For some experiments, it may not be realistic to run all treatments in all blocks. In this case, you may need to run an incomplete block design. If you need to run an incomplete block design, it is strongly recommended that you run a balanced incomplete design. A balanced incomplete block design has the following properties:
The Friedman test is the most common nonparametric test for complete block designs. The Durbin test is a nonparametric test for balanced incomplete designs that reduces to the Friedman test if you in fact have a complete block design).
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
Then the Durbin test is
Note that T1 was the original statistic proposed by Durbin. 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(Xij). Dataplot prints the value of both the T1 and T2 statistics, but it only prints the critical values for the T2 statistic (you can compute the appropriate critical value for T1 by using the CHSPPF function).
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
where <y> is the response variable;
<block> is a variable that identifies the block;
<treat> is a variable that identifies the treatment;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
DURBIN TEST Y X1 X2
DURBIN TEST Y BLOCK TREATMENT SUBSET BLOCK > 2
Rows where no data are available are omitted.
If your data are in a format similar to that given in the DESCRIPTION section (i.e., you have colums Y1 to Yk, each with b rows where missing values are identified with a specific numeric value), you can convert it to the format required by Dataplot with the commands (the value for MV should be modified to match what you use to identify missing rows):
LET NBLOCK = SIZE Y1
LET BLOCKID = SEQUENCE 1 1 NBLOCK
LET RESPONSE TREAT BLOCK = REPLICATED STACK Y1 Y2 ... YK BLOCKID
RETAIN RESPONSE BLOCK TREAT SUBSET RESPONSE <> MV
DURBIN TEST Y2 BLOCK TREAT
The treatment ranks and multiple comparisons are written to the file dpst2f.dat in the current directory. Comparisons that are statistically significant at the 90% leverl are flagged with a single asterisk, comparisons that are statistically significant at the 95% level are flagged with two asterisks, and comparisons that are statistically significant at the 99% level are flagged with three asterisks.
. Following data from page 391 of the Conover text READ Y BLOCK TREAT 2 1 1 3 1 2 1 1 4 3 2 2 1 2 3 2 2 5 2 3 3 1 3 4 3 3 6 1 4 4 2 4 5 3 4 7 3 5 1 1 5 5 2 5 6 3 6 2 1 6 6 2 6 7 3 7 1 1 7 3 2 7 7 END OF DATA READ CONOVER.DAT Y BLOCK TREAT DURBIN Y BLOCK TREATThe following output is generated:
The following output is generated. DURBIN TEST FOR IDENTICAL TREATMENT EFFECTS: TWO-WAY BALANCED, INCOMPLETE BLOCK DESIGNS 1. STATISTICS NUMBER OF OBSERVATIONS = 21 NUMBER OF BLOCKS = 7 NUMBER OF TREATMENTS = 7 NUMBER OF BLOCKS FOR EACH TREATMENT = 3 A (SUM OF SQUARES OF RANKS) = 98.00000 C (CORRECTION FACTOR) = 84.00000 DURBIN TEST STATISTIC T1 (UNCORRECTED) = 12.00000 DURBIN TEST STATISTIC T2 (CORRECTED) = 8.000000 2. PERCENT POINTS OF THE F REFERENCE DISTRIBUTION FOR DURBIN TEST STATISTIC 0 % POINT = 0.000000 50 % POINT = 0.9711078 75 % POINT = 1.650838 90 % POINT = 2.668334 95 % POINT = 3.580580 99 % POINT = 6.370685 99.9 % POINT = 12.85810 99.50956 % Point: 8.000000 3. CONCLUSION (AT THE 5% LEVEL): THE 7 TREATMENTS DO NOT HAVE IDENTICAL EFFECTS