FMPH 102 Study Notes: Correlation of Two Continuous Variables

FMPH 102: Biostatistics in Public Health

Course Overview

• Focus on the analysis of correlation between two continuous variables in health statistics.

Topics Covered

• Continuous Variables Inference for Means:

  • One sample

  • Two paired samples

  • Two independent samples

  • Three or more independent samples

• Correlation and Linear Regression

• Binary Variables Inference for Proportions:

  • One sample

  • Two paired samples

  • Two independent samples

  • Three or more independent samples

Understanding Correlation

Definition

• Correlation assesses the relationship between two continuous variables.

• Key Principle: "Everybody who went to the moon has eaten chicken"—implies correlation can sometimes suggest spurious relationships.

Correlation of Continuous Variables

• Correlation examines how two continuous variables for the same sample of individuals vary together.

• Examples of Correlation Investigations:

  • Is Body Mass Index (BMI) correlated with total exercise time per week?

  • Is LDL cholesterol associated with the total amount of cholesterol in the diet?

  • Is blood pressure related to total sodium intake?

Properties of Correlation

Form of Correlation

• Investigates whether the relationship is linear, curved, or random.

Direction of Correlation

• Positive Association: Upward trend indicating that as one variable increases, the other does too.

• Negative Association: Downward trend indicating that as one variable increases, the other decreases.

• Flat Trend: No correlation, indicating no association between the variables.

Strength of Correlation

• Measured by how closely data points adhere to the trend line or curve in a scatterplot.

• Example:

  • Values of variable x for person A

  • Values of variable y for person A

Visualization and Example

Jackson Study Example

• Utilized a scatterplot of percentage of body fat (PBF) and BMI with n = 655 participants.

• Observed a positive association between PBF and BMI:

  • Individuals with high PBF tend to exhibit high BMI.

  • Relationship appears approximately linear but likely has a quadratic form expressed as:
    $y = a(x+b)^2$

Strength of Correlation

Pearson’s Correlation Coefficient (r)

• Definition:

  • The population correlation coefficient, denoted by $corr(X,Y) = \rho$, estimated by the sample correlation $corr(x,y) = r$.

• Assumptions:

  • Both X and Y are normally distributed.

  • There must be a linear association between X and Y.

• Range:

  • Values lie between -1 and 1:

  • If X and Y are independent: $\rho = 0$ or $r \approx 0$.

  • Positively correlated: \rho > 0 (or r > 0).

  • Negatively correlated: \rho < 0 (or r < 0).

  • Strong correlation: $|\rho| \geq 0.5$ or $|r| \geq 0.5$.

    • Often calculate $r^2$: strong correlation if $r^2 \geq 0.25$.

Visualization of Pearson's Correlation Coefficients

• Various scatterplots visualize correlation coefficients where:

  • $r = 0.9$, $r = -0.8$, $r = 0.7$, $r = -0.6$, etc.

Hypothesis Testing with Pearson’s Correlation Coefficient

Process

• As with means and proportions, hypothesis tests can be conducted using the correlation coefficient.

• Null Hypothesis (H0): No correlation between variables $X$ and $Y$ ($\rho = 0$).

• Alternative Hypothesis (HA): There is a correlation between variables $X$ and $Y$.

Covariance

• Definition:

  • $cov(X,Y)$ or $S_{XY}$ measures how random variables X and Y relate to one another, capturing the degree to which two variables change together.

• Positive Covariance: Indicates direct relationship.

• Negative Covariance: Indicates inverse relationship.

Formula

• $COV(X,Y) = S<em>{XY} = \frac{1}{N-1}\sum</em>{i=1}^{N} (x<em>i - \bar{x})(y</em>i - \bar{y})$.

• Covariance can range from negative to positive infinity.

Covariance Characteristics

• Scale Effect:

  • Covariance is sensitive to the scale of the variables—the unit is the product of the units of both variables.

• Adjustment Example:

  • If the y-values are doubled: changing $(xi, yi)$ to $(xi, 2yi)$ affects covariance.

• Important conclusion: Covariance is not standardized, meaning it retains the scale of measurement.

Correlation vs. Covariance

• Correlation is a scaled form of covariance, making it unit-less and allowing comparison across different datasets.

Assumptions of Pearson's Correlation Coefficient r

Assumptions

• Normal distribution of both variables X and Y.

• Linear relationship between X and Y must be true.

Cases When Assumptions Fail

• If assumptions do not hold, use a nonparametric (rank-based) test such as Spearman's Correlation.

Spearman’s Correlation Coefficient $\rho<em>S, r</em>S$

• Definition:

  • Spearman's correlation coefficient considers the ranks of X and Y, rather than their raw values.

• Use Cases:

  • For non-linear relationships.

  • When data are not normally distributed.

  • Robust in various conditions.

• Hypothesis for rank correlation:

  • Null Hypothesis ($H<em>0: \rho</em>S = 0$).

  • Test statistic follows the Student's t-distribution with $df = n - 2$.

Conducting Pearson & Spearman Correlations in SPSS

Step-by-Step Process

1. Access the dataset of interest (e.g., weight & energy intake from Dennison.sav).

2. Generate scatter plots:

   • SPSS: Graph > Legacy Dialogs > Scatter/Dot > Simple Scatter > Define

     • Set Y-Axis to weight and X-Axis to energy.

3. Statistical Correlation Analysis:

   • SPSS: Analyze > Correlate > Bivariate…

     • Input weight and energy, choose Pearson and Spearman correlations.

Examples and Consistency of Results

Case of Inconsistent Results

• When testing correlation between bilirubin and creatinine:

  • Pearson showed significant correlation $r = 0.198$, p = 0.004, while Spearman suggested no correlation $r_S = 0.021$, p = 0.76.

• Conclusion: Pearson correlation should be validated for normal distribution; otherwise, rely on Spearman's coefficient.

Summary of Key Concepts

Final Insights

• Correlation of continuous variables can be assessed using:

  • Pearson correlation coefficient (for normally distributed data)

  • Spearman rank correlation (for non-normal data or non-linear relationships).

• Statistical tests apply to both Spearman and Pearson correlations via:

  • $tstat = \frac{r}{SE_r}$

  • $tstat = \frac{r<em>S}{SE</em>{r_S}}$

Hypothesis Testing

• Effective statistical test design requires defining assumptions and ensuring methods are appropriate for the underlying data characteristics, validating results, and correctly interpreting correlation as not implying causation.