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 \( T = c(c-1) \frac{\sum_{i=1}^{c}{(C_j - \frac{N}{c})^2}} {\sum_{i=1}^{r}{R_i(c - R_i)}} \) 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:	\( \alpha \)
Critical Region:	T > \( \chi_{1-\alpha,c-1}^{2} \) where \( \chi^{2} \) 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.

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

Analysis of Binary Two-Way Randomized Block Designs

2005/12

 
. Following example from p. 253 of Conover
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 Y1
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 Test for Two-Way Randomized Complete Block Designs
                           (Dichotomous Data)
 
Response Variable: Y
First Group-ID Variable: BLOCK
Second Group-ID Variable: TREAT
 
H0: Treatments Have Identical Effects
Ha: Treatments Do Not Have Identical Effects
 
Summary Statistics:
Total Number of Observations:                36
Number of Blocks:                            12
Number of Treatments:                        3
 
Test:
Cochran Test Statistic:                        2.800000
CDF of Test Statistic:                        0.7534030
P-Value:                                      0.2465970
 
 
Percent Points of the Chi-Square Reference Distribution
-----------------------------------
  Percent Point               Value
-----------------------------------
            0.0    =          0.000
           50.0    =          1.386
           75.0    =          2.773
           90.0    =          4.605
           95.0    =          5.991
           97.5    =          7.378
           99.0    =          9.210
           99.9    =         13.816
 
 
Conclusions (Upper 1-Tailed Test)
----------------------------------------------
  Alpha    CDF   Critical Value     Conclusion
----------------------------------------------
    10%    90%            4.605      Accept H0
     5%    95%            5.991      Accept H0
   2.5%  97.5%            7.378      Accept H0
     1%    99%            9.210      Accept H0