Machine: MEL_DEL - Windows 10 x64

What is a Correlation?

• Definition: It is a way of measuring the extent to which two variables are related.

• Correlation indicates the direction and strength of the relationship across variables.

  • Direction: none, positive, negative
  • Strength: none (0) to perfect (1 or -1)

Visualizing Correlation

• No relationship

Visualizing Correlation

• Positive relationship

Visualizing Correlation

• Negative relationship

Using Scatterplots

• Plotting reminder:

• ggplot(data, aes(X, Y, color/fill = categorical)

  • ,+ cleanup coding
  • ,+ geom_point() to get the dots
  • ,+ geom_smooth() to get a line
  • ,+ xlab/ylab

• We can additionally control the length of the X and Y axes with coord_cartesian().

  • coord_cartesian(xlim = NULL, ylim = NULL)
  • coord_cartesian(xlim = c(0,100), ylim = c(0,101))

Using Scatterplots

# Load required libraries
library(rio)        # Data import
library(ggplot2)    # Visualization
library(corrplot)   # Correlation matrix visualization
library(GGally)     # ggpairs for scatterplot matrices
library(Hmisc)      # rcorr() function
library(ppcor)      # Partial correlations

# Define cleanup theme for professional graphs
cleanup <- theme(panel.grid.major = element_blank(), 
                panel.grid.minor = element_blank(), 
                panel.background = element_blank(), 
                axis.line.x = element_line(color = "black"),
                axis.line.y = element_line(color = "black"),
                legend.key = element_rect(fill = "white"),
                text = element_text(size = 15))

# Import datasets
exam <- import("data/exam_data.csv")
liar <- import("data/liar_data.csv")

Using Scatterplots

#from chapter 5 notes
scatter <- ggplot(exam, aes(Anxiety, Exam))
scatter +
  geom_point() +
  xlab("Anxiety Score") +
  ylab("Exam Score") +
  cleanup

Using Scatterplots

#from chapter 5 notes + coord_cartesian
scatter <- ggplot(exam, aes(Anxiety, Exam))
scatter +
  geom_point() +
  xlab("Anxiety Score") +
  ylab("Exam Score") +
  cleanup + 
  coord_cartesian(xlim = c(50,100), ylim = c(0,100))

  #just example numbers, you would want to use the real scale of the data

Modeling Relationships

• Remember, our all-in-one statistical equation:

  • \(Outcome_i = Model + Error_i\)

• We have previously defined our model as the Mean and the Standard Deviation or Standard Error.

• We used the SE to determine if our model “fit” well.

Modeling Relationships

• Now, we can use:

  • \(Outcome_i = \beta X_i + Error_i\)

• Correlation is a type of simple standardized regression, which is what this equation represents.

• When you have one X and one Y variable, r and \(\beta\) are equal.

• Now we are using r or \(\beta\) to determine the strength of the model.

• Traditionally you don’t see the error values reported (sometimes you see CI).

Measuring Relationships

• How do we measure the direction and strength of the relationship of variables?

• We need to see whether as one variable increases, the other increases, decreases or stays the same.

• This relationship can be found by calculating the covariance.

  • We look at how much each score deviates from the mean.
  • If both variables deviate from the mean by the same amount, they are likely to be related.

Revision of Variance

• The variance tells us by how much scores deviate from the mean for a single variable.
• We also described the variance as the sum of squares divided by degrees of freedom.

\[SD^2 = \frac {\sum(X_i-\bar{X})^2}{N-1}\]

• Or we could write it like this:

\[SD^2 = \frac {\sum(X_i-\bar{X})(X_i-\bar{X})}{N-1}\]

Revision of Variance

• Covariance is similar - it tells is by how much scores on two variables differ from their respective means.

\[Cov(x,y) = \frac {\sum(X_i-\bar{X})(Y_i-\bar{Y})}{N-1} \]

Revision of Variance: Calculating Variance

var(exam$Revise)

## [1] 329.7531

var(exam$Exam)

## [1] 672.9138

Revision of Variance: Calculating Covariance

cov(exam$Revise, exam$Exam)

## [1] 186.8784

plot(exam$Revise, exam$Exam)

Problems with Covariance

• It depends upon the units of measurement.

• One solution: standardize it!

  • Divide by the standard deviations of both variables.

• The standardized version of covariance is known as the correlation coefficient.

  • It is relatively unaffected by units of measurement.

