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 Precision in net benefit analy...
 Representation of risk
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Title: Unification of statistical and economic analysis : training working document no. 1
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Table of Contents
    Front Cover
        Front Cover
    Title Page
        Title Page
    Preface
        Preface
    Precision in net benefit analysis
        A-1
        A-2
        A-3
        A-4
        A-5
        A-6
    Representation of risk
        B-1
        B-2
        B-3
        B-4
        B-5
        B-6
        B-7
        B-8
    Discrete and continuous analysis
        C-1
        C-2
        C-3
        C-4
        C-5
        C-6
        C-7
        C-8
        C-9
        C-10
        C-11
        C-12
        C-13
        C-14
    Back Cover
        Back Cover
Full Text















UNIFICATION OF STATISTICAL
AND ECONOMIC ANALYSIS

Training Working Document No. I




15;69 /


UNIFICATION OF STATISTICAL
AND ECONOMIC ANALYSIS

Training Working Document No. 1




Prepared by
Roger Mead
Consultant
in collaboration
with CIMMYT staff












CIMMYT
Lisboa 27
Apdo. Postal 6-641,
06600 M6xico, D.F., Mexico







PREFACE


This is one of a new series of publications from CIMMYT entitled Training Working
Documents. The purpose of these publications is to distribute, in a timely fashion,
training-related materials developed by CIMMYT staff and colleagues. Some Training
Working Documents will present new ideas that have not yet had the benefit of extensive
testing in the field while others will present information in a form that the authors have
tested and found useful for teaching. Training Working Documents are intended for
distribution to participants in courses sponsored by CIMMYT and to other interested
scientists, trainers, and students. Users of these documents are encourage to provide
feedback as to their usefulness and suggestions on how they might be improved. These
documents may then be revised based on suggestions from readers and users and
published in a more formal fashion.

CIMMYT is pleased to begin this new series of publications with a set of six documents
developed by Professor Roger Mead of the Applied Statistics Department, University of
Reading, United Kingdom, in cooperation with CIMMYT staff. The first five documents
address various aspects of the use of statistics for on-farm research design and analysis,
and the sixth addresses statistical analysis of intercropping experiments. The documents
provide on-farm research practitioners with innovative information not yet available
elsewhere. Thanks goes out to the following CIMMYT staff for providing valuable input
into the development of this series: Mark Bell, Derek Byerlee, Jose Crossa, Gregory
Edmeades, Carlos Gonzalez, Renee Lafitte, Robert Tripp, Jonathan Woolley.

Any comments on the content of the documents or suggestions as to how they might be
improved should be sent to the following address:

CIMMYT Maize Training Coordinator
Apdo. Postal 6-641
06600 Mexico D.F., Mexico.







Document 1A


PRECISION IN NET BENEFIT ANALYSIS

This document is intended to be used in parallel with the Cimmyt Economics Training Manual "From
Agronomic Data to Farmer Recommendations". References to tables or figures in that manual will be in the
form Table M3.1

1. Basic Precision

All the agronomic data which are used in economic analysis are derived from experimental data.
Consequently all such data is to some degree imprecise. That is, the values calculated as mean yields for
particular treatments are estimates of the population mean yields for those treatments. The precision of the
estimates is represented by the standard errors of the mean yields, which are usually obtained from an
analysis of variance of the experimental data. For the data from the weed control experiment of Table M3.1
the analysis of variance is shown in Table 1.

Table 1: Analysis of variance of weed control data


Yields (kg/ha)
Location 1


Treat 1
Treat 2
Treat 3
Treat 4


Block 1
2180
3030
2440
3200


Analysis of Variance
Source SS df MS


Block 2
2220
2570
2180
3060


Location 2

Location 3

Location 4


Blocks
Treatments
Error


84000
282300
66100


22033


Error Mean Square = 13200

Error Mean Square = 34900

Error Mean Square .= 22900


Location 5 Error Mean Square = 37000

Since these error mean squares are fairly homogeneous we can calculate a combined analysis as follows;


Source SS df MS


Blocks(for
(each location)
Treatments

Locations

Treat x Location

Combined Error


184350

2507920

32045540


553272

174100


36870

835973

8011385

46106

11607









We note the evidence of very large treatment and location effects and the non-negligible treatment by
location interaction. However our only purpose now is to use the error mean square to estimate the
precision of treatment means. The error standard deviation is 108 which is less than 5% of the overall
mean. This is unusually low; in most on-farm experiments we might expect values between 10% and 20%
and even higher values do occur.

The standard error of a difference between two means (derived from a total of 10 plots) is calculated as

4(2(11607)/10) = 48.

If we compare the treatment means using this standard error we have


Treatment 1 2 3 4


1994 2444 2084 2600


Standard error
of difference
based on 15 df
48


Comparison of the treatment means shows that all treatment means differ significantly at the 5% level
except for the difference between treatments 1 and 3 which is close to significance.

2. Precision of Net Benefits

All the calculations for Net Benefit Analysis and Marginal Analysis are based on the initial yield data and
the precision of derived quantities may be calculated from the standard errors of the initial mean yields in
the same way that the derived quantities are calculated. Thus in the calculations for the Partial Budget we
first calculate the adjusted yields (80% of the initial mean yields for the weed control example); then the
gross field benefits(x$8/kg for the example); then the net benefits by subtracting the total costs that vary.
The corresponding calculated standard errors for the example would be:


1) for adjusted yields;
2) for gross field benefits;
3) for net benefits;


80% of that for initial yields
x$8 of that for adjusted yields
unchanged since the total costs that vary are not subject
to precision estimation.


