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Skillings-Mack Test

Includes: Skillings–Mack test (handles missing values).
Purpose: A rank-based test for repeated-measures / block designs (like the Friedman test), but it allows missing observations within blocks.

Overview

The Skillings–Mack test can be seen as a generalization of the Friedman test to incomplete block designs (not every block needs to contain every treatment/condition).

When to use it

Use the Skillings–Mack test when:

  • The same subjects/blocks are measured under 3 or more conditions (treatments in columns).
  • Some blocks have missing measurements for one or more conditions.
  • You want a nonparametric alternative to repeated-measures ANOVA.

Null hypothesis: all treatments have the same distribution (equivalently: no systematic treatment effect in ranks).
Alternative: at least one treatment tends to have systematically higher/lower ranks.

Example dataset

Example data: 023skillingsmack.csv

  • Columns = treatments/conditions (A–D)
  • Rows = blocks/subjects
  • Missing values are allowed (empty cells / NaN).

Input dialog

Skillings–Mack input

  • Data: Select the full rectangular range (treatments in columns, blocks in rows).

Options

The Skillings–Mack dialog has no additional options (the method is defined by the selected data range).

Output

Skillings–Mack results

BESHStatNG writes the results to a new worksheet (default) or to the selected output location.

Skillings–Mack section

  • Test Statistic — the Skillings–Mack statistic \(T\).
  • two-sided P-value — asymptotic p-value from a chi-square distribution with \(k-1\) degrees of freedom (\(k\) = number of treatments/columns).

Analyzed table

The output includes an Analyzed table showing the data actually used by the test:

  • Blocks with fewer than two valid observations are omitted.
  • Missing values within retained blocks are kept as blanks.

Mathematical details

Let \(k\) be the number of treatments (columns) and \(b\) the number of blocks (rows).
Within block \(i\), let \(k_i\) be the number of non-missing observations (\(k_i \ge 2\) for the block to contribute).

1) Ranks within each block (ties averaged)

For each block \(i\), compute ranks \(r_{ij}\) among the observed values only (average ranks for ties).
Missing observations are treated as having the block mid-rank, so they contribute 0 after centering.

2) Standardized centered rank scores

BESHStatNG forms standardized scores

\[ s_{ij}=\sqrt{\frac{12}{k_i+1}}\left(r_{ij}-\frac{k_i+1}{2}\right). \]

For a missing value in block \(i\), the add-in sets \(r_{ij}=(k_i+1)/2\), so \(s_{ij}=0\).

Then the treatment score totals are

\[ S_j = \sum_{i=1}^{b} s_{ij}. \]

3) Covariance matrix for incomplete blocks

Let \(c_{j\ell}\) be based on co-occurrence of treatments within blocks:

  • If treatments \(j\) and \(\ell\) are both observed in a block, it contributes +1 to the diagonal and −1 to the corresponding off-diagonal (Laplacian-style structure).
  • Summed over blocks, this yields a \(k\times k\) matrix \(\mathbf{C}\).

4) Test statistic and p-value

BESHStatNG computes the quadratic form (using a \((k-1)\times(k-1)\) invertible submatrix of \(\mathbf{C}\)):

\[ T = \mathbf{S}_{1:(k-1)}^{\mathsf{T}}\,\mathbf{C}_{1:(k-1),1:(k-1)}^{-1}\,\mathbf{S}_{1:(k-1)}. \]

Under \(H_0\), \(T\) is approximately chi-square distributed:

\[ T \;\overset{\cdot}{\sim}\; \chi^2_{k-1}. \]

Relation to Friedman: If there are no missing values (all \(k_i=k\)), Skillings–Mack is closely related to the Friedman test; with missing data, Skillings–Mack remains applicable while Friedman requires complete blocks.

R reference code

In R, the Skillings–Mack test is available in the Skillings.Mack package.

# install.packages("Skillings.Mack")
library(Skillings.Mack)

dat <- read.csv("023skillingsmack.csv")

# Ensure the data are treated as a numeric matrix with NA for missing values
x <- as.matrix(dat)

res <- Skillings.Mack(x)
res

Why results may differ slightly

Different implementations can differ slightly due to:

  • Tie handling (average ranks is standard, but corner cases can differ),
  • how missing values are represented/filtered (e.g., blocks with too few observations),
  • numerical details of the matrix inversion / generalized inverse used for the covariance structure.

For this dataset, the R package result should match the add-in very closely (same \(T\) and p-value up to rounding).

See also