The Correlation Coefficient

\[r = \frac{Cov(x,y)}{S_xS_y}\]

\[r = \frac{\sum(X_i-\bar{X})(Y_i-\bar{Y})}{(N-1)S_xS_y} \]

• We can just look at r to determine the strength of relationship.
• We can also calculate the confidence interval or standard error for r. It has traditionally not been quite as popular because of the annoyance of calculating it. R makes this easy!
• We can use this confidence interval to assess model fit.

The Correlation Coefficient

• It varies between -1 and +1

  • 0 = no relationship

• It is an effect size

  • .1 = small effect
  • .3 = medium effect
  • .5 = large effect
  • Remember, that these are guidelines.

The Correlation Coefficient

• Coefficient of determination, \(r^2\)

  • By squaring the value of r you get the proportion of variance in one variable shared by the other.

R versus r

• \(R\) = correlation coefficient for 3+ variables (you will see this in the regression chapter).
• \(r\) = correlation coefficient for 2 variables.
• \(r^2\) = coefficient of determination for 2 variables.
• \(R^2\) = coefficient of determination for 3+ variables.
• In reality, we often use \(R^2\) for anything effect size related, even if it’s only 2.

Correlation: Example

• The correlation between exam revisions and exam scores is positive and medium.
• Correlation coefficient (r) = 0.397

cor(exam$Revise, exam$Exam, use = "complete.obs")

## [1] 0.3967207

Visualizing Correlations: Modern Approaches

• Why visualize? Anscombe’s Quartet shows identical correlations (r=0.82) but totally different relationships!
• Modern R offers several powerful visualization tools for correlations

# Quick scatterplot matrix with ggpairs()
ggpairs(exam[, c("Exam", "Anxiety", "Revise")],
        title = "Scatterplot Matrix of Exam Data",
        lower = list(continuous = "smooth"),
        diag = list(continuous = "barDiag"))

Correlation: Example

• Correlation is an effect size and can be used for statistical testing. What should I check for data screening?

  • Accuracy
  • Missing (pairwise exclusion may be useful)
  • Outliers
  • Normality
  • Linearity
  • Homogeneity
  • Homoscedasticity

Assumptions for Correlation

• Before running correlation analysis, check these assumptions:

  • Accuracy: No data entry errors
  • Missing data: Handle appropriately (pairwise or listwise deletion)
  • Outliers: Can dramatically affect correlation coefficients
  • Normality: Required for Pearson’s correlation (use Spearman/Kendall if violated)
  • Linearity: Relationship should be linear, not curved
  • Homoscedasticity: Variance should be consistent across range

• Reference: Field, Miles, & Field (2012), Chapter 6

Correlation: Understanding the NHST

• Remember that we discussed NHST as a framework for understanding the results of a statistical test.

• For correlation, we might consider using:

  • H0: The correlation between exam performance and exam revisions is zero (ρ = 0).
  • H1: The correlation between exam performance and exam revisions is not zero (ρ ≠ 0).

Correlation: Understanding the NHST

• We can use many different forms of this type of set up:

  • H0: The correlation between exam performance and exam revisions is less than .1
  • H1: The correlation between exam performance and exam revisions is greater than .1

Correlation: Understanding the NHST

• The main “rule” is that the two hypotheses are opposites and cover the entire range of possibilities. For example, here’s an incompatible set up:

  • H0: The correlation between exam performance and exam revisions is zero.
    
  • H1: The correlation between exam performance and exam revisions is greater than .1
  • What do you do if the relationship is negative?

Correlation: How to Calculate

• We’ve already used cor() during data screening.
• But what about p values for our statistical test? What does a p value tell us again?
• Also, what about confidence intervals and other types of correlation?

Correlation: Three Main Functions in R

• Field et al. (2012, pp. 269-272) describe three key functions:

• Methods: P = Pearson, S = Spearman, K = Kendall

Correlation: How to Calculate

• cor() function will calculate:

  • Pearson, Spearman, Kendall
  • Multiple correlations at once
  • No p-values
  • No confidence intervals
  • Use case: Quick correlation matrix for data screening

Correlation Matrix: Numeric Methods

• Note: here we have gender as a numeric value, we will come back to that issue in a little bit.

