workshops

Correlation and Causation

• Correlation:
  • A statistical measure indicating the extent to which two or more variables fluctuate together.
  • Values range from -1 (perfect negative correlation) to +1 (perfect positive correlation), with 0 indicating no correlation.
• Causation:
  • Implies that a change in one variable is responsible for a change in another.
  • This relationship establishes a cause and effect between variables.
• Differentiation:
  • Correlation does not imply causation; two variables can be correlated without one causing the other to change.
  • Causation explicitly requires a cause-and-effect relationship, often established through controlled experiments.
• Importance of Visualizations:
  • Visual tools like scatter plots and correlation matrices help in identifying patterns and relationships between variables.
  • Effective visualization aids in distinguishing between mere correlations and potential causative relationships, guiding further statistical analysis or experimental design.

if (!requireNamespace("ggplot2", quietly = TRUE)) install.packages("ggplot2")
if (!requireNamespace("corrplot", quietly = TRUE)) install.packages("corrplot")

# Load packages
library(ggplot2)
library(corrplot)

corrplot 0.92 loaded

Visualize Correlation with Synthetic Data

# Generate data with correlation
set.seed(123)
n <- 100
x <- rnorm(n)
y <- x + rnorm(n)

correlation_coeff <- cor(x, y)

# ggplot
library(ggplot2)
df <- data.frame(x = x, y = y)
ggplot(df, aes(x, y)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE, color = "red") +
  labs(title = "Scatterplot showing correlation", x = "X", y = "Y") +
  annotate("text", x = 1, y = 4, label = paste("Correlation coefficient:", round(correlation_coeff, 2)))

`geom_smooth()` using formula = 'y ~ x'

Visualize Correlations in MTCARS data

data("mtcars")
head(mtcars)

A data.frame: 6 × 11

	mpg	cyl	disp	hp	drat	wt	qsec	vs	am	gear	carb
	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>
Mazda RX4	21.0	6	160	110	3.90	2.620	16.46	0	1	4	4
Mazda RX4 Wag	21.0	6	160	110	3.90	2.875	17.02	0	1	4	4
Datsun 710	22.8	4	108	93	3.85	2.320	18.61	1	1	4	1
Hornet 4 Drive	21.4	6	258	110	3.08	3.215	19.44	1	0	3	1
Hornet Sportabout	18.7	8	360	175	3.15	3.440	17.02	0	0	3	2
Valiant	18.1	6	225	105	2.76	3.460	20.22	1	0	3	1

cor_matrix <- cor(mtcars)
corrplot(cor_matrix, method = "circle")

cor_matrix <- cor(mtcars)

corrplot(cor_matrix, method = "circle", type = "upper", order = "hclust",
         addCoef.col = "black", # Color of the correlation coefficients
         tl.col = "black", tl.srt = 45, # Color and rotation of the labels
         title = "Correlation Matrix of Mtcars Dataset")

cor_matrix <- cor(mtcars, method="spearman")
corrplot(cor_matrix, method = "circle", type = "upper", order = "hclust",
         addCoef.col = "black",
         tl.col = "black", tl.srt = 45,
         title = "Correlation Matrix of Mtcars Dataset")

Choosing Between Spearman’s and Pearson’s Correlation

Pearson’s Correlation Coefficient

• Appropriate for Continuous Data: Best suited for data measured on an interval or ratio scale.
• Assumes Normal Distribution: The data should be approximately normally distributed.
• Linear Relationships: Used to assess the strength and direction of a linear relationship between two variables.
• Sensitivity to Outliers: Can be significantly affected by outliers.
• Homoscedasticity Required: Assumes that the variance of one variable is constant at all levels of the other variable.

Spearman’s Rank Correlation Coefficient

• Non-Parametric: Does not assume a specific distribution for the data.
• Ordinal Data: Suitable for ordinal data or when ranking variables.
• Monotonic Relationships: Effective for detecting both increasing and decreasing relationships, not limited to linear.
• Robust to Outliers: Less sensitive to outliers than Pearson’s correlation because it uses rank orders.
• Nonlinear Relationships: Can handle nonlinear relationships well, provided they are monotonic.

Summary

• Use Pearson’s when dealing with continuous data that are normally distributed and you’re interested in linear relationships, but be wary of outliers.
• Use Spearman’s for ordinal data, non-normal distributions, or nonlinear relationships, and when dealing with outliers or non-homoscedastic data.

