Spearman correlation in SciPy - Step-by-Step Execution

Concept Flow - Spearman correlation

Start with two data lists

↓

Rank each list separately

↓

Calculate difference of ranks

↓

Compute Spearman correlation coefficient

↓

Output correlation and p-value

↓

End

Spearman correlation measures how well two sets of data relate by comparing their ranks, not the raw values.

Execution Sample

SciPy

from scipy.stats import spearmanr
x = [10, 20, 30, 40, 50]
y = [12, 24, 33, 48, 55]
correlation, pvalue = spearmanr(x, y)
print(correlation, pvalue)

Calculate Spearman correlation between two lists of numbers and print the correlation coefficient and p-value.

Execution Table

Step	Action	x ranks	y ranks	Rank differences (d)	d^2	Correlation	p-value
1	Input data	[10, 20, 30, 40, 50]	[12, 24, 33, 48, 55]
2	Rank x	[1, 2, 3, 4, 5]
3	Rank y		[1, 2, 3, 4, 5]
4	Calculate d = x_rank - y_rank			[0, 0, 0, 0, 0]
5	Calculate d^2				[0, 0, 0, 0, 0]
6	Compute Spearman correlation					1.0	1.4042654220543672e-24
7	Output result					1.0	1.4042654220543672e-24

💡 All ranks match perfectly, so correlation is 1.0 and p-value is very small indicating strong correlation.

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	After Step 5	Final
x	[10, 20, 30, 40, 50]	[1, 2, 3, 4, 5]	[1, 2, 3, 4, 5]	[1, 2, 3, 4, 5]	[1, 2, 3, 4, 5]	[1, 2, 3, 4, 5]
y	[12, 24, 33, 48, 55]	[12, 24, 33, 48, 55]	[1, 2, 3, 4, 5]	[1, 2, 3, 4, 5]	[1, 2, 3, 4, 5]	[1, 2, 3, 4, 5]
d				[0, 0, 0, 0, 0]	[0, 0, 0, 0, 0]	[0, 0, 0, 0, 0]
d^2					[0, 0, 0, 0, 0]	[0, 0, 0, 0, 0]
correlation						1.0
p-value						1.4042654220543672e-24

Key Moments - 3 Insights

Why do we rank the data instead of using the raw values?

Spearman correlation measures how well the order of data matches, not the exact values. Ranking converts values to their order positions, which is why in execution_table rows 2 and 3 we rank x and y.

What does a correlation of 1.0 mean here?

A correlation of 1.0 means the ranks match perfectly with no differences (d=0 for all), as shown in execution_table rows 4 and 5 where all d and d^2 are zero.

Why is the p-value so small?

The p-value shows the chance of seeing this correlation by random chance. Since correlation is perfect, p-value is very small, indicating strong evidence of correlation (execution_table row 6).

Visual Quiz - 3 Questions

Look at the execution table at step 4. What are the rank differences (d) between x and y?

A [0, 0, 0, 0, 0]

B [1, 1, 1, 1, 1]

C [-1, -1, -1, -1, -1]

D [5, 4, 3, 2, 1]

💡 Hint

Check the 'Rank differences (d)' column in execution_table row 4.

At which step does the code calculate the Spearman correlation coefficient?

A Step 3

B Step 5

C Step 6

D Step 7

💡 Hint

Look at the 'Correlation' column in execution_table to find when the value appears.

If the y data were reversed, how would the correlation value change in the table?

A It would stay 1.0

B It would become -1.0

C It would become 0

D It would become a random value

💡 Hint

Spearman correlation measures monotonic relationship; reversing order flips sign.

Concept Snapshot

Spearman correlation measures how well two variables relate by comparing their ranks.
Use scipy.stats.spearmanr(x, y) to get correlation and p-value.
Ranks replace raw values to focus on order, not magnitude.
Correlation ranges from -1 (perfect opposite) to 1 (perfect match).
P-value shows significance of the correlation.

Full Transcript

Spearman correlation is a way to measure how two sets of data relate by looking at their order, not their exact values. We start with two lists of numbers. Then, we rank each list separately, turning the numbers into their positions in order. Next, we find the difference between these ranks for each pair. We square these differences and use them to calculate the Spearman correlation coefficient. This coefficient tells us how closely the two lists match in order. A value of 1 means perfect match, -1 means perfect opposite, and 0 means no relation. The p-value tells us if this correlation is likely due to chance. In our example, the ranks match perfectly, so the correlation is 1 and the p-value is very small, showing a strong relationship.

Practice

1. What does the Spearman correlation measure between two variables?

easy

A. The variance of a single variable

B. The strength and direction of a monotonic relationship based on ranks

C. The difference in means of two variables

D. The linear relationship between two variables

Solution

1. Step 1: Understand correlation types

   Spearman correlation measures how two variables move together based on their rank order, not their actual values.

