• TABLE OF CONTENTS
HIDE
 Front Cover
 Title Page
 Table of Contents
 Introduction
 Farm experiments or research station...
 Selecting a plan
 Deciding on the number of...
 Field procedure
 The plans






Group Title: Guide to extensive testing on farms
Title: A guide to extensive testing on farms
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 Material Information
Title: A guide to extensive testing on farms
Alternate Title: Extensive testing on farms
Physical Description: 4 pts. in 1 v. : illus., tables. ; 26 cm.
Language: English
Creator: Hopp, Henry, 1911-
United States -- Foreign Agricultural Service
Publisher: Foreign Agricultural Service, U.S. Dept. of Agriculture
Place of Publication: Washington
Publication Date: 1954
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Subject: Field experiments   ( lcsh )
Agriculture -- Statistics   ( lcsh )
Genre: federal government publication   ( marcgt )
non-fiction   ( marcgt )
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Funding: Electronic resources created as part of a prototype UF Institutional Repository and Faculty Papers project by the University of Florida.
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Table of Contents
    Front Cover
        Front Cover 1
        Front Cover 2
    Title Page
        Page i
    Table of Contents
        Page ii
    Introduction
        Page 1
    Farm experiments or research station experiments? - What is meant by design?
        Page 2
    Selecting a plan
        Page 3
        Decide on the number of treatments - Classify the experiment
            Page 4
            Page 5
        Decide on the number of plots on each farm
            Page 6
        Select the appropriate plan
            Page 7
            Page 8
    Deciding on the number of farms
        Page 9
        Estimate the minimum difference
            Page 9
        Estimate the error
            Page 10
            Page 11
            Page 12
            Page 13
            Page 14
        Determine the number of repetitions
            Page 15
            Page 16
        Determine the number of farms - Choose among several plans
            Page 17
            Page 18
            Page 19
            Page 20
    Field procedure
        Page 21
        Select the regions and the farms
            Page 21
        Assign the blocks to the farms
            Page 21
            Page 22
            Page 23
            Page 24
            Page 25
        Collect the data
            Page 26
        Analyze the data
            Page 27
            Page 28
            Page 29
            Page 30
    The plans
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
Full Text



extensive
testing
on farms

in four parts

PAR/T III.
Foreign Agricultural Service April 1954
U.S.DEPARTMENT OF AGRICULTURE


/1.-7fq






















































This publication has
been prepared for use in
the technical cooperation
program of the Foreign
Operations Administration.














a guide to


EXTENSIVE TESTING ON FARMS




by Henry Hopp


in 4 parts


Part III:


Farm Experiments


Foreign Agricultural Service
UNITED STATES DEPARTMENT OF AGRICULTURE
Washington, D. C.
April 1954








CONTENTS


Page
Introduction . .... . . . 1

Farm experiments or research-station experiments? . 2

What is meant by design? . . . . 2

Selecting a plan .. . . . .... 3

Step 1 Decide on the number of treatments .. .. 4

Step 2 Classify the experiment . . 4

Step 3 Decide on the number of plots on each farm .. 6

Step 4 Select the appropriate plan . .. .

Deciding on the number of farms ..... . 9

Step 1 Estimate the minimum difference . ... 9

Step 2 Estimate the error . . . .. 10

Step 3 Determine the number of repetitions . 15

Step 4 Determine the number of farms . . 17

Step 5 Choose among several plans . . . 17

Field Procedure . .. . ..... 21

Step 1 Select the regions and the farms . .. 21

Step 2 Assign the blocks to the farms . . 21

Step 3 Collect the data . . . 26

Step 4 Analyze the data . . . ... 27

Appendix: The Plans . . . ... 31








INTRODUCTION


In Part I of this Guide we discussed the purposes and principles of extensive
testing on farms. We said that when you are concerned with regional applica-
bility of an agricultural practice, you need tests on farms as the next step
after tests at the research station. Also, we distinguished two classes of
extensive tests--result tests and farm experiments. The result test, which
was discussed in Part II, helps the technician and the community learn how
well an improved practice works under farm conditions. But it does not take
care of all the needed tests of applicability on farms. For a result test
involves just a single practice and has as its purpose the testing of the
benefit of this practice alone.

Often, however, the technician has a more complicated problem: he does not
know which of several alternative practices is the best under farm conditions.
To find out, he must undertake an actual experiment on farms; and the way in
which he can do so is the subject of Part III of this Guide.

Conducting farm experiments requires great care both in planning and in exe-
cution. In addition to the usual techniques of good experimental design,
two requirements must be observed in designing farm experiments:

1. Keep the number of plots on any one farm to a minimum, so that no one
farm is overburdened with plots.

2. Spread the test out over enough farms to make results representative of
the region.

In order to show how the first of these requirements can be met, we give
some designs in this part of the Guide that permit the plots per farm to be
fewer than the number of practices under test. Most of these designs have
been adapted from those that appear in the book Experimental Designs by
Cochran and Cox 1/

To help you design a farm experiment, we have outlined the procedure step
by step. The first steps help you select an appropriate plan. Then an
approximate but simple method is given for determining how many farms to
have in the test. Following this, a procedure is shown for deciding on one
plan when you have several to choose from. Finally, the field procedure
itself is described.

Before you start using this part of the Guide, be sure to read Part I,
which presents the principles of extensive testing.


1/ William G. Cochran and Gertrude M. Cox. Experimental Designs. John
Wiley and Sons, Inc., New York, 1950.









FARM EXPERIMENTS OR RESEARCH-STATION EXPERIMENTS?


An experiment on farms is likely to be more expensive than an experiment
at the research station, even though the cost may be partly reduced by
spreading the work out among more people. Besides, an experiment is more
inconvenient to operate on several dispersed farms than at a single re-
search station. Hence, before starting an experiment on farms,2 you
should be sure that this approach is what the problem really requires.

You might keep this simple rule in mind as a starting point for deciding
which type of experiment is required:

When you do not know whether the'practices under test are
effective, conduct the experiment at a research station
(or on just one or two experimental sites); when you are
reasonably sure that the practices are effective but do
not know their regional applicability, conduct the experi-
ment on farms.

A more detailed presentation of the idea of regional applicability is
given in Part I of this Guide.


WHAT IS MEANT BY DESIGN?

Let us consider what is meantby "design."' What are you trying to
accomplish? After all, you might ask, what else do we need to do be-
sides trying out the treatments on each farm?

In some cases, that is all there is to the problem of design: merely a
testing of different practices on several farms. 'More often than not,
however, such a simple procedure is not practical. For example, what
if y want to test 8 treatments? Considering the difficulty of con-
ducting cooperative tests on farms,you may not want this many plots on
one farm. You can overcome the difficulty by selecting a plan tha, for
instance, permits you to test 8 treatments and yet have no more than two
plots on any one farm. In the Appendix, beginning on page 31, are
various plans that you can use when you want to test more treatments
than you can put on a single farm.

Biometricians have developed plans also that will help increase the
precision of the test.. By using these plans you can reduce the number
of'replications, or repetitions, that are required and therefore reduce
the cost of the extensive test.

But design means more than selecting a plan. It also means determining
the number of farms required for the test, and the number of replications
that should be used. For you must have replication of farms ifyou are to
get a representation of farms and a measure of the consistency of the
results over a region






SELECTING A PLAN


Most of the plans shown here are for testing 10 treatments or less.
Those calling for a larger number are used in testing varieties of a
crop, or testing factors in different combinations and at different
levels--such as nitrogen, phosphorus and potassium in different combi-
nations and amounts. Most of the plans are confined also to just a
few plots--four or fewer--on any one farm. All the plans are to be
followed without repetition in any one location.

If you desire to iav'e more treatments or more plots per farm than in
the plans included in this Guide, you will have to refer to a more
exhaustive source of information on experimental design, such as the
book by Cochran and Coxs-. However, before you do that, it might be
well to reconsider your objectives, recognizing that extensive tests
are intended to determine the applicability of practices that are
already known to be effective.

The practices, in the first place, should be fairly well screened to
eliminate those that are obviously not worthwhile. If you have a
large number of practices to test, possibly the screening has not been
adequate. It would be better, then, not to conduct an extensive test,
but rather to begin with a screening experiment at one or two locali-
ties, probably at a research station, using a conventional research
design. This research test will permit you to eliminate those treat-
ments that are likely to be ineffective, impractical, or definitely
inferior. Thereafter you will be ready to conduct the extensive test,
using only practices that are likely to be effective and for which, in
the region under consideration, you need the information on relative
applicability.

All the plans shown call for only a few plots on each farm. It is
generally inadvisable to have a large number of plots on any one farm
because there is too much danger that the technician and the cooperat-
ing farmer will get the plots confused. If you are thinking of a large
number of plots on each farm, reconsider the idea. Avoid getting in-
volved in a complicated study when you are making cooperative tests.

There is a further disadvantage in having a large number of plots on
any one farm: it will tend to concentrate the test at too few locations.
Since the principal purpose of an extensive test is to get a repre-
sentative answer for a region, it is better to have a few plots per
farm and more farms than to have many plots on only a few farms.



2/ O. cit.
















The treatments are the different practices you are testing: treatments
applied to the land or crop, or different varieties of a single crop.
Often the purpose is to compare improved practices with a current
practice. The current practice is called the check, or control, and
it counts as one of the treatments. Thus, if you want to test 4 new
varieties of wheat in comparison with the native variety of wheat,
you will have a total of 5 treatments. Sometimes the check is not
actually the same practice on every farm: the native wheat, for
instance, may comprise as many different varieties as there are farms.
But having exactly the same variety as a check on every farm is not
necessary, for the purpose of the test is to compare new practices
with a current practice, whatever that may be.




