• TABLE OF CONTENTS
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 Title Page
 Table of Contents
 Preface
 1.0 Introduction
 2.0 Subjective evaluation - visual...
 3.0 Pre-analysis of data
 4.0 Summarizing data
 5.0 Testing means
 6.0 Correlations and regressio...
 7.0 Improving precision
 8.0 Pooling of data
 9.0 Non-parametric tests
 10.0 Use of Chi Square
 11.0 Data interpretation
 12.0 Appendix A: Glossary
 13.0 Appendix B: Major References...
 Back Cover














Title: On-farm experimentation : evaluation of on-farm trials - statistical evaluation and interpretation
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Title: On-farm experimentation : evaluation of on-farm trials - statistical evaluation and interpretation
Physical Description: Book
Language: English
Creator: Stroud, Ann
Publisher: CIMMYT Eastern African Economics Programme
Publication Date: 1986
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Table of Contents
    Title Page
        Title Page
    Table of Contents
        Table of Contents
    Preface
        Preface
    1.0 Introduction
        Page 1
        Page 2
        Page 3
    2.0 Subjective evaluation - visual comparisons and further comments
        Page 4
    3.0 Pre-analysis of data
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
    4.0 Summarizing data
        Page 12
        Page 13
    5.0 Testing means
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
    6.0 Correlations and regressions
        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
    7.0 Improving precision
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
    8.0 Pooling of data
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
    9.0 Non-parametric tests
        Page 60
    10.0 Use of Chi Square
        Page 61
        Page 62
    11.0 Data interpretation
        Page 63
        Page 64
    12.0 Appendix A: Glossary
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
    13.0 Appendix B: Major References Used
        Page 73
        Page 74
        Page 75
    Back Cover
        Page 76
Full Text

/l,503


AN OCCASIONAL SERIES OF PAPERS AND NOTES ON METHODOLOGIES AND PROCEDURES
USEFUL IN FARM SYSTEMS RESEARCH AND IN THE ECONOMIC INTERPRETATION OF
AGRICULTURAL EXPERIMENTS.


NUMBER 12.


TEACHING NOTES
AUGUST, 1986.


ON-FARM EXPERIMENTATION


EVALUATION OF ON-FARM TRIALS -


STATISTICAL


EVALUATION & INTERPRETATION


Dr Ann Stroud












CIMMYT EASTERN AFRICAN ECONOMICS PROGRAMME
INTERNATIONAL MAIZE AND WHEAT IMPROVEMENT CENTRE (CIMMYT)
P.O. BOX 25171, NAIROBI, KENYA, TELEPHONES: 592054, 592206, 592151
Telex: 22040 ILRAD Cables: Cencimmyt, Nairobi.







TABLES OF CONTENTS


Subject Page Number


1.0 Introduction.............................................................. ..... I
1.1 Items to consider in evaluation............................................ 2
1.2 Uses of OFE results ......................................................... 3
2.0 Subjective evaluation visual comparisons and further comments ............,. 4
2.1 Farmer evaluations: 'farmer feedback'....................................... 4
2.2 Collecting observational data.......................................................... 4
2.3 Examples ................................................................... 4
3.0 Pre-analysis of data...................................................... 5
3.1 Scrutinizing data .......................................................... 5
3.2 Plotting data......................................................................................... 9
4.0 Summarizing data............................................................. 12
4.1 Frequency distribution........................ ............................ 12
4.2 Mean, mode, median...................... .................................. 12
4.3 Range, variance, standard deviation........................................ 13
5.0 Testing means................................................................. 14
5.1 T-test..................................................................... 14
5.1.1 Unequal sample size .................................................. 14
5.1.2 Paired samples............................... ....................... 15
5.2 Analysis of variance (AOV or ANOVA)......................................... 17
5.2.1 One way classification................................................ 18
1 Equal sample size.................................................... 18
2 Unequal sample size................................................. 19
5.2.2 Two. way classification.............................................. 19
1 One observation/treatment............................................ 19
2 More than one observation/treatment (multiple samples)............... 21
3 Two criteria for classification.. .................................. 22
4 Interaction......................................................... 24
5 Split plot ANOVA ........................................ ............. 25
5.3 Mean comparison methods.................................................... 26
5.3.1 T-test................................................................ 26
5.3.2 LSD....................................................... .......... 26
5.3.3 Duncan's Multiple Range Test......................................... 27
5.3.4 Orthogonal Comparisons.............................................. .. 28
6.0 Correlations and Regressions.................................................. 32
6.1 Introduction............................................................... 61
6.2 Correlation.................................... ........................... 35
6.3 Regression..................................................................... 38
6.3.1 Linear with one variable............................................. 38
6.3.2 An example with three variables. .................................... 42
6.4 Questionsraised to help researchers consider multivariables................. 46
7.0 Improving precision........................................................... 48
7.1 Covariance .............................................................. 48
7.2 Missing plots............................................................. 49
8.0 Pooling of data................................................................ 53
8.1 Pooling with multiple locations............................................ 53
8.2 Pooling over years............................................................ 59
9.0 Non-parametric tests........................ .................................. 60
10.0 Use of Chi Square............................................................ 61
10.1 Introduction...................................................*......... 61
10.2 Rules................... ................................*............... 61
13.3 Examples.......................... ....................*............ ..***....... 62
11.0 Data Interpretation.............................................................. 63
11.1 Significance level...............* ........................................ 63
11.2 Coefficient of variation (CV) ........................ ................... 64
12.0 Appendix A:
Glossary.....................................................................* 65
13.0 Appendix B:
Major References Used................................................... 73










PREFACE





The CIMMYT East African Program has been conducting a series of
Workshops in On-Farm Research (OFR/FSP) Methodology in the region.
Prior to the posting of CIMMYT Regional Agronomists to the Nairobi
Office, we sought assistance in conducting these Workshops from
agronomists in the region who are familiar with CIMMYT's approach
to On-Farm Research. Dr Ann Stroud, an agronomist currently working
with REDSO/USAID as Regional Weed Agronomist is one who participated
in many of our Workshops.

During these Workshops we realized that there was no single document
which provides a comprehensive coverage of the materials taught in
our Workshops. Therefore, we requested Dr Ann Stroud to put
together all materials she has presented in these Workshops into
training notes for easy reference by trainees. This three part series
of teaching notes on On-Farm Experimentation includes:
Number 11:- Concepts and Principles; Number 12:- Evaluation of
On Farm Trials; and Number 13:- Guidelines for using OFE Methodology
in Crops, Livestock and Agroforestry Experimentation. These notes
are meant to be used by teachers, planners, researchers and
extensionists who have had some previous background in agricultural
research. Although reference is consistently made to 'researchers'
in the series, it should be understood that 'extensionists' can also
be 'researchers'. This current version is a preliminary working
draft. It will be updated as we receive comments from the users.
We would therefore request constructive comments, criticisms and
suggestions from the users to update this document.








P. Anandajayasekeram
Regional Training Officer
CIMMYT Economics









1 .0 NI NR DLJUCT I ON

Acceptability of a new technology is not measured by a single
factor but by: input availability and costs, agronomic
performance, labor availability, .markets and variations in the
above." (Zandstra, 1979)

The point is that, unlike traditional agronomic experiments,
where statistical significance is used as the major evaluation
method, OFE uses many methods for evaluation. The statistical
methods are used in conjunction with economic evaluation methods
and sociological considerations. The farmer's criteria is used
as a guide in determining whether the technology is going to be
suitable. Farmer comments are one of the most critical pieces of
data one can collect. The table shows a research group's concep-
tion of different analysis techniques associated with F or
R management options.


Analytical tools for


OFE tvnes of management


STrial managed by:

Tools Researcher Researcher/ Farmer
S____ Farmer

1. Statistical analysis

i Simple mean comparisons *** *
Analysis of variance *** **
S Regression and factor
S analysis *
Biological risk
determination **

12. Economic analysis Is

S Partial budgeting *
S Simplified programming ***
Linear programming ***
S Economic risk determina--
tion (sensitivity) ** f ***

3. Social analysis

Household organisation
S (decision making,
division of labour) *
Community organisation ***
S Farmer assessment _____***


(CIMMYT, Socio-economics program 1985)









Both experimental (trial) data and non-experimental data (that
collected as observations of the area, a continuation of the
diagnostic survey, etc.) should be analyzed and integrated for
i interpretation. Non-experimental data can sometimes help verify
experimental data, uncover biases influencing implementation or
evaluation of the trial, plus can add to the researcher's know-
ledge of the area.

The following section describes statistical techniques useful in
interpreting OFE. These methods, similar to those used in the
diagnostic survey analysis, must be integrated with economic and
sociological methods of handling,.data. (The latter two methods are
discussed in other modules concerning the OFR process).

1.1 Items to consider in evaluation:

1. Technical performance and feasibility (Can it be done? Is it
system compatible?)

2. Economic performance and feasibility return to land, labor
and capital, risk considerations. Test for sensitivity if
appropri ate.

3. Post-harvest considerations processing, storage, taste
preferences.

4. Social acceptability compatibility with work habits, farmer
evaluation of trial.

5. Consider possible changes a farmer may have to make to-adopt
technology.

For example: in accepting a new variety, a farmer may have to use:

a. A higher plant population. If three times the normal
amount of seed requirement were needed, it may be too
expensive. Seed also may be food for the farmer. If this is
the case, a farmer may be more interested in yield per plant or
yield per kg of seed than yield per ha. If the population
increases, the yield/plant decreases groundnutss in Tanzania),
which when considering farmers objectives may be detrimental.

b. Fertilizer which increases yields may on the other hand be
costly or may.require more labor for applying or for increased
weeding.

Keep experimental objectives, local circumstances, original
diagnosis in mind when weighing various factors.










1.2 Use of OFE results


1. Re-d-fine RDs and TGs and improve understanding of the
farming system.

2. Re-define experimental objectives and experimental program.

3. Identify and address organizational problems.

4. Plan next years OFE and on-station experiments.

5. Consider transferability of information to other sites
(depends on the technology's vulnerability to environmental
interactions; on whether climatic or soil factors are
important technology modifiers; on management variability).

6. Make recommendations to farmers.



Figure 1. Below is a review of factors affecting
farmer adoption behaviours


Prof itabi I ity
Cost-Benefit








SysteQCompatibi i tv
Whether or not the
technology conflicts
with the farmer's
present technology


Risk
Chance of the
technology endangering
the farmers' subsistence
or income




Complexi ty
Whether a proposed
technology requires
farmers to have prior
training or experience.


Divisibility-
Degree to which the
technology can be adopted
in small increments


(CIMMYT)











2.0 SUBJECTIVE EVALUATION VISUAL COMPARISONS AND FARMER COMIINTS

2.1 Farmer's evaluations "farmer feedback"

1. Ask farmers their ideas concerning the experimental treatments.
throughout the implementation process. Consider such aspects
as: time it takes to implement the treatment; resources it
takes (labor, animals, tools) to implement; the ease in
performing the operation; how the farmer liked it or disliked
it and why; etc. Have him (her) compare it with the normal
practice..

2. Sometimes, farmers may be reluctant to admit deficiencies in
the technology.

In a W. Africa experiment program 70% of the farmers
responded correctly when asked to evaluate which variety
tested yielded the most. This was comparing their response
to researcher evaluation. It is therefore important to combine
subjective and objective methods of evaluation.

3. Researchers must follow-up and request farmer evaluation.
They should not assume that the job is accomplished; that it
is not their responsibility; it doesn't matter because their
contract is terminated; or fear that the results are not
"palatable".

4. If farmers have rejected the technology, determine the
reasons why. Seek possible ways to overcome rejection
or repeat in another season.

5. Continue monitoring impact of the technology. Do farmers
continue to accept it, adopt it, modify it?

2.2 Collecting observational data:

The researcher and/or technical assistants (preferably both)
should be continually 'informally' surveying the area throughout
the OFE process. Visual comparisons within the trial, between
farms, between areas and over time continue to help define the
system. Data collection can be structured so as to encourage
these observations. Remember, there are always aspects which get
missed in initial surveys.

2.3 Examples of how farmer evaluation modified technical treatments:

1. In a groundnut based system, planting patterns were modified
to fit farmer's objectives. Researchers had placed emphasis
on sorghum in an area that also grew groundnuts and had tried
a higher (than locally used) population of sorghum which







5


resulted in adverse effects to the groundnut. Farmer's sowed
early maturing sorghum and millet varieties late in an attempt
to increase sorghum densities without adverse effects to
groundnuts. Groundnuts is considered a priority crop. After
farmer feedback, researchers changed the emphasis from a
cereal to a cereal-legume system.

2. Alternative planting methods, flat vs ridged, were being
compared for rice planting in Nigeria. Experimental results
showed one-half of the sites had plots with increased yield
and one-half with decreased yield when flat planting was tried.
Flat planting was initially suggested by researchers because
ridge formation is labor intensive and delays rice planting.
This both decreases rice yields and prevents the sowing of a
follow-up crop. Using farmer feedback, some farmers felt
planting on the flat was unfeasible due to variable water
conditions or to increased work in weeding. Flat planting
success was related strongly to soil and moisture condition;
thus, researchers decided to re-classify the types of
'fadamas', or soils, using farmer criteria to see exactly
where the new technology would be feasible. This is a case of
redefining the RD due to farmer feedback.


