Our first form of bivariate analysis.

Correlation always involves two or more variables (columns).

The correlation coefficient measures the extent to which two variables are related, from -1 to 1.

Pearson’s correlation coefficient multiplies the deviations from the mean for two variables, and divides by the product of the standard deviation.

Tells us the strength of a correlation.

Pearson’s correlation, r, is the default in Pandas.

# Load cars sample dataset
cars = data.cars()

# Calculate correlations between all columns in a dataframe 
cars.corr(numeric_only=True)

# Calculate correlation between just two variables
cars.Miles_per_Gallon.corr(cars.Displacement)

Scatterplots show potential correlation between two variables

The y-axis shows the dependent variable, while the x-axis shows the independent variable.

Make a scatterplot with Altair

alt.Chart(cars, title="Fuel Efficiency and Engine Displacement").mark_point().encode(
    x=alt.X("Displacement:Q", title="Engine Displacement (liters)"),
    y=alt.Y("Miles_per_Gallon:Q", title="Fuel Efficiency (mpg)")
).interactive()

Add a line of best fit to make a regression plot

scatter = alt.Chart(cars, title="Fuel Efficiency and Engine Displacement").mark_point().encode(
    x=alt.X("Displacement:Q", title="Engine Displacement (liters)"),
    y=alt.Y("Miles_per_Gallon:Q", title="Fuel Efficiency (mpg)")
).interactive()

scatter + scatter.transform_regression('Displacement','Miles_per_Gallon').mark_line()

Avoid overplotting with heatmaps or kernel density estimation.

# Heatmap example
alt.Chart(cars, title="Fuel Efficiency and Engine Displacement").mark_rect().encode(
    x=alt.X("Displacement:Q", title="Engine Displacement (liters)").bin(),
    y=alt.Y("Miles_per_Gallon:Q", title="Fuel Efficiency (mpg)").bin(),
    color=alt.Color("count():Q")
)

Correlation matrix shows all possible correlations.

# Re-arrange correlation matrix data
cars_corr = (cars.corr(numeric_only=True)
             .stack()
             .reset_index()
             .rename(columns={0:'corr','level_0':'var1','level_1':'var2'})
            )
# Create correlation heatmap
base = alt.Chart(cars_corr, title="Cars Correlation Matrix").mark_rect().encode(
    x=alt.X("var1:N",title=None),
    y=alt.Y("var2:N",title=None),
    color=alt.Color("corr",title="Correlation coefficient").scale(scheme='blueorange')
).properties(width=300,height=300)
# Add text labels for coefficients
text = base.mark_text(baseline='middle').encode(
    alt.Text('corr:Q', format=".2f"),
    color=alt.condition(
        (alt.datum.corr < -0.5) | (alt.datum.corr > 0.5),
        alt.value('white'),
        alt.value('black')
    )
)
base+text # Display visualization

How do we know if a correlation coefficient is statistically significant?

There are standard parametric approaches to this, but we can use permutation!

We need a new simulation function for correlation.

def simulate_correlation(df,var1,var2):
    shuffled = df[var1].sample(frac=1).reset_index(drop=True)
    corr = shuffled.corr(df[var2])
    return corr

Let’s Try It!

Using the function from the previous slide, run 5000 permutations of the correlation between engine displacement and miles per gallon.

Graph the results as a histogram and calculate a p-value. Is this a statistically significant correlation?

Always use summary statistics and visualization together.

If we have the same mean, standard deviation, and correlation we might expect the data sets to be similar…

But they could be very clearly and visually distinct!

Data Challenge

Use pandas to find the summary statistics for each dataset in the datasaurus_dozen.

• Find mean, standard deviation, and correlation for both x and y of each dataset. (You may need to group things by the “dataset” column.)
• When you’re done, try making scatter plots! (You may need to use the column encoding.)