STEP 2
CLASSIFY THE EXPERIMENT


Table 1 lists the various plans from which you will select the one
best suited to your testing problem. To facilitate your choice,
first classify the experiment into one of three types:

Type I

Experiments are put in Type I when they involve 10 or fewer treatments;
treatments should not be combinations of factors at several levels.
For example, you may be comparing several different insecticides. Each
insecticide is a treatment, and the check is an additional treatment.
The purpose of the experiment, then, is to compare the several different
insecticides with the check, and determine which insecticide gives the
best results; and the experiment is simply a comparison among several
separate items.

Type II

Experiments are put in Type II when they involve more than 10 treatments;
as in Type I, they should not be combinations of factors at several
levels. Such tests apply most often to varieties of a crop. As a rule
you will not want to test a very large number of varieties in an ex-
tensive test. You can usually eliminate the unadapted varieties by a







Table 1. Key to selecting plan for an extensive test.


TYPE I
10 treatments or less
Number of Number of Plan Number of Number of Plan
treatments plots on to treatments plots on to
each farm use __each farm use


A
B

A

C
D-1
B

A
C
{D-2
E-1
B

A



F-l

A

D-4
E-3
F-2


A
C{
D-5
E-4
F-3

A

fD-6
F-4

A

{D-7
E-5
F-5

A

F-6
-6
F-6


TY PE II T Y P E III
More than 10 treatments Combination of factors at
several levels
Number of Number of Plan Number of Number of Plan
treatments plots on to Factors Levels plots on to
__each farm use each farm use

16 4 G-1 2 H-1
14 H-2
25 5 G-2 2 f H-3
\9 H-4
27 3 G-3 4 4 H-5

36 6 G-4 f4 H-6
5 Jf8 H-7


4 / 2 H-9
________ __________ 4 ____ ____ 18 1-10









research test at the experiment station. Occasionally, however, you
may have more than 10 varieties to test. With such a large number of
varieties, you will test only a few of them at any one location so as
not to have too many plots on each farm.

Type III

Experiments are put in Type III when the treatments are combinations
of factors at several levels. An example is a fertilizer test in
which the effects of nitrogen, phosphorus, and potash are being deter-
mined alone and in various combinations. The factors are the indi-
vidual fertilizer elements; levels are the amounts or rates at which
they are used.

When a fertilizer element is used at two rates, such as present and
absent, or high and low, we say that the factor is at two levels.
If we go on to testing two different rates of a fertilizer in addition
to none, we say that the factor is at three levels. It is not often
advisable to test more than three levels (plans given here are for
tests up to 4 factors at 2 levels). If you test the minimum, average,
and maximum amounts of fertilizer that are practical, you will be able
to draw a curve of the response, and thereby get a reasonably good
idea of the most economic level.

Although determining the optimum economic level of various fertilizer
combinations is a common problem in extensive testing, experiments in
Type III need not be confined to fertilizers. One of the factors
might be a practice, such as applying green manure or an insecticide.

Each of the Type III plans referred to in Table 1 tests an equal
number of levels for all factors. Some factors are tested at two
levels; some, at three; one, at four. If you want to test an un-
equal number of levels in the same experiment, you need more compli-
cated designs: consult chapters 5 and 6 of the book by Cochran and
coxZ/.




STEP 3
DECIDE ON THE NUMBER OF PLOTS ON EACH FARM


Generally you will want to have only a few plots at each farm. In
deciding on the number, consider the time available to each field
technician involved and the land available on the cooperating farms.
Plans are shown for as few as one and two plots per farm.


i/ Op. cit.








The number of plots per farm is limited somewhat by the number and
type of treatments. For Type I tests, plans are given here that
permit you to have up to 4 plots on a farm with almost any number
of treatments from 2 to 10.




STEP 4
SELECT THE APPROPRIATE PLAN


You are now ready to consult Table 1 for selecting the appropriate
plan. In the column under "Plan to use" you will see 8 series, A to
H. General characteristics of each series are as follows:

A. Each treatment is on a different farm, with only 1 plot
pcr farm.
B. All treatments are on each farm.
C. Each treatment is on a different farm, as in series A;
but each farm also has a check plot, making 2 plots
per farm.
D. Two treatments are on each farm, but there are 3 or more
treatments in all.
E. Three treatments are on each farm, but there are 4 or
more treatments in all.
F. Four treatments are on each farm, but there are 5 or
more treatments in all.
G. There is a large number of treatments or varieties (16
up to 36).
H. Treatments are factorial; i.e., two or more factors are
applied in combination at different levels.

Suppose you want to test 3 separate treatments and have decided on 1
plot for each farm: these specifications will lead you, in the third
column of table 1, to select Plan A,. For detailed layout of a single
repetition of that plan, see page 31. Or maybe you want to test 4
separate treatments and have decided on 3 plots for each farm. Such
specifications call for plan E-l; layout is shown on page 42.

Occasionally you will find you have a choice of plans. For example,
with 4 treatments and 2 plots at each farm, there are two possibilities:
Plans C and D-2. We will have more to say on pages 17-20 on how to
choose among plans when there are several possibilities.

Suppose your test involves a large number of separate treatments and so
falls in Type II. Plans given here for this type of test cover numbers
of treatments between 16 and 36. If the number of treatments you want







is not included in the list, you can follow one of two alternatives.
The first is to repeat several of the treatments to round out the
total number to fit into one of the designs shown. For example, if
you have 23 varieties to test, you might repeat two, for a total of
25, and then use plan G-2. The second alternate vme is to refer to
Cochran and Cox's book on experimental designs,-/ where more plans
are shown. Most of these plans, however, are more complicated.

But suppose your test involves combinations of factors at different
levels and so falls in Type III. Here the specifications become a
bit more complex: instead of just the number of treatments, you must
specify both the number of factors under test and the number of
levels.

For example, if you are testing nitrogen, phosphorus, and potash, you
have three factors; and if you are testing just the absence and
presence of each element, you have two levels. With 3 factors at 2
levels, you will have 8 possible treatment combinations ("l" means
absent or the lower level; "2" means present or the higher level):

1. NP1K1 5. NiPIK2

2. N2P1K1 6. N2P1K2

3. N1P2K1 7. NIP2K2

4. N2P2K1 8. N2P2K2


There are one or more possible plans for each combination of factors
and levels, according to the number of plots on a farm. With 5
factors at 2 levels, there is a plan for 4 plots per farm (H-6, on
page 63) and another for 8 plots per farm (H-7, on page 64). If it
is immaterial whether you have 4 or 8 plots per farm, you might refer
to Step 5 of the next section (pages 17-20) for guidance.

If you want to find plans for a larger number of factors or for
factors at diff~gent levels, you can find suggestions in the book by
Cochran and Cox L. However, you should be wary of tests that involve
a large number of variables and complicated designs. In extensive
testing it should be possible to answer most problems with relatively
simple designs.




4/ p. cit.. Table 9.5

5/ Op. cit. Table 6.19






DECIDING ON THE NUMBER OF FARMS


The four steps you have just taken have led you to one or more
possible plans for the experiment. You now have to answer the question:
How many farms should be included in the experiment? That is, how many
repetitions of a plan are required to get an adequate test of the
region? By following the first four steps in this section, you will be
able to determine the number of farms and repetitions that are required.
If you have several possible plans, you must choose among them; the
fifth step tells you how to do that.




STEP 1
ESTIMATE THE MINIMUM DIFFERENCE


The decision to be made now is this: How small an improvement can a
treatment give and yet prove worthwhile; in other words, what is the
minimum difference youwant to test? At first this may appear to be a
difficult decision, but it is in fact quite simple for any experiment
that has a practical objective.

It is easy to see why this decision must be made. the smaller the
effect, the more repetitions are needed. If you are dealing with a
practice that increases yield to a spectacular extent, you can get by
with a rather small extensive test. But if the practice produces
only a small effect, you will need a much more thorough test. Fortu-
nately, most extensive tests involve practices that give large benefits
or large increases in yield.

One of two methods can be used to determine the minimum difference to
be tested: One is based on the cost of applying the practice; the
other on the effort required to put the practice into effect.

On the basis of cost

Let us say that you are undertaking an extensive test to determine
whether it is beneficial to apply fertilizer and have celar;d an
application rate of 300 pounds to the acre. If fertilizer costs $55
a ton, the cost of the practice would be approximately $8 per acre
for the fertilizer and perhaps $2 more for applying it, or a total
of $10 per acre. If the crop under test is corn, worth $2 a bushel,
the farmer will have to get an increased yield of 5 bushels per acre
to pay the cost of the practice. If the average yield of corn with-
out fertilizer is 30 bushels to the acre, the minimum increase needed
would be 1/6, or approximately 16 percent. For the farmer to show a
profit from the practice, an even larger increase in yield would be
needed. Let us say, then, that you set the minimum difference at
something above 16 percent, perhaps 20 percent.






DECIDING ON THE NUMBER OF FARMS


The four steps you have just taken have led you to one or more
possible plans for the experiment. You now have to answer the question:
How many farms should be included in the experiment? That is, how many
repetitions of a plan are required to get an adequate test of the
region? By following the first four steps in this section, you will be
able to determine the number of farms and repetitions that are required.
If you have several possible plans, you must choose among them; the
fifth step tells you how to do that.