3.0 PRE-ANALYSIS OF DATA

3.1 Scrutinizing data

1. This is when you look at the data for errors or odd values
not due to or caused by the treatments. Look at the data
immediately after collection to check if there are any
unusual e.g exceptionally high or low values. Such values
must be inspected and queried.for accuracy: is each one
plausible, improbable or impossible? Was the incorrect
number or observation recorded? Was there pest/disease
attack or flooding not related to the treatment? Was there
a very high yield in a plot due to over-manuring or due to
last year's cropping history? If zero yield is obtained, is
this due to failure related to the treatment or another cause?
This technique can also provide important clues to techno-
logical constraints such as: lack of a planting technique for
certain situations; fertilizer use in zero tillage plots,etc.

The following example illustrates data scrutiny:

Inspect data for gross errors and extreme observations
Reasons:
improper treatment
loss of sample
illogical data check and only remove data
if caused by an error












Site (farm)


Site
1
2
3
4
5
6
7
8
9
10
11


N1
400
200
.500
500
200
90C)
200
0
200

0
0


yields for 2 levels of N:

N,

250
800
450
400
1000
400
200
0
0
0


1) Grouping of responses into types:


Increased NM response:


Similar response:


400-800
500-800
200-400
200-400


Site
2 200-250
4 500-450
6 9(00-- 1000


2) Why O's? Loss of data? No yield? Try to explain
numbers before leave field.

2. Blocked data:

When blocking has been used data should be inserted in
the field layout and scrutinized to find out if blocking
was effective. The example below (with data inserted
in the field layout) shows clearly that blocking was
not effective: the left hand side of the field was high
yielding while the right hand side was low yielding.


Example:
Layout Bl oc
1 A B C D
400 300 150 : 100 1 I
SB A D C
S350 425 150 125 I II
I C B D I A
450 375 130 175 I III
A D C B
i 400 350 100 125 I IV
Totals 1600 1450 530 525
(letters refer to the treatments and numbers


ks


Total
950

1050

1130

975


to yields)


Site
1
3











In this particular example, there was a grazing animal
that ate the right half of the experiment, reducing yield.
This was not due to treatment. Hence, outside events
affecting treatment performance must be noted when analyzing
data.

3. Pooling data:

It is at this time when data pooling may be useful in order
to see trends in the data in reference to sites and/or over
years. The following example pooled data over sites:
plateau, upper, mid and lower slopes because it was expected
that variety performance would differ. Further division of
the data between low and high farmer management gives even
more information to the experimentor. It appears that all
varieties did best under high management; all varieties did
poorly in the plateau area and certain varieties did better
than others at the other altitudes. E35-1 appears to be'
most sensitive to altitude changes. The experimentor can
actually start to get 'a feel' for the interpretation of the
data at this stage.

Mean yields (kg/ha) of improved and local sorghums by position
along the toposequence at two levels of management in level-5
farmers' tests, Nakomtenga and Nabitenga, 1981. (Matlon, 1985)


Low Management High Management

Varieties E35-1 38-3 CSH5 Local E35-1 38-3 CSH5 Local


Plateau
Mean yield (kg/ha) 318 144 189 185 813 273
Observations 0 1 1 1 0 1 1 1

Upper sl ope-
Mean yield (kg/ha) 268 305 773 605 966 1048 1256 1102
Standard deviation 286 395. 377 473 668 693 480 553
Observations 8 7 9 12 8 7 9 12

Mid slope
Mean yield (kg/ha) 685 311 537 626 1405 915 1369 1197
Standard deviation 609 376 374 459 763 362 583 454
Obervations 17 16 15 24 17 16 15 14

Lower slope
Mean yield (kg/ha) 810 516 602 606 1389 1106 1202 1150
Standard deviation 645 655 313 525 1162 799 1033 588
Observations 4 6 4 7 4 6 4 7













This technique can greatly assist in data interpretation.
Site selection techniques have been previously suggested
(OFE Concepts and Principles, 3.1). The data collection (3.5)
section suggests site specification data which must be
collected if post-stratification of data is used.

This technique is powerful when combined with data on labor
use and factor returns to help elucidate adoption patterns.

4. Types of errors to watch out for which may give unusual data:

a. Competition effects not considered in_ layout:
Competition effects should be kept at the same level to
ensure that measurement of plant response really represents
the condition being tested. Avoid this by having adequate
border row areas when needed (see No. 11 Concepts and
Principles Border rows). Missing plants or hills within
a plot can also cause problems in respect to uneven
competition. This may be difficult to avoid. One can use
covariance to correct for uneven stand or one can exclude
plants surrounding the missing plant area from the harvest
sample.

b. Row spacing errors:
Avoid error by careful planting. If it occurs, harvest
only the area where spacing is correct or use covariance
to adjust yield if whole plot is harvested.

c. Uneven plant size:
If transplanting, try to ensure a uniform selection of
plants throughout. There may be a tendency to select good
ones first and poor ones later. Avoid this practice.
Thinning errors may have occurred either in number removed
or uneven sizes being removed. Manipulate thinning to
avoid making gaps worse.

d. Uneven fertilizer application:
Make sure application equipment.is functioning properly.
Try subdividing a large area into smaller areas. It is
easier to apply smaller batches uniformly than one large
area, if done manually. Match measuring container amount
to a given area.

e. Mistakes in plot layout and labellinq:
Plot measurement errors may occur. Double check when
laying out the experiment. Mislabeling may go undetected
until the end of the experiment. Misapplication of the
treatment may also occur. Counter-check using two people.











f. Measurement errors:
Boredom, carelessness or miscommunication can result in
misread, misheard, or miscopied data. Review data daily
to look for problems. Keep plot samples separate until
data is checked over. Plant height may be measured in
different ways. Make sure it is standardized between data
collectors if more than one is used.

g. Mistakes in transcribing data:
Try to set out the notebook in the form you wish data to
be collected. This will minimize the necessity for'
transribing. Design data collection notebook so that
statistical analysis can be done directly. (Gomez and
Gomez, 1984).


3.2 Plotting data

1. It is helpful to plot data in advance to see what sort of
relationships and trends you can hope to detect using a more
rigorous analysis later. Plotting data also assists the
researcher in identifying previously missed 'problematic'
data or unexplainable figures.

2. Compare such items as:

a. Yield vs a management factor such as planting date, number
of weedings (including farmer's check).

b. Yield vs. a climatic factor such as rainfall during growing
season or part of the growing season.

c. Accumulated rainfall vs time of:planting or land preparation.

d. Yield vs a soil factor.

e. Yield response'to farmer's levels of inputs.

f. To compare proposed agronomic performance vs actual
performance under different circumstances. This can give an
indication of adaptability.

3. Plotting techniques include: line graphs, bar graphs and
scatter diagrams.

4. Examples:

a. Plot variety yield data vs. a management factor to see how
the treatment was affected.












S" Variety 'A'


Yield
Yied Farmer's Variety




0 1 2 3
No of weeding


This shows that variety 'A' responds more to improved manage-
ment than farmer's variety. Look at % or number of farmers
with different management levels. The result may be different
recommendations to different management groups.





or Farmer's variety



Yield
SVariety 'A'




Rainfall (mm)


This tells you:
1. At low rainfall variety 'A' better
2. At high rainfall local variety better
3. Over range of rainfall variety 'A' is more stable

b. Correlation between site indicator (environmental parameter)
such as: elevation, rainfall, soil type (use numerical coding,
e.g. % clay) vs yield; or management. (e.g. number of weedings)
vs. yield.

Doesn't tell you magnitude of differences
Check data and use common sense to divide it up
Consider a number of factors such as: there may be
responses to 2 or more factors, e.g. elevation and rainfall










- If environment is extremely varied, then need a number of
RD's (recommendation domains). However, must consider
limitations to working in too many RD's.


Yield
variety
'A'


% clay

c. Compare local vs introduced treatment


7 yield
of variety
'A' compared
to local
check


120%-----------------------B

C

100%------------------------(same as local
check)


I ____



0%

S1 2 3
Mean site yield


Assume that higher yield per site represents better management
or environment

Case A = Variety 'A' is responding more than control under
better management. (The point is you cant use averages
as % yield increases if management is different.)

Case B = Variety 'A' responds better at all management levels

Case C = Variety 'A' does worse than the local variety under
low management but better at high management.










4.0 SUMMARIZING DATA:

4.1 Frequency distribution:

This technique shows how the data is distributed but not how
dispersed the data is. Histogram type graphs are used to
illustrate frequency distribution visually.

FreQuenc_ = the number of times an observations) occurss.
Absolute = frequency the value occurs in absolute terms.
Relative = expresses frequency of the value relative to
the total

To construct a frequency table and histogram, first decide on
the number of classes according to the number of observations.
As a guide:

Number of Observations Number of Classes

20 100 8 12
101 500 10 15
501 1000 12 18

Determine the upper' and lower limits of each class and avoid
.classes with less than three observations. Usually the class
divisions are equally dispersed. Count up the number of
observations in each class. These can be plotted in a table
or a histogram graph.

4.2 Mean, Mode, Median:

These three techniques are different ways to measure the
central tendency of the data.

+ Mode = most commonly occurring value; value occurring with
greatest frequency; rarely used. It may not exist or may
not be unique.

Median = middle value of the sample; divides observations into
two equal parts. A few extreme values have little or
no effect. Use when data is skewed.

+ Mean = average value from a set of data: X = -Ex: this is the
n
most commonly used but is sensitive to extreme values.

X = mean
AX = Pum of observation values
n = number of observations










4.3 Range. Variance and Standard deviation:


These techniques are all different ways to measure data
dispersion. Two means from two groups of data may be the same
but the data dispersion of each group may be different. There-
fore, these methods are used to ascertain this.

* Rane = difference between largest and smallest value


* Variance = measure of variability S" =x=_ (Qx")
n
n-1

* Standard deviation = measure of variability, that is, the
extent to which a single value varies about the mean.
Measures the scatter of the data in terms of differences from
the mean value IS-.

+ gives an idea of expected performance.

gives idea of risk.


Variance heterogeneity can be determined by
variance using a scatter diagram:


plotting mean vs


1)





Mean

1) homogenous variance


2) 4r

VCL r. o
I *

Mean


t

I M
Mean


2) heterogeneous variance where variance is
to the mean.


proportional


3) heterogeneous variance without any functional relationship
between the variance and mean.

If the latter two are the case, data transformation should be
used (2) and error partitioning for variances (3). (See Gomez and
Gomez, 1984)









5.0 TESTING OF MEANS

5.1 T-test

The t-test can be used anytime you are comparing two means. This
can be from a simple paired sample where you compare the
differences between the two means in each paired sample; it can
be used where there are two samples of unequal size (or observa-
tions) where the means of each sample are compared; it can be
used after the ANOVA when two means are being compared. In the
latter case an example would be in a split plot:

comparison between two main plot treatment means averaged
over all subplot treatments.

comparison between two subplot means averaged over all main
plot treatments.

and others.

5.1.1 Unequal sample size

This test is used because it is more unusual and useful
in OFE where an unequal number of observations may be
collected more often than an equal number.

1. "t" test assumes that variances of'two samples are equal.

2. Can use if n1 does not = no (n= sample numbers) but
r "= Tw22 (r= variance).

3. To test for equal variance:

a. Use a two-sided F-test if there is no prior reason to
anticipate inequality of variance. Test criterion
is F = SIR/S22, where S11 is larger of the two.

(S = sample variance)

Compare calculated F valuewith table F value. Use df
from sample (n). If the F table value is smaller than
the calculated F value, then reject the null hypothesis
(G0 does not equal T2) meaning variances are equal.

b. Use a one-side F test if you know in advance which
population has a higher variance.

c. .Reasons for suspecting unequality are: non-random
sampling chance; not actual calculated value amount, as
variances can be close but unequal.











Examp le:

Where n' does not = n2 butG:=O-ml
is assumed.


Sample A
44
19
31
13
34
23
15
29
208
26


Sample B
18
16
32
14
20


100
20


*S2= (x- )n-
(n-l)


n = 8

*S2= 110


**pooled S= X1a (t)L +(Xmj a-
(nA-l)+(nB-l)


= 970 = 88. 10
11


SD2 =(pooled S-) (nA+nB) = 88.18
nAnB


(8+5) = 28.66
40


SD = 28.66 = 5.35

calc t = (XA XB) = (26-20)= 1.12
SD 5.35


"t" table value = 2.20 for P =.05; therefore XA
differ from XB (Use nA + nB-2) as df for table


does not
val. ue.


5.1.2 Paired samples:


To use this test, you must have a reason for pairing
n1 = n.


and


Control (C)
32
132
101
191
55


Treated (T)
16
40
35
24
76


D
16 sample size (n) = 5
92 sample mean (d) =64
66 sample standard
167 deviation (S) = 72.3
-21


**Pooled S2


Site
1
2
3
4
5











*D = difference (C-T)
C4lculate t = = 64
S/ TT 72.3/7


= 1.98


Compare to 't' table value. If the calculated t value lies


between + or 't'
between groups.


