COCHRAN TEST

COCHRAN TEST

Analysis Command

Perform a Cochran test that c treatments have identical effects.

The Cochran test is a non-parametric test for analyzing randomized complete block designs where the response variable is a binary variable (i.e., there are only two possible outcomes, which are coded as 0 and 1).

The Cochran test assumes that there are c experimental treatments (c >= 2). The observations are arranged in r blocks, that is

	Treatment
Block	1	2	...	c
1	X11	X12	...	X1c
2	X21	X22	...	X2c
3	X31	X32	...	X3c
...	...	...	...	...
r	Xr1	Xr2	...	Xrc

The Friedman test is the usual non-parametric test for this kind of design. The Cochran test is applied for the special case of a binary response variable (i.e., it can have only one of two possible outcomes).

Then the Cochran test is

H0:	The treatments are equally effective.
Ha:	There is a difference in effectiveness among treatments.
Test Statistic:	The Cochran test statistic is with c, Ci r, Ri and N denoting the number of treatments, the column total for the ith treatment, the number of blocks, the row total for the ith block, and the grand total, respectively.
Significance Level:
Critical Region:	T > 1-,c-1 where is the chi-square percent point function. Note that this is based on a large sample approximation. In particular, it assumes that r is "large".
Conclusion:	Reject the null hypothesis if the test statistic is in the critical region.

COCHRAN 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.

COCHRAN TEST Y BLOCK TREATMENT
COCHRAN TEST Y X1 X2
COCHRAN 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	...
X1c	1	c
X21	2	1
X22	2	2
...	2	...
X2c	2	c
...	...	...
Xr1	r	1
Xr2	r	2
...	r	...
Xrc	b	c

If your data are in a format similar to that given in the DESCRIPTION section (i.e., you have colums Y1 to Yc, each with r rows), you can convert it to the format required by Dataplot with the commands:

LET NTREAT = 5
LET NBLOCK = SIZE Y1
LET NTOTAL = K*NBLOCK
LET BLOCK = SEQUENCE 1 1 NBLOCK FOR I = 1 1 NTOT
LET Y2 TREAT= STACK Y1 Y2 Y3 Y4 Y5
COCHRAN TEST Y2 BLOCK TREAT

The Cochran test is based on the following assumptions:
1. The blocks were randomly selected from the population of all possible blocks.

2. The outcomes of the treatments can be coded as binary responses (i.e., a "0" or "1") in a way that is common to all treatments within each block.

The case where there are exactly two treatments is equivalent to the McNemar test. The McNemar test is equivalent to a two-tailed sign test.

If the Cochran test rejects the null hypothesis of equally effective treatments, pairwise multiple comparisons can be made by applying the Cochran test on the two treatments of interest. For example, to test treatments 3 and 5, you can do something like the following
COCHRAN TEST Y BLOCK TREATMENT SUBSET TREATMENT = 3 5

None

None

FRIEDMAN TEST	= Perform a Friedman test.
KRUSKAL WALLIS TEST	= Perform a Kruskal Wallis test.
ANOVA	= Perform an analysis of variance.
SIGN TEST	= Perform a sign test.
MEDIAN POLISH	= Carries out a robust ANOVA.
T TEST	= Carries out a t test.
RANK SUM TEST	= Perform a rank sum test.
SIGNED RANK TEST	= Perform a signed rank test.

"Practical Nonparametric Statistics", Third Edition, Wiley, 1999, pp. 250-256.

Analysis of Binary Two-Way Randomized Block Designs

2005/12

 
 . Following example from p. 253 of Conovover
 READ Y1 Y2 Y3
 1 1 1
 1 1 1
 0 1 0
 1 1 0
 0 0 0
 1 1 1
 1 1 1
 1 1 0
 0 0 1
 0 1 0
 1 1 1
 1 1 1
 END OF DATA
 LET N1 = SIZE X1
 LET NTOT = 3*N1
 LET BLOCK = SEQUENCE 1 1 N1 FOR I = 1 1 NTOT
 LET Y TREAT = STACK Y1 Y2 Y3
 COCHRAN Y BLOCK TREAT
    

This example generates the following output

 
        COCHRAN NON-PARAMETRIC TEST FOR RANDOMIZED COMPLETE BLOCK DESIGN
        FOR DICHOTOMOUS DATA
     
    1. STATISTICS
          NUMBER OF SUBJECTS (ROWS)           =       11
          NUMBER OF TREATMENTS                =        3
          COCHRAN TEST STATISTIC             =    2.800000
     
    2. PERCENT POINTS OF THE LARGE SAMPLE CHI-SQUARE REFERENCE DISTRIBUTION
       FOR COCHRAN TEST STATISTIC
          0          % POINT    =    0.000000
          50         % POINT    =    1.386294
          75         % POINT    =    2.772589
          90         % POINT    =    4.605170
          95         % POINT    =    5.991464
          99         % POINT    =    9.210342
          99.9       % POINT    =    13.81554
     
     
             75.34030       % Point:     2.800000
     
    3. CONCLUSION (AT THE 5% LEVEL):
          THE        3 TREATMENTS HAVE EQUAL EFFECTS