STEP 1
ESTIMATE THE MINIMUM DIFFERENCE


The decision to be made now is this: How small an improvement can a
treatment give and yet prove worthwhile; in other words, what is the
minimum difference youwant to test? At first this may appear to be a
difficult decision, but it is in fact quite simple for any experiment
that has a practical objective.

It is easy to see why this decision must be made. the smaller the
effect, the more repetitions are needed. If you are dealing with a
practice that increases yield to a spectacular extent, you can get by
with a rather small extensive test. But if the practice produces
only a small effect, you will need a much more thorough test. Fortu-
nately, most extensive tests involve practices that give large benefits
or large increases in yield.

One of two methods can be used to determine the minimum difference to
be tested: One is based on the cost of applying the practice; the
other on the effort required to put the practice into effect.

On the basis of cost

Let us say that you are undertaking an extensive test to determine
whether it is beneficial to apply fertilizer and have celar;d an
application rate of 300 pounds to the acre. If fertilizer costs $55
a ton, the cost of the practice would be approximately $8 per acre
for the fertilizer and perhaps $2 more for applying it, or a total
of $10 per acre. If the crop under test is corn, worth $2 a bushel,
the farmer will have to get an increased yield of 5 bushels per acre
to pay the cost of the practice. If the average yield of corn with-
out fertilizer is 30 bushels to the acre, the minimum increase needed
would be 1/6, or approximately 16 percent. For the farmer to show a
profit from the practice, an even larger increase in yield would be
needed. Let us say, then, that you set the minimum difference at
something above 16 percent, perhaps 20 percent.








A similar method is used when you are testing factors at several
levels, that is, making Type III tests. Continuing the illustration,
suppose that you wish to test 5 levels of fertilizer--0, 150, and 300
pounds to the acre. Each level is 150 pounds higher than the preced-
ing one; and 150 pounds, with the labor of applying it, costs, let us
say, $5. Going through the same computations as in the previous para-
graph, you find that an increase of 8 percent in yield would pay this
cost; a little more than 8 percent, then, is the minimum difference
you will be interested in testing.

On the basis of effort required

Sometimes the method based on cost is impractical or gives unrealistic
results and must be tempered by judgment. The minimum increase that
would be profitable by the cost method--say 5-percent for a certain
practice--might be too small to compensate for the effort that farmers
would have to put forth to make the change. You might then decide
that unless the practice results in, say, a 20 percent benefit, it has
no promising basis for an extension program.

Thus, you see, you may decide on the minimum difference quite sub-
jectively. The decision may even involve program policy. Several
alternative extensive tests may be proposed; yet your organization
may have facilities for incorporating only a few of the practices into
its technical program. If so, practices that give only a small benefit
will not be of much interest. Of course, if the practice is not likely
to produce the minimum difference, you should not consider it further.




STEP 2
ESTIMATE THE ERROR


When you try out a practice at different places, you do not always get
the same result. Sometimes the practice is good, sometimes not. This
random, accidental, or unexplained variability in different places is
called the error of the test. After the test is completed, you will
find the actual error and you will use it to set up confidence limits
for recommendations to farmers. But before you start the test, you
have to make an "informed guess"--or estimate--of what the error will
be. You need to do so because the greater the error, the more repe-
titions you will need for precision.

A simple, approximate method will be described here to estimate the
error. It is based on your knowledge of the area and the practice
being tested. There are more advanced methods, in which you use
actual data from previous surveys or research tests and make statisti-
cal computations on a calculating machine. These methods are more









exact. If you do not feel satisfied with the method given in this
chapter, you might want to use the more exact methods. These are
given in Part IV of this Guide.

To determine the error of the test, you must make some guesses on
the three factors of variability:

1. Plot variability: How much do adjacent plots on the
same farm vary in yield?

2. Location variability: How much do farms in the same
region vary in yield?

3. Treatment variability: How much does the benefit
from a new treatment or practice vary from farm to
farm?

If you intend to use Plan A, you will need to guess at all three of
these. But for Plans B to H, you can skip the second--location varia-
bility.

Estimating plot variability

Consider two plots of the crop in question. They are side by side
and are treated the same way. Sometimes the two plots will actually
have the same yield, but more often they will not. Now answer the
question, "What is the maximum difference in yield that we might
obtain between these two plots?" Note that we want the maximum
difference, not the usual difference.

You see that the decision does not require actual data; it is reached
by guess. It might be wise to discuss your guess with several of your
associates; perhaps they can even help you make it.

As an example, let us say that the test is with 1/4-acre corn plots.
Further, let us say that the average yield of corn in the area is 30
bushels per acre. The question, then, is this: "If two plots, side
by side and treated alike, gave this average yield, what would be the
extreme, or maximum, difference that we might anticipate between them?"
You now answer this question, perhaps after consulting with several
competent people. You might conclude that two 1/4-acre plots side by
side, with an average yield of 30 bushels per acre, might, at the ex-
treme, have a difference of 20 bushels per acre, one plot yielding 40
bushels andmtteother 2.0. This is your estimate of the mLximum difference.






Having arrived at a maximum difference, you obtain the value for plot
variability by dividing the maximum difference by 6:6/

Plot variability =Maximum difference
6

20
T

= 3-1/3 bushels

A slightly different way of making the same guess is by thinking of
the average, rather than the maximum, difference. You might try this
procedure as a check on the reasonableness of the first procedure.
Now you ask the question this way: "If we take the yields on two
adjacent plots, what on the average will be the difference between
them, when the two plots together have a mean yield of 30 bushes per
acre?"

Clearly, two adjacent plots might have almost the same yield or they
might be quite divergent. But, after considering the question, and
discussing it with your associates, you may decide that, on the average,
two adjacent plots might differ by 5 bushels. The plot variability is
now obtained by dividing this difference by the number 1.4:I/

Plot variability Average difference
Plot variability = 1.4


-A

= 5.6 bushels

Using the two procedures, you might decide on a plot variability of
3.5 bushels. This value is now expressed as a percent of the mean
yield, which is 50 bushels per acre:

Plot variability = 3.5 x 100
30

= 12%

Estimating location variability

Skip this section if you are considering plans B to H. You need
determine location variability only for Plan A tests.

Location variability is merely the variability in yield between farms
that are handled by usual farming methods, that is, the differences in
yield that farmers in the region obtain. Location variability combines
differences due to climate, soil fertility, and farm practice.


6/ George W. Snedecor, Statistical Methods, 4th ed., 1946, p. 96.
I/ Ibid., p. 49.








To determine location variability, you use a method very much like the
one for plot variability: you draw upon your own experience and upon
the judgment of other competent technicians or farmers in the area.
Answer this question: "For the crop and for the region in which the
extensive test will be conducted, what is the highest and lowest yield
per acre that farmers obtain?" Note, again, that we want the maximum
difference.

As an example, let us say that the test will be conducted on corn;
that your estimate of the highest and lowest yields is 100 and 10
bushels per acre, respectively; and that the mean yield is 30 bushels
per acre. The location varia blity is obtained simply by dividing
this maximum difference by 6:T/
Maximum difference
Location variability = 6

100 10


= 15 bushels

Now convert this value to percent simply by multiplying by 100 and
dividing by the mean:
15 x 100
Location variability = 15 x 00
30

= 50%

Estimating treatment variability

In some places a given practice will certainly work better than in
others. Sometimes it will produce a very large effect, sometimes
little or perhaps none. Now, answer this question: "Considering that
a treatment will not always have the same effect, what will be the
very most and the very least effects that we can expect?"

If the treatment produced exactly the same increase in yield every-
where it was applied, the treatment variability would be zero. Such
a result, however, is uncommon. More often, results will be somewhat
inconsistent among the different locations. The decision you must
now make is this: What is the probable inconsistency of the results?

You doubtless have a pretty good idea what the average effect of the
treatment will be; let us say that you anticipate an average increase
in yield of 50 percent. Of course there will be some variation; not
all locations can be expected to show this much improvement. In some


8/ Snedecor, op. cit., p. 98






places, the treatment may work well, in others, not so well.

You may be quite certain, for example, that in no place will you
get more than 100 percent increase from the treatment. This, then,
will be the upper limit of the treatment effect. The lower limit
will be the least effect that you anticipate--perhaps a 20-percent
increase. Or you may expect that in some cases you will get no
increase in yield at all. Then the lowest treatment effect would
be zero.

By this means, you arrive at two extreme values: the highest and
lowest yield increases that you can anticipate from the treatment.
The difference between these two extreme values, divided by 6,
gives the treatment variability:
Maximum difference
Treatment variability 6

100 -0


=17%


Another way of making the same estimate is in actual
than in percent. You start by making estimates like
the following example:


Average yield of
Average,,increase
Maximum increase
Minimum increase


check plot - - -
in yield due to treatment - -
in yield due to treatment - -
in yield due to treatment - -


measures rather
those shown in


30 bu. per acre
20 bu. per acre
30 bu. per acre
0 bu. per acre


Then the treatment variability is computed as follows:


Treatment variability =


Maximum yield minimum yield


60 o


= 5 bushels

This value is multiplied by 100 and divided by the mean yield without
treatment to get the treatment variability in percent:

Treatment variability a x100
30
= 17%

When you are testing several treatments in one extensive test, make the
estimate of treatment variability for the treatment or treatments you
were thinking of when you estimated the minimum difference (page 9).