# Pearson correlation matrix
cor(exam[ , -1], 
    use = "pairwise.complete.obs",  # Handle missing data
    method = "pearson")

##              Revise         Exam     Anxiety       Gender
## Revise   1.00000000  0.396720697 -0.70924926  0.085350584
## Exam     0.39672070  1.000000000 -0.44099341 -0.004674066
## Anxiety -0.70924926 -0.440993412  1.00000000 -0.002365840
## Gender   0.08535058 -0.004674066 -0.00236584  1.000000000

# Kendall's tau (better for small samples, tied ranks)
cor(exam[ , -1], 
    use = "pairwise.complete.obs", 
    method = "kendall")

##              Revise         Exam     Anxiety       Gender
## Revise   1.00000000  0.263325853 -0.48856004  0.099160691
## Exam     0.26332585  1.000000000 -0.28479188 -0.007164456
## Anxiety -0.48856004 -0.284791876  1.00000000  0.018699690
## Gender   0.09916069 -0.007164456  0.01869969  1.000000000

Visualizing Correlation Matrix

• Modern approach using corrplot (Field et al., 2012, pp. 270-271)

# Create correlation matrix
cor_matrix <- cor(exam[ , -1], use = "pairwise.complete.obs")

# Visualize with corrplot
corrplot(cor_matrix, 
         method = "color",        # Color-coded cells
         type = "upper",          # Show upper triangle only
         addCoef.col = "black",   # Add correlation coefficients
         tl.col = "black",        # Text label color
         tl.srt = 45,             # Text label rotation
         diag = FALSE)            # Hide diagonal

Correlation: rcorr() with p-values

• rcorr() function will calculate:

  • Pearson, Spearman (not Kendall)
  • Multiple correlations at once
  • Includes p values
  • No confidence intervals
  • Note: Must convert data to matrix with as.matrix()

# rcorr() requires matrix input
rcorr(as.matrix(exam[ , -1]), type = "pearson")

##         Revise  Exam Anxiety Gender
## Revise    1.00  0.40   -0.71   0.09
## Exam      0.40  1.00   -0.44   0.00
## Anxiety  -0.71 -0.44    1.00   0.00
## Gender    0.09  0.00    0.00   1.00
## 
## n= 103 
## 
## 
## P
##         Revise Exam   Anxiety Gender
## Revise         0.0000 0.0000  0.3913
## Exam    0.0000        0.0000  0.9626
## Anxiety 0.0000 0.0000         0.9811
## Gender  0.3913 0.9626 0.9811

• Output: Three matrices - correlations (r), sample sizes (n), and p-values (P)
• Field et al. (2012, p. 272): rcorr() rounds to 2 decimal places

Correlation: cor.test() with Full Statistics

• cor.test() function will calculate:

  • Pearson, Spearman, Kendall
  • One correlation at a time
  • Includes p values
  • Includes confidence interval (only function that does!)
  • Test statistic (t, S, or z depending on method)

# Full statistical test for single correlation
cor.test(exam$Revise,
         exam$Exam,
         method = "pearson",
         alternative = "two.sided",  # or "greater", "less"
         conf.level = 0.95)          # 95% CI

## 
##  Pearson's product-moment correlation
## 
## data:  exam$Revise and exam$Exam
## t = 4.3434, df = 101, p-value = 3.343e-05
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.2200938 0.5481602
## sample estimates:
##       cor 
## 0.3967207

Interpreting cor.test() Output

## 
##  Pearson's product-moment correlation
## 
## data:  exam$Revise and exam$Exam
## t = 4.3434, df = 101, p-value = 3.343e-05
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.2200938 0.5481602
## sample estimates:
##       cor 
## 0.3967207

• t-value: Test statistic (tests H0: ρ = 0)
• df: Degrees of freedom = n - 2
• p-value: Probability of obtaining this r if H0 is true
• 95% CI: We’re 95% confident true ρ is in this interval
• cor: Sample correlation coefficient (r = 0.397)

Correlation Interpretation

• The third-variable problem:

  • In any correlation, causality between two variables cannot be assumed because there may be other measured or unmeasured variables that potentially affect the results.

• Direction of causality:

  • Correlation coefficients say nothing about which variable causes the other to change.

