of Miss America correlating with murders by steam and hot objects.

    • Cheese consumption correlating with deaths through bed sheet tangling.

Interpreting Correlation

  • Range of Correlation Values

    • Values range from -1 to +1.

    • Notations:

    • Lowercase italicized r for sample correlations.

    • Greek letter ρ (rho) for population parameter correlations.

    • Pearson correlation: Most common type, measuring linear relationships.

    • Sensitive to whether as one variable increases, the other does too (positive correlation) or decreases (negative correlation).

    • Zero Correlation: Indicates no linear relationship; non-linear relationships (curvilinear) can exist.

    • Curvilinear Relationships:

    • Not detected by Pearson correlation; relationships often exist outside linear models.

  • Covariance and its Limitations:

    • Covariance measures relationship strength but is scale-dependent, making interpretation difficult.

    • Calculation involves:

    • Differences between individual scores and means of each variable, multiplied and summed up, then divided by sample size minus one to form covariance value.

  • Calculating Correlation from Covariance:

    • Correlation is derived by dividing covariance by the product of standard deviations of both variables.

    • Produces a value between -1 and 1, indicating strength and direction of correlation.

Practical Application of Correlation

  • Calculating Correlation in Statistical Software (e.g., SPSS):

    • Use the correlation matrix to analyze relationships between variables (e.g., trait anxiety pre and post treatment).

    • Null Hypothesis: $
      ho = 0$, suggesting no correlation in the population.

    • Alternative Hypothesis: $
      ho \neq 0$, suggesting a correlation exists.

  • Significance Levels:

    • Example: A correlation statistically significant at $p < 0.05$ leads to rejecting the null hypothesis.

    • Credible intervals indicate the range of expected values for population correlation; intervals not containing zero point to significant results.

Visualizing Correlations

  • Scatter Plots:

    • Effective for visual correlation understanding, with each dot representing paired scores across variables.

    • Higher correlation strength produces tighter clusters along the trend line, whether positive or negative.

Effect Size Considerations in Correlation

  • Understanding Effect Sizes:

    • Correlations serve as effect sizes, indicating the strength of relationships in a more practical manner.

    • Guidelines for interpretation of correlation strength:

    • < 0.1 = very small

    • 0.1 - 0.29 = small

    • 0.3 - 0.49 = moderate

    • > 0.5 = strong

Conclusion and Future Topics

  • Closing emphasis on credible intervals in correlation interpretation.

    • Understanding statistical significance is crucial for research outcomes.

  • Next class will address bivariate regression as an extension of correlation concepts.