Combining the estimates of variability

Once you have the estimates of variability, putting them together to
get an error for the extensive test is just a matter of simple arith-
metic. You merely square the variabilities, add the squares, and
then take the square root of this sum. For Plan A, you use all three
variabilities:

1. Plot variability = 12%; squared = 400
2. Location variability = 50%; squared = 2500
3. Treatment variability = 17%; squared = 289

Total of the squares 2933
Extensive-test error (square root of 2933) 54%

For Plans B to H, you use just the plot and treatment variabilities;

1. Plot variability = 12%; squared 144
2. Treatment variability = 17%; squared 289

Total of the squares 433
Extensive-test error (square root of 433) 21%




STEP 3
DETERMINE THE NUMBER OF REPETITIONS


After you have selected a plan and have estimated the minimum differ-
ence and the error, you can determine the number of repetitions of
the plan by simple arithmetic. You can see how to do it by an example.

Let us say that you have chosen Plan E-2 (page 43); that is, you are
testing 5 treatments with 3 plots per farm. Also, you have made the
following estimates:

Minimum difference to be tested - 30
Error - - - --- 21%

Now divide the difference by the error:

30 = 1.43
21

Then, in table 2,2/ find 1.43 or the numbers closest to it in the
first column (1.25 and 1.50). The number of replications required


2/ Cochran and Cox, op. cit., Sec. 2.21










Table 2. Number of replications required
for extensive tests on the basis
of the ratio between minimum
difference and error.


Ratio: Number
difference of
divided replications
by error


10

7.5


3.0

2.5

2.0

1.75

1.50

1.25

1.00

.90

.80

.70

.60

.50








is estimated in the other column. About 12 replications are required
for the whole test.

Finally, look at the description at the head of Plan E-2 (page 43)
to find that it has 6 replications for each repetition. Therefore--

Times to repeat the plan = Number of replications required for test
Number of replications in each repetition

12
6

S2

Now you know that you will need to repeat Plan E-2 twice.

If you look at Plan G-1, you will note that it, like some others in
the Appendix, provides several arrangements for each repetition. If
you have to repeat the plan, use as many of the arrangements as you
can. Thus, if 2 repetitions are required, use arrangements 1 and 2
once each, rather than arrangement 1 twice.



STEP 4
DETERMINE THE NUMBER OF FARMS


Suppose again that you are using Plan E-2 and have determined that
you need 2 repetitions. As is shown on page 43, 1 repetition calls
for 10 farms; therefore you will need 20 farms in all.




STEP 5
CHOOSE AMONG SEVERAL PLANS


Now you come to the matter of choosing when Table 1 offers more than
one plan for a given number of treatments. For example, you may
wish to have 7 treatments but are undecided as to the number of plots
per farm. Table 1 offers you 5 possible plans for 7 treatments: A,
1 plot per farm; C, 2 plots; D-5, also 2 plots; E-4, 5 plots; and F-3,
4 plots.

Which of these designs should you use? Begin by summarizing in
tabular form, Steps 1 to 4 of this section for the several possible
designs. Table 3 is such a summary for the example under consideration.







Table 3. Example of a summary to aid in choosing among several possible plans.

SPECIFICATIONS
No. of treatments: 7 (i is a check)
No. of plots at each location: 1, 2, 3, or 4
Minimum difference: 30
Anticipated error, Plan A: 54%
Plan B to H: 21%


-(1) (2) ?(3) ) (5) (6) (7) (8) (9) (10)
Possible Plots Replications Replications Repetitions Farms Plots Total Requirement
plans on each Difference required per required per per Farms Plots
farm 4 repetition (Cols. repetition repetition (Cols. (Cols.
(Table 1) (Table 1)i Error. (Table 2) (Appendix) i 4 5) (Appencix) (Appendit) 6 x 7) 6x 8)
A 1 0.56 68 1 68 7 7 476 476
C 2 1.43 12 1 12 6 12 72 144

I
D-5 2 1.43 12 6 2 21 42 42 84

E-4 5 1.43 1 12 3I 4 7 21 28 84
F-5 4 1.43 12 4 3 7 1 28 21 84
_______ 1 ____________ I _______I____ I-----









At the head of the table are listed the specifications for the design:
7 treatments and 1, 2, 3, or 4 plots per farm. The minimum difference
to be tested has been determined, by Step 1 of this section, as 30
percent. The error, as determined by Step 2, is 54 percent for Plan A
and 21 percent for Plans B to H.

You are now ready to fill out the 10 columns of the table. In column
1 copy the possible plan numbers offered in table i and in column 2
list the number of plots per farm that each plan calls fore In column
5 list the ratio of difference to error: For Plan A, it is 50/54, or
0.56; for the other plans it is 50/21, or 1.435 In column 4 give the
number of replications that are required; this you will get from table
2, which indicates 68 replications for a difference-error ratio of
0.56, and 12 for a ratio of 1.45. In column 5 list the number of
replications in each plan. In column 6 give the number of repetitions
required--a value you will arrive at by dividing the number in column
4 by its corresponding number in column 5. Columns 7 and 8 you will
get from the appendix. Column 9, which gives the total number of
farms required for the extensive test, is obtained by multiplying
column 6 by column 7. Column 10, the total number of plots in the
test, is obtained by multiplying column 6 by column 8.

The last two columns contain the information that you need for making
a choice among the possible designs. Note that the number of farms
(column 9) decreases as the number of plots per farm (column 2) in-
creases. With 1 plot per farm (Plan A) you will need 476 farms; with
4 plots per farm (Plan F-5) you will need only 21.

Obviously, it is better to have a few plots per farm, provided the
required number of farms is not too great. But, in this example,
Plan A calls for an excessive number of farms. Almost surely you
cannot operate an extensive test that requires 476 farms; therefore
you will eliminate Plan A.

So you move on to Plan C, which has 2 plots per farm, It requires
72 farms, still a fairly large number. Plan D-5 also has 2 plots
per location, but it requires only 42 farms.

The one advantage of Plan C, however, is that it has both a treat-
ment and a control on each farm. This fact may give the test better
demonstration value, for the farmer will be able to see the effect
of the new practice more easily. However, to get this added demon-
strational value, you have to use 72 locations instead of 42, or
approximately 75 percent more locations.

If both these plans require too many locations, you might go on to Plan
E-4, which calls for 3 plots per farm but only 28 farms. Finally, Plan
F-5, with 4 plots per farm, reduces the number of locations to 21.
Plans E-4 and F-3 require the same number of total plots, 84, and you









choose between them on the basis of your preference for fewer farms or
for fewer plots per farm. You might decide, for example, that 4 plots
are too many for each location, but that 3 would be satisfactory.
Then you choose Plan E-3. Or if you think the additional 7 farms will
be a serious handicap, you will decide in favor of Plan E-4.







FIELD PROCEDURE


In laying out an extensive test, you will first have to decide whether
results are desired separately by regions within the area where the
practice is to be applied. The point is discussed in Part II of this
Guide, where you will find help in making the decision. If you decide
to get results separately by regions, lay out a complete design in
each region.

With more than one region to test. you may find that your estimates of
the minimum difference or of the error may differ among regions. If
so, determine the required number of farms separately for each region.

Methods of selecting the farms and locating the plots on each farm
also have been discussed in Part II.




STEP 2
ASSIGN THE BLOCKS TO THE FARMB


Note that all plans shown in the Appendix are drawn up in blocks,
with one plot or more in each block. Each block goes on a different
farm. Your next job is to assign the blocks to the farms.

In each plan the blocks are numbered consecutively. You will have
to rearrange the blocks; for, if you assign them in the order shown,
each treatment will tend to be concentrated in one section of the
region and the effects will not be representative for the entire region.

There are two schemes for assigning the blocks to the farms. Scheme 11-
is to subdivide the region and put a separate repetition in each sub-


10/ Scheme 1 permits measurement of subdivision variability. Selection
of plans suitable for this purpose is made somewhat arbitrarily. In
general, it has been drawn at 20 degrees of freedom for error when there
are two repetitions. Table 6 shows that the degrees of freedom for error
depend on the number of repetitions. But some plans will have a large
number of degrees of freedom for error, even with Just 2 repetitions;
with such plans we have an opportunity to get more information without
extra work. However, for the plans that are not likely to have 20 de-
grees of freedom in the error, such a subdivision of the region would
not be advisable and Scheme 2 is indicated.







FIELD PROCEDURE


In laying out an extensive test, you will first have to decide whether
results are desired separately by regions within the area where the
practice is to be applied. The point is discussed in Part II of this
Guide, where you will find help in making the decision. If you decide
to get results separately by regions, lay out a complete design in
each region.

With more than one region to test. you may find that your estimates of
the minimum difference or of the error may differ among regions. If
so, determine the required number of farms separately for each region.

Methods of selecting the farms and locating the plots on each farm
also have been discussed in Part II.




STEP 2
ASSIGN THE BLOCKS TO THE FARMB


Note that all plans shown in the Appendix are drawn up in blocks,
with one plot or more in each block. Each block goes on a different
farm. Your next job is to assign the blocks to the farms.

In each plan the blocks are numbered consecutively. You will have
to rearrange the blocks; for, if you assign them in the order shown,
each treatment will tend to be concentrated in one section of the
region and the effects will not be representative for the entire region.

