DURBIN TEST

DURBIN TEST

Analysis Command

Perform a Durbin test that k treatments in a two-way balanced incomplete block design have identical effects.

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 is arranged as follows:

	Treatment
Block	1	2	...	k
1	X11	X12	...	X1k
2	X21	X22	...	X2k
3	X31	X32	...	X3k
...	...	...	...	...
b	Xb1	Xb2	...	Xbk

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:

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

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(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

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

Then the Durbin test is

H0:	The treatment effects have identical effects
Ha:	At least one treatment is different from at least one other treatment
Test Statistic:	\( T_{2} = \frac{T_{1}(t-1)}{(b(k-1) - T_{1})/(bk -b -t + 1)} \) where T1 = \( \frac{(t-1)\{\sum_{j=1}^{t}{R_{j}^2} - rC\}}{A - C} \) t = the number of treatments k = the number of treatments per block b = the number of blocks r = the number of times each treatment appears A = \( \sum_{i=1}^{b}{\sum_{j=1}^{t}{\{R(X_{ij}\}^2}} \) C = \( \frac{bk(k+1)^2}{4} \)
Significance Level:	\( \alpha \)
Critical Region:	\( T_{2} > F_{(\alpha,k-1,bk - b -b + 1)} \) where FPPF is the F percent point function. \( T_{1} > \chi_{(\alpha,t-1)}^{2} \) where \( \chi^2 \) is the percent point function of the chi-square distribution.
Conclusion:	Reject the null hypothesis if the test statistic is in the critical region.

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(X_ij). 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

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

DURBIN TEST <y> <block> <treat>
                              <SUBSET/EXCEPT/FOR qualification>
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 BLOCK TREATMENT
DURBIN TEST Y X1 X2
DURBIN TEST Y BLOCK TREATMENT SUBSET BLOCK > 2

In Dataplot, the variables should be given as:

Y	BLOCK	TREAT
X11	1	1
X12	1	2
...	1	...
X1k	1	k
X21	2	1
X22	2	2
...	2	...
X2k	2	k
...	...	...
Xb1	b	1
Xb2	b	2
...	b	...
Xbk	b	k

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 MV = 1
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 response, ranked response, block, and treatment are written to the file dpst1f.dat in the current directory.

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.

The Durbin test is based on the following assumptions:
1. The b blocks are mutually independent. That means the results within one block do not affect the results within other blocks.

2. The data can be meaningfully ranked (i.e., the data have at least an ordinal scale).

None

None

ANOVA	= Perform an analysis of variance.
FRIEDMAN TEST	= Perform a Friedman test.
COCHRAN TEST	= Perform a Cochran test.
KRUSKAL WALLIS TEST	= Perform a Kruskal Wallis test.
VAN DER WAERDEN TEST	= Perform a Van Der Waerden test.
MEDIAN POLISH	= Carries out a robust ANOVA.
BLOCK PLOT	= Generate a block plot.
DEX SCATTER PLOT	= Generates a dex scatter plot.
DEX ... PLOT	= Generates a dex plot for a statistic.
DEX ... EFFECTS PLOT	= Generates a dex effects plot for a

"Practical Nonparametric Statistics", Third Edition, Wiley, 1999, pp. 388-395.

Analysis of Two Way Tables

2006/1

 
.  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 TREAT
    

The 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