Correlational Research

• Correlation


  Describing the relationship between two variables
• Examples of correlations


  • Academic performance and hours of study
  • Level of physical activity and general happiness
• Correlation


  • Indicates the relative degree of consistency
  • Indicates the variability in Y scores paired with X scores
• Regression analysis


  • Indicates how closely the scatterplot fits the regression line (line of best fit)
  • Indicates the relative accuracy of a prediction
• Positive relationship


  Increasing one variable leads to an increase in the other variable
• Negative relationship


  Increasing one variable leads to a decrease in the other variable
• Positive correlation


  • Considered weak, why?
• Pearson Product Moment Correlation (PPMC)


  • The "r" value
  • Determines strength of a linear relationship between 2 variables
  • Greater r, the more "predictive" the relationship
  • Indicates both the direction and strength of the relationship
  • A high correlation does not prove causation
  • Relies on the concept of co-variance
• Relationship strength


  • 0.00 to 0.25 no - weak
  • 0.25 to 0.50 weak - moderate
  • 0.50 to 0.75 moderate - strong
  • Above 0.75 good - excellent

  
• Depends on the field of study, general guideline in research
• The ideal r-value? r2 > 0.49
• Non-linear relationships


  Relationships that are not linear
• Example of non-linear relationship


  • Muscle strength and age (childhood to older adults)
• Pearson Product Moment Correlation calculation

  r = (N*XY - X*Y) / sqrt((N*X^2 - X^2)*(N*Y^2 - Y^2))
• X


  Predictor variable (independent variable)
• Y


  Criterion variable (dependent variable)
• N


  Total number of participants (or scores), XY pairs
• Limitations of PPMC


  • Influence of data range, if range is low (variability), hard to detect relationship
  • Two groups with no relationship - together produce strong correlation. No cause and effect, coincidental occurrence of scores
  • Extreme data points can exert large effect. Outliers will reduce the strength of a correlation
  • Relationship could be present but not evident from PPMC since not linear relationship. PPMC assumes a linear relationship between 2 variables
• Coefficient of determination


  • Measures the proportion of variability in one variable that can be determined from the relationship with the other variable
  • Based on magnitude of rXY
  • rXY^2 is the coefficient of determination
  • The raw scores will deviate vertically (i.e., on the scatterplot) from the regression line
  • How much can the X-variable explain the degree of variation away from the regression line?
• Linear regression


  • ŷ = a + bX
  • ŷ = predicted Y value
  • X = measured X value
  • a = Y intercept (X = 0)
  • b = slope of the regression line
• Short term forecast model for projected SARS CoV2 infections (Canada)
• Randomized Controlled Trial (RCT)


  Establishes "cause and effect", gold standard in research designs, study volunteers randomly assigned to drug vs placebo groups, volunteers followed up for months, outcome variable is efficacy
• Correlational studies can examine historical data to address the question of whether covid19 vaccination works
• Hypothesis testing


  Null hypothesis: r = 0 (no relationship)

  Alternative hypothesis: r != 0 (relationship exists)
• There is a statistically significant positive relationship between a country's vaccination rate and the 28-day new SARS CoV-2 infection rate. A greater infection rate is related to higher vaccination rates.
• Correlational Research


  Examines how two (or more) variables relate to each other, looking for patterns and associations
• Correlational Research


  Does NOT prove one thing causes another
• Correlational Research Questions


  • Is there a relationship between hours studied and academic performance?
  • Does increased physical activity relate to lower levels of depression symptoms?
• Correlation Coefficient (r)


  The statistic that tells us both the direction and strength of the relationship
• Correlation Coefficient (r)


  • Range: -1.0 to 1.0
  • Positive: Both variables increase or decrease together
  • Negative: One variable increases while the other decreases
  • Closer to 1 or -1 = stronger relationship
  • Closer to 0 = weaker relationship
• Scatter Plots


  Visualize correlations, each dot is one person's data
• Scatter Plots


  • Strong positive: Dots form a clear upward line
  • Strong negative: Dots form a clear downward line
  • Weak correlation: Dots are scattered with less clear pattern
  • Zero correlation: Just random scatter
• Correlation ≠ Causation


  • Third variables: An unseen factor could be influencing both variables
  • Direction of causation: We don't know which variable came first
• Importance of Correlational Research

  • Identifying patterns of how things relate
  • Starting point for further research to investigate causation
  • Helping to make predictions, though not with complete certainty
• Finding no correlation matters, as it challenges assumptions and tells us there's more to be understood
• Correlation


  Beyond simple linear models
• Relationships in research


  • Not all are simple, straight lines
  • Assuming linearity can lead to misinterpretation and missed insights
• Age and Muscle Strength


  • Scatterplot shows strength increasing in childhood, peaking in adulthood, then declining
  • This is clearly NOT a linear relationship
• Linear Correlation (Pearson's r)


  If we force a linear model, we'd likely get a weak correlation, even though there's an obvious pattern
• Nonlinear Regression


  The right tool here! It can model the curve, revealing the true nature of the relationship