There are two schemes for assigning the blocks to the farms. Scheme 11-
is to subdivide the region and put a separate repetition in each sub-


10/ Scheme 1 permits measurement of subdivision variability. Selection
of plans suitable for this purpose is made somewhat arbitrarily. In
general, it has been drawn at 20 degrees of freedom for error when there
are two repetitions. Table 6 shows that the degrees of freedom for error
depend on the number of repetitions. But some plans will have a large
number of degrees of freedom for error, even with Just 2 repetitions;
with such plans we have an opportunity to get more information without
extra work. However, for the plans that are not likely to have 20 de-
grees of freedom in the error, such a subdivision of the region would
not be advisable and Scheme 2 is indicated.







FIELD PROCEDURE


In laying out an extensive test, you will first have to decide whether
results are desired separately by regions within the area where the
practice is to be applied. The point is discussed in Part II of this
Guide, where you will find help in making the decision. If you decide
to get results separately by regions, lay out a complete design in
each region.

With more than one region to test. you may find that your estimates of
the minimum difference or of the error may differ among regions. If
so, determine the required number of farms separately for each region.

Methods of selecting the farms and locating the plots on each farm
also have been discussed in Part II.




STEP 2
ASSIGN THE BLOCKS TO THE FARMB


Note that all plans shown in the Appendix are drawn up in blocks,
with one plot or more in each block. Each block goes on a different
farm. Your next job is to assign the blocks to the farms.

In each plan the blocks are numbered consecutively. You will have
to rearrange the blocks; for, if you assign them in the order shown,
each treatment will tend to be concentrated in one section of the
region and the effects will not be representative for the entire region.

There are two schemes for assigning the blocks to the farms. Scheme 11-
is to subdivide the region and put a separate repetition in each sub-


10/ Scheme 1 permits measurement of subdivision variability. Selection
of plans suitable for this purpose is made somewhat arbitrarily. In
general, it has been drawn at 20 degrees of freedom for error when there
are two repetitions. Table 6 shows that the degrees of freedom for error
depend on the number of repetitions. But some plans will have a large
number of degrees of freedom for error, even with Just 2 repetitions;
with such plans we have an opportunity to get more information without
extra work. However, for the plans that are not likely to have 20 de-
grees of freedom in the error, such a subdivision of the region would
not be advisable and Scheme 2 is indicated.








division. You can use this scheme only if you need repetitions, and
then only with the following plans:

D-4 to D-8
E-2, E-3, E-5, E-6
F-2 to F-6
G-1 to G-4
H-5, H-8 and H-9

For the other plans--and for all plans when they are not repeated--
use scheme 2.

Scheme 1.


To show you how to fit a plan that permits subdivision
region, we will use Plan G-1 (page 54) as an example.
calls for 4 blocks per repetition and gives 5 possible
ments, each one of which counts as a repetition.


of the
This plan
arrange-


Let us say you have determined by the procedure given on page 17
to use 4 arrangements repeated twice. This calls for a total
of 32 farms. The arrangements and repetitions are assigned
to different subdivisions of the region, but each set of 4
blocks must be in the same subdivision. Hence, you will have
8 subdivisions and 4 farms per subdivision. Figure 1 shows
how the subdividing may be done; here it is partly physio-
graphic, partly political, and partly by extension districts.


Tuscarora township


2. East part of int
river plain



1. West part of
inter-river
plain




4. Extension District A


Figure 1.


7. North part of
Extension
District B
27
*28
y- = 6. South part of
3 JExtension
District B

5. West part of
Extension
District B
. Plateau


Numbering the farms for Plan G-1 in a
test region that has been partitioned
into subdivisions.








The rest of the procedure is shown in Table 4. The columns
for subdivisions, farms, and arrangements (each repeated twice)
are listed in consecutive order. But within each subdivision
the blocks are assigned at random--"out of the hat." For plans
that do not have a choice of arrangements, substitute a repe-
tition column for the arrangement column.

Scheme 2.

When you are using one of the plans for which subdivision is
not advisable, or when you are not repeating a plan, follow
the scheme shown in Figure 2. To sharpen the contrast between
the two schemes, Figure 2 has been made for the same area as
Figure 1. Now, though, you pay no attention to subdivisions.
The farms are numbered consecutively over the region. As you
might expect, the numbering order corresponds approximately to
the geographic order. This time we are fitting Plan E-l (page
42)to the region. It calls for 3 plots at each farm and re-
quires 4 farms for a single repetition. Let us say that you
have determined to use 8 repetitions, or a total of 32 farms.
The assignment of treatments to the farms is shown in Table 5;
you list the blocks and repetitions in consecutive order, but
list the farms in randomized order.


Figure 2.


Numbering the farms for Plan E-l in a
test region that has not been partitioned
into subdivisions.









Table 4. A convenient scheme for fitting Plan G-1 (4 blocks
per arrangement) when 4 arrangements are repeated
twice and the 52 farms are numbered in approximate
geographic order in the test region.


1. Inter-river
plain (west)


2. Inter-river
plain (east)



3. Tuscarora
township



4. Extension
District A



5. Extension
District B
(west)


6. Extension
District B
(south)


7. Extension
District B
(north)


8. Plateau


S 5
6
7
8

S 9
10
11
12

* 13
14
15
16

17
18
19
20

21
22
23
24

25
26
27
28










Table 5. A convenient scheme for fitting Plan E-l
(4 blocks per repetition) when there
are 8 repetitions and the 32 farms are
numbered in approximate geographic
order in the test region.

Farm
Block Repetition (randomized)

1 1 3
2 1 16
5 1 12
4 1 18

1 2 26
2 2 23
3 2 31
4 2 27

1 5 29
2 3 11
3 3 14
4 3 20

1 4 5
2 4 7
3 4 17
4 4 9

1 5 24
2 5 22
3 5 1
4 5 21

1 6 32
2 6 28
3 6 19
4 6 8

1 7 15
2 7 4
3 7 13
4 7 30

1 8 25
2 8 10
8 6
4 8 2








Once the blocks are assigned to the farms you will be able to tell
from the plan just what treatments go on the plots of each farm. Your
next job is to lay out the plots and assign the treatments to them.
In selecting the plots, one would ILk to select areas that are exactly
alike in all respects, but it is impossible to do so. No set of plots
will ever be found exactly alike. There are sure to be some differ-
ences in productivity among them. If you attempt to select plots that
are of equal fertility, you are up against the same problem you facq
in attempting to select 'typical farms."

The safest as well as the simplest procedure is merely to assign the
treatments at random to the plots. This assures that the effects of a
treatment will be fully representative of the results farmers will
obtain when they use the practice at random in their fields.

You must be careful not to bias your results. If a new practice is
tested on land that has been selected because it is more fertile than
the land !n which the old practice is used, the result of the test
will be biased and may be very much in favor of the new practice. It
will not be a valid measure of the true effect of the practice. Rather,
th- effect will be confused, or, as the biometrician says, "confounded,"
with the natural fertility differences between the two selected plots of
land. Such selection may be considered a good idea for the purposes of
demonstration. But if the purpose of the work is to find out the true
effect of the practice so that you can be correctly guided in your
recommendations to farmers, it is essential that the treatment effects
should not be biased by natural differences in soil fertility. If you
lny out the plots and then simply assign the treatments at random to
the plots, you will assure against this bias.




STEP 5
COLLECT THE DATA


The data for each plot are obtained separately. At times you will feel
that the yields are unrepresentative, especially when the new practice
fails to yield as well as the check practice. But be slow to eliminate
or "correct" plot yields solely because they are not in line with
expectations. Just remember that some variability is to be expected;
in fact, one purpose of the extensive test is to get a true idea of
variability in the effectiveness of the practice. Eliminate a plot
only if you are certain that something was done wrong during the test.
Even if some of the plots have to be eliminated, or are missing, a
proper analysis can still be made.















Analyze the data by the usual statistical methods. The analysis will
elimi-i;: farm differences so that you can estimate the true, unbiased
effects of the treatments. The analysis also gives the confidence
limits, that is, tells how much the benefit from each treatment may be
expected to vary. Conclude your analysis with some statement like this:

As a result of the test we recommend variety A for
this region. On the average, an increase in yield
of 50 percent can be expected. Three-fourths of the
time, the increase will be at least 37 percent.
Only about 1 farmer out of every 100 can expect no
increase in yield.

Such an objective, straightforward, and conclusive statement can be
made only if you have completed a statistical analysis. The methods
need not be repeated here; they are fully described in several books
on statistical methods. For your convenience, however, we are referr-
ing only to the book by Cochran and Cox.ll/ Table 6 will be useful to
the technician who does the statistical analysis. The degrees of
freedom to use in the analysis of variance, and an exact reference in
Cochran and Cox, are listed for each plan. If the data from a farm
experiment are sent to a consulting biometrician for analysis, the
pertinent information from this table, together with a copy of the
plan, should be included.

One word more about the statistical analysis: Remember that in most
of the plans a complete set uf treatments is not tested on every farm.
Therefore, before you can make comparisons, you must equalize treatment
values to take into account the differences among farms. Do not make
the mistake of taking the degrees of freedom shown in Table 6 and pro-
ceeding as if the tests were simply designed as randomized blocks.











11/ Cochran and Cox, op. cit.








rabie 6. Notes for statistical analy,;is: Degrees of freedom for
the analysis of variance and the pertinent reference
in Cochran and Cox, Experimental Designs.