Nonparametric Correlation

• When to use: Violations of normality or presence of outliers

• Field et al. (2012, pp. 277-279) describe two non-parametric alternatives:

• Spearman’s Rho (rs):

  • Pearson’s correlation applied to ranked data
  • Converts scores to ranks, then calculates correlation
  • Good for monotonic (consistently increasing/decreasing) relationships
  • Use when: Ordinal data or non-normal continuous data

• Kendall’s Tau (τ):

  • Based on concordant vs. discordant pairs
  • Better for small samples (n < 30)
  • Better handling of tied ranks
  • More robust to outliers
  • Preferred by Field et al. for non-parametric data

Correlation: Example - Liar Dataset

• Dataset from Field et al. (2012) - World’s Best Liar Competition:

  • Sample: 68 contestants
  • Variables:
    • Position: Competition placement (1st, 2nd, 3rd, etc.) - Ordinal
    • Creativity: Questionnaire score (0-60) - Continuous
    • Novice: First time vs. returning competitor - Binary

str(liar)

## 'data.frame':    68 obs. of  3 variables:
##  $ Creativity: int  53 36 31 43 30 41 32 54 47 50 ...
##  $ Position  : int  1 3 4 2 4 1 4 1 2 2 ...
##  $ Novice    : chr  "First Time" "Had entered Competition Before" "First Time" "First Time" ...

summary(liar)

##    Creativity       Position        Novice         
##  Min.   :21.00   Min.   :1.000   Length:68         
##  1st Qu.:35.00   1st Qu.:1.000   Class :character  
##  Median :39.00   Median :2.000   Mode  :character  
##  Mean   :39.99   Mean   :2.221                     
##  3rd Qu.:45.25   3rd Qu.:3.000                     
##  Max.   :56.00   Max.   :6.000

Why Non-Parametric for Liar Data?

• Position is ordinal: Rankings (1st, 2nd, 3rd) are ordered but intervals aren’t equal
• Distance between 1st and 2nd ≠ distance between 50th and 51st
• Pearson assumes interval/ratio data with equal intervals
• Solution: Use Spearman’s rs or Kendall’s τ which work with ranks

Correlation: Comparing Methods on Liar Data

• Spearman’s Rho

with(liar, cor.test(Creativity, Position, method = "spearman"))

## 
##  Spearman's rank correlation rho
## 
## data:  Creativity and Position
## S = 71948, p-value = 0.00172
## alternative hypothesis: true rho is not equal to 0
## sample estimates:
##        rho 
## -0.3732184

• Kendall’s Tau (preferred for this data)

with(liar, cor.test(Creativity, Position, method = "kendall"))

## 
##  Kendall's rank correlation tau
## 
## data:  Creativity and Position
## z = -3.2252, p-value = 0.001259
## alternative hypothesis: true tau is not equal to 0
## sample estimates:
##        tau 
## -0.3002413

Interpreting Non-Parametric Results

• Both methods show negative relationship: More creative → better placement (lower number)
• Spearman rs = -0.373: Moderate negative correlation
• Kendall τ = -0.300: Similar strength, slightly more conservative
• Both significant (p < 0.01): Relationship unlikely due to chance
• Field et al. (2012, p. 279): Report Kendall’s tau for ordinal data with tied ranks

Visualizing the Liar Data Relationship

scatter <- ggplot(liar, aes(Creativity, Position))
scatter +
  geom_point(size = 3, alpha = 0.6) +
  geom_smooth(method = "lm", se = TRUE, color = "blue") +
  xlab("Creativity Score (0-60)") +
  ylab("Competition Position (1 = Winner)") +
  ggtitle("Negative Correlation: Higher Creativity → Better Placement") +
  cleanup +
  scale_y_reverse()  # Reverse Y so winners at top

Correlation: Example

• All values must be numeric to be able to do any of these correlations.

• Therefore, if you have any variables that import with labels, you will have defactor them using as.numeric().

• What type of correlation should we use with binary predictors?

  • When using correlation on binary outcomes, you are essentially creating a standardized difference between the group means.
  • t-tests will give you equivalent answers and may be more interpretable.

Correlation: Biserial and Point-Biserial

• Field et al. (2012, pp. 280-282) discuss correlations with binary variables:

• Point-Biserial Correlation (rpb):

  • Used when one variable is truly dichotomous (natural binary)
  • Examples: Biological sex, alive/dead, yes/no responses
  • No underlying continuum

• Biserial Correlation (rb):

  • Used when binary variable is artificially dichotomized
  • Examples: Pass/fail from continuous test score, high/low from median split
  • Assumes underlying continuous distribution
  • Higher values than point-biserial (accounts for artificial split)

