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Understanding Relationships: Correlation Analysis

Source: vignettes/correlation-analysis.Rmd
correlation-analysis.Rmd

Overview

Correlation measures how two variables move together. mariposa provides three methods for different situations:

Method Function Best for
Pearson’s r pearson_cor() Linear relationships between continuous variables
Spearman’s \rho spearman_rho() Monotonic relationships, ordinal data, or data with outliers
Kendall’s \tau kendall_tau() Ordinal data, small samples, or many tied values

All three support survey weights, multiple variables (correlation matrices), and grouped analysis.

Pearson Correlation

Basic Usage 

survey_data %>%
  pearson_cor(age, income)
#> Pearson Correlation: age x income
#>   r = -0.007, p = 0.761 , N = 2186

Interpretation of r:

  • r = 1: Perfect positive correlation
  • r = 0: No linear relationship
  • r = -1: Perfect negative correlation

With Survey Weights 

survey_data %>%
  pearson_cor(age, income, weights = sampling_weight)
#> Pearson Correlation: age x income [Weighted]
#>   r = -0.005, p = 0.828 , N = 2201

Correlation Matrix

Pass multiple variables to get all pairwise correlations:

survey_data %>%
  pearson_cor(trust_government, trust_media, trust_science,
              weights = sampling_weight)
#> Pearson Correlation: 3 variables [Weighted]
#>   trust_government x trust_media: r = 0.012, p = 0.582  
#>   trust_government x trust_science: r = 0.031, p = 0.145  
#>   trust_media x trust_science:   r = 0.024, p = 0.259  
#>   0/3 pairs significant (p < .05), N = 2242

Detailed Output 

result <- survey_data %>%
  pearson_cor(age, income, weights = sampling_weight)

summary(result)
#> 
#> Weighted Pearson Correlation 
#> -----------------------------
#> 
#> - Weights variable: sampling_weight
#> - Missing data handling: pairwise deletion
#> - Confidence level: 95.0%
#> 
#> 
#>   Correlation: r = -0.005
#>   p-value: p = 0.828 
#>   N = 2201
#>   95% CI: [-0.046, 0.037]
#>   r-squared: 0.000
#> 
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05

Grouped Analysis 

survey_data %>%
  group_by(region) %>%
  pearson_cor(age, income, weights = sampling_weight)
#> [region = 1]
#> Pearson Correlation: age x income [Weighted]
#>   r = 0.050, p = 0.293 , N = 449
#> [region = 2]
#> Pearson Correlation: age x income [Weighted]
#>   r = -0.019, p = 0.427 , N = 1751

Spearman Correlation

Use Spearman’s \rho when the relationship is monotonic but not necessarily linear, or when working with ordinal data or data with outliers:

survey_data %>%
  spearman_rho(political_orientation, environmental_concern,
               weights = sampling_weight)
#> Spearman Correlation: political_orientation x environmental_concern [Weighted]
#>   rho = -0.576, p < 0.001 ***, N = 2207

Multiple Variables 

survey_data %>%
  spearman_rho(political_orientation, environmental_concern,
               life_satisfaction, trust_government,
               weights = sampling_weight)
#> Spearman Correlation: 4 variables [Weighted]
#>   political_orientation x environmental_concern: rho = -0.576, p < 0.001 *** 
#>   political_orientation x life_satisfaction: rho = -0.004, p = 0.833  
#>   political_orientation x trust_government: rho = -0.055, p = 0.011 * 
#>   environmental_concern x life_satisfaction: rho = 0.003, p = 0.904  
#>   environmental_concern x trust_government: rho = 0.067, p = 0.002 ** 
#>   life_satisfaction x trust_government: rho = 0.002, p = 0.942  
#>   3/6 pairs significant (p < .05), N = 2207

Kendall’s Tau

Use Kendall’s \tau for ordinal data, small samples (n < 30), or data with many tied values. It is more robust than Spearman but typically produces smaller absolute values:

survey_data %>%
  kendall_tau(political_orientation, life_satisfaction,
              weights = sampling_weight)
#> Kendall's Tau: political_orientation x life_satisfaction [Weighted]
#>   tau = -0.005, p = 0.747 , N = 2241

Partial Correlation

partial_cor() (SPSS PARTIAL CORR) removes the influence of control variables from a correlation. Comparing the partial against the zero-order correlation shows how much of the original association the controls explain:

# Does the satisfaction-income correlation survive controlling for age?
survey_data %>%
  partial_cor(life_satisfaction, income, controls = age,
              weights = sampling_weight)
#> Partial Correlation: life_satisfaction, income | controlling for age [Weighted]
#>   life_satisfaction x income: partial r = 0.450, p < 0.001 *** (zero-order r = 0.450), N = 2130
#> Use summary() for detailed output.