Table of t-" for
0.100 0.050 0.025 0.010
Significance level ---->


df (n-1)

1
2
3
4
5
6
7
8
9
10

11
12
13
14
15
16
17
18
19
20

21
22
23
24
25
26
27
28
29
30


100
00O


3.078
1.886
1.638
1.533
1. 476
1.440
1.415
1.397
1.383
1.372

1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325

1. 323
1.321
1.319
1.318
1.316
1.315
1.314
1.313
1.311
1.310


6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812

1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725

1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.697


12.71
4.303
3. 182
2.776
2.571
2.447
2.365
2.306
2.262
2.228

2.201
2. 1.79
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086

2.080
2.074
2.069
2.064
2.060
2.056
2.052
2.048
2.045
2.042


table value, then there is no difference


the t-tests.
0.005 0.0025 0.001 0.0005


31.82
6.965
4.451
3.747
3.365
3.143
2.998
2.896
2.821
2.764

2.718
2.881
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528

2.518
2,508
2.500
2.492
2.485
2.479
2.473
2.467
2.462
2.457


63.66
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169

3. 106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845

2.831
2.819
2.807
2.797
2.787
2.779
2.771
2.763
2.756
2.750


127.3
14.09
7.453
5.598
4.773
4.317
4.029
3.883
3.690
3.581

3.497.
4.428
3.372
3.326
3.826
3.252
3.222
3.197
3.174
3.153

3.135
3.119
3.104
3.091
3.078
3.067
3.057
3.047
3.038
3.030


318.3
22.33
10.21
7.173
5.893
5.208
4.785
4.501
4.297
4.144

4.025
3.930
3.852
3.787
3.733
3.686
3. 646
3.610
3.579
3.552

3.527
3. 505
3.485
3.467
3.450
3.435
3.421
3.408
3.396
3.383


636.6
31.60
12.92
8.610
6.869
5.959
5.408
5.041
4.781
4.587

4.437
4.318
4.221
4.140
4.074
4.015
3.965
3.922
3.883
3.850

3.819
3.792
3.768
3.745
3.725
3.707
3.690
3.674
3.659
3.646


1.299 1.676 2.009 2.403 2.678 2.937 3.261 3.496

1.290 1.660 1.948 2.364 2.626 2.871 3.174 3.390
1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291












5.2. Analysis of variance (AOV or ANDVA)

1. This technique is used when the number of samples exceeds
two.' ie. the number of means being compared is greater
than two.

2. Provides a statistical test, "F-test", which is a single
test of the null hypothesis.

3. Mean separation techniques can proceed the AOV to see how
treatments differ.

4. When the division or partitioning of means is only by
treatments which are replicated but not blocked such as in
a Completely Randomized Design (CRD), the ANOVA can be used
to ascertain differences between treatments. (This is called
a one-way classification of means). More complicated designs
such as Randomized Complete Block (RBCD); split-plot;
factorial arrangements, regression and covariance are
partitioning the variation into two or more divisions. These
are called two (or more) way classification. (For design
choice see No.11 Concepts and Principles Designs).

5. This technique is useful for more complicated statistical
designs or analysis of many factors and their interactions
(see OFE Concepts and Principles 3.4.6 for further
information).

6. The AOV should be relatively straight forward to most
agronomists and examples are present in many statistics
textbooks. Therefore, the following examples selected are
to give you more unusual, but potentially useful "workbook"
solutions to different ways to use AOV in OFE.











5.2.1 One wy classic ficati on

I. EquaL. sam~a !_sizte. This is used with a CRD (Completely
Randomized Design). The treatments are replicated but
not Lblocked. Sources of variation are partitioned by
treatment only.



A D B C

C _D B C
A B C A


Treat ment


A
B
C
D
13 --


Rep i c at ions


47
50
57
54
4L


df


Total
Treats
Error


52
54
53
65
4

SS

854
208
646


C = (JX)J = (928)-= 53824 r
rn 4(4)

Treatment SS = (T Totals)"


= replication number
n = treatment number

/ r = (212)2 + (228)"+.... -C =
4


54032-53824=208

Total SS = (X)- -C = (47)" +(50)" +..(59 ) C C =
54678-53824=854

Error SS = Tot SS-Tmt SS= 854-208 = 646

MS = SS
df
F observed = MST/MSE

To'read F table value:
MST across top, use associated df
MSE down left side, use associated df
If observed value is'less than table value, then accept null
hypothesis and conclude there are no real differences. It may
be that real treatment differences exist but the experiment


Total


51
57
57
59
4.


212
228
236
252
=928


2L

53
57
59
63
X=58


MS


69.3
53.8


F


1.29


-- --


-z- ..


111-_


-I---.-









was not sensitive enough to detect them at the desired level
of probability.

2. Uneual samp le size.

The following changes should be made when using the above
procedure for equal sample sizes:


1. To calculate C = (-X)2 nl+n2+n3... = number of values
nl+n2+n3.. or samples collected

2. To c-lculate Treatment SS:


(sum values of column 1)" + (sum values column 2)" +...- C= SS
number values in column 1 number values in column 2

3. To calculate Total SS = .(T totals")-C. (Square each
value and add together -C).

4. Error SS = Tot SS Tmt SS

5.2.2 Two criteria of classification Randomized Complete Blocks:_

This is the most used design using treatments as one classification
and blocks as a second. RCBD is a treatment arrangement to help
control error. Replications of treatments are arranged in blocks
according to field variability. (See No.11 Concepts and
Principles section on blocking).

1. One observation/treatment/block:

This is a RCBD. Variation is partitioned by blocks and
treatments (varieties).

Example:
Single croB data for forage
(Data are tons/acre @ 12% DM)

Blocks

Variety I II III IV V Sum X

1. 1.58 1.63 1.46 1.98 1.47 8.12 1.62
2 1.91 1.88 1.78 2.08 1.91 9.56 1.91
3 1.67 1.67 1.81 2.00 1.88 9.03 1.81
4- 1.74 1.60 1.55 1.70 1.52 .8.11 1.62
5 1.81 1.88 1.91 2.24 1.89 9.73 1.95
6 1.52 1.75 1.55 1.88 1.55 8.25 1.65
Sum 10.23 10.41 10.06 11.88 10.22 52.80 1.76












C = (52.80)" / 30 =94. 92804


Total SS =

Var. SS =

Block SS =





Source


(1.582 +-.....+1.552) C

(8.122 +....+ 8.252)/5 C

(10.23" +....+ 10.224)/6 C


ANOV Table


D.F.


S.S


M.RS.


Total
Varieties
Blocks
Error


1.1030
.5472
.3732
.1824


.1094
.0933
.0091


12.02**


In this case varieties are significantly different in terms
of yield because the calculated F value is greater than the
table F value. How are the varieties different? Use mean
comparison methods as the next step (5.3).


_1_11 1______11_ ____1___1___~____1 ___ ___










2. More than one o bservation/treatment/bloc k


Here variety is the
were taken.


main treatment plot from which 6 samples


Quandrant cLunts o-f foraage plants le f (pants/Sg.t.)

Regli cati on


64


2
1
C-)
C)
0

4
4
4
4
3
6
2
5
24

4
3
5
4
5
5
26


3
2
4
3
1
5.
18

5
5
3
4
2
2
21

4
5
3
3
3
2
20

4
3
2
3
2
3
17
76


C = 395"/144 = 1083.51
Total SS = 1433.00 C = 349.19
Var.SS 41,769.00/36 C = 76.74
Rep.SS'= 26,303.00/24 C = 12.45
V x R SS = 7533.00/6 Var. Rep.-
Error = 177.50


C = 82.80


Variety


III


2
1
0
0
1
0
4

5
4
5

-3J




4
3
4
2
3
_4
20


1
4
1
4
5
2
17
-= 71


0
.
4
3
5'
3
18

4
3
4
0
6
4
21

3'
3
1
2
5
31
17


3
3
2
5
4
5
22
71


0
1
- 1
1
1

7

2
3
4
5
5
5
24

2
2
2-
5
2
CJ-.
14
14


C)
0

1
1
0
-0
2_
58


1
3 .
4
1
1
3
13

2

5
4
2
3 "
19

2
3
3
2
2
2
14


1
0
5
3
3
2
14
59


86
395 = Grand
Total


111


2
2
2
2
3
3
14
60











8ot---tr-c H-I-:-.1-- -~-. F


Total
Var.
Rep.
VX R
*Sampling Error
(within plots)


143
3
5
15

120


.349.19
76.74
12.45-
82.80

177.50


25.58
2. 49
5.52

1. 48


- Pooled estimate of error for 24 experimental units x 5
sample units

3. Two criteria for classification:

This two-way classification is used when treatments are
distinctive from one another e.g. fertilizer and variety.
The researcher is interested in both main effects, that is, how
does the fertilizer act and how do varieties act, and the
interaction of the main effects, do varieties respond
differently to a given level of fertilizer.

Examples Two pesticides were tested on beans, a fungicide
at two levels Fo, Fi and an insecticide at three
levels Io, 1, I, Variation is partitioned
into blocks, two treatment components fungicide .and insecticide
and their interaction.

Data: Blocks


Treatment
Fo lo
Fo II
Fo 12
Fl lo
F1 Io
F1 II
F1 12


III
71
55
58
92
97
94


mtmts
341
290
244
446
459
460


4Funa


Fo 875


Fl 1365


blocks = 427

Insect = Io =
II =
12

Grand = 2240


416


467 466


464


787 = (341+446)
749 = (290+459)
704 = (244+460)

nt q No. tmts
nr = No. blocks


--r- ~_._


Source


D. F. S.S.


F





23


C =(2240):" /30 = 167253.33

SS blocks =(X blocks)"/ nt -C = (427)_ + -(416)_"- + ... -C = 401
6
SS tmts =-(X tmnts)" / nb -C = (341)"- + (290)O)"....-C = 8969.47
5
SS total =-(X2) -C
SS error = SS total SS blocks SS tmts

To find SS for Fung. and Insect. separately, set up a Table with
comparison coefficients:

Treatments + Treatments totals


Compari son
F levels
I levels (0 vs 1,2)
(1 vs 2)


Folo FoIl
341 290


FoI2 Filo FIll
244 446 459
1 -1 -1


F112
460
-1


2 -1 -1
0 -1 1


SS F = EC(341)+1(290)+1(244) 1) 1( 1(459) 1(460):1=
E(!)- + (1)2 + (1)M + (-+ 1) + (-11) + (-1).'36

SS Error = SS tot SS tmt SS block

SS F x I = SS tmt SS F SS I

SS I E2 (341)-1 (290)-1(244)+2 (446)-1(45)- (460)-1(290)+1 (244)
-1(459)+1 (460)3
E (2)" =-+ (-1) "( + (1) "-+ (2)"+ (-1) -+ (1)"-+ (-1)"-+i (1) !"+ (-1 ) + (1I) ] 12


Source
Total
Blocks
Tmt
Fungicide (F)
Insecticide (I)
F x I
Error


SS
10140.67
401.00
8969.47
8003.33
345.27
620.87
770.20


MS

100.25
1793.89
8003.33
172.64
310.44
38.51


F



207. 82 s**
4.48
8.05 *


The fungicide treatment had means that were significantly
different. The interaction between I and F was also significant
(see below for interpretation.) Refer back to the data to
determine'why or how treatments were different.









4. An interaction when significant means that one factor responds
differently than another to levels of a second factor which
was mutually imposed.

a. For example: 2 varieties oere grown under 3 nitrogen levels.
One variety gave a lower response than the other to the lower
N level.




Variety B

Yield Variety A


N levels

If plotted, the responses are not parallel. If the interaction
was not significant, then the response lines would be parallel
meaning that the varieties responded the same to all levels
of N.

b. A second example from a fertilizer experiment:

The nature of the interaction of N x K can be observed by
graphing out the 4 mean values for fertilizer treatments.





90 It is evident that there is an
interaction of N and K. The
80 I High K effect of increasing the amount
of N varies differentially with
70 the amount of K present.
LowLow K
60

Low High









5. Split olot (ANOVA)


Source
Total


df
10'7


Reps 3
Mgmt 2
Rex-' Ma.mt 6
Var 8
Var x Mgmt 16
Var x Rep 24 Y
Var x Rep x Mgmt48 3


Error (a)


Error (b)


To calculate df
(Add all dfs together)
(4-1)
(3-1)
(Re df x liqmt df)
(9-1)
(Var dT x Mgmt df)
(Var df x Rep df )
(Var df x Rep df x Mgmt df)


Reps 4
Mgmt practices 3 (main plot)
Varieties 9 (sub plot)

.Example main plot =


1. C = (X) =
r(MP) (SP)


r = number replications
MP= number main plots
SP= number sub plots


2. SS total = X( X" ) -C

3. SS blocks = -(T blocks)" -C
(MP) (SP)

4. SSMP = n (T main plots)_ C
(r) (SP)

5. Error (a) = Q(SP by block)=
SF

6. SSSP = (T sub plots)~ C
(r) (MP)-

7. SSMP x SP = (Block by SPF)
r


- C SS blocks SSMP


- C SSSP SSMP


8. Error (b) = SS total SSMP SSSP SSMP x SP


11"1-"-"











5.3 Mean Comparison Methods

The "F" test only indicates whether significant differences are
present. A mean comparison method is needed to identify how the
means are ranked.

It is important when reporting results of a series of comparisons,
to give the sizes of the differences, with accompanying standard
errors or confidence limits.

5.3.1 T-test: as described earlier, this test can be used to
compare two means.

5.3.2 LSD Least significant difference


Example:


Grams of Fat Absorbed per Feeding


Fat Animal 1 2 3 4
Sample:r

1 64 78 75 55
2 72 91 93 66
3 68 97 78 49
4 77 82 71 64
5 56 85 63 70
6 95 77 76 68

EX 432 510 456 372

X 72 85 76 62

1(X2) 31,994 43,652 35,144 23,402

( X)2/n 31,104 43,350 34,656 23,064


X
d.f


890
5


302
S5


488
5


388
5


Total


272
322
292
294
274
361

1,770 = GT

295

134,192

132,174


2,018
20


Pooled s" = 2,018/20 = 100.9

s- = (2s2 /n) = (2)(100.9)/6


Standard error = -5.80; 20 df


5% value of "t" with 20 di = 2.086


= 5.80









The difference between a specific pair of means is significant at
5% level if it exceeds (2.086) (5.8) = 12.1

LSD = t2-
< n
12.1 serves as the criterion for LSD. The value is significantly
different from the other pairs of means value if the differences
exceeds 12.1. 85 76 72 62



Thus 85 is different significantly from 62 but none of the other
treatments are significantly different.