Abbreviations: T = No. of treatments
R = No. of repetitions of plan
L = No. of locations
I = No. of arrangements used
X = No. of times the selected number
of arrangements are repeated

Plan Degrees of freedom** Ref rence
lan in Cochran
Treatments Locations Error Total and Cox

A T-1 T(R 1) L-1 Sect. 4.1

B T-l L-1 (T-1)(L-1) TL-1 Sect. 4.2

C T -T(R-1) L Sect. 441*


Dl 2 3R-1 3R-2 6R-1 Sect. 11.54

D2 3 6R-1 6R-3 12R-1 Plan 11.1

D3 4 10R-1 10R-4 20R-1 Plan 11.2

D4 5 15R-1 15R-5 50R-1 Plan 11.3

D5 6 21R-1 21R-6 42R-1 Table 11.3

D6 7 28R-1 28R-7 56R-1 Plan 11.9

D7 8 36R-1 36R-8 72R-1 Table 11.3

D8 9 45R-1 45R-9 90R-1 Plan 11.14


See footnotes at end of table.










Table 6. Notes for statistical analysis: Degrees of freedom for
the analysis of variance and the pertinent reference
in Cochran and Cox, Experimental Designs--Continued.


Reference
Plan __ Degrees of freedom** in Cocran
Treatments Locations Error Toal and Cochran
Treatments Locations Error Total and Cox


3

4

5

6

8

9


4R-1

10R-1

10R-1

7R-1

12R-1

30R-1


5R-1

15R-1

7R-1

14R-1

18R-1

15R-1


4AX-1

5AX-1

27R-1

6AX-1


See footnotes at


8R-3

20R-4 '

20R-5

14R-6

24R-8

60R-9


15R-4

45R-5

21R-6

42R-7

54R-8

45R-9


12AX-15

20AX-24

54R-26

30AX-55


12R-1

50R-1

30R-1

21R-1

36R-1

90R-1


20R-1

60R-1

28R-1

56R-1

72R-1

60R-1


16AX-1

25AX-1

81R-1

36AX-1


Table 11.3

Table 11.5

Plan 11.4

Plan 11.7

Plan 10.1

Plan 11.15


Table 11.5

Plan 11.6

Plan 11.8

Plan 11.10

Plan 11.11

Plan 11.16


Plan 10.2

Plan 10.5

Sect. 10.4

Plan 10.7


end of table.









Table 6. Notes for statistical analysis: Degrees of freedom for
the analysis of variance and the pertinent reference
in Cochran and Cox, Experimental Designs--Continued.

Degrees of freedom** Reference
Plan in Cochran
Treatments Locations Error Total and Cox

H1 5 6R-1 6R-5 12R-1 Plan 11.1

H2 3 R-l 3R-3 4R-1 Sect. 5.1

H3 8 6R-1 12R-8 18R-1 Sect. 6.15

H4 8 R-l 8R-8 9R-1 Sect. 5.26

H5 15 12R-1 36R-15 48R-1 Plan 6.12

H6 6*** 2R-1 6R-6 8R-1 Plan 6.1

H7 7 R-l 7R-7 8R-1 Sect. 5.23

H8 26 3AX-1 24AX-26 27AX-1 Plan 6.7

H9 15 4AX-1 i2AX-15 16AX-1 Plan 6.4

H10 14*** 2R-1 14R-14 16R-1 Plan 6.2


* Analysis is made
treated plots.


on differences at each location between check and


** For designs that involve subdivision of a region (see Step 2, pages
21-26), locations should be partitioned into subdivisions and farms
in subdivisions, while the error should be partitioned into treat-
ments x subdivisions and treatments x farms in subdivisions.

*** One degree of freedom confounded with locations.










APPENDIX: THE PLANS


Plan A


2 or more separate treatments.
1 treatment and 1 plot on each farm.
A single repetition of the plan requires as many farms as there are
treatments and contains 1 replication.


Farm 1


Farm 2


. and so forth









Plan B


2 or more acparate treatments.
A block of all treatments on each farm.
A single repetition of the plan appears on each farm and contains 1
replication.











Block 1 Block 2 Block 3 Block 4

1
1 1 1 and so f

2 2 2 2 and so f rth.


etc.


etc.


etc.
iLL


etc.








Plan C


2 or more separate treatments, each compared with a check.
A block of 2 plots on each farm--one for a treatment and the other
for the heck.
A single repetition of the plan requires as many farms as there
are treatmentss and contains 1 replication.


Block 1
1
check


Block 2
2
check
7;-


. and so fcrth.









Plan D-l


3 separate treatments.
A block of 2 plots (treatments) on each farm.
A single repetition of the plan requires 3 farms and contains 2
replications.













Block 1 Block 2 Block
IF EI I -









Plan D-2


4 separate treatments.
A block of 2 plots (treatment) on each farm.
A single repetition of the plan requires 6 farms and contains 3
replications.














Block 1 Block 2 Block

1 1 1
2 3 4









Plan D-3


5 separate treatments.
A block of 2 plots (treatments) on each farm.
A single repetition of the plan requires 10 farms and contains 4
replications.


Block 2 Block 3

1 1

3 4


Block 4

1
5


Block 9

3

5


Block

2

5


Block 10

4

5


Block 1

1

2


Block 6

2

4


Block 7

2
5


Block 8


B








Plan D-4


6 separate treatments.
A block of 2 plots (treatments) on each farm.
A single repetition of the plan requires 15 farms and contains
5 replications.


7
2
4


12

3
6


8

2

5


9
2

6


13 14

4 4

5 6


10


4


15

5
6


Block 6

2
LI


Block 11


5









Plan D-5


7 separate treatments
A block of 2 plots (treatments) on each farm.
A single repetition of the plan requires 21 farms and contains
6 replications.









Block Block Block Block Block Block Block
1 2 3 4 5 6 7

1 1 1 1 1 1 2
2 .5 4 5 6 7 3


Block
8 9 10 11 12 13 14

2 2 2 2 5 3 5

4 5 6 7 4 5 6


Block
15 16 17 18 19 20 21

3 4 4 4 5 5 6

7 5 6 7 6 7 7








Plan D-6


8 treatments.
A block of 2 plots (treatments) on each farm.
A single repetition of the plan requires 28 farms and contains
7 replications.





Block Block Block Block Block Block Block
1 2 3 4 5 6 7



2 5 4 5 6 7 8


Block
8 9 10 11 12 13 14

2 2 2 2 2 2 3

5 4 5 6 7 8 4


Block
15 16 17 18 19 20 21

5 5 5 3 4 4 4

5 6 7 8 5 6 7


Block
22 23 24 25 26 27 28

4 5 5 5 6 6 7

8 6 7 8 7 8 8










Plan D-7


9 treatments.
A block of 2 plots (treatments) per farm.
A single repetition of the plan requires 36 locations and contains
8 replications.





Block Block Block Block Block Block Block Block Block
1 2 5 4 5 6 7 8 9

1 1 1 1 1 1 1 1 2

2 3 4 5 6 7 8 9 3


Block
10 11 12 13 14 15 16 17 18

a 2 2 2 2 2 3 3

4 5 6 7 8 9 4 5 6


Block
19 20 21 22 23 24 25 26 27

3 3 3 4 4 4 4 4 5

7 8 9 5 6 7 8 9 6


Block
28 29 50 31 32 33 34 35 36

5 5 5 6 6 6 7 7 8

7 8 9 7 8 9 8 9 9
-67 7









I Plan D-8


10 separate treatments.
A block of 2 plots (treatments) on each farm.
A single repetition of the plan requires 45 farms and contains
9 replications.


Block Block Block Block
1 2 3 4

1 1



B oc.
10 11 12 13



4 5 6

Block
19 20 21 22





Block
28 ,9 30 31


Block Block Block Block Block
5 6 7 3 9

11111

6 7 8 9 10


14 15 16 17 18



7 8 9 10 4


23 24 25 26 27




2 3 34 35 36

32 33 34 35 36


4 4 4 5 5 5 5 5 6

8 9 10 6 7 8 9 10 7

Block
37 38 39 40 41 42 43 44 45

6 6 6 7 7 7 8 8 9

8 9 10 8 9 10 9 10 10










Plan El


4 separate treatments.
A block of 5 plots (treatments)
A single repetition of the plan
3 ieplications.


on each farm.
requires 4 farms and contains


Block 2


Block 4


Block 1


Block 3










Plan E-2


5 separate treatments.
A block of 3 plots (treatments) on each farm.
A single repetition of the plan requires 10 farms and contains
6 replications.







Block 1 Block 2 Block 3 Block 4 Block 5

1 1 1 1 1

2 2 2 3 3

3 4 5 4 5



Block 6 Block 7 Block 8 Block 9 Block 10

1 2 2 2 5

S4 45

5 4 5 5 5








Plan E-3

6 separate treatments.
A block of 5 plots (treatments) on each farm.
A single repetition of the plan requires 10 farms and contains
5 replications







Block 1 Block 2 Block 3 Block 4 Block 5

1 1 1 1 1

2 2 3 5 4

5 6 4 6 5


Block 6 Block 7 Block 8 Block 9 Block 10

2 2 2 3 4

5 3 4 5 5
4 5 6 6 6









Plan E-4


7 separate treatments.
A block of 3 plots (treatments) on each farm.
A single repetition of the plan requires 7 farms and contains
3 replications.












Block Block Block Block Block Block Block
1 2 3 4 5 6 7

1 2 3 4 5 6 7

2 3 5 6 7

4 5 6 7 1 2 3









Plan E-5

9 separate treatments.
A block of 3 plots (treatments) on each farm.
A single repetition of the plan requires 12 farms and contains
4 replications.