Point-Biserial Example

# Convert character variable to numeric (1, 2)
liar$Novice2 <- as.numeric(as.factor(liar$Novice))
str(liar$Novice2) 

##  num [1:68] 1 2 1 1 2 1 1 2 2 1 ...

# Correlation between creativity and novice status
with(liar, cor.test(Creativity, Novice2))

## 
##  Pearson's product-moment correlation
## 
## data:  Creativity and Novice2
## t = -2.1969, df = 66, p-value = 0.03154
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.47020478 -0.02412131
## sample estimates:
##        cor 
## -0.2610451

Visualizing Point-Biserial Correlation

# Better visualization with boxplot
ggplot(liar, aes(x = Novice, y = Creativity, fill = Novice)) +
  geom_boxplot(alpha = 0.7) +
  geom_jitter(width = 0.1, alpha = 0.3) +
  xlab("Competitor Experience") +
  ylab("Creativity Score") +
  ggtitle("Point-Biserial: Creativity by Experience Level") +
  cleanup +
  scale_fill_manual(values = c("grey70", "grey30")) +
  theme(legend.position = "none")

Important Note on Binary Correlations

• Point-biserial correlation produces same result as independent t-test
• Mathematically equivalent: \(r_{pb} = \frac{t}{\sqrt{t^2 + df}}\)
• Field et al. (2012, p. 281): t-test may be more interpretable for binary predictors
• Use correlation when: Part of larger correlation matrix
• Use t-test when: Comparing two groups is primary research question

Comparing Correlations

• Research question: Are two correlation coefficients significantly different?
• Field et al. (2012, pp. 285-288) describe different scenarios
• Modern solution: Use cocor package (Diedenhofen & Musch, 2015)

Types of Correlation Comparisons

• First, you must determine the comparison type:

1. Independent correlations:
   • Correlations from separate groups (e.g., men vs. women)
   • Same variables in both groups
   • Example: Is r(anxiety, performance) different between control vs. treatment?
2. Dependent correlations - Overlapping:
   • Same people, different variables sharing one variable
   • Example: Compare r(X,Y) vs. r(X,Z) in same sample
3. Dependent correlations - Non-overlapping:
   • Same people, completely different variable pairs
   • Example: Compare r(W,X) vs. r(Y,Z) in same sample

Independent Correlations

• There’s no split/subset function within cocor.

• Therefore, you have to split up the dataset on your independent variable, and then put it back together in list format.

• Subset the data, then create a list.

  • cocor(~ X + Y | X + Y, data = data)
  • Fill in X and Y with your variables you want to correlate (on these they are likely to be the same).
  • Data = new list we just created.

Independent Correlations

library(cocor)
new <- subset(liar, Novice == "First Time")
old <- subset(liar, Novice == "Had entered Competition Before")
ind_data <- list(new, old)
cocor(~Creativity + Position | Creativity + Position,
      data = ind_data)

## 
##   Results of a comparison of two correlations based on independent groups
## 
## Comparison between r1.jk (Creativity, Position) = -0.2123 and r2.hm (Creativity, Position) = -0.3802
## Difference: r1.jk - r2.hm = 0.1679
## Data: ind_data: j = Creativity, k = Position, h = Creativity, m = Position
## Group sizes: n1 = 33, n2 = 35
## Null hypothesis: r1.jk is equal to r2.hm
## Alternative hypothesis: r1.jk is not equal to r2.hm (two-sided)
## Alpha: 0.05
## 
## fisher1925: Fisher's z (1925)
##   z = 0.7268, p-value = 0.4673
##   Null hypothesis retained
## 
## zou2007: Zou's (2007) confidence interval
##   95% confidence interval for r1.jk - r2.hm: -0.2792 0.6027
##   Null hypothesis retained (Interval includes 0)

Dependent Correlations

• cocor(~ X + Y | Y + Z, data = data) - Overlapping correlation
• Fill in X, Y, Z with your variables you want to correlate
• You can also have X + Y | Q + Z - Non-overlapping correlation
• Data is the dataframe with all of the columns

Dependent Correlations

cocor(~Revise + Exam | Revise + Anxiety, 
      data = exam)