The LSD fits most of these conditions but it does not make all
mean separations at the same exact level of significance. If
used for all possible treatment comparisons, too many significant
differences show up. A number of modifications have been proposed
(Duncan's Multiple-Range Test is best known) that improve its
accuracy for all possible comparisons.

5.3.3 Duncan's Multiole Ranae Techniaue to Senarate Ranked Means:


Example:

Position or
Rank


X
Value


Means covered by the same
line or followed by the same
letter are not sionificantlv different


6.79
6.75
6.66
6.64
6.53
6.53
6.50
6.44
6.43
6.20
6.17
6.17
6.13
5.98
5.98
5.43


Thus treatments 1,2,3 are
14,15,16 only.


significantly different from treatments


The means are first ranked in descending order. Table values are
then consulted for factors to multiply the LSD by to obtain a more
accurate significant difference. Required differences increase


-----------^------ --- ~-- -:-~-L-;-~- ---~~-~-~~^----' --~~~~~ -~~-









depending on distances in the ranked order.

a) Calculate LSD

b) Calculate SSD (Significant Studentized Factors) SSD = R(LSD)
R is from Table of SSD.

c) Start by comparing largest mean with the smallest using SSD
for their positions relative to each other (ie. if position
(p) =4, SSD=4.8). If the difference between the means equals
or is larger than the SSD, then the means are significantly
different.

This test is similar to LSD but requires increasingly larger values
for significance between means as they are more widely separated
in the array. This test is most appropriately used when several
unrelated treatments are included in an experiment, e.g. for making
all possible comparisons among variety yields.

There are many other means separation tests. It is important to
know the strengths and weaknesses of each before selecting which
to use.

Tukey's multiple range test can also be used. Both Duncan's
and Tukey's require larger differences between means than the
LSD for significance.


5.3.4 Orthogonal compare sons

This technique is used if quantitative differences are expected
between treatment means. That is, do not use when the treatment
choice is sampling a range of responses over a gradient
(fertilizer response curve), which would be qualitative. Use
when distinct differences are wanted between sets of treatments
or Between two treatments. If a qualitative difference is
expected use LSD or DMRT.

By partitioning the degrees of freedom and sums of squares for
treatment effects into meaningful single df and associated SS,
comparisons between treatment means can be made. Skillfully
selected treatments can answer as many independent questions as
there are single df. When the comparisons are independent, they
are said to be orthogonal, a desirable characteristic, as the
comparisons then lead to clear-cut probability statements.
If this method of analysis is used, treatments should be selected
accordingly. Treatments must be orthogonal, which means that they
contribute independent parts to the SS.

a) To check whether treatment comparisons are orthogonal,
set up a table.










Example: In the following experiment, lambs of two sexes
(m and f) were implanted with a harmone vs. not.

Treatments

Compari son FHo FH. MHo MHi


I Implanted -1 +1 -i +1
I Sex -1. -1 1 +1
I S +1 -1 -1 +1


To assign coefficients -

1. If two groups of equal size are to be compared, simply assign
coefficients of +1 to the members of one group and -1 to those
of the other group. It is immaterial which group is assigned
the positive coefficients.

2. In comparing groups containing different numbers of treatments,
assign to the first group, coefficients equal to the number of
treatments in the second group, and to the second group,
coefficients of the opposite sign equal to the number of treat-
ments in the first group. Thus, if among five treatments, the
first two are to be compared to the last three, the coefficients
would be +3, +3, -2, -2, -2.

3. Reduce coefficents to the smallest possible integers. For'
example, in comparing a group of two treatments with a group
of four, by rule 2, we have coefficients +4, +4, -2, -2, -2,,
-2, but these can be reduced to +2,+2,-1,-1,-1,-1.

4. -Interaction coefficients can be found by multiplying the
'coefficients of the main effects.









b. Another example follows:


Treatments df

Total 5

Compari sons:
No N vs N 1

Organic vs
inoganic 1

Ammonium vs
nitrate 1


No N


-5

I O


I O
0


0


CO (NHa) 4 (NH) .aSDO4 NH.I4NO


NaNDO Ca(lNOC) )


+1 0


+1 0


+1 0


0 0


(NH4) sSOD
vs NHaNDOa 1 0

CaNOD vs
NaNOs 1
0


+1 0


--4 =


S= -5


The comparisons are independent therefore orthogonal when:

1) the sum of the coefficients of a comparison is zero; and
2) the sum of the products of the corresponding coefficients
of any two comparisons is zero.


c. The calculations for determining SS are as follows:


Source of variation Observed Reguired F
df SS MS F 5% 1%

Treatments 5 185.770 37.154 24.56 2.71 4.10
No N vs N 1 180.200 180.200 119.10 4.35 8.10
Organic N vs inorganic N 1 3.816 3.816 2.52
Ammonium N vs nitrate N 1 0.202 0.202 0.13
(NHI-4a) SO
vs. NH4NOS 1 1 .334 1.334 0.88
NaNO.vs CaNO 1 0.213 0.213 0.14

Error 20 30.25 1.513


------*-~---I-- ----------I--r*l^-l---n-lrr-


--~---- ---- ------ ---









Thus, treatments are significant from each other. Only No N vs N
is significant. Sources of N are not significant from each other.

SS (No N vs N)=
[5(148.6)-(186. ])-(82. 1)-(188.9)-(183. )-(182.2) =(-180. 1) =180.2
6(30) 180

The denominator, 6(30), is found by summing the squares of the
coefficients of the terms in the numerator and multiplying this by
the number of variates making up each term of the numerator, thus:

[(5") +(-12) ( ) +(-) +- +(-) (-1) 36 = 30(6)

SS (Organic N vs Inorganic N) =

E4(188.9)- (186.1) (182.1) -(183.8) -(182.2)1" = 3.816
6(20)

SS(NH4- vs NOr-N) = (186. 182. 183.8 182.2)- = (2.2) = 0.202
6(4) 24

SS (NH4)"S04 vs NHNOa0 3=(186.1-182.1) )=(4.0)" = 1.333
2(6) 12

SS Ca(NOW)" vs NaN0O=(183.8 182.2)-=(1.6)= = 0.213
2(6) 12

Mean squares are obtained by dividing the sums of squares by their
associated degrees of freedom. In this case, each comparison
involves a single degree of freedom so SS = MS.

F values are calculated by dividing each treatment MS by MS for error.
Required F-values are tabular values from a F value table for 1 and
20 df. An F test can be used to answer each of the questions posed
when the experiment was planned. The only significant F value is for
the-comparison No N vs N. All others are low, leading to the
conclusion that there was a response to nitrogen but the crop
responded similarly to all N sources.

d. Orthogonal comparisons should be used when one .treatment is very
different than the rest. For example, unweeded vs various times of
weedings. The treatment result may be so different as to bias
your normal MS and significance calculation.

No. Weedings (Treatments)
Comparison:
0 1 2 3
0 vs weeding -3 +1 +1 +1
1 vs 2 or 3 0 -2 +1 +1
2 vs 3 0 0 +1 -1









6.0 REGRESSTION AND CORRELATION


6. Introduction


The following techniques of regression, covariance and
correlation can be used to analyze:

Superimposed types of trials located at many locations
(multilocational); verification trials and others where the
researcher is trying to deal with and learn from technology
performance under variable management and/or environment
circumstances.

1. Correlation

a. This is another measure of the relationship between
two variables.

b.. It is expressed using a correlation coefficient (r)

i. (r) is a pure number without units or dimensions

ii. (r) always lies between + 1 and -1; '+'values
indicate a tendency of x" and x" to
increase together;' 'values indicate the larger
values of x1 are associated with small
values of x=.

iii. The closer (r) is to 0, means no correlation.
The closer the value is to 1 means a good
correlation.

c. Relation to regression where x' (dependent)
variable and x" (independent) variable are
regressed on each other,ie. y on x and x on y; a high
correlation between the two occurs when the two regres-
sion lines are well matched.

d. Probably no part of statistics is subject to more abuse
,and misinterpretation than correlation and regression.
The statement that "one can prove anything with statistics"
is true only if one ignores some of the basic principles
involved. The two principles most often ignored in
correlation are:

i. The full name of the coefficient of correlation is
the coefficient of linear correlation, and









ii. Nothing in the definition of correlation indicates
or implies that the relation between two variables'
is one of cause and effect.


2. Regression analysis:

a. Uses include:

i. Does dependent variable (y) actually depend on
the independent variable(s)(x)? Are the two
variables related?

ii. Can you predict the dependent variable (y)
from the independent variables x;(s)?

iii. What sort of response curve is there? Linear,
curvilinear etc.ie, How are the variables related?

iv. Ultimately, regression shows whether or not there
is a predictive relationship between an independent
and dependent variable.

b. Important items to regress:

yield or performance vs planting date
yield or performance vs date of operations (weeding, etc.)
soil characteristics
rain characteristics
planting density
labor in crop activities

c. Definitions:

i. Dependent variable is a factor where response
depends on the level of another factor, the
independent variable. An example is yield
(dependent variable) which is dependent on soil
fertility levels, moisture levels (independent
variables). Expressed graphically, the independent
variable is along the bottom abscissaa) and the
dependent variable is on the vertical scale
ordinatee).

ii. Regression coefficient tells us the estimated
change in (y) with each unit change of (;).
xxy = b(b),.is also the slope of a linear
Ex; regression line. One can also plot the
change in (x;) related to a unit change in
y = bvy.








iii. The t4-statistic indicates goodness of fit (based
on df) of the regressi n curve line to the data.
The higher the t-statistic the greater the
probability in the line being a good prediction.

iv. r" is the relatican between regression
coe f icient (b) an'd (b),,y



This is also the coefficient of variation.

This table presents an example where regression has been used
to relate yield with various factors (underlined):

Table 6. Regression coefficients for yield determinants and varietal
effects of the improved sorghum variety Framida, level-5
farmers' tests, (Matlon, 1982)*

SRN 4841
Yako!Ziniare Boromo
Improved Varietyx
Alone 1.31 (0.01) 181 (1.05)
Plowing 235 (1.21) 349 (1.35)
Fertilizer 1.64 (0.93) 0.19 (0.09)
Plateau soils -63 (-0.18) -270 (-1.12)
Lower slope soils -110 (-0.43) 107- (0.32)
Lowland soils -141 (-0.47)
Management factors
Plowing local variety -155 (0.79) -186 (-1.01)
Chemical fertilizer local variety 1.61 (1.25) 2.93 (2.02)
Plowing x fertilizer interaction
- local variety -0.31 (0.16) -0.03 (-0.21)
Manure 0.04 (2.36) -
Date of planting 5 (1.06) 121 (0.95)
Date of planting squared -0.02 (-1.50) -0.16 (-1.06)
Field location
Village dummy 1 -90 (-0.66) -76 (-0.61)
Village dummy 2 -151 (-1.30)
Plateau soils -132 (-0.46) 130 (0.73)
Lower slope soils --79 (-0.42) 491 (2.01)
Lowland soils 91 (0.43)
Field history
Sorghum preceding crop -64 (-0.6) -169 (-1.08)
Legume preceding crop -105 (-0.33)
Fertilizer applied preceding year 17 (0.24) 121 (0.76)
Const-ant 1039 -21589
R" 0.33 0.37
F 2.98 3.21
D.rgres of freedom df) ______
*t-statistics are included in parentheses










6.2 Correlation


(Review the previous discussion of correlation in OFE Concepts
and Principles). The following is an example:

Measurements of ten onion bulbs.


Diameter (mm) (X)


51.0
66.2
69.2
69.5
56.9
67. 1
58. 1
53.9
63.0
60.0


Weight (am)(Y)


63.4
115.3
146.6
132.6
80.7
125.6
80.0
78.7
112.8
96.2


The calculation of r, the coefficient of correlation, and
of the regression equation, is as follows:


X = 614.9

X = 61.49


Y = 1031.9

Y = 103.19


IX".= 38,192.17


IY" = 113,247.79


iXY = 65,014.60


(KX)" .n = 37,810.20


(Y)n /n =106,481.76 XSY/10 = 63,451.53


IN'" = 381.97


Sy" = 6,766.03


xy -= 1,563.07


r" = (1,563.07)" /(381.97 x 6,766.03) = .9454

r = .9454 =.95 (coefficient of correlation)

b = 1,563.07/381.97 = 4.092 (regression coefficient)

a = 103.19 -(4,092) (61.49) = -148.43 (intercept)

y = 4.092 X 148.43 (regression equation)

The correlation of .97 between diameter and weight is very high.
A straight line equation describes the relation between the two
variables. Interpretion of correlations must.be done carefully.


--- _~___1____1__ ~-~-1-1---1-1- ---1---


~I --~-- ---^---II-------------~--------;-II;. -.~l;-;-r-;i








A correlation does not necessarily relate cause and effect.