Block Block Block Block
7 8 9 10

1 7 4 1

5 2 8 8

9 6 5 6
















46


Block
4

1

4

7


Block
5

2

5
8
Li


Block
6

3

6

9


Block Block
11 12

4 7

2 5

9 3


Block
1

1

2

3


Block
2

4

5
6


Block
3

7

8

9








Plan E-6


10 separate treatments.
A block of 3 plots (treatments) on each farm.
A single repetition of the plan requires 30 farms and contains
9 replications.





Block Block Block Block Block Block Block Block Block Block
1 2 3 4 5 6 7 8 9 10

1 1 1 1 1 1 1 1 1 2

2 2 3 4 5 6 7 8 9 3

3 4 5 6 7 8 9 10 10 6


Block
11 12 13 14 15 16 17 18 19 20

2 2 2 2 2 2 5 5 3 3

4 5 5 6 7 8 4 4 5 7
10 8 9 7 9 o10 7 8 6 10


Block
21 22 25 24 25 26 27 28 29 30

3 3 4 4 4 4 5 5 6 6

8 9 5 5 6 7 6 7 7 8
9 10 9 10 9 8 10 8 10 9








Plan F-1


5 separate treatments.
A block of 4 plots (treatments) on each farm.
A single repetition of the plan requires 5 farms and contains
4 replications.










Block 1 Block 2 Block 3 Block 4 Block 5

1 1 1 1 2

2 2 2 3 3

3 3 4 4 4

4 5 5 5 5
























48








Plan F-2


6 separate treatments.
A block of 4 plots (treatments) on each farm.
A single repetition of the plan requires 15 farms and contained
10 replications



Block 1 Block 2 Block 3 Block 4 Block 5

1 1 1 1 1

2 2 2 2 2

33 3 4 4

4 5 6 5 6

Block 6 7 8 9 10

1 1 1 1 1

2 5 5 4

5 4 4 5 5

6 6 6 6

Block 11 12 13 14 15

2 2 2 2 3

S3 4 4

4 4 5 5 5

56 6 6 6











49








Plan F-3


7 separate treatments.
A block of 4 plots (treatments) on each farm.
A single repetition of the plan requires 7 farms and contains
4 replications.










Block Block Block Block Block Block Block
1 2 3 4 5 6 7

1 1 1 1 2 2 3

2 2 3 4 3 4 5

3 5 4 6 4 5 6

6 7 5 7 7 6 7








Plan F-4

8 separate treatments.
A block of 4 plots (treatments) on each farm.
A single repetition of the plan requires 14 farms and contains
7 replications.








Block Block Block Block Block Block Block
1 2 3 4 5 6 7

1 5 1 3 1 3 1
2 6 2 4 2 4 3

3 7 5 7 7 5 5
4 8 6 8 8 6 7


Block
8 9 10 11 12 13 14

2 1 2 1 2 1 2

4- 3 4 4 3 4 3

6 6 5 5 6 6 5

8 8 7 8 7 7 8









Plan F-5


9 treatments.
A block of 4 plots (treatments) on each farm.
A single repetition of the plan requires 18 farms and contains
8 replications.



Block Block Block Block Block Block
1 2 3 4 5 6

1 1 1 1 1 1

2 2 2 3 4 3

3 5 7 5 6 6
4 6 8 7 8 9

Block
7 8 9 10 11 12

1 1 2 2 2 2

4 5 5 4 6 3

8 7 8 5 7 4

9 9 9 9 9 7

Block
13 14 15 16 17 18

2 3 4 3 3 4

5 5 6 4 6 5

6 8 7 5 7 7

8 9 9 6 8 8






52








Plan F-6


10 separate treatments.
A block of 4 plots (treatments) on each farm.
A single repetition of the plan requires 15 farms and contains
6 replications.


Block 1

1

2

3

4


Block 6

1

6

8

10


Block 11

3

5

9

10


7

7-

5

6

9


BlocK

1

3

7

8


8
I I

4

7
10


13

3

4

5

8


Block 4

1

4

9

10


9


Block 5


15

4

6

8

9







Plan G-1


16 treatments.
A block of 4 plots (treatments) on each farm.
A single repetition of the plan requires 4 farms and contains
1 replication.

Note that for each repetition 5 arrangements are possible. First
determine the number of repetitions you need (use Step 3, pages
15-17), and then get them in by using as many arrangements as
you can, repeating them as many times as you need to. For example--

1. For 15 repetitions, use 5 arrangements 3 times--not 3
arrangements 5 times.
*2. For 7 repetitions, use 4 arrangements 2 times--not 2
arrangements 4 times.
3. For 2 repetitions, use 2 arrangements 1 time--not 1
arrangement 2 times.


Arrangement 1
Block
1234
1 2 3 4


2 A 6 10 14
3 7 11 15
4 8 12 .161


Arrangement 4


Block
1 2 5


14 2 10

7 11 5
12 8 16
L.


Arrangement 2

Block
1 2 4
'i 2i35

6 7 8
9 I30 111 12

115 14 15 16


Arrangement 5

Block
1 2 4
1 9 i153 5

10 2 6 14

15 7 11
8 16. 12 4


4
9 !

61

15
4


Arrangement 5

Block
.1 2 4
i 5 9 13
r'-

6 2J 4i 10
11 15 53 7

16 il2 8 4
I-I-


* Note: Number of repetitions is raised to 8 because 7 is prime.








Plan G-2


25 separate treatments.
A block of 5 plots (treatments) on each farm.
A single repetition of the plan requires 5 farms and contains
1 replication.

Note that for each repetition 6 arrangements are possible. First
determine the number of repetitions you need (use Step 3, pages
15-17), and then get them in by using as many arrangements as you
can, repeating them as many times as you need to. For Example--

1. For 18 repetitions, use 6 arrangements 3 times--not 3


arrangements 6 times, or 2 arrangements
*2. For 7 repetitions, use 4 arrangements 2
arrangements 4 times.
3. For 2 repetitions, use 2 arrangements 1
arrangement 2 times.


9 times.
times--not 2

time--not 1


Ar agement 1


Arrangement 2


Arrangement 3


Block
1 2
1 2

6 71

11 12 1

16 17 1

21 22 2


4
4

9

14

19

24


Block
1 2
1 21 16

7 2 22

15 8

19 4 9

2 20 15


Arrangement 4


Arrangement 5


Arrangement 6


Block Block Block
1 2 5 4 5 1 2 3 4 5 1 2 3 4
1 16 6 2111 1 1 21 6 16 1 6 11 16 2

12 2 17 7 22 172 12 22 7 22 2 7 12 1

23 13 3 1 8 8 18 3 15 23 18 23 3 5 8 1.

9 24 14 4 19 24 9 19 4 14 14 19 24 4 5

20 10 25 15 5 15 25 10 20 5 1015 20 25 5

* Note: Number of repetitions is raised to 8 because 7 is prime.

55


Block
2 3
6 11

7 12

8 13

9 14

10 15









Plan G-3


27 separate treatments.
A block of 5 plots (treatments) on each farm.
A single repetition of the plan requires 27 farms and contains
5 replications.





Block Block Block Block Block Block Block Block Block
1 2 3 4 5 6 7 8 9

1 4 7 10 13 16 19 22 25

2 5 8 11 14 17 20 23 26

5 6 9 12 15 18 21 24 27


Block
10 11 12 15 14 15 16 17 18

1 2 ) 10 11 12 19 20 21

4 6 13 14 15 22 23 24

7 8 9 16 17 18 25 26 27


Block
19 20 21 22 25 24 25 26 27

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

19 20 21 22 25 24 25 26 27






Plan 0-4


36 separate treatments.
A block of 6 plots (treatments) on each farm.
A single repetition of the plan requires 6 farms and contains only
1 replication for each treatment.

Note that for each repetition 3 arrangements are possible, and that
each arrangement constitutes 1 repetition. First determine the
number of repetitions you need (use Step 3, pages 15-17), and then
get them in by using as many arrangements as you can, repeating
them as many times as you need to. For example--

1. For 12 repetitions, use 3 arrangements 4 times--not 2
arrangements 6 times.
*2. For 5 repetitions, use 3 arrangements 2 times--not 2
arrangements 3 times.
3. For 2 repetitions, use 2 arrangements 1 time--not 1
arrangement 2 times.


Arrangement 1


Arrangement 2


Block
1
1

7

135

19

25

51


5
5

11

17

23

29

55


Arrangement 3


Block
1
1

2



5
4 "



6



Block
1
1

8

15

22

29

36


* Note: Number of repetitions
is raised to 6 because 5
is prime.







Plan H-1


4 treatments: All possible combinations of 2 factors (a, b) at 2
levels (lower (1) and upper (2)). The lower level of a factor
may be the complete absence of it.
A block of 2 plots (treatments) on each farm.
A single repetition of the plan requires 6 farms and contains
3 replications.


Block
1


Block
2


Block
3


Block
4


albi al 1 alb, a2bl a2bl alb2

a2b alb2 a2b2 alb2 a2b2 a2b2
___^B Bb -. ** -___-_


Block
5


Block
6







Plan H-2


4 treatments: All possible combinations of 2 factors (a, b) at 2
levels (lower (1) and upper (2)). The lower level of a factor
may be the complete absence of it.
A block of treatments (4 plots) at each farm.
A single repetition of the plan requires 1 farm and, for main
effects, contains 2 replications.