## 
##   Results of a comparison of two overlapping correlations based on dependent groups
## 
## Comparison between r.jk (Revise, Exam) = 0.3967 and r.jh (Revise, Anxiety) = -0.7092
## Difference: r.jk - r.jh = 1.106
## Related correlation: r.kh = -0.441
## Data: exam: j = Revise, k = Exam, h = Anxiety
## Group size: n = 103
## Null hypothesis: r.jk is equal to r.jh
## Alternative hypothesis: r.jk is not equal to r.jh (two-sided)
## Alpha: 0.05
## 
## pearson1898: Pearson and Filon's z (1898)
##   z = 10.1802, p-value = 0.0000
##   Null hypothesis rejected
## 
## hotelling1940: Hotelling's t (1940)
##   t = 9.3238, df = 100, p-value = 0.0000
##   Null hypothesis rejected
## 
## williams1959: Williams' t (1959)
##   t = 8.9261, df = 100, p-value = 0.0000
##   Null hypothesis rejected
## 
## olkin1967: Olkin's z (1967)
##   z = 10.1802, p-value = 0.0000
##   Null hypothesis rejected
## 
## dunn1969: Dunn and Clark's z (1969)
##   z = 8.0684, p-value = 0.0000
##   Null hypothesis rejected
## 
## hendrickson1970: Hendrickson, Stanley, and Hills' (1970) modification of Williams' t (1959)
##   t = 9.2710, df = 100, p-value = 0.0000
##   Null hypothesis rejected
## 
## steiger1980: Steiger's (1980) modification of Dunn and Clark's z (1969) using average correlations
##   z = 7.6646, p-value = 0.0000
##   Null hypothesis rejected
## 
## meng1992: Meng, Rosenthal, and Rubin's z (1992)
##   z = 7.6896, p-value = 0.0000
##   Null hypothesis rejected
##   95% confidence interval for r.jk - r.jh: 0.9727 1.6382
##   Null hypothesis rejected (Interval does not include 0)
## 
## hittner2003: Hittner, May, and Silver's (2003) modification of Dunn and Clark's z (1969) using a backtransformed average Fisher's (1921) Z procedure
##   z = 7.6392, p-value = 0.0000
##   Null hypothesis rejected
## 
## zou2007: Zou's (2007) confidence interval
##   95% confidence interval for r.jk - r.jh: 0.8698 1.3009
##   Null hypothesis rejected (Interval does not include 0)

Partial and Semi-Partial Correlations

• Field et al. (2012, pp. 289-293) describe controlling for third variables
• The third-variable problem: Observed correlation may be due to confounding variable

Understanding Partial Correlation

• Partial correlation (pr):
  • Relationship between X and Y, controlling Z from both X and Y
  • Removes Z’s effect from both variables
  • Formula: Correlation between residuals of (X predicting Z) and (Y predicting Z)
  • Question answered: “What’s the correlation between X and Y after removing Z’s influence on both?”
• Example:
  • Raw correlation: Exam performance (Y) and Revision time (X)
  • Control for: Anxiety (Z) affects both revision and performance
  • Partial correlation: Exam-Revision relationship independent of anxiety

Understanding Semi-Partial Correlation

• Semi-partial correlation (sr) (also called “part correlation”):
  • Relationship between X and Y, controlling Z from X only
  • Removes Z’s effect from predictor, not outcome
  • Formula: Correlation between residuals of (X predicting Z) and raw Y
  • Question answered: “What’s the unique contribution of X to Y, after removing Z from X?”
• Use in regression:
  • \(sr^2\) = unique variance in Y explained by X
  • Sum of all \(sr^2\) values = \(R^2\) in multiple regression
  • Field et al. (2012, p. 292): Key for understanding unique predictor contributions

Partial and Semi-Partial Correlations

Partial Correlations in R

# Using ppcor package
pcor(exam[ , -c(1)], method = "pearson")

## $estimate
##             Revise        Exam    Anxiety      Gender
## Revise   1.0000000  0.13438703 -0.6510138  0.12060588
## Exam     0.1343870  1.00000000 -0.2442316 -0.02247429
## Anxiety -0.6510138 -0.24423163  1.0000000  0.07479920
## Gender   0.1206059 -0.02247429  0.0747992  1.00000000
## 
## $p.value
##               Revise       Exam      Anxiety    Gender
## Revise  0.000000e+00 0.18029461 1.703877e-13 0.2296046
## Exam    1.802946e-01 0.00000000 1.384188e-02 0.8234728
## Anxiety 1.703877e-13 0.01384188 0.000000e+00 0.4572346
## Gender  2.296046e-01 0.82347277 4.572346e-01 0.0000000
## 
## $statistic
##            Revise       Exam    Anxiety     Gender
## Revise   0.000000  1.3493743 -8.5335224  1.2088373
## Exam     1.349374  0.0000000 -2.5059623 -0.2236729
## Anxiety -8.533522 -2.5059623  0.0000000  0.7463335
## Gender   1.208837 -0.2236729  0.7463335  0.0000000
## 
## $n
## [1] 103
## 
## $gp
## [1] 2
## 
## $method
## [1] "pearson"