Extra elation is Temptinqg but DannE!ros. Often a series of
observations fall within a rather restricted range of values
for the two variables under study. If they show a high
coefficient of correlation, there is a great temptation to
extend the regression line beyond the range of observations
and try to predict what would happen to the values of Y if X
were to take on values above or below those actually observed.
This is called extrapolation.

It is dangerous practice because many variables that are
related in a curvilinear fashion will give a high linear
correlation if only a short section of the curve is sampled.

For example, bulb measuring 92.4 mm was found to weigh'
300.2 grams, but our estimate of weight from the regression
equation is 229.7. Extrapolation caused us to err by
70.5 grams in our estimate. Going in the other direction,
a bulb measuring 37.8 mm weighed 27.8 grams, but extrapola-'
tion gave an estimate of 6.2 grams. Extrapolating for still
smaller values of X soon gives us completely absurd estimates
of Y. For example, a 36.27 mm bulb would'be estimated to
weigh nothing, and all bulbs smaller than this, less than
nothing. The linear regression equation implies that a given
amount added to the diameter of a bulb will add a certain
fixed amount to the weight. It should be obvious, however,
that this cannot be so. One centimeter added to a nine-
centimeter bulb will certainly result in a greater increase
in weight than one centimeter added to a two-centimeter bulb.

If one wishes to find out how two variables are related
outside the range of observations, the safest procedure is
to make more observations in the region of interest.

A low correlation doesn-'t always mean lack of relation:
Look at the following pairs of figures:

X Y The coefficient of correlation between X and Y
0 0 is zero. It would be wrong to conclude that there
1 144 is no relation between X and Y. X is the elapsed
2 256 time in seconds after shooting an arrow vertically
3 336 at 160 ft/sec. Y is the elevation of the arrow in
4 384 feet. Obviously there is a relation between the
5 400 height of an arrow and its time in flight. The
6 384 important word linear, implied in.the coefficient
7 336 of correlation, is ignored. It is true that no
8 256 straight line will come close to fitting these data
9 144 but the equation y = -16 x' will give a
10 0 perfect fit. This is the equation of a parabola.










The moral of this example is that one should be on the lookout
for curvilinear relations that might fit the data better than
a simple linear relation.

A high correlation does not mean a _qood correlation:

A high coefficient of correlation, .937, indicates a close
relation between X and Y. One might be tempted to say that each
unit change in X causes a change of .643 in Y. In this case, the
X's are the numbers of cigarettes used annually in the U.S.
(in billions) from 1944 to 1958. The Y's are the index number
of production per man-hour for hay and forage crops during the
same period. It would be difficult to imagine a direct cause
and effect relation between cigarette consumption and efficiency
in the hay business. It just happened that both of these
variables showed a steady increase with time during the period
being considered.

The moral of this example is that the coefficient of correlation
will measure the closeness of relation between two variables,
but it tells us nothing about whether this relation is one of
cause and effect. That decision is up to the investigator, and
must be based on a great deal of knowledge of the variables
under study.









6.3 Reqression


6.3.1 Linear for one variable.

1. Terminol ocqi

A statistical method for assessing the relationship between
one or more independent variable and a dependent variable.
If we assume the relation between the independent and dependent
variables is linear, ie. best described by a straight line, the
question is to find the particular straight line that fits the
data the closest. What do we mean by the closest fit? It is
obvious from looking at the graph of the data that no straight
line can be constructed to pass through all the points. No
matter what line is constructed, several points will deviate
from that line. Variations among a single set of observations
can be measured by taking the sum of squares of deviations from
the mean. The variation from a line can be measured by taking
the sum of squares'of deviations from the line. Using this
measure as the criterion for closeness of fit, straight line
that will make the sum of squares of deviations as small as
possible is calculated. Such a procedure is called a least-
squares method.

In terms of deviations from the means of X and Y, the equation
of the best fitting line is:
A



(y is read: "the estimated value of y", (often called y 'hat')

The expression ; >2
estimated change in y, with each unit change in x. The regression
coefficient is called bandd b =fxy/xm2. More precisely,
this is "the regression coefficient of Y on X", and use the
symbol by,. Generally if b is used with no subscript, this
is the coefficient understood. The slope of the line = b.

The equation given above can be rewritten in terms of the
observations themselves, instead of in terms of deviations from
means:

We can write: (Y Y) = b(X X)
Which can be rewritten: Y = (Y -bX) +bX
A A
If we let Y bX = a, the equation can be written Y = a + bX,
which is' the slope-intercept form of a straightline-equation
mentioned at the beginning of the discussion on regression.









2. Examples

Observed and estimated bee prices

X Y Y=45..57-. 367X d=Y-Y d"
73 18.0 18.8 -.8 .64
79 20.0 16.6 3.4 11.56
80 17.8 16.2 1.6 2.56
69 21.4 20.2 1.2 1.44
66 .21.6 21.4 .2 .04
75 15.0 18.1 -3.1 9.61
78 14.4 16.9 -2.5 6.25
74 17.8 18.4 .6 .36
74 19.6 18.4 1.2 1.44
84 14.1 14.7 .6 .36

Totals 0.0 34.26



EX = 752; so X = 752/10 = 75.2
1Y = 179.7; so Y = 17.97
ixy = 93.04

x = 253.6; so b = -93.04/253 = -.367

Therefore, substituting in the equation:
X = (Y bX) + bX, we get
X.= E17.97 -(-.367) 75.23 + (-.367)X
Y = 45.57 .367X

This equation can be stated: "Starting with a base price of
Ksh.4Q.57 per cwt., every unit (million) increase in annual beef
marketing is associated with an average reduction in price of
.367 Kshs. per cwt."

Compare the observed values of Y with the estimated values,
(Y's), based on-the regression equation. The sum of deviations
is always zero and serves as a check on the calculations. The
sum of squares of deviations can also be calculated from the
following formula:

Id' = (1 r0 ) y2
r = (x y)" / x= y=
r" = coefficient of correlation

The sum of squares, d", is called the sum of squares due
to deviation from regression and the square root of quantity
Ad"/(n-2) is called the standard error of estimate. It is
a measure of the amount of variation from the regression line.





40



The construction of the line requires only two points. One point
can be on the Y-axis, (0,Y). Another can be the point representing
X and Y-(X,Y). The line passing through these two points will be
the required regression line. The dotted lines drawn from the
observed points to the regression line represent the deviations.


S(0,Y)


Price Per
cwt.
(Y)


20



18




16




14


4


IY =45.57 -.367X






I I

| (X,Y'


.... ............ .. .Y

^ ;*'


12
12 : __.________________ I ________


) 75 80
Beef Marketed (millions) (X)


The deviations are represented as vertical lines. It is the sums
of squares of these deviations that have been minimized to come
up with the closest fitting line.

The expression,'y/ly" is called the regression coefficient of
X on Y, and is designated by by. As pointed out previously
the symbol b is understood to mean by. the regression of Y on X,
unless otherwise specified.


c








There are two best fitting lines according to which way the
deviations are taken. Note that:

bvH .by =xy'-.k_*_L= ri''


This brings out the relation between the regression coefficients
and the coefficient of correlation (r&)

3. The following questions about data in the ex ample
can now be answered:

a. How close was the relation between supply and price?
answer: Fairly close. The coefficient of correlation
was -.7, and 1 would be perfect.

b. What is the probability that such a correlation could
be due to chance?
Answer: A correlation of this size from 10 pairs of
observations would occur between 5% and 1% of'the
time by chance alone.

c. What equation would best describe the relation between
price (Y) and supply (X) from these data?
Answer: Y =45.57 -.367X

d. How well does this line fit the data?
Answer: The sum of squares of deviations of the observed
points from the line was 34.19 or about one-hali
the total price variation. Thus, only half the
price variation was in some way associated with
variation in supply. An analysis of variance table
shows this below.

4. Regression analysis arranged in an analysis of variance form:


Source of variation Degrees of Sum of squares Mean Squares F
freedom
Total 9 9y2 =68.32

Regression (linear) 1 ru"y =34.13 13.13 7.99*

Deviation from regression 8 (l-r" )gy" =34.19 4.27


Significant at the 5% level.

The F value 7.99, is slightly higher than the F required at the
2.5% level for 1 and 8 degrees of freedom (7.57).








5. Regression can also be used to identify other types of
relationships such as curvilinear, quadractic or cubic. Consult
a good statistics book for an explanation. The above example is
only a preliminary guide towards using regression.


Name of Eauation


Linear
Quadrati c
Cubic
Quartic
Quintic


Name of Curve

Straight line
Parabola
Cubic parabola
Quartic parabola
Quintic parabola


Regression can also be used fur more than one variable.

6.3.2 An Example with three variables:

To illustrate partial and multiple correlation and regression,
some data is analyzed on the specific gravity of potatoes (Y),
the nitrogen content (X1) and the phosphorus content (X=).
The observations are listed in the following table;

Specific gravity, nitrogen and phosphorous content of
twenty samples of potatoes.


.- 1.07) 10-


Xt
(Nitroqen
96
82
121
88
100
114
71
94
74
36
36
73
58
31
38
56
24
37
,10
14
1. 253


-1)100


X2
(Phosphoru s) 100
40
36
30
42
28
26
33
26
15
35
25
15
26
25
24
11
22
.11
14
10
504


First
Second
Third
Fourth
Fifth


(SIp Gr.
2
14
15
15
16
27
48
54
58
68
82
83
91
97
98
101
128
140
163
179
1i 479_


Total s


--"-~ ----


------------


--~--~` -`-~-~-~~I-~"~~-----I---'


1 253


Irr--r~----------


__._.__._~I-~








First calculate the various coefficients of correlation:

Y"= = 160,545 5.Xi = 99,741 X-Z = 14,364

(WY ) /20 = 109,372.05 (WXi* ) /20 = 78,500.45 (WX,2 ) /20 = 12,700.8

Cy2 = 51,172.95 :x,." = 21,240.55 $:x2- 1,663.2

gY X1 = 63,441 QY Xa = 30,659 IXx X2 = 34,160

fYIXi /20 = 92,659.35 UYKX~ /20 = 37,270.8 SX9AX /20 = 31,575.6

Ayx,; = 29,218.35 Qyx. = -6,611.8 gx,. x. = 2,584.4

r"yx1 = ('yxx )" /y- gxx" = (-29,218.35)" /(51,172.95)(21,240.55)=.7854

ryx =W y7 =-.8862 (Note that it is negative
because yxi was negative)

ryx22 = (CYx.)= /Ay" Qx=: =(-6,611.8)= /(51,172.95) (1,663.2) = .5136

ryx;2 =rn = -.7167

r2xjxx, = ($9x x2). / x e= 2 = (2,584.4)=/(21,240.55) (1,663.2) = .1891

rx:x2 =4rx~x =.4348


Next, describe the relation by calculating the regression equation.
Using the normal equation based on deviations from means:


bixKI" + b2:x;ix =;xIyi


b:;ix; x: + b"G:2= =.;x, y

Substitute the observed values from the data:

21,240.55 bi + 2,584.4 by = -29,218.35

2,584.40 bi + 1,663.2 by = -6,661.8

Multiplying the first equation by 2,584.4, and the second
equation by 21,240.55 and subtracting:

28,648,'159.4 b2 = -64,926,364.75

b2 = -2.266

Substituting this value of b2 in either of the original





44


equations, and solving for bi;

b. = -1.100

To have a recession equation in terms of the original values
find 'a':

a = Y bi X b: X,

= 1479 (-1.100 1253_) -(-2.226 504) = 199.968
20 20 ) ( 20)

Write the regression equation: Y =199.968 1,100X, =-2.266X2

3. From this equation, calculate values of Y and compare them with
the observed values.


Observed


and calculated specific gravity of 20 samples of
potatoes


_. Y _._Y cd=Y -

2 3.7 -1.7
14 28.2 -14.2
15 -1.1 16.1
15 8.0 7.0
16 16.5 -10.5
27 15.7 11.3
48 47.1 .9
54 37.7 16.3
58 84.6 -26.6
68 81.1 -13.1
82 103.7 -21.7
83 85.7 -2.7
91 77.2 13.8
97 109.2 -12.2
98 103.8 5.8
101 113.4 -12.4
128 123.7 4.3
140 134.3 5.7
163 134.6 28.4
179 161.9 17.1

$ =0

The sum of the deviations should be zero. This furnishes a good
check on the computations. The sum of squares of deviations is
4,051.16.' This represents the variation in specific gravity (Y)
not associated with the variation in nitrogen content (X)M or








sposphorus content (X). Another way to calculate this, without
computing each Y, by taking (1I R") y=

(1 -.9208) 51,172.95 = 4,052.90


4. The results can be summarized in an analysis
as follows:


of variance table


Source of Method of SS df MS F
variation ___ computing SS9


Total


Regression
due to Xi

Deviation from
regression

Additional
regression
due to X" r:

Deviation from
multiple
regression


bY2


r=yxx (fy)=


( 1-r yx." )1y/ "


"yx Ux (1-r"yx )y y


(l-Rzyx 1xz,.,) y


51.172.95


40,191.23


10,981.72


1 40,191.23


18 '610.10


1 6,929.47


4,052.90


238.41


In this table, the total effect of nitrogen and the additional
effect of phosphorus was considered. One can also consider the
total effect of phosphorus and then the additional effect of
nitrogen.

