Correlations
Illya Mowerman, Ph.D.
Why correlations?
It brings satistical significance when describing a realtionship
between two continuous variables.
What is the realtionship between: MPG vs HP?
Can you see the relationship?
ggplot(mtcars, aes(x = hp, y = mpg)) +
geom_point() +
labs(title = "Scatter Plot: MPG vs. Horsepower",
x = "Horsepower",
y = "Miles per Gallon")
What is the realtionship between: MPG vs HP?
Can you see the relationship better?
ggplot(mtcars, aes(x = hp, y = mpg)) +
geom_point() +
geom_smooth(method = "lm", se = FALSE, color = "red") +
labs(title = "Scatter Plot: MPG vs. Horsepower",
x = "Horsepower",
y = "Miles per Gallon")
What is the realtionship between: MPG vs HP?
This is how to prove it statistically
cor_test_result <- cor.test(mtcars$mpg, mtcars$hp)
cor_test_result
##
## Pearson's product-moment correlation
##
## data: mtcars$mpg and mtcars$hp
## t = -6.7424, df = 30, p-value = 1.788e-07
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.8852686 -0.5860994
## sample estimates:
## cor
## -0.7761684
Introduction to Correlation
- Correlation measures the strength and direction of relationship
between variables
- Values range from -1 to 1
- Types: Pearson (linear), Spearman (monotonic), Kendall
(ordinal)
Interpreting P-values
- P-value: Probability of observing such a correlation by chance if
there’s no real correlation
- Typically, we use a significance level of 0.05 for small datasets
(<150 observations)
- If p-value < 0.05, we say the correlation is statistically
significant
- Remember: Statistical significance ≠ Practical significance
Interpreting Correlation Coefficients
- Perfect positive correlation: +1
- Strong positive correlation: +0.7 to +0.9
- Moderate positive correlation: +0.4 to +0.6
- Weak positive correlation: +0.1 to +0.3
- No correlation: 0
- Weak negative correlation: -0.1 to -0.3
- Moderate negative correlation: -0.4 to -0.6
- Strong negative correlation: -0.7 to -0.9
- Perfect negative correlation: -1
What is the realtionship between: MPG vs Weight?
cor.test(mtcars$mpg, mtcars$wt)
##
## Pearson's product-moment correlation
##
## data: mtcars$mpg and mtcars$wt
## t = -9.559, df = 30, p-value = 1.294e-10
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.9338264 -0.7440872
## sample estimates:
## cor
## -0.8676594
Visualizing Correlations: MPG vs Weight
ggplot(mtcars, aes(x = wt, y = mpg)) +
geom_point() +
geom_smooth(method = "lm", se = FALSE, color = "red") +
labs(title = "Scatter Plot: MPG vs. Weight",
x = "Weight (1000 lbs)",
y = "Miles per Gallon")
Interpretation: We can clearly see that as weight increases, fuel
efficiency (MPG) decreases.
What is the realtionship between: Horsepower vs Quarter-mile
Time?
cor.test(mtcars$hp, mtcars$qsec)
##
## Pearson's product-moment correlation
##
## data: mtcars$hp and mtcars$qsec
## t = -5.4946, df = 30, p-value = 5.766e-06
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.8475998 -0.4774331
## sample estimates:
## cor
## -0.7082234
Visualizing Correlations: Horsepower vs Quarter-mile Time
ggplot(mtcars, aes(x = hp, y = qsec)) +
geom_point() +
geom_smooth(method = "lm", se = FALSE, color = "blue") +
labs(title = "Scatter Plot: Quarter-mile Time vs. Horsepower",
x = "Horsepower",
y = "Quarter-mile Time (seconds)")
Interpretation: Cars with more horsepower generally complete the
quarter-mile in less time.
Layman’s Explanation of Key Correlations
- MPG (Miles per Gallon) and Weight: -0.87
- As cars get heavier, they tend to have worse fuel efficiency
- This is a strong negative correlation
- Horsepower and Quarter-mile time: -0.71
- Cars with more horsepower tend to complete the quarter-mile
faster
- This is a strong negative correlation (remember, faster times are
lower numbers)
Computing Correlations
cor_matrix <- cor(mtcars)
round(cor_matrix, 2)
## mpg cyl disp hp drat wt qsec vs am gear carb
## mpg 1.00 -0.85 -0.85 -0.78 0.68 -0.87 0.42 0.66 0.60 0.48 -0.55
## cyl -0.85 1.00 0.90 0.83 -0.70 0.78 -0.59 -0.81 -0.52 -0.49 0.53
## disp -0.85 0.90 1.00 0.79 -0.71 0.89 -0.43 -0.71 -0.59 -0.56 0.39
## hp -0.78 0.83 0.79 1.00 -0.45 0.66 -0.71 -0.72 -0.24 -0.13 0.75
## drat 0.68 -0.70 -0.71 -0.45 1.00 -0.71 0.09 0.44 0.71 0.70 -0.09
## wt -0.87 0.78 0.89 0.66 -0.71 1.00 -0.17 -0.55 -0.69 -0.58 0.43
## qsec 0.42 -0.59 -0.43 -0.71 0.09 -0.17 1.00 0.74 -0.23 -0.21 -0.66
## vs 0.66 -0.81 -0.71 -0.72 0.44 -0.55 0.74 1.00 0.17 0.21 -0.57
## am 0.60 -0.52 -0.59 -0.24 0.71 -0.69 -0.23 0.17 1.00 0.79 0.06
## gear 0.48 -0.49 -0.56 -0.13 0.70 -0.58 -0.21 0.21 0.79 1.00 0.27
## carb -0.55 0.53 0.39 0.75 -0.09 0.43 -0.66 -0.57 0.06 0.27 1.00
Correlation Matrix Visualization
corrplot(cor_matrix, method = "color", type = "upper", order = "hclust",
tl.col = "black", tl.srt = 45)
- Blue: Positive correlations
- Red: Negative correlations
- Darker colors: Stronger correlations
Statistical Significance: MPG vs Weight
cor_test_result <- cor.test(mtcars$mpg, mtcars$wt)
cor_test_result
##
## Pearson's product-moment correlation
##
## data: mtcars$mpg and mtcars$wt
## t = -9.559, df = 30, p-value = 1.294e-10
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.9338264 -0.7440872
## sample estimates:
## cor
## -0.8676594
Interpretation: - Correlation coefficient: -0.8677 (strong negative
correlation) - P-value: 1.294e-10 (much smaller than 0.05, so
statistically significant) - We can be very confident that this
correlation isn’t due to chance
Multiple Variable Relationships
pairs(mtcars[, c("mpg", "disp", "hp", "wt")],
main = "Scatter Plot Matrix of Selected mtcars Variables")
This matrix shows relationships between multiple variables at
once.
Layman’s Explanation of Overall Patterns
- Fuel efficiency (MPG) is negatively correlated with weight, engine
displacement, and horsepower
- Bigger, more powerful cars tend to use more fuel
- Weight is positively correlated with engine displacement and
horsepower
- Heavier cars tend to have larger, more powerful engines
- These relationships make intuitive sense in the context of car
design and physics
Best Practices and Pitfalls
- Correlation ≠ Causation
- Check for non-linear relationships
- Consider the context and domain knowledge
- Be cautious with small sample sizes
- Remember that outliers can strongly influence correlation
Conclusion
- Correlations help understand relationships between variables
- Interpretation involves considering the strength, direction, and
significance
- Always interpret results in the context of your data and research
question
- The mtcars dataset shows how car characteristics are interrelated in
ways that align with our understanding of automotive engineering
Thank You!
Questions? Comments?