Multiple analysis variables and multiple controls work together; with three or more variables, summary() adds the partial-correlation matrix:

result <- survey_data %>%
  partial_cor(trust_government, trust_media, trust_science,
              controls = c(age, political_orientation))
summary(result)
#> 
#> Partial Correlation Results
#> ---------------------------
#> - Variables: trust_government, trust_media, trust_science
#> - Controlling for: age, political_orientation
#> - Missing: Listwise deletion
#> 
#> Partial Correlation Matrix:
#> --------------------------- 
#>                  trust_government trust_media trust_science
#> trust_government            1.000       0.019         0.033
#> trust_media                 0.019       1.000         0.009
#> trust_science               0.033       0.009         1.000
#> --------------------------- 
#> 
#> Pairwise Results:
#>   -------------------------------------------------------------------------------------- 
#>   Variable 1           Variable 2  Partial r  Zero-order r    df      t     p     n  sig 
#>   -------------------------------------------------------------------------------------- 
#>   trust_government    trust_media      0.019         0.019  1960  0.859  .390  1964      
#>   trust_government  trust_science      0.033         0.030  1960  1.447  .148  1964      
#>   trust_media       trust_science      0.009         0.010  1960  0.407  .684  1964      
#>   -------------------------------------------------------------------------------------- 
#> 
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05

Interpreting Correlations

Strength Guidelines

Absolute value of r Interpretation
0.0 – 0.3 Weak
0.3 – 0.7 Moderate
0.7 – 1.0 Strong

These are general guidelines. In survey research, correlations of 0.2–0.4 between different constructs are common and often substantively meaningful.

Choosing the Right Method

  1. Both variables continuous, relationship is linear → Pearson
  2. Monotonic but non-linear relationship, or outliers present → Spearman
  3. Ordinal data, small sample, or many ties → Kendall

Comparing Methods

If Pearson and Spearman give very different results, the relationship may be non-linear:

pearson_result <- survey_data %>%
  pearson_cor(life_satisfaction, income, weights = sampling_weight)

spearman_result <- survey_data %>%
  spearman_rho(life_satisfaction, income, weights = sampling_weight)

kendall_result <- survey_data %>%
  kendall_tau(life_satisfaction, income, weights = sampling_weight)

comparison <- data.frame(
  Method = c("Pearson", "Spearman", "Kendall"),
  Correlation = c(pearson_result$correlations$correlation[1],
                  spearman_result$correlations$correlation[1],
                  kendall_result$correlations$correlation[1]),
  P_Value = c(pearson_result$correlations$p_value[1],
              spearman_result$correlations$p_value[1],
              kendall_result$correlations$p_value[1])
)
print(comparison)
#>     Method Correlation       P_Value
#> 1  Pearson   0.4501535 8.997507e-107
#> 2 Spearman   0.4501535 2.322944e-113
#> 3  Kendall   0.4501535 6.963954e-132

Correlation Does Not Imply Causation

A significant correlation does not mean one variable causes the other:

  • A third variable may drive both (e.g., warm weather → both ice cream sales and drowning)
  • The direction of causality is unknown
  • There may be no causal link at all

Complete Example 

# 1. Correlation matrix for key variables
cor_result <- survey_data %>%
  pearson_cor(age, income, life_satisfaction,
              political_orientation, environmental_concern,
              weights = sampling_weight)
cor_result
#> Pearson Correlation: 5 variables [Weighted]
#>   age x income:                  r = -0.005, p = 0.828  
#>   age x life_satisfaction:       r = -0.029, p = 0.150  
#>   age x political_orientation:   r = -0.029, p = 0.168  
#>   age x environmental_concern:   r = 0.024, p = 0.244  
#>   income x life_satisfaction:    r = 0.450, p < 0.001 *** 
#>   income x political_orientation: r = -0.034, p = 0.125  
#>   income x environmental_concern: r = 0.015, p = 0.503  
#>   life_satisfaction x political_orientation: r = -0.004, p = 0.836  
#>   life_satisfaction x environmental_concern: r = -0.003, p = 0.866  
#>   political_orientation x environmental_concern: r = -0.584, p < 0.001 *** 
#>   2/10 pairs significant (p < .05), N = 2201

# 2. Identify the strongest correlations
significant <- cor_result$correlations %>%
  filter(p_value < 0.05) %>%
  arrange(desc(abs(correlation)))
print(significant)
#>                    var1                  var2 correlation       p_value
#> 1 political_orientation environmental_concern  -0.5843752 1.558234e-203
#> 2                income     life_satisfaction   0.4501535 8.997507e-107
#>   conf_int_lower conf_int_upper    n sig r_squared
#> 1     -0.6111168     -0.5563011 2221 *** 0.3414944
#> 2      0.4156264      0.4833852 2130 *** 0.2026382

# 3. Check if relationships vary by region
survey_data %>%
  group_by(region) %>%
  pearson_cor(age, income, weights = sampling_weight)
#> [region = 1]
#> Pearson Correlation: age x income [Weighted]
#>   r = 0.050, p = 0.293 , N = 449
#> [region = 2]
#> Pearson Correlation: age x income [Weighted]
#>   r = -0.019, p = 0.427 , N = 1751

Reporting Results (APA Style)

Include the correlation coefficient, confidence interval, p-value, and sample size:

“There was a moderate positive correlation between age and income (r = .34, 95% CI [.29, .39], p < .001, Neff = 2,341), indicating that older respondents tended to report higher incomes.”

Practical Tips

  1. Always check the scatterplot. Correlations can miss non-linear relationships, and a single outlier can inflate or deflate the coefficient.

  2. Use Spearman when in doubt. It makes fewer assumptions than Pearson and works with both continuous and ordinal data.

  3. Report the magnitude, not just significance. With large samples, even r = .05 can be significant but is practically meaningless.

  4. Use correlation matrices to prioritize. Before regression, check which variables are most strongly related to your outcome.

Summary

  1. Pearson measures linear relationships between continuous variables
  2. Spearman measures monotonic relationships and works with ordinal data
  3. Kendall is the most robust choice for ordinal data and small samples
  4. All three support weights, correlation matrices, and group_by()
  5. Correlation does not imply causation

Next Steps