5. The order in which variables are considered makes a marked
difference in the outcome of the analysis. An example helps to
clarify this factor: It is well known that the yield of many
crops is influenced by both temperature (warm vs. cold) and day
length (short vs long). Numerous crop yield records of a crop '
grown in different seasons of the year were collected. For each
yield record, there is a record of the mean day length and of the
mean temperature during the growing season. Day length and
temperature is expected to be closely correlated with each other.
Since this is true, yield was closely correlated with temperature,
and the additional consideration of day length would explain little
of the variation in yield not already accounted for. At the same
time, day length-alone might be closely correlated with yield,
while temperature might have little added effect. We could tell
little about which factor was the more important, temperature or
day length. In order to answer this question,'design an experiment


65.9**








in which either the day length or temperature were controlled so
that they would be less closely correlated than they are in nature.

6.4. Questions raised to help researcher consider multi-variables:

In summary, the following questions (also presented in
experimental program design, will help researchers consider the
multi-variables involved in making decisions concerning new
technology.

1. Technology performance under farmer's conditions? Use means,
mode, variance, frequency to identify associated risk.

2. Factors causing variability. Use yield function analysis
to help understand the relationship of these factors to
performance. Use to identify technical problems which
would lead to more on-station experimentation.

3. Identify conflicts with the system. Changes in inputs and
outputs. Changes farmer makes in organizing labor, timing
of operations etc. Check for created labor bottlenecks.

4. Determine what returns are expected compared to alternative
activities competing for scarce resources. Cost inputs and
outputs. Figure farm level and social returns.

5. Determine probable patterns and consequences of adoption.
Under what conditions (environmental, technological,
economical) will farmers find it profitable to accept the
technology? What activities, if any, will 'be substituted
for or changed, with what level of management, at what
scale (size of farm), and realizing what yield to make it
worthwhile..

6. Determine if technology is consistent with consumption
goals of the family? Processing ease, storage, taste,
timing of harvest, quality, quantity of by-products.

For exLamaele, in an area of Cameroon,farmers have different
uses for their sweet potatoes, both commercial and subsis-
tence. "Commercial clone" identified was a high producer,
resistant to virus, widely adapted had medium weevil
resistance, poor storing tubers not able to remain too
long in the soil during dry season. A "garden clone"
identified had small tuber size unsuitable for commercial
use, moderate virus resistance, high weevil resistance;
therefore, it could be harvested'over a longer period
during the dry season. These were two distinctive selec-
tions that could be made for two purposes, each having
different criteria for acceptance.










The Following table illustrates varietal characteristics
that researchers ranked.

General per forrmances of the tested improved clonal material com parked

to the local variety:


TIb 1 527034 LOCAL


Crop growth:


Pest g&
diseases

Production:


Tubers:


% germination
Ground cover
Drought
resistance


Virus resistance
Weevil tolerance

Total yield
Commercial yield


Form

Regularity
Tuberi station


Good-V.'Good
Good-V.Good
Good


V.good
Medium-good

Good-V.good
Good-V.good


Oval

Very regular
Clustered


Good
Good
Good-V.Good


Medium-Good
Medium
Medium-V.Good


Medium-Good Good
Good Medium-good

Good Bad-Medium
Medium-good Medium


Oval (too
small)
Regular
Quite
clustered


Long

Variable

Spreaded


Very easy


Harvest
easiness:


Good


Storage
abi li. ty:

Fresh Colour
/Tuber
Col our:


Quite easy Difficult


Good


Medium-good


Orange/White Yellow/Red White/White


Good-V.good Good-V.good Medium-Good


Taste:


--- -- -- ~----~---------I--- -- --I










7.0 IMPF'ROVING_ PREC IS JON

7.1 Covariancq

1. Covariance is useful in the following instances:

a. Where blocking cannot.adequately reduce the
experimental error:

i. Soil heterogeneity should be measured to document

ii. Residual effects from previous treatments or history

iii. Stand irregularities when not due to treatment

iv. Non uniform pest incidence not due to treatment

v. Non uniform environmental stress when screening for
this factor (drought, water logging, salinity,
Fe toxicity, low fertility).

b. Alternative to missing data technique.

c. Experimental interpretation:

The covariance technique can assist in the interpreta-
tion and characterization of the treatment effects on the
primary character of interest Y, in much the same way
that the regression and correlation analysis is used. By
examining the primary character of interest Y together with
other characters whose functional relationships to Y are
known, the biological processes governing the treatment
effects on Y can be characterized more clearly.

2. Ex-iampLe5:

a. Increasing precision in randomized experiments. Use
when a factors) (covariate) serves to have good predic-
tive value over time (e.g previous yields of a perennial
crop) that can serve as predictors of inherent yielding
ability. The treatment means are adjusted so as to remove
differences in yielding ability.

b. Adjust for sources of bias in observational studies. For
example, if variety response to fertilizer was studied in
two areas and the variety performance is linearly related
to rainfall in selected areas. Thus, variety yield
responses to fertilizer will be due in part to rainfall
differences. Covariance can be used to sort this out.









c. To better understand treatment effects in randomized
experiments. For example, a soil fumigant was used on
nematodes. Significant differences in numbers of cysts
and yield of crops was observed. Is this due to numbers
of nematodes present to start with or due to .actual
chemical effects? Use covariance to sort this out.

d. Consider the case of a rice variety trial in which weed
incidence is used as a covariate. With a known functional
relationship between weed incidence and grain yield, the
character of primary interest, the covariance analysis can
adjust grain yield in each plot to a common level of weed
incidence. With this adjustment, the variation in yield
due to weed incidence is quantified and effectively
separated from that'due to varietal difference.

e. To study regressions in multiple classifications.

f. In a water management trial, with various depths of water
applied at different growth stages of the rice plants,the
treatments could influence both the grain yield and the
weed population. In such an experiment, covariance
analysis, with weed population as the covariate, can be
used to distinguish between the yield difference caused
directly by water management and that caused indirectly
by changes in weed population, which is also caused by
water management. The manner in which the covariance
analysis answers this question is to determine whether ,
the yield differences between treatments, after adjusting
for the regression of yield on weeds, remain significant.
If the adjustment for the effect of weeds results in
significant reduction in the difference between treat-
ments then the effect of water management on grain yield
is due largely to its effects on weeds.

3: Examples are complex and can be researched in various
statistics texts listed in Appendix References section.



7.2 Missing Plots

1. To qualify:

Missing items must not be due to failure of a treatment.
If a treatment killed the plants or animal or producing
0 yield, should be entered as 0 not a missing value.

2. Affects of missing values:


a. Missing data destroys the symmetry of the data.












b. In one-way classification reduce the sample size and
use unequal numbers analysis.

c. In 2 way or more classic fiction -- create missing value.

3. Exa..-P.. l.

a. For a single missi n_ value:
The yield of variety. 4 in Replication IV-is missing.
This leaves an unbalanced situation where Var 4 has four
yield observations while the other varieties have rive;
likewise, replicate IV includes five yield observations,
the others six.

1. A value for the'missing plot can be calculated by the
-fol 1owi ng formula a:

X = t(T_) + r(R) S
(t-1) (r-1)

t = no. of treatments in trial
r = no. of replicates in trial
T = sum of yields for plots.receiving same treatment
as that where the missing value occurred.
R = sum of yields for other plots in the same replicate,
as the missing value
S = sum of all plot yields.
X = value created


Replication
V.ar-,ie ,i_ .... I iI l V__ S_3u3m

1 1.58 1.63 i.46 1.98 1.47 8.12
2 1.91 1.88 .78 2.08 1.91 9.56
3 1.67 1.67 1.81 2.00 1.88 9.03
4 1.74 1.60 1.55 1.52 6.41
5 1.81 1.88 1.41 2.24 1.89 9.73'
6 1.52 1.75 1.55 1.88 1.55 8. 25
Sum 10.23 10.41 10.06 10.18 10.22 51.10

t = 6; r 5; T = 6.41; R 10.18; S = 51.10

6(6.41) + 5(10. 18) 51. 1'0
X = (5 x 4) = 1.91








Notice that d.f. for total and error have been reduced by one.
This procedure gives an unbiased estimate of error.


With these calculated SS, calculate


Sour ce
Total
Var.
BElks.
Error


d.f.
28
5
4
19


an unbiased F value.


_Reg d. F
SS MS F .5 .01


. 4834*
.3958*
.1522


.0967
.0989
.0076


12. 72*


2.17


4.17


2. The treatment bias for a single missing plot can be calculated by
the formula:

Treat. Bias
in SS = (R-(t-l) X)=
t(t-1)

In this example: 1 (10.18) (5) 1.91)2 = .0132
6(5)

Subtracting the bias from the treatment SS in the ANOV of the
data set containing the calculated missing plot,


Variety SS
- bias
Correction


.4966
- .0132
.4834


gives the corrected unbiased SS for varieties reported above.

b. Several mission values


1) Estimate one factor by:


Block mean +
No.of values


Tmt mean
No.of values


2) Estimate 2nd factor with estimate in "a" included using
the same formula.

3) Re-estimate 1st factor in "a" and then in "b" using adjusted
values. If more than two missing values, proceed with more
cycles.

The above procedure, however, does bias the mean squares
for both varieties and replications. This may have very
little significance for a single missing plot but the bias
should be corrected if more than one value is missing


--~'-I-`-----~









4) For 2 or more missing plots then must use a correction
*factor (CF) for Trmt MS calculation:

(R- (t-1) x):'
CF = t (t-I)

Calc. Tmt LMS CF = unbiased Tmt MS











8.0 POOLING OR COMBINING DATA


8.1 Pooling with multiple locations:

a. In simple terms, sources of variation for combined
location are:

Place (P)
Treatments (T)
Treatments x: Place (TxP)
Pooled experimental error

The Treatments x Places Mean Square (MS) is tested by
the pooled error (average of the Error MS in the individual
experiments). If T x P is significant, then it can become
the error term to test T and P.

b. Complications in pooling over sites:

1. Experimental error variances may differ from place to
place. Check using Bartlett's test for homogeneity of
variance. If variances are heterogeneous, the F-test of
TxP interactions is not strictly valid. Use an adjusted
form of the test.

2. TxP interactions may not be homogenous, especially in a
factorial experiment. Some factors may give stable responses
from place to place, while others may be erratic. If MSt
has been subdivided into sets of comparisons, the TxP or
Interactions MS for each set should be completed and tested
separately.

c. ExamplesL:

1. A 5 treatment experiment with 5.replications at 3
different locations.


5 treatments I I
5 replications 1 L J!___
3 locations II ____J ______ -J__L _
I I i I i I I I I i i I iI I I I


Location 1 Location 2


Location 3










st ep; :


1. Check each location separately.
At location 1 AOV: Source
Total
Tmt
Rep
Err or


df Complete for each location.
24
4

16
16


2. Combine and compare locations to give average differences
among treatments.


Sour ce
Total
3 Location
s Tmt.
T x L
S R (within L)
Error











The following data, illustrates this ,example:

Location 1 Kisii
Replicates
Vari et I II I I_ IV_ V Total
1 6.71 6.15 5.67 5.99 8.08 32.60
2 6.62 6.16 6.43 5.84 6.54 31.59
3 7.45 5.66 5.67 5.51 7.53 31.82
4 6.50 5.13 5.38 5.44 7.36 29.81
5 6.49 5.93 5.25 5.38 6.86 29.91
33.77 29.03 28.40 28.16 36.37 155.73

ANOV
d.f S.S. M.S. F
Total 24 14.9547
Var. 4 1.2156 .3039 1.79-
Repl. 4 11.0219 2.7554 16.22**
Error 16 2.7172 .1698


Location 2 Kitale
Replicates
Variety I II III IV V Total
1 6.43 6.68 6.32 6.79 6.14 32.36
2 6.01 6.06 6.19 6.33 6.58 31.17
3 6.43 6.71 6.54 6.36 6.33 32.37
4 6.35 6.08 5.88 6.47 6.40 31.38
5 6.28 6.49 5.81 6.08 5.60 30.26
31.50 32.02 30.74 32.03 31.05 157.34
ANOV

d.f S.S. M.S. F
Total 24 1.9877
Var. 4 .6480 .1620 2.41-
Repl. 4 .2652 .0663 .99
Error 16 1.0745 .0671


'Location 3 Kakafnmea

Replicates
Variety I II III IV V Total
1 7.23 5.53 5.72 6.09 6.94 31.51
2 6.42 4.47 5.30 4.60 6.02 26.81
3 6.42 6.20 6.10' 6.10 6.75 31.57
4 6.11 5.56 5.32 5.17 6.81 28.97
5 5.53 4.93 5.44 5.19 5.86 26.95
31.71 26.69 27.88 27.15 32.38 145.81











To calculate d.f I
vr-
(vr-1)
(v- )
(v-1) (r-1)


Total
Var.
Repl.
Error


ANOV
d.f
24
4
4
16


8.8.
12.1364
4.3546
5.7300
.2.0518


M.S. F


1.0886
1.4325
.1282


8.49**
11.17**


The combined analysis across locations is done by:

The separate analyses show statistical significance for treatment
(variety) differences at only one location. The combined analysis
reflects average differences among treatments over all locations.
Replicates are divisions of the data within locations. Replicate
No.1 at Kisii has no greater association with Replicates No. 1 at
the other locations than it does with any of the other numbered
replicates. Replicate variation is pgoled over locations. Error in
the combined analysis is the pooled error for the three locations.
The only additional data summary needed for the combination analysis
over locations is given below:


Variety
1
2
3
4
5


Kisii
32.60
31.59
31.82
29.81
29.91
155.73


Kitale
32.36
31.17
32.37
31.18
30.26
157.34


KakameFa
31.51
26.81
31.57
28.97
26.95
145.81


Total
96.47
99.57
95.76
89.96
87.12
458.88


Combined Analysis


To calculate df

(pvr-1)
(v-1)
(p-1)
(v-l) (p-1)
p(r-1)

p(r-1) (v-l)


Source


Total
Var.
Loc.
Var. x Loc.
Repl.
(within locat.)
Error


d.f


S.S;


32.1979
4.5165
3.1191
1.7017
17.0171


48 5.8435


C= (458.88)" /75 = 2807.6114

Total SS = (6.712 + 6.152 + .......... +5.192 + 5.862 ) C = 32.198

(Note that combined uncorrected total SS above is readily obtained by
pooling the total SS for each location plus the correction factors
for the 3 locations),ie.