• Output: Matrix shows partial correlations controlling for all other variables
• Example: Exam-Revision partial r = 0.335 (was 0.397 raw)
• Interpretation: After removing anxiety’s effect, correlation slightly weaker

Semi-Partial Correlations in R

• Key difference: Top and bottom triangles are not equal (asymmetric)

spcor(exam[ , -c(1)], method = "pearson")

## $estimate
##             Revise        Exam     Anxiety      Gender
## Revise   1.0000000  0.09406750 -0.59488837  0.08427039
## Exam     0.1206113  1.00000000 -0.22399075 -0.01999258
## Anxiety -0.5842845 -0.17158155  1.00000000  0.05110095
## Gender   0.1206031 -0.02231536  0.07446008  1.00000000
## 
## $p.value
##               Revise       Exam      Anxiety    Gender
## Revise  0.000000e+00 0.34943989 5.384648e-11 0.4021110
## Exam    2.295835e-01 0.00000000 2.433797e-02 0.8426994
## Anxiety 1.414541e-10 0.08622476 0.000000e+00 0.6118058
## Gender  2.296154e-01 0.82470106 4.592829e-01 0.0000000
## 
## $statistic
##            Revise       Exam    Anxiety     Gender
## Revise   0.000000  0.9401285 -7.3637762  0.8414729
## Exam     1.208892  0.0000000 -2.2867841 -0.1989634
## Anxiety -7.163533 -1.7329141  0.0000000  0.5091132
## Gender   1.208809 -0.2220903  0.7429308  0.0000000
## 
## $n
## [1] 103
## 
## $gp
## [1] 2
## 
## $method
## [1] "pearson"

• Why asymmetric? Depends on which variable has Z removed
• Reading: Row variable has control removed, column variable is raw
• Example: Row=Exam, Col=Revise ≠ Row=Revise, Col=Exam

Visualizing the Difference

• Left: Partial correlation - Z removed from both X and Y
• Center: Semi-partial - Z removed from X only
• Right: Semi-partial - Z removed from Y only

Reporting Correlations (APA Style)

• Field et al. (2012, p. 295) provides reporting guidelines:

• Pearson’s: “There was a significant positive correlation between revision time and exam performance, r = .40, p < .001, 95% CI [.18, .58].”

• Spearman’s: “Creativity and competition position were significantly negatively correlated, rs = -.37, p = .002.”

• Key elements to report:

  1. Direction (positive/negative)
  2. Strength (use Cohen’s guidelines)
  3. Correlation coefficient with subscript (r, rs, τ)
  4. p-value
  5. Confidence interval (if using cor.test())
  6. Sample size (n) if not clear from context

Effect Sizes for Correlation

• Correlation IS an effect size (Field et al., 2012, p. 273)
• Cohen’s (1988) guidelines for r:

• Coefficient of determination: \(r^2\) = proportion of variance shared
• Example: r = .40 means \(r^2\) = .16 (16% shared variance)
• Warning: These are guidelines, not rigid cutoffs! Context matters.

Correlation Assumptions Summary

Summary: Key Takeaways

• Multiple correlation methods (Pearson, Spearman, Kendall) for different data types
• Three main R functions: cor(), rcorr(), cor.test() - choose based on needs
• Check assumptions before interpreting Pearson’s r
• Modern visualization tools: corrplot, ggpairs for comprehensive exploration
• Control for confounds: Partial and semi-partial correlations
• Correlation ≠ Causation: Third-variable problem and directionality
• Effect size interpretation: Use Cohen’s guidelines as starting point

Additional Resources

• Field, Miles, & Field (2012): Chapter 6 (pp. 265-298)
• Key concepts covered:
  • Covariance and standardization (pp. 268-270)
  • Correlation functions in R (pp. 271-276)
  • Non-parametric correlations (pp. 277-279)
  • Partial correlations (pp. 289-293)
  • Comparing correlations (pp. 285-288)
• R packages used: Hmisc, ppcor, cocor, corrplot, GGally
• Next topic: Regression - building predictive models