Durbin test

Not to be confused with Durbin–Watson statistic or Durbin–Wu–Hausman test.

In the analysis of designed experiments, the Friedman test is the most common non-parametric test for complete block designs. The Durbin test is a nonparametric test for balanced incomplete designs that reduces to the Friedman test in the case of a complete block design.

Background

See also: Blocking (statistics)

In a randomized block design, k treatments are applied to b blocks. In a complete block design, every treatment is run for every block and the data are arranged as follows:

	Treatment 1	Treatment 2	$\cdots$	Treatment k
Block 1	X₁₁	X₁₂	$\cdots$	X_1k
Block 2	X₂₁	X₂₂	$\cdots$	X_2k
Block 3	X₃₁	X₃₂	$\cdots$	X_3k
$\vdots$	$\vdots$	$\vdots$	$\ddots$	$\vdots$
Block b	X_b1	X_b2	$\cdots$	X_bk

For some experiments, it may not be realistic to run all treatments in all blocks, so one may need to run an incomplete block design. In this case, it is strongly recommended to run a balanced incomplete design. A balanced incomplete block design has the following properties:

Every block contains k experimental units.
Every treatment appears in r blocks.
Every treatment appears with every other treatment an equal number of times.

Test assumptions

The Durbin test is based on the following assumptions:

The b blocks are mutually independent. That means the results within one block do not affect the results within other blocks.
The data can be meaningfully ranked (i.e., the data have at least an ordinal scale).

Test definition

Let R(X_ij) be the rank assigned to X_ij within block i (i.e., ranks within a given row). Average ranks are used in the case of ties. The ranks are summed to obtain

R_j = \sum_{i=1}^b R(X_{ij})

The Durbin test is then

H₀: The treatment effects have identical effects

H_a: At least one treatment is different from at least one other treatment

The test statistic is

T_2 = \frac{T_1/\left(t - 1\right)}{\left(bk-b - T_1\right)/\left(bk - b - t + 1\right)}

where

T_1 = \frac{t - 1}{A-C}\sum_{j=1}^t \left(R_j^2 - rC\right)

A = \sum_{i=1}^b\sum_{j=1}^j R(X_{ij})^2

C = \frac{1}{4}bk\left(k+1\right)^2

where t is the number of treatments, k is the number of treatments per block, b is the number of blocks, and r is the number of times each treatment appears.

For significance level α, the critical region is given by

T_2 > F_{\alpha,k-1,bk-b-t+1}

where F_{α, k − 1, bk − b − t + 1} denotes the α-quantile of the F distribution with k − 1 numerator degrees of freedom and bk − b − t + 1 denominator degrees of freedom. The null hypothesis is rejected if the test statistic is in the critical region. If the hypothesis of identical treatment effects is rejected, it is often desirable to determine which treatments are different (i.e., multiple comparisons). Treatments i and j are considered different if

|R_j - R_i| > t_{1-\alpha/2,bk-b-t+1}\sqrt{\frac{2\left(A-C\right)r}{bk-k-t+1}\left(1-\frac{T_1}{b\left(k-1\right)}\right)}

where R_j and R_i are the column sum of ranks within the blocks, t_{1 − α/2, bk − b − t + 1} denotes the 1 − α/2 quantile of the t-distribution with bk − b − t + 1 degrees of freedom.

Historical note

T₁ was the original statistic proposed by James Durbin, which would have an approximate null distribution of $\chi_{t-1}^2$ (that is, chi-squared with $t-1$ degrees of freedom). The T₂ statistic has slightly more accurate critical regions, so it is now the preferred statistic. The T₂ statistic is the two-way analysis of variance statistic computed on the ranks R(X_ij).

Related tests

Cochran's Q test is applied for the special case of a binary response variable (i.e., one that can have only one of two possible outcomes). Cochran's Q test is valid for complete block designs only.

References

Conover, W. J. (1999). Practical Nonparametric Statistics (Third ed.). Wiley. pp. 388–395. ISBN 0-471-16068-7.

This article incorporates public domain material from websites or documents of the National Institute of Standards and Technology.