2. Step 2: Compare with other correlations

   Unlike Pearson correlation, which measures linear relationships, Spearman focuses on monotonic relationships using ranks.

3. Final Answer:

   The strength and direction of a monotonic relationship based on ranks -> Option B

4. Quick Check:

   Spearman = monotonic rank correlation [OK]

Hint: Spearman uses ranks, not raw values, to find monotonic links [OK]

Common Mistakes:

• Confusing Spearman with Pearson correlation
• Thinking it measures linear relationships only
• Assuming it compares means or variances

2. Which of the following is the correct way to import the Spearman correlation function from scipy?

easy

A. from scipy.stats import spearmanr

B. import scipy.spearmanr

C. from scipy import spearmanr

D. import spearmanr from scipy.stats

Solution

1. Step 1: Recall scipy.stats module structure

   The Spearman correlation function is inside the stats submodule of scipy, so it must be imported from scipy.stats.

2. Step 2: Check correct import syntax

   The correct Python syntax to import a function is 'from module import function', so 'from scipy.stats import spearmanr' is correct.

3. Final Answer:

   from scipy.stats import spearmanr -> Option A

4. Quick Check:

   Correct import syntax = from scipy.stats import spearmanr [OK]

Hint: Use 'from scipy.stats import spearmanr' to import correctly [OK]

Common Mistakes:

• Trying to import spearmanr directly from scipy
• Using incorrect import syntax like 'import spearmanr from ...'
• Missing the stats submodule in import

3. What is the output of this code snippet?

from scipy.stats import spearmanr
x = [1, 2, 3, 4, 5]
y = [5, 6, 7, 8, 7]
result = spearmanr(x, y)
print(round(result.correlation, 2))

medium

A. 0.5

B. 0.9

C. 1.0

D. 0.82

Solution

1. Step 1: Understand the data ranks

   x is increasing from 1 to 5, y mostly increases but has a slight drop at the end (7 instead of 9), so ranks are similar but not perfect.

2. Step 2: Calculate Spearman correlation

   Spearman correlation measures rank similarity; here it is less than 1 but still strong, about 0.82 after rounding.

3. Final Answer:

   0.82 -> Option D

4. Quick Check:

   Spearman correlation ≈ 0.82 [OK]

Hint: Spearman close to 1 means strong rank similarity [OK]

Common Mistakes:

• Assuming perfect correlation (1.0) without checking data
• Confusing Pearson and Spearman correlation values
• Ignoring the slight rank difference in y

4. Identify the error in this code that calculates Spearman correlation:

from scipy.stats import spearmanr
x = [1, 2, 3]
y = [4, 5]
result = spearmanr(x, y)
print(result.correlation)

medium

A. Missing import for numpy

B. spearmanr function is misspelled

C. Lists x and y have different lengths

D. result.statistic attribute does not exist

Solution

1. Step 1: Check input data lengths

   Spearman correlation requires both input lists to have the same length. Here, x has 3 elements, y has 2.

2. Step 2: Understand function requirements

   Because lengths differ, spearmanr will raise an error. Other options are incorrect as spelling and attribute usage are correct.

3. Final Answer:

   Lists x and y have different lengths -> Option C

4. Quick Check:

   Input length mismatch causes error [OK]

Hint: Ensure input lists have equal length before correlation [OK]

Common Mistakes:

• Ignoring length mismatch of input lists
• Thinking spearmanr needs numpy import explicitly
• Assuming result.statistic is invalid

5. You have two variables with many tied ranks and some missing values (NaNs). Which approach using scipy.stats.spearmanr is best to get a valid Spearman correlation?

hard

A. Use spearmanr with nan_policy='omit' to ignore NaNs and handle ties automatically

B. Replace NaNs with zeros and then use spearmanr without extra parameters

C. Remove all tied values manually before using spearmanr

D. Use spearmanr with nan_policy='raise' to get an error for NaNs

Solution

1. Step 1: Handle missing values properly

   Spearman correlation in scipy can handle NaNs if nan_policy='omit' is set, which ignores NaNs safely.

2. Step 2: Understand tied ranks handling

   Spearman correlation automatically manages tied ranks, so no manual removal is needed.

3. Step 3: Evaluate other options

   Replacing NaNs with zeros or removing tied values manually can bias results; raising errors is not helpful for analysis.

4. Final Answer:

   Use spearmanr with nan_policy='omit' to ignore NaNs and handle ties automatically -> Option A

5. Quick Check:

   nan_policy='omit' + automatic tie handling = Use spearmanr with nan_policy='omit' to ignore NaNs and handle ties automatically [OK]

Hint: Use nan_policy='omit' to skip NaNs safely in spearmanr [OK]

Common Mistakes:

• Replacing NaNs with zeros causing bias
• Trying to remove tied ranks manually
• Not setting nan_policy causing errors