COCHRAN TEST
Name:
Type:
Purpose:
Perform a Cochran test that c treatments have identical
effects.
Description:
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.
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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.
|
Syntax:
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.
Examples:
COCHRAN TEST Y BLOCK TREATMENT
COCHRAN TEST Y X1 X2
COCHRAN TEST Y BLOCK TREATMENT SUBSET BLOCK > 2
Note:
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
Note:
The Cochran test is based on the following assumptions:
- The blocks were randomly selected from the population of
all possible blocks.
- 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.
Note:
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.
Note:
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
Default:
Synonyms:
Related Commands:
Reference:
Conover (1999), "Practical Nonparametric Statistics", Third Edition,
Wiley, pp. 250-256.
Applications:
Analysis of Binary Two-Way Randomized Block Designs
Implementation Date:
Program:
. 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
Date created: 12/05/2005
Last updated: 12/11/2023
Please email comments on this WWW page to
alan.heckert@nist.gov.
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