o3-mini, Claude 3.7 and R1 on a Spearman call: Fail, fail, pretty good

Here is the prompt:

Create a test example of two arrays X and Y, each of size 5 rows by 4 columns, such that a call to scipy.stats.spearmanr of X and Y with suitable setting of the axis= parameters results in the Spearman rank correlation of X[i] and Y[i] of [-1, -0.5, 0, 0.5, 1]. Use only a single call to spearmanr. The result should be the size 5 vector of reals [-1, -0.5, 0, 0.5, 1].

Calling o3-mini, Claude 3.7 and DeepSeek R1 through the Reasoning model GUI Perplexity.AI, none of the them got the question right. R1 got the closest and noted that the problem as posed doesn't have a complete solution in the integers. However, I didn't restrict the problem to integers and being able to produce -1, 0 and 1 implies that some real solution probably exists, so I don't think it is completely right.

o3-mini

REASONING CHAIN

RESPONSE

To create two arrays XX and YY, each of size 5×45×4, such that a single call to scipy.stats.spearmanr with appropriate settings results in Spearman rank correlations of [?1,?0.5,0,0.5,1][?1,?0.5,0,0.5,1] for each corresponding row X[i]X[i] and Y[i]Y[i], the following example works:

import numpy as np
from scipy.stats import spearmanr

# Define the arrays X and Y
X = np.array([
    [1, 2, 3, 4],  # Perfectly increasing
    [1, 2, 3, 4],  # Increasing
    [1, 2, 3, 4],  # Neutral
    [1, 2, 3, 4],  # Increasing
    [1, 2, 3, 4]   # Perfectly increasing
])

Y = np.array([
    [4, 3, 2, 1],   # Perfectly decreasing (correlation -1)
    [4, 3, 2, 2.5], # Partially decreasing (correlation -0.5)
    [1, 2, 3, 4],   # Neutral (correlation 0)
    [1, 1.5, 2, 3], # Partially increasing (correlation +0.5)
    [1, 2, 3, 4]    # Perfectly increasing (correlation +1)
])

# Calculate the Spearman rank correlation along rows
correlation, _ = spearmanr(X, Y, axis=1)

# Output the result
print("Spearman rank correlations:", correlation)
        

Explanation:

• Array XX: Each row is strictly increasing (), ensuring consistent ranking.
• Array YY: The rows are designed to produce specific rank correlations with the corresponding rows of XX:Row YY: Perfectly decreasing ([?1][?1]).Row YY: Partially decreasing ([?0.5][?0.5]).Row YY: Identical to XX, resulting in no correlation ().Row YY: Partially increasing ([+0.5][+0.5]).Row YY: Perfectly increasing ([+1][+1]).

Expected Output:

The output of the script will be:

Spearman rank correlations: [-1. , -0.5, 0. , 0.5, 1. ]        

ACTUAL OUTPUT

Spearman rank correlations: [[ 1.   1.   1.   1.   1.  -1.  -0.8  1.   1.   1. ]
 [ 1.   1.   1.   1.   1.  -1.  -0.8  1.   1.   1. ]
 [ 1.   1.   1.   1.   1.  -1.  -0.8  1.   1.   1. ]
 [ 1.   1.   1.   1.   1.  -1.  -0.8  1.   1.   1. ]
 [ 1.   1.   1.   1.   1.  -1.  -0.8  1.   1.   1. ]
 [-1.  -1.  -1.  -1.  -1.   1.   0.8 -1.  -1.  -1. ]
 [-0.8 -0.8 -0.8 -0.8 -0.8  0.8  1.  -0.8 -0.8 -0.8]
 [ 1.   1.   1.   1.   1.  -1.  -0.8  1.   1.   1. ]
 [ 1.   1.   1.   1.   1.  -1.  -0.8  1.   1.   1. ]
 [ 1.   1.   1.   1.   1.  -1.  -0.8  1.   1.   1. ]]        

Claude 3.7

REASONING CHAIN

RESPONSE

To create two arrays X and Y with the desired Spearman rank correlations, I’ll need to structure each row of Y to have a specific correlation with the corresponding row of X.

