PAGE_TREND_TEST

Overview

The PAGE_TREND_TEST function performs Page’s L trend test, a non-parametric statistical test for detecting ordered alternatives across multiple treatments. It is used when you have repeated measures (e.g., ratings or scores) for several treatments and want to test whether there is a monotonic trend (e.g., increasing or decreasing) in the means of the treatments. The test is more powerful than the Friedman test for detecting trends and is especially useful when you have a specific hypothesis about the order of the treatments.

The calculation is based on ranking the data within each subject, summing the ranks for each treatment, multiplying by the predicted ranks, and summing the products to obtain the L statistic:

L = \sum_{j=1}^n R_j P_j

where $R_j$ is the sum of ranks for treatment $j$ and $P_j$ is the predicted rank for treatment $j$ . The p-value is computed either exactly or asymptotically, depending on the data size and method.

For more details, see the scipy.stats.page_trend_test documentation .

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

Usage

To use the function in Excel:


=PAGE_TREND_TEST(data, [ranked], [predicted_ranks], [method])

data (2D list, required): Table of shape (m, n), where each row is a subject and each column is a treatment. Must have at least 2 rows and 3 columns.
ranked (bool, optional, default=False): If True, data is already ranked; otherwise, it will be ranked within each row.
predicted_ranks (2D list or None, optional, default=None): 1-row 2D list of predicted ranks for each treatment. If None, defaults to [1, 2, …, n].
method (str, optional, default=“auto”): Method for p-value calculation: “auto”, “asymptotic”, or “exact”.

The function returns a 1-row 2D list: [L_statistic, p_value, method], or an error message (string) if the input is invalid.

Examples

Example 1: Standard Page Trend Test

Inputs:

data			ranked
3	4	3	FALSE
2	2	4
3	3	5
1	3	2
2	3	2
2	4	5
1	2	4
3	4	4
2	4	5
1	3	4

Excel formula:


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

Expected output:

L_statistic	p_value	method
133.5	0.001819116195	exact

This means the L statistic is 133.5 and the p-value is approximately 0.00182 using the exact method.

Example 2: Asymptotic Method

Inputs:

data			ranked	method
3	4	3	FALSE	asymptotic
2	2	4
3	3	5
1	3	2
2	3	2
2	4	5
1	2	4
3	4	4
2	4	5
1	3	4

Excel formula:


=PAGE_TREND_TEST({3,4,3;2,2,4;3,3,5;1,3,2;2,3,2;2,4,5;1,2,4;3,4,4;2,4,5;1,3,4}, FALSE, , "asymptotic")

Expected output:

L_statistic	p_value	method
133.5	0.001269343369	asymptotic

Example 3: Data Already Ranked

Inputs:

data			ranked
1.5	3	1.5	TRUE
1.5	1.5	3
1	3	2
1.5	3	1.5
1	2	3
1	2.5	2.5
1	2	3
1	2	3
1	2.5	2.5
1	2	3

Excel formula:


=PAGE_TREND_TEST({1.5,3,1.5;1.5,1.5,3;1,3,2;1.5,3,1.5;1,2,3;1,2.5,2.5;1,2,3;1,2,3;1,2.5,2.5;1,2,3}, TRUE)

Expected output:

L_statistic	p_value	method
133.5	0.001819116195	exact

Example 4: Custom Predicted Ranks

Inputs:

data			ranked	predicted_ranks
4	3	3	FALSE	2,3,1
2	4	2
3	5	3
3	2	1
3	2	2
4	5	2
2	4	1
4	4	3
4	5	2
3	4	1

Excel formula:


=PAGE_TREND_TEST({4,3,3;2,4,2;3,5,3;3,2,1;3,2,2;4,5,2;2,4,1;4,4,3;4,5,2;3,4,1}, FALSE, {2,3,1})

Expected output:

L_statistic	p_value	method
133.5	0.001819116195	exact

This demonstrates the use of custom predicted ranks for the treatments.

Python Code


from scipy.stats import page_trend_test as scipy_page_trend_test
 
 
def page_trend_test(data, ranked=False, predicted_ranks=None, method="auto"):
    """
    Perform Page’s L trend test for monotonic trends across treatments.
 
    Args:
        data: 2D list of shape (m, n), each row is a subject, each column is a treatment.
        ranked: If True, data is already ranked; otherwise, it will be ranked within each row (default: False).
        predicted_ranks: 2D list (1 row) or None. Predicted ranks for each treatment (default: None, i.e., [1,2,...,n]).
        method: Method for p-value calculation: "auto", "asymptotic", or "exact" (default: "auto").
 
    Returns:
        1-row 2D list: [L_statistic, p_value, method], or error message (str) if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Validate data
    if not isinstance(data, list) or len(data) < 2 or not isinstance(data[0], list) or len(data[0]) < 3:
        return "Invalid input: data must be a 2D list with at least 2 rows and 3 columns."
    try:
        arr = [[float(x) for x in row] for row in data]
    except Exception:
        return "Invalid input: data must contain only numeric values."
    # Validate ranked
    if isinstance(ranked, list):
        if len(ranked) == 1 and isinstance(ranked[0], bool):
            ranked = ranked[0]
        elif len(ranked) == 1 and isinstance(ranked[0], str):
            ranked = ranked[0].strip().lower() == "true"
        else:
            ranked = bool(ranked)
    # Validate predicted_ranks
    pr = None
    if predicted_ranks is not None:
        if isinstance(predicted_ranks, list):
            if len(predicted_ranks) == 1 and isinstance(predicted_ranks[0], list):
                pr = [float(x) for x in predicted_ranks[0]]
            else:
                pr = [float(x) for x in predicted_ranks]
        else:
            return "Invalid input: predicted_ranks must be a 2D or 1D list."
    # Validate method
    if method is None:
        method = "auto"
    if isinstance(method, list):
        if len(method) == 1:
            method = str(method[0])
        else:
            return "Invalid input: method must be a string."
    try:
        res = scipy_page_trend_test(arr, ranked=ranked, predicted_ranks=pr, method=method)
    except Exception as e:
        return f"scipy.page_trend_test error: {e}"
    return [[float(res.statistic), float(res.pvalue), str(res.method)]]

Example Workbook

Link to Workbook