WEIGHTEDTAU

Overview

The WEIGHTEDTAU function computes a weighted version of Kendall’s tau correlation coefficient. The weighted tau gives more influence to exchanges involving high-importance elements, with importance determined by rank. The default parameters compute the additive hyperbolic version (τ_h), which provides the best balance between important and unimportant elements. The weighting is defined by a rank array that assigns importance to each element, and a weigher function that assigns weights based on ranks:

\tau_h = \frac{\sum_{i<j} w_{ij} \cdot \text{sign}((x_i - x_j)(y_i - y_j))}{\sum_{i<j} w_{ij}}

where $w_{ij} = 1/(r_i+1) + 1/(r_j+1)$ for the additive hyperbolic case, and $r_i$ , $r_j$ are the ranks of elements $i$ and $j$ . For more details, see the scipy.stats documentation .

This example function is provided as-is without any representation of accuracy.

Usage

To use the function in Excel:


=WEIGHTEDTAU(x, y, [rank], [additive])

x (2D list, required): Array of scores. Must be a column vector (single column).
y (2D list, required): Array of scores of the same length as x. Must be a column vector (single column).
rank (bool, optional, default=True): If True, use decreasing lexicographical rank averaging (x,y) and (y,x). If False, use element indices as ranks.
additive (bool, optional, default=True): If True, compute weights by addition; if False, by multiplication.

The function returns the weighted tau correlation coefficient (float). The p-value is not available as the null distribution is unknown, so it returns NaN for statistical significance.

Examples

Example 1: Basic Weighted Tau (Default Parameters)

This example calculates the weighted tau using default additive hyperbolic weighting with rank averaging.

Inputs:

x	y	rank	additive
12	1	True	True
2	4
1	7
12	1
2	0

Excel formula:


=WEIGHTEDTAU({12;2;1;12;2}, {1;4;7;1;0})

Expected output:

Result
-0.567

This shows a moderate negative weighted correlation between the two variables.

Example 2: Multiplicative Weighting

This example uses multiplicative weighting instead of additive weighting.

Inputs:

x	y	rank	additive
12	1	True	False
2	4
1	7
12	1
2	0

Excel formula:


=WEIGHTEDTAU({12;2;1;12;2}, {1;4;7;1;0}, True, False)

Expected output:

Result
-0.622

The multiplicative weighting produces a slightly different correlation value.

Example 3: Index-Based Ranking

This example uses element indices directly as ranks instead of data-based ranking.

Inputs:

x	y	rank	additive
12	1	False	True
2	4
1	7
12	1
2	0

Excel formula:


=WEIGHTEDTAU({12;2;1;12;2}, {1;4;7;1;0}, False)

Expected output:

Result
-0.516

Using index-based ranking produces a different correlation pattern.

Example 4: Different Data Set

This example demonstrates the function with a different data set showing positive correlation.

Inputs:

x	y	rank	additive
1	2	True	True
2	3
3	4
4	5
5	6

Excel formula:


=WEIGHTEDTAU({1;2;3;4;5}, {2;3;4;5;6})

Expected output:

Result
1.000

This shows perfect positive weighted correlation for monotonically increasing data.

Python Code


from scipy.stats import weightedtau as scipy_weightedtau
 
def weightedtau(x, y, rank=True, additive=True):
    """
    Compute a weighted version of Kendall's tau correlation coefficient.
 
    Args:
        x: 2D list containing scores as a column vector.
        y: 2D list containing scores as a column vector, same length as x.
        rank: If True, use decreasing lexicographical rank averaging (x,y) and (y,x).
              If False, use element indices as ranks.
        additive: If True, compute weights by addition; if False, by multiplication.
 
    Returns:
        The weighted tau correlation coefficient (float), or an error message (str) if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Handle scalar inputs by converting to 2D lists
    if not isinstance(x, list):
        x = [[x]]
    if not isinstance(y, list):
        y = [[y]]
    
    # Validate x input
    if not isinstance(x, list) or len(x) == 0:
        return "Invalid input: x must be a non-empty 2D list."
    
    # Check if x is a 2D list and extract column vector
    x_values = []
    try:
        if isinstance(x[0], list):
            # 2D list - should be a column vector
            if any(len(row) != 1 for row in x):
                return "Invalid input: x must be a column vector (single column)."
            x_values = [float(row[0]) for row in x]
        else:
            # 1D list - convert to column vector format
            x_values = [float(val) for val in x]
    except (ValueError, TypeError):
        return "Invalid input: x must contain numeric values."
    
    # Validate y input
    if not isinstance(y, list) or len(y) == 0:
        return "Invalid input: y must be a non-empty 2D list."
    
    # Check if y is a 2D list and extract column vector
    y_values = []
    try:
        if isinstance(y[0], list):
            # 2D list - should be a column vector
            if any(len(row) != 1 for row in y):
                return "Invalid input: y must be a column vector (single column)."
            y_values = [float(row[0]) for row in y]
        else:
            # 1D list - convert to column vector format
            y_values = [float(val) for val in y]
    except (ValueError, TypeError):
        return "Invalid input: y must contain numeric values."
    
    # Check if x and y have the same length
    if len(x_values) != len(y_values):
        return "Invalid input: x and y must have the same length."
    
    # Check minimum length requirement
    if len(x_values) < 2:
        return "Invalid input: x and y must have at least 2 elements."
    
    # Validate rank parameter
    if not isinstance(rank, bool):
        return "Invalid input: rank must be a boolean value."
    
    # Validate additive parameter
    if not isinstance(additive, bool):
        return "Invalid input: additive must be a boolean value."
    
    try:
        # Call scipy's weightedtau function
        result = scipy_weightedtau(x_values, y_values, rank=rank, additive=additive)
        return result.statistic
    except Exception as e:
        return f"scipy.stats.weightedtau error: {e}"

Example Workbook

Link to Workbook