Durbin test is a non-parametric statistical test for balanced incomplete designs that reduces to the Friedman test in the case of a complete block design. In the analysis of designed experiments, the Friedman test is the most common non-parametric test for complete block designs.
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 | … | Treatment k | ||
---|---|---|---|---|---|
Block 1 | X11 | X12 | … | X1k | |
Block 2 | X21 | X22 | … | X2k | |
Block 3 | X31 | X32 | … | X3k | |
Block b | Xb1 | Xb2 | … | Xbk |
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:
The Durbin test is based on the following assumptions:
Let R(Xij) be the rank assigned to Xij 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
Rj=
b | |
\sum | |
i=1 |
R(Xij)
H0: The treatment effects have identical effects
Ha: At least one treatment is different from at least one other treatmentThe test statistic is
T2=
T1/\left(t-1\right) | |
\left(bk-b-T1\right)/\left(bk-b-t+1\right) |
T1=
t-1 | |
A-C |
t | |
\left(\sum | |
j=1 |
2 | |
R | |
j |
-rC\right)
A=
k | |
\sum | |
j=1 |
R(Xij)2
C=
1 | |
4 |
bk\left(k+1\right)2
For significance level α, the critical region is given by
T2>F\alpha,k-1,bk-b-t+1
|Rj-Ri|>t1-\alpha/2,bk-b-t+1\sqrt{
2\left(A-C\right)r | \left(1- | |
bk-k-t+1 |
T1 | |
b\left(k-1\right) |
\right)}
T1 was the original statistic proposed by James Durbin, which would have an approximate null distribution of
2 | |
\chi | |
t-1 |
t-1
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.