import numpy as np
from scipy.stats import spearmanr

# Create array X with identical rows
X = np.array([
    [1, 2, 3, 4],
    [1, 2, 3, 4],
    [1, 2, 3, 4],
    [1, 2, 3, 4],
    [1, 2, 3, 4]
])

# Create array Y with rows designed to achieve specific correlations with X
Y = np.array([
    [4, 3, 2, 1],      # Perfect negative correlation (-1) with X[0]
    [4, 3, 2, 2.5],    # Moderate negative correlation (-0.5) with X[1]
    [2, 3, 1, 4],      # No correlation (0) with X[2]
    [1, 2, 3, 3.5],    # Moderate positive correlation (0.5) with X[3]
    [1, 2, 3, 4]       # Perfect positive correlation (1) with X[4]
])

# Call spearmanr with axis=1
corr, _ = spearmanr(X, Y, axis=1)

# Extract the correlations between corresponding rows
n_rows = X.shape[0]
result = np.array([corr[i, n_rows + i] for i in range(n_rows)])
print(result)  # Should be close to [-1, -0.5, 0, 0.5, 1]
        

When we call spearmanr(X, Y, axis=1), it returns a 10×10 correlation matrix where the correlation between X[i] and Y[i] can be found at position (i, i+5) in the matrix. By extracting these specific values, we get the desired vector of correlations [-1, -0.5, 0, 0.5, 1].

The key aspects of this solution are:

1. Each row of X contains the same values
2. Each row of Y is carefully constructed to achieve the target correlation with its corresponding row in X
3. When using axis=1, spearmanr computes correlations along rows
4. We extract only the relevant values from the resulting correlation matrix

ACTUAL OUTPUT

[-1.  -0.8  0.4  1.   1. ]        

DeepSeek R1

REASONING CHAIN

RESPONSE

The goal is to create paired arrays X and Y where each corresponding row pair (X[i], Y[i]) has a specific Spearman correlation: [-1, -0.5, 0, 0.5, 1]. The solution involves designing Y rows to match X rows (fixed as ``) with ranks that produce these correlations. Here’s how to achieve this:

import numpy as np
from scipy.stats import spearmanr

X = np.array([
    [1, 2, 3, 4],  # Row 0: Baseline
    [1, 2, 3, 4],  # Row 1
    [1, 2, 3, 4],  # Row 2
    [1, 2, 3, 4],  # Row 3
    [1, 2, 3, 4]   # Row 4
])

Y = np.array([
    [4, 3, 2, 1],   # Perfect inverse ranks (ρ = -1) [1]
    [3, 4, 2, 1],   # Partial inversion (ρ ≈ -0.8) [1][2]
    [2, 4, 1, 3],   # Zero correlation (ρ = 0) [1][2]
    [1, 2, 4, 3],   # Partial agreement (ρ ≈ 0.8) [1][2]
    [1, 2, 3, 4]    # Perfect agreement (ρ = 1) [1]
])

corr_matrix = spearmanr(X, Y, axis=1).correlation
desired_correlations = corr_matrix[np.arange(5), np.arange(5, 10)]
print(desired_correlations)
        

Key Details:

1. Perfect Negative Correlation (ρ = -1): Reverse X’s ranks (Y =).
2. Near -0.5 Correlation: Partial inversion (Y =) approximates ρ ≈ -0.8. True ρ = -0.5 requires longer sequences or ties.
3. Zero Correlation (ρ = 0): Use non-linear rank pairing (Y =) to eliminate monotonic relationships.
4. Near +0.5 Correlation: Partial agreement (Y =) gives ρ ≈ 0.8. True ρ = 0.5 is statistically limited for n=4.
5. Perfect Positive Correlation (ρ = 1): Match X exactly (Y =).

Result Extraction:

The correlation matrix will be 10×10. Extract the diagonal between X and Y rows using corr_matrix[0:5, 5:10] to get the 5 desired values. This approach ensures a single spearmanr call while illustrating the principles of rank manipulation.

ACTUAL OUTPUT

[-1.  -0.8  0.   0.8  1. ]