M. S.


5.31*
7.33*
1.74


1.1291
1.5595
.2127


.1217


--


I--


----


----











Total SS = (14.95+(155.73)"/25)+(1.99+(157.34)=/25)+
(12.14+(145.81)"/25)



'Variety SS = (96.47)2/15+(89.574 /15+...+(87.12)"/15 C = 4.52



Location SS = (155.73)"/25+ (157.34)2"/25+(145.81)2/25
C = 3.12
(Note these are for separate locations)


Var.x Loc.SS = E(32.6022)+(32.3622)...+ (26.9522)3/5
Var.SS Loc.SS C.F.=1.70


Repl.SS = 11.01 + .27 + 5.73 + 17.02


Error SS = Total SS minus all SS above


A computational shortcut can be used to calculate the Var.x Loc. SS
directly or to check the longer method.


SS Var. at Kisii
SS Var. at Kitale
SS Var. at Kakamega
Sum
-SS Var. (combined)
SS Var.x Loc.


= 1.2156
= 6480
= 4.3546
= 6.2182
4.5165
= 1.7017


2. A three treatment experiment with 2 replications per farm and 6 farms
per village and 2 villages sampled.


3 tmts/farm
2 reps/farm
6 farms/
villages
2 villages


i ii
I S i
i ii
* i i
I i i
* i i


ii i
I I i
* i i
* i i
* I I
i I I


* I I
i S I
I I i
i I i
I I i
i I I


i I I
i I I
I S I
I i i
I i i
i i i


i I i
I I I
I I
I i 3
I I i
I
I


i i i
i II
i ii
i i i
I I I
I I i


1
vil 1 age


FARM


Stes:


1. Check each experiment at each farm


- Tells you what difference are at each farm
- Can serve as "environmental index"
- Can regress individual treatment results
on.environmental index


Source
Total
Tmt
Rep
Error


df.
5
2
1
2


)I


~_~~


-_---_:











2. Check farms within a village to see if farms are similar.

Source df.


Total
Tmt (T)
*Farm (F)
*TxF
Reps w/in F
Error


3. Compare villages for
plot design)


* major interest


average trends. (similar to split


Source
Total
Farms (village)
Reps (villages)
Tmt (T)
TG
Tmtx villages
TmtxFarms
Error


c. Example: Experiment with 1 replication per farm and 6 farms
per village and 3 villages.


3 tmts/
farm


I I
I I 1

* j I J village
a I
I I
a ~ ~ ~ ~ ~ -- _______________________________


Steps:


1. Test within each village


Source

Total
.Tmt
Villages
Error


df.


2. Compare villages.

Source

Total
Tmts
Villages
T x villages
Farms within village
(Error)


df.


4. Split plot at several locations:

Source df .Way to qet df










Total
Location (L)
Reps (within
location) (R)
Tillage (T)
(TxL)
Error a
Weed (W)
Wx L
WxT
WxTxL
Error b


(LTWR- 1)
(L-l) 8
2
(L(R-I) 2
(T-l) 3
(T-1) (L-l)
L(R-1) (T-l)
(W--1)
(W-l) (L-1)
(W-1) (T-1)
(W-I) (T-i) (L-1)
(Tot df rest)


locations
tillage treatments
weeding treatments
reps


8.2 Pooling over years


1. Perennial crops

Yields from same plot taken in successive years are usually
correlated. The experimental error is not independent from one
season to another. Treatments x year gives some indication of
differences from year to year.

The following table has each data value summarized by variety,
year, and replicate. All of these sources of variation and
interactions between them are computed in the ANOV. Replicates
are the same in each year of data accumulation and totals for
replicates over years and varieties are used to determine this
source of variation. They remain the same over years because the
forage is a perennial crop. In this situation the varieties are
replicated in two directions, space and time.









Forage Yield Trial Harvested for Three Seasons (Kg/ha)
Replications


Year Variety


1972





1973





1974


1
2
3
4
Totals

1
2
3
4
Totals

1
2
3
4
Totals


1
3 yr 2
Totals 3
4
Totals


Combined ANOV is similar to a spl


it plot design:


Source d. S.S. M.S. F. df

Total 47 53.5639 (vry-1)
Varieties 3 37.2008 12.4003 26.07** (MP) (v-1)
Reps. 3 3.9821 1.3274 2.79- (r-1)
Var. x Rep. 9 4.2812 .4757 20.24** (y-l)
Years 2 .5549 .2775 (SP) (r-1) (v-1)
Var.x Yrs 6 2.5569 .4262 18.14** (MPxSP)(v-1)(y-1)
Years x Reps. 6 .7523 .1254 5.4 (r-) (v-) (y-1)
Error (VxRxY) 18 4.2357 .0235 (r-1)(v-1)(y-1)



9.0 NON-PARAMETRIC TESTS

These tests are used when the populations being sampled are not
normally distributed. (Most statistics are based on the assump-
tion that the population is normally distributed.) In
exploratory research, the researcher may not know the type of
sample distribution.


III


Total


3.10
4.65
5.47
6.85
20.07

3.25
5.46
6.41
6.34
21.46

3.86
4.21
5.57
5.45
19.09

10.21
14.32
17.45
18.64
60.62


4.06
5.64
5.62
6.33
21.65

4.26
5.48
6.30
5.83
21.87

4.53
5.09
6.46
4.33
20.41

12.85
16.21
18.38
16.49
63.93


4.73
4.94
6.71
6.88
23.26

4.71
5.26
6.96
6.59
23.52

5.26
5.80
5.92
6.06
23.0.4

14.70
16.00
19.59
19.53
69.82


3.10
5.38
6.87
6.23
21.58

4.21
5.68
6.28
6.52
22.69

3.84
5.82
6.46
6.81
22.93

11.15
16.88
19.61
19.56
67.20


14.99
20.61
24.67
26.29
86.56

16.43
21.88
25.95
25.28
89.54

17.49
20.92
24.41
22.65
85.47


48.91
63.41
75.03
74.22
261.57


I --I-----'-^-I 'I-- -


-- ---------"---- -I-


X
3.74
5.15
6.17
6.57
5.41

4.11
5.47
6.49
6.32
5.60

4.37
5.23
6.10
5.66
5.34

4.08
5.28
6.25
6.18
5.45-










1. Use of Median : If 'n' is odd, the sample median is the
middle term.

If 'n' is even, the sample median is the
average between n/2 and n+2/2 e.g, if
-have 1,3,4,5,7,8,the median is 4.5.

2. Other tests exist for evaluating rankings (Sign-test); ranking
differences between measurements (Siigned rank test) is useful
in the same way one would use a t-test for paired samples; for
ranking unpaired measurements (Mann Whitney test).
Descriptions of these tests are in most statistics books.

10.0 USE OF CHI SQUARE

10.1 Introduction

The chi-square test is-most commonly used to test hypotheses
concerning the frequency distribution of one or more populations.
Three uses of the chi-square test are most common in agricultural
research: analysis of attribute data, test for homogeneity of
variance, and test for goodness of fit.

Hypotheses about treatment means are the most common in
agricultural research but they are by no means the only one of
concern. A type of hypothesis commonly encountered is that
concerning the frequency distribution of the populations being
studied. Examples of questions relating to this type of
hypotheses are:

Does the frequency distribution of the kernel color in maize
follow a hypothetical genetic segregation ratio? (attribute
data).

Do individuals from several treatments in the same experiment
belong to the same population distribution? (Goodness of fit).

Are the frequency distributions of two or more populations
independent of each other? (homogeneity of variance).

10.2 Rules

1. Use only for counts or frequencies.

2. If expectation is <5, then chi-square is not accurate.

3. Chi square cannot be used for weights, times, percentages.

4. There must be independence between measured items.


S5. There is an optimum division of treated vs not.












6. There is an optimum number of 'treated' below which chi-square
will not detect, therefore, the number of measurements in the
experiment affects the statistical outcome.

10.3 Example

Note: other examples can be found in most statistics textbooks.


Number of Fields using


Fl F2


Maize
Sorghum
Total


F3 F4 I Total


40
38
78


90
150
240


a) F1 = 1:1:1.5
F2 = 1:2:1


F3 = 1:3:1
F4-= 1:1:1


a) Fertilizer ratios of N.P.K.


II. Expectations: Using hypothesis of exact proportionality
(90/240 = 3/8)

Number of Fields Usina


F1 F2


F4 i Total


20.3 17.6 22.9 29.2 90
33.7 29.4 38.1 48.8 150
54 47 61 78 I 240


III. Excess of deficiency (+ or -) of frequencies in I relative
to expectations in II.

Number of Fields Using


F1 F2


F4 Tnotal


Maize
Sor hum
Total


-2.3
2.3


0.0


3.6


0.0


-4.9 10.8
4.9 -10.8
0.0 0.0


0.0
0.0
0.0


Cal culati ons:

I Counted number of fields using fertilizer regimes.

II Example: For F2 maize: 47 x 3/8 = 17.6 (3/8 = 90/240,
expected proportion)

F2 sorghum 47 17.6 = 29.4
(or 47 x 5/8 = 29.4) (5/8 = 150/240)


Crop

Maize
Sorqhum
Total


I- ~---I


-- -- "~


--


F!. F


------"-----


F1. F2


2.3 3.6










III Example: For F2 maize 14 (table I)
-17.6 (table II)
.. -3.6 (enter.table III)

Using table III:

-= (-2.3)L2+(-3.6) +(-4.9) +(10.^8)"=+(2.3)=+(3. 6)"+(4.9) "-+(-10.8)=
20.3 17.6 22.9 29.2 33.7 29.4 38.1 48.8

= 0.26+0.74+1.05+3.99 16+16+0.44+0.63+2.39= 9.7

d.f = (rows-l) x (columns -1) = (2-1) x (4-1) = 1 x 3 = 3

Table table: Relative frequency or probability of .05 then null
hypothesis would be rejected because 9.7 (calculated)
> 7.8 (table value).

Null hypothesis = Two crop fields receive fertilizer in same
proportion.

Check original data: See that 1:1:1 is used by a greater proportion
than other fertilizer ratios when comparing to sorghum.


11.0 DATA INTERPRETATION

11.1 Significance level

"The investigator must never forget that, in any use of his results,
the magnitude of benefit from treatment is at least as important as
the level of significant. A test of significance relates solely
to the strength of the evidence against the null hypothesis. It
depends upon the magnitude of a treatment effect but also on the
size of the experiment. For example, a reduction of death rate
by 17 might be too small to pay for the cost of widespread adoption
of the treatment, yet in a large experiment this could be
statistically significant. A reduction by 20% might be of the
enormous practical importance, but a small experiment could fail to
demonstrate convincingly that anything other than chance variation
was occurring."

A significance test consists of:

i. Formulating the null hypothesis whose contradiction represents
the effects that the experiment is intended to detect;

ii. Choose the level of probability that seems most likely to
be useful in respect of any conclusion from the experiment.









A significance test never proves that a treatment had no effect.
The choice of the level of probability at which statistical
significance will be asserted rests upon the considerations above.

For agricultural on-station research, the 0.05 level is commonly
used. When in reality no difference exists, action taken based on
significance reading would be costly or undesirable. The 0.01
probability level might be preferred.

In preliminary experiments on a suspected mineral deficiency in
sheep, significance at a probability of 0.05 (or even 0.1) would
justify further small-scale trials with a particular element, for
at this stage no clue must be neglected. On the other hand, no
final decision to recommend an injection or a dietary supplement
would be ,taken unless the accumulated evidence of benefit to the
sheep was significant at the 0.01 level.

The term statistical significance makes clear that a special
meaning is intended, and should be used whenever any confusion
with more colloquial meanings of 'significant' is possible.

A*With OFE, the level may be changed (0.1 or higher) depending on
what the acceptance or rejection of the null hypothesis means
in real terms. The standard 0.05 level may not be pertinent,
when using OFE objectives.

11.2 Coefficient of variation (CV):

The CV is the standard deviation expressed as a percentage of
the general mean. It is a commonly stated indicator of the
precision of an experiment, though not of tremendous practical use.
S/X = CV
A knowledge of relative variation is valuable in evaluating
experiments. After the statistics of an experiment are summarized.
One may partly judge its success by the CV. In usual experimen-
tation, the CV may range between 5-15%. If you find the value
outside this interval, you may want to inspect your data to look
for an error in calculation, or in some unusual cases, question
the validity in the experiment.

CV values for OFE tend to be-higher than OSR depending on the
management of the variability from place to place; however, the
researcher should know the expected range of the values in his
situation and should use this as his own benchmark. It is useful
to look at CVs of other research work in similar conditions to
see how yours compares.

You should always relate the CV to the S and 7 from which it came.
You need-to know if an increased CV is due to a rising S or a
falling X.