# Scatter plot of mpg vs. wt
ggplot(mtcars, aes(x = wt, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm", color = "blue") +
  ggtitle("MPG vs. Weight") +
  theme_minimal()

`geom_smooth()` using formula = 'y ~ x'

How Does Sample Size Effect Correlation

# Simulate data and calculate correlation
simulate_correlation <- function(n, rho = 0.5) {
  x <- rnorm(n)
  y <- rho * x + sqrt(1 - rho^2) * rnorm(n)
  cor(x, y)
}

# Different sample sizes
sample_sizes <- seq(10, 1000, by = 10)

# Calculate correlation for each sample size
set.seed(123)
correlations <- sapply(sample_sizes, simulate_correlation)

# Create a dataframe
data_for_plot <- data.frame(sample_size = sample_sizes, correlation = correlations)

# Plotting
ggplot(data_for_plot, aes(x = sample_size, y = correlation)) +
  geom_line() +
  geom_hline(yintercept = 0.5, linetype = "dashed", color = "red") +
  labs(title = "Effect of Sample Size on Correlation",
       x = "Sample Size",
       y = "Estimated Correlation Coefficient") +
  theme_minimal() +
  annotate("text", x = 500, y = 0.5, label = "True Correlation (0.5)", hjust = 0, color = "red")

Causation Analysis Guide

• Identify Variables:
  • Determine the independent (predictor) and dependent (outcome) variables for your analysis.
• Collect Data:
  • Gather data that is relevant to your hypothesis. Preferably use data from randomized controlled trials (RCTs) to minimize bias.
• Statistical Models:
  • Employ statistical models such as regression analysis to estimate the relationship between variables. Include potential confounders.
• Control for Confounders:
  • Identify and adjust for confounding variables that might influence both the independent and dependent variables.
• Use Appropriate Tests:
  • Apply statistical tests (e.g., t-tests, ANOVA) to determine the significance of the observed relationships.
• Consider Experimental Design:
  • If possible, design experiments to randomly assign subjects to treatment and control groups to establish causality.
• Check Assumptions:
  • Ensure your data and model comply with the assumptions required for your statistical tests and models.
• Interpret Results Carefully:
  • Analyze the output of your statistical tests, considering the size and significance of effects. Be cautious about drawing causal conclusions.
• Validate Findings:
  • Use external data, replication studies, or different statistical methods to validate your findings.
• Report with Transparency:
  • Clearly document your methodology, analysis, and limitations. Acknowledge any uncertainties in establishing causation.

library(tidyverse)

── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
✔ tibble  3.2.1     ✔ dplyr   1.1.1
✔ tidyr   1.3.0     ✔ stringr 1.5.0
✔ readr   2.1.3     ✔ forcats 0.5.2
✔ purrr   1.0.1     
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()

Hypothetical Scenario:

Imagine we have a dataset from examining the effect of a new educational program on student performance, where:

• treatment is a binary variable indicating participation in the program (1 for participants, 0 for control group).
• pre_test is a score on a standardized test before the program starts.
• post_test is a score on a standardized test after the program ends.
• The dataset is called edu_data.

set.seed(123)
n <- 100
edu_data <- data.frame(
  treatment = sample(c(0, 1), size = n, replace = TRUE, prob = c(0.5, 0.5)),
  pre_test = rnorm(n, mean = 75, sd = 10),
  post_test = NA
)

# Assuming the treatment has a positive effect
edu_data$post_test[edu_data$treatment == 1] <- edu_data$pre_test[edu_data$treatment == 1] + rnorm(sum(edu_data$treatment == 1), mean = 5, sd = 5)
edu_data$post_test[edu_data$treatment == 0] <- edu_data$pre_test[edu_data$treatment == 0] + rnorm(sum(edu_data$treatment == 0), mean = 0, sd = 5)

head(edu_data)

A data.frame: 6 × 3

	treatment	pre_test	post_test
	<dbl>	<dbl>	<dbl>
1	1	77.53319	86.47188
2	0	74.71453	77.43050
3	1	74.57130	83.41651
4	0	88.68602	86.61432
5	0	72.74229	70.36106
6	1	90.16471	96.82572

# Calculate average improvement by group
avg_improvement <- edu_data %>%
  mutate(improvement = post_test - pre_test) %>%
  group_by(treatment) %>%
  summarize(mean_improvement = mean(improvement))

print(avg_improvement)

# A tibble: 2 × 2
  treatment mean_improvement
      <dbl>            <dbl>
1         0           -0.391
2         1            5.49

ggplot(edu_data, aes(x = factor(treatment), y = post_test - pre_test, fill = factor(treatment))) +
  geom_boxplot() +
  labs(x = "Treatment Group", y = "Improvement in Test Scores", fill = "Group") +
  theme_minimal() +
  ggtitle("Effect of Educational Program on Test Score Improvement")

t.test(post_test ~ treatment, data = edu_data)

	Welch Two Sample t-test

data:  post_test by treatment
t = -4.0852, df = 97.985, p-value = 9.011e-05
alternative hypothesis: true difference in means between group 0 and group 1 is not equal to 0
95 percent confidence interval:
 -13.50354  -4.67358
sample estimates:
mean in group 0 mean in group 1
       72.37132        81.45988