Block 1



a2b1



ab
22







Plan H-3


9 treatments: All possible combinations of 2 factors (a, b) at 3
levels lowest (1), middle (2), and highest (3)). The lowest
1 1 of c factor may be the complete absence of it and may
corr, pond to a check treatment.
bock of 3 plots (treatments) on each farm.
Use at least 2 repetitions of this plan. A single repetition
requires 6 farms and, for main effects, contains 6 replications.


Block 1


Block 4

alb1
a2b

a b2


Block 2

alb2

a2b3
bl
V1



Block 5

alb2

a2bI
ab 3


Block 5

alb3

a2bI

aBb2



Block 6

alb3

a2b2

a3bl







Plan H-4


9 treatments: All possible combinations of 2 factors (a, b) at 3
levels (lowest (1), middle (2), and highest (3)). The lowest
level of a factor may be the complete absence of it and may
correspond to a check treatment.
A block of treatments (9 plots) on each farm.
Use at least 2 repetitions of this plan. A single repetition
requires 1 farm and, for main effects, contains 3 replications.









Block 1

ab
11
a2b1

a3bl

alb2
a2b2
a3b2

alb3

a2b3

ab
33






Plan H-5


16 treatments: All possible combinations of 2 factors (a, b) at 4
levels. The lowest level of a factor may be the complete absence
of it and may correspond to a check treatment.
A block of 4 plots (treatments) on each farm.
A single repetition of the plan requires 12 farms and, for main
effects, contains 12 replications.


Block 1 Block 2 Block 3 Block 4

a4b4 a4bB a4b2 a4b1

a3b3 a3b4 a3b1I a3b2
ab aby asb1 a b2
a2b1 a2b2 a2b3 a2b4

a2 1l a4


Block 5 6 7 8

a4b4 a4b1 a4b a4b2
a*b2 a3b a3bl a3b4

a2b3 a2b2 a2b4 a2b1

albl alb4 alb2 alb3


Block 9 10 11 12

a4b4 a4b2 a b3 a4bl
ab ab a b2 ab4

a2b2 a2b4 a2b1 a2b
alb3 ab alb4 alb2







Plan H-6


8 treatments: All possible combinations of 3 factors (a, b, c) at
2 levels (lower (1) and upper (2)). The lower level of a factor
may be the complete absence of it and may correspond to a check
treatment.
A block of 4 plots (treatments) on each farm.
Use at least 2 repetitions of this plan. A single repetition requires
2 farms and, for main effects, contains 4 replications.











Block 1 Block 2

a2blc1 a2b2cl

alb2C1 a2blc2

alb1c2 alb2c2
a2b2c2 alb1c1







Plan H-7


8 treatments: All possible combinations of 3 factors (a, b, c) at
2 levels (upper (1) and lower (2)). The lower level of a factor
may be the complete absence of it and may correspond to a check
treatment.
A block of all treatments (8 plots) on each farm.
Use at least 2 repetitions of this plan. A single repetition requires
1 farm and, for main effects, contains 4 replications.










Block 1

alblc1

a2blcl1

alb2c1

a2b2cl

aibico

a2blc2








Plan H-8


27 treatments: All possible combinations of 3 factors (a, b, c) at
3 levels lowestt (1), middle (2), and highest (3)). The lowest
level of a factor may be its complete absence and may correspond
to a check treatment.
A block of 9 (treatments) on each farm.
A single repetition of the plan requires 3 farms and, for main
effects, contains 9 replications.

Note that for each repetition 4 arrangements are possible and that
each arrangement constitutes 1 repetition. First determine the
number of repetitions you need (use Step 3, pages 15-17), and then
get them in by using as many arrangements as you can (use at least
the first 2), repeating them as many times as you need to. For
example--

1. For 7 repetitions, use 4 arrangements 2 times*--not 2
arrangements 4 times, or 1 arrangement 7 times.
2. For 3 repetitions, use 3 arrangements 1 time--not 1
arrangement 3 times.


Arrangement 1

Block 1 Block 2 Block 3


al bc1

alb2c2

alb 3C3


a2b2c3

a2bcl
a3blc3

a b2c1

a~b c2


alble
alb2C1

alb3c2
a2blc1

a2b2c2

abc3

a3blC2

ab2c,

a3b3c1


Arrangement 2

Block 1 Block 2 Block 3


al blC1

alb2c3

alb3c2

a2blce
a2b2c2

a2b c1

ajblc2


a3b 3c
33,!


alblc2
alb2c1

alb3C3
a2blc1

a2b2c3
a2b c2

a blc3

a b2c2
a b3cl


alble3
Sa1b2c2

alb3cl
a blc2

a5b2c1

a2b c

a3b1c1

a~b2c3

a3b3c2


* Note: Number of repetitions is raised to 8 because 7 is prime.




65








Plan H-8 -- Continued


Arrangement 3

Block 1 Block 2 Block 3


alb c1





a2b2c1

a2b3c2
a blC2

a b2c3
a3b3cl


alblc2

alb2c3
alb3c1

a2blcl

a2b2c2
a2b3c3

apblc3

a b2cl
a~b c2


alblc3

alb2cl
a1lbc2

a2blc2
a2b2c3

a2b cl
a3blcI

a b2c2
a3b3c3


Arrangement 4

Block 1 Block 2 Block 3


alblcI
alb2c3

alb3c2
a2blc2

a2b2c1

a2b9c,



a3b2c2
a3b3cl
ab,,


alblc2

alb2cl
alb,3C
a2blc3

a2b2c2

a2b3cI
a3blcl

a b2c3

a3b3c2


alblC3

alb2c2

albjci

a2blc1
a2b2c

a2b3c2
a3blc2

a3b2cl

a3b3c3








Plan H-9


16 treatments: All possible combinations of 4 factors (a, b, c, d)
at 2 levels (lower (1) and upper (2)). The lower level of a
factor may be its complete absence and may correspond to a check
treatment.
A block of 4 plots (treatments) on each farm.
A single repetition of the plan requires 4 farms and, for main
effects, contains 8 replications.

Note that for each repetition 6 arrangements are possible and that
each arrangement constitutes 1 repetition. First determine the
number of repetitions you need (use Jtep j, pages 15-17), and then
get them in by using as many arrangements as you can (use at least
the first 2), repeating them as many times as you need to. For
example--

1. For 11 repetitions, use 6 arrangements 2 times*--not 4
arrangements 5 times, 3 arrangements 4 times, 2 arrangements
6 times, or 1 arrangement 11 times.
2. For 4 repetitions, use 4 arrangement 1 time--not 1 arrange-
ment 4 times.
Arrangement 1


Block 1
alblcldl
a2blc2d2

alb2c2d2
a2b2c1dl


Block 2
alblc2d2
a2blcld1

alb2cld1

a2b2c2d2
|l~


Arrangement 2


alb1cldl
a2b2c2d1

a2b2cld2
albl2d2
a~bjjj
1gg~d


a2b2cldI
alblc2dl

alblcld2

ab cd
2222jdj


a2blcld1
alb2c2d
alb2cl d2

a 2blc2d2
222


a lb2c1dl
a~blc2dI
a2blcld2

alb2c2d2
1222


* Note: Number of repetitions is raised to 12 because 11 is prime.



67








Plan H-9 -- Continued


Arrangement 3


Block 2
a b2cld2

a2b1cldl
alb c2d1

a2b2c2d2


Block 3
alb2cldI
a2blcld2

alblc2d2

a2b2c2dI


Block 4
alb1cld2

a2 2Cldl
alb2c2dI

a2blc2d2


Arrangement 4


Block 2
alb2c2d1

a2b1cldl

alblcld2

a2b2c2d2


Block 3
a lb2cld ,
a2blc2dl

alb1c2d2

a2b2cld2


Arrangement 5


* Block 2
a2blc1d2

alb2cld1

alblc2d1
a2b2c2d2


Block 3
a2blcldI
alb2c1d2

alblc2d2

a2b2c2dI


Block 4
alblc2d1

a2b2cldl

alb2cld2

a2b1c2d2



Block 4
alblcld2
a2b2cld1

a2b1c2dl

a1b2c2d2


Arrangement 6


Block 1
alblcldI

a2b2c2d,

alb2c2d2

a2blcld2



Block 1
alblCld1

a2b2cld2

a2blc2d2
alb2c2d,



Block 1
alblcld1

a2b2c2d
a2b1c22

aib2cl 2


Block 3
a2blc1dl

alb2c2dl
alblc2d2

a2b2cld2


Block 4
alblc2d1

a2b2cldl
a2bc1ld2

alb2C2 2


Block 1
alb. cld

a2b2cld2
alb2c2d2

a2blc2dI


Block 2
a2blc2d1

alb2cldI

a1b1c1d2

a2b2c2Q2









Plan H-10


16 treatments: All possible combinations of 4 factors (a, b, c, d)
at 2 levels (lover (1) and upper (2)). The lower level of a factor
may be its complete absence and may correspond to a check treatment.
A block of 8 plots (treatments) on each farm.
A single repetition of the plan requires 2 farms and, for main
effects, contains 8 replications. Use at least 2 repetitions.


Block 1


a2blcldi

alb2cldl

alblc2d1
alblcld2

a2b2c2d 1

a2b2cld2
a2blc2d2
alb2c2d2
1222


Block 2


alblcldl

a2b2cld1

a2blc2d1

alb2c2d1

a2bl cld 2

alb2cld2


ab
alblc22
a2b2c2 2




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