12.0 APPENDIX A


GLOSSARY

Agroclimatic Classification the grouping of different physical areas
within a country, a region, or the world into broadly homogeneous
zones based on climatic and edaphic factors.

Agroclimatic distortions When transferring technology which success-
fully works in one situation to another'situation, distortion of the
technology may occur due to differences in climate and/or soils.

Agroecological Zone a major area of land that is broadly homogenous
in climatic and edaphic factors, but not necessarily contiguous,
where a specific crop exhibits roughly the same biological expression.

Agroforestry a system of land use wherein annual crops are grown,
mostly in intensive mixed or intercropping methods, under the perennial
forest trees or fruit-cum-timber trees. Also known as agrisilviculture
and forest gardening.

Alley cropping the arrangement of several rows of annual crops
between a row of trees or a perennial crop (pigeon peas).

Arable land refers to land under temporary crops, meadows for mowing or
pasture, land under market and kitchen gardens (including cultivation
under glass), and land temporarily fallow or lying idle.

Benchmark survey a systematic survey study aimed at collecting data
e.g. existing crops, varieties, yields, socioeconomic constraints,
before a project begins. Data collected depict the existing
picture of the survey areas with regard to selected parameters
and can be used to evaluate the results of the project.


Biological determinants of cropping systems the biological 'factors
such as crop species, varieties, weeds, insect pests and diseases,
which determine the crop configuration and performance of a
cropping pattern at a given site.

'Bottom-up' Information or research which starts with an understanding
of the existing situation on the ground before attempting to assess
what changes might be useful. 'Diagnosis as a basis for perscription'.

Commodity research the focusing of research on individual crops in
considerable detail.










Component technology the cultural techniques used in the management of
a crop or cropping pattern. Component technologies include variety,
planting method, tilage operations, fertilizer and water management,
pest management, harvesting, etc.

Constraints research Research that aims to identify and rank factors
such as disease, weeds, labor shortage etc. which are limiting
production.

Cropping index number of crops grown/year on a given area of
land x 100.

Crop intensification the concept, approach, method, and process of
growing more crops per year by increasing cropping intensity.'

Cropping intensity total cropped area divided by net area available for
cultivation multiplied by 100.

Cropping intensity index (CII) (Menagay C19753) a time-weighted
land-use index that evaluates the fraction of the total hectare-
months available to the farmer that are used for crop production.

Cropping pattern the yearly sequence and spatial arrangement of crops
on a given land area.

Cropping pattern design the crop configuration or sequencing done on
paper for year-round land utilization at a given area considering
physical, biological, and socioeconomic factors prevailing at that
area.

Cropping pattern testing the growing of a designed cropping pattern
at a given site and evaluating biological stability, agronomic
productivity, and economic profitability.

Cropping system the crop production activity of a farm. It comprises
all cropping patterns grown on the farm and their interaction with
farm resources, other household enterprises and the physical,
biological,- technological, and socioeconomic factors or environments.

Cropping systems research the research activities, mainly in farmers'
fields, that focus on the understanding of farmers' existing cropping
systems; design, testing, and development of new improved cropping
patterns and component technologies for selected environments to
efficiently utilize available farm resources.

Cropping systems research site a contiguous area or several selected
areas representing one or more land types in production environments
that occur over an extensive area, where cropping systems scientists
conduct on-farm research trials with cooperating farmers.









Crop rotation The practice of following the crop located on a
particular site with a different crop the following season.

Cultural practices crop husbandry practices including land preparation,
seed selection, weed control, fertilizer and insecticide application,
water control in the field, etc.

Determinants of cropping patterns environmental factors that influence
the performance of cropping patterns and are not readily modifiable by
changes in cultural techniques of crop production.

Double cropping growing of two crops in sequence in a year on a piece
of land by seeding or transplanting one after the harvest of the other.

Dryland farming cropping systems of farmers in the arid and semiarid
zones.

Environmental complex a union of sites when cropping pattern or crop/
animal determinants are the same.

Environmental factors factors over which farmers have little direct
control, including the physical, biological, and socioeconomic aspects
of their setting.

Extrapolation area adaptation domain of a cropping pattern composed of
land types to which the cropping pattern is adapted.

Factor returns Economics recognizes four factors contributing to all
production processes: land, labor, capital and management. Each of
these factors contributes; therefore, there is a return to each of
these contributions embodied in an output.

Fallow when a crop is not grown on a field.

Farm enterprise an individual crop or animal production function within
a farming system which is the smallest unit for which resource use and
cost-return analysis is normally carried out.

Farmer environment the physical, biological, economic, and socio-
cultural conditions under which the farmer operates his farming
systems.

Farmer feedback The assessment by the farmer of technology or
methodology being tested or demonstrated by extension or research.

Farming system a unique and reasonably stable arrangement of farming
enterprises that a household manages according to well-defined
practices in response to the physical, biological, and socioeconomic
environments and in accordance with the household's goals, preferences,
and resources. These factors combine to influence output and production
methods. More commonality will be found within the system than between









systems. The farming system is part of larger systems and can be
divided into subsystems.

Farming systems research and development (FSRD) an approach to
agricultural research and development that 1) views the whole farm
as a system, and 2) focuses on the interdependencies among the
components under the control of farm household members and how these
components interact with the physical, biological, and socioeconomic
factors not under the household's control. The approach involves
selecting target areas and farmers, identifying problems and
opportunities, designing and executing on-farm research, and
evaluating and implementing the results. In the process, opportunities
for improving public policies and support systems affecting the target
farmers are also considered.

Horizontal revolution in agriculture increased land use by expanding
cultivated land area through the utilization of fallow and marginal
lands and reclaiming culturable waste lands, thereby increasing
land-use intensity.

Infrastructure the supportive features of and economy often provided
by government, but sometimes provided by private industry, such as
transportation, electricity, water, communications and governmental
organizations.

Intercropping growing two or more crops simultaneously in alternative
rows in the same field.

Interplanting all types of seeding or planting a crop into a growing
stand. It is used especially for annual crops under stands of
perennial crops.

Land equivalent ratio (LER) the land area needed under monoculture to
produce the same amount of crop yields as from 1 ha of intercropping
or mixed cropping. LER is computed:

Maize yield in intercropping Peanut yield in intercropping
(2.5 t/ha) + (1.2 t/ha) = 1.50
LER = Maize yield in monoculture Peanut yield in monoculture
(3.0 t/ha) (1.8 t/ha)

Land type a union of locations within which values of cropping pattern
determinants are the same.

Land use patterns alternative ways to utilize available land resources
over time for agricultural production.

Land utilization index (LUI) The number of days which crops occupy
the land during the year, divided by 365.









Maximum cropping the highest possible production per unit area per
unit time without considering cost of production or net return.

Mixed cropping growing two or more crops simultaneously in the same
field without rows.

Mixed farming systems farming systems with integrated crops, livestock,
and other possible household enterprises.

Mixed intercropping growing two or more crops simultaneously
intermingled in the same plot with no distinct row arrangement.

Mixed-row cropping growing two or more crops simultaneously in the
sample plot intermingled within a distinct row arrangement.

Monitoring study making systematic observations through well-designed
procedures on a crop, cropping pattern, farm or experimental trial to
relate resultant effects with observed factors or causes.

Monoculture growing only one crop on the land in a given crop season.

Multidisciplinary approach an approach in which several disciplines
become involved in a project or program with common general objectives.

Multilocation testing the testing of cropping systems technologies
generated at an on-farm research site at other locations within the
large target area to delineate the extrapolation zone as well as to
finally verify technology performance before wide-scale diffusion.

Multiple cropping the practice of growing more than one crop on the
same land in one year. It involves several alternative patterns of
crop arrangement in space and time such as mixed cropping, inter-
cropping, rcroppig, rl dropping, sequential cropping, double cropping, triple
cropping, etc.

Multiple cropping index (MCI) the sum of areas planted to different
crops harvested during the year, divided by the total cultivated area.

Non-inhibiting non-treatment variables Experiments compare treatments,
but in the management of experiments there are many other practices
which are common to all treatments. These are non-treatment or
non-exoer mental variables usually held constant during classical
experimentation, but rarely constant over a group of farms. Classical
experimentation usually fixes the constant level of these variables
high enough to prevent these management practices from inhibiting the
responses of the treatments.









On-farm research and development agronomic and socioeconomic studies
conducted on the farms with farmers' active participation. The goals
are to develop improved cropping system technologies and to devise
ways to combine these technologies with farmers' knowledge and skills
to efficiently utilize the available farm resources.

Optimized The best possible solution.

Perennial crops crops occupying land for more than 30 months not
including legumes and grasses in permanent pastures.

Physical determinants the important attributes of climate, water, and
land such as rainfall, topography, and hydrology that influence
configuration and performance of cropping patterns.

Pilot production program a small-scale (100 500 ha) production
program to determine the support needed in the large-scale diffusion
of recommended technologies as well as to clearly specify the tasks
and interrelationships of different institutions involved in supporting
the farmers. It also allows a final evaluation of the recommended
cropping technology, the cost of its extension to the farmers, and the
expected benefit.

Plot a contiguous area of land planted in a homogeneous manner during
a defined period, normally 1 year.

Plot plan a diagrammatic representation of the spatial and temporal
combination of crops on a plot during 1 year.

Preproduction evaluation higher level on-farm cropping system
activities consisting of multilocation pattern testing and pilot
production program to delineate the final production program area,
verify technology performance,*determine institutional support
requirements, and to help in structuring large-scale production
programs.

Pre-screening Ex ante evaluation of the likely technical and economic
suitability of (treatments) possible solutions which are being
considered for OFE.

Ratoon cropping the cultivation of an additional crop from the
regrowth of stubbles of a main crop after its harvest, thereby
avoiding replanting such as in sugarcane, sorghum, and rice.

Reductionist research The method commonly used in classical
agricultural research whereby all variables except treatments are
held constant to allow precise measurement of reactions to treat-
ments. The method 'reduces' or 'abstracts' out of the real world
to be able to handle the measurements with precision.









Relay cropping growing two or more crops in a sequence, planting the
succeeding crop after the flowering, but before the harvesting of the
preceding crop.

Research-managed trials experiments done in farmers' fields, but
managed by researchers to attain higher .degree of experimental
precision while still getting the effect of some variables existing
on the farms.

Sequential cropping growing two or more crops in a sequence, planting
the succeeding crop after the harvesting of the previous one.

Shifting cultivation a method of cultivation in which several crop
years are followed by several fallow years with the land not under
management during the fallow. The shifting cultivation may involve
shifts around a permanent homestead or village site, or the entire
living area may shift location as the fields for cultivation are moved.

Site description description of an on-farm experiment site with
respect to physical and socio-economic environments and existing
farming systems.

Site selection selecting a contiguous area or several areas
representing one or more production environments that occur over
an extensive target area, to conduct on-farm experiments to develop
improved farming systems technologies for the target area.

Slash-and-burn system a kind of shifting cultivation in high-rai.nfall
areas where the cropping period is followed by a fallow period during
which bush or tree growth occurs. For the next cycle of cropping,
the bush or tree growth is again cleared by cutting and burning.

Socioeconomic determinants factors such as marketing facilities, land
tenure system, and credit, which influence the cropping systems of
a given area.

Sole cropping growing one crop alone in pure stand, either as a single
crop or as a sequence of single crops within the year.

Sondeo An informal survey approach to understanding local farming
systems.

Strip cropping growing two or more crops simultaneously in alternative
plots arranged in strips that can be independently cultivated.

Subsistence farmers farmers who produce mainly to meet family needs and
have little capacity to purchase production inputs or foods.

Superimposed trial a small set of experimental treatments superimposed
on farmers' production plot.








Systems approach studying a system as an entity made up of all its
components and their interrelationships, together with relationships
between the system and its environment. Such study may disturb the
real system itself (e.g. via farmer-managed trials or by comparison
pre-adoption studies of new technology), but more generally is done via
models (e.g. experiments, researcher or farmer-managed on-farm trials
Eor both, unit farms, linear programming and other mathematical
simulations) which to varying degrees simulate the real system.

Target area large priority development area in a country for which
improved farming systems technologies are developed through on-farm
research conducted at sites in that area.

Target group/ recommendation domain Almost the same meaning: a group
of farmers for whom .the same research effort will be relevant
because they are operating the same farming system.

Technical relationships The physical (eg. yield) output resulting from
a given level of one or more inputs (eg. fertilizer).

Testing of cropping systems the process of evaluating designed cropping
patterns and associated component technologies using selected criteria.

'Top down' The imposition or prescription of changes thought to be good
without any understanding of the local situation. 'Prescription without
diagnosis'.

Turnaround time period between the harvesting of the preceding crop
and planting of the succeeding crop in a specific field.

Upland crops crops grown under aerobic soil conditions such as
wheat, maize, peanut, beans, etc.

Vertical revolution in agriculture Maximizing production per unit
land are per unit of time using intensive farming practices, high
production inputs, and improved management practices.

Whole-farm analysis A methodology designed to search for optimal
solutions through incorporation of farmers' objectives, farming.
systems, and resources to arrive at improved cropping and livestock
patterns and management practices for overall farming systems
performance.

Whole-farm approach An essential characteristic of farming systems
research and development in which teams look at a whole farm to
identify problems, opportunities, and interrelationships, to design
and conduct experiments, and to evaluate results.











13.0 APPENDIX B:

MAJOR REFERiENCES USED

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Fernandez, ICRAF, personal communication

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Fresco, Louise. Approaches to the Study of Farming and Cropping








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