For the weed control example the results would appear as:


Treatment 1 2 3 4 Standard
error(15df)
Mean yield (kg/ha) 1994 2444 2084 2600 48
Adjusted yield (kg/ha) 1595 1955 1667 2080 38
Gross field benefits ($/ha) 12760 15640 13336 16640 307
Total variable costs (S/ha) 2400 3875 3200 4675 0
Net benefits($/ha) 10360 11765 10136 11965 307


Notice that although the standard error in this example is small the significance of differences between
treatments changes for the different estimated quantities. For the initial mean yields all differences are
significant at 5% except for that between treatments 1 and 3. The same is true for the adjusted yields and
the gross field benefits. However the net benefits show a different pattern, the differences between








treatments I and 3 and between treatments 2 and 4 both being small compared with the standard error. The
net benefits for each of treatments 2 and 4 are still significantly different (5%) from those for each of
treatments I and 3.

This change of significance for different quantities is very common and indeed we should expect it. The
difference between treatments 2 and 4 for gross field benefits is 1000$/ha. The same difference for net
benefits is reduced by 800S/ha because of the differential costs but the precision remains unchanged. Such
a change in significance will appear surprising only if we allow ourselves to interpret "significance" as
implying a real difference while "non-significance" is assumed to imply no difference. This is not a valid
interpretation of significance which measures the strength of the evidence for a difference. In the example
we should be strongly convinced that the gross field benefits for treatments 2 and 4 are different, but the
size of the difference is not much more than the difference in the variable costs and so we have little
grounds for believing that the net benefits for these two treatments are much, if at all, different.

3. Precision of Marginal Rates of Return.

The standard errors for marginal rates of return may also be derived from those for net benefits though now
the standard errors for MRR comparing different treatments will be different. The MRR irom treatment 1
to treatment 2 is calculated as the marginal net benefit divided by the marginal cost. We know the standard
error of the marginal net benefit(the standard error of the difference between any two net benefit values)
and so the standard error of the MRR is simply the standard error of the marginal net benefit divided by the
marginal cost.

The MRR for treatment I to treatment 2 is calculated as

(11765 -10360) / (3875 -2400) = 1405/1475 = 0.95.

The standard error of this MRR is calculated as

307 / (3875 -2400) 0 307/1475 = 0.21.

The corresponding calculations for the MRR for treatment 2 to treatment 4 are:

MRR = (11965 -11765) / (4675 -3875) = 200/800 = 0.25

Standard error = 307 / (4675 -3875) = 307/800 = 0.38.

We can also calculate the MRR for treatment 1 to treatment 4

MRR = (11965 10360) / (4675 -2400) = 1605/2275 = 0.71

Standard error = 307 / (4675 -2400) = 307/2275 = 0.13.

In significance terms we should be strongly convinced that the MRR for treatments 1 to 2 is greater than
zero and moreover is also just about significantly(5%) greater than a critical MRR value of 0.5.

We can construct confidence limits for the MRR. For example the 95% confidence limits for the MRR for
treatments I to 2 are

0.95 2.13 x 0.21 giving (0.50, 1.40).

The 90% confidence limits for the same MRR are

0.95 1.75 x 0.21 giving (0.58, 1.32).








The corresponding 95% limits for the other MRR values are:


MRR(2 to 4) 0.25 2.13 x 0.38 giving (-0.56,1.06)

MRR(1 to 4) 0.71 2.13x 0.13 giving (0.43, 0.99).

Note that the MRR(1 to 4) is the most precise because it is based on the largest cost difference. However
although it is even more convincingly different from zero than is the MRR(I to2) it is not so convincingly
different from a critical MRR value of 0.5.

4. Graphical Representation of Precision

Much of the interpretation of precision information, and more generally of net benefits and marginal rates
of return, is simplified by the use of graphical presentation of information. The basic structure of the
graphical presentation is the net benefit/total variable costs diagram (as used in figure M4.1). For simplicity
we first consider, in Figure 1, the case of two treatments, using treatments 1 and 4 of the weed control
example. The precision information is presented as 95% confidence limits (other % confidence limits or
simply standard errors could be used provided it is clear which form is being used).

The confidence limits for net benefits are shown in the form for marginal net benefits; that is for
differences between the net benefits for two treatments. When we now consider the confidence limits for
the MRR between treatments I and 4 we draw lines from the point value for the lower net benefit to the
95% confidence limits set about the upper net benefit. This correctly allows for the precision of the
marginal net benefit and shows the slopes of the MRR corresponding to the confidence limits (0.43 and
0.99) calculated in the previous section.

More usefully we can display the MRR rates and their confidence limits for the sequence of intermediate
(non-dominated) treatments, showing the MRR rates for each section of the net benefit curve. In Figure 2
the MRR rates for treatments 1 to 2 and for treatments 2 to 4 are shown, each with its corresponding 95%
confidence limits We can see how the range of credible MRR values is narrower when the cost difference
is larger (compare also with Figure 1, where the cost difference is largest). The MRR(Ito2) is sufficiently
well estimated that the whole confidence interval is above the critical level of 0.5. In contrast, the
MRR(2to4) is poorly estimated and includes a substantial range of negative MRR values.

5. Use of Critical MRR Lines

Another addition to the net benefit/variable costs diagram is the inclusion of lines representing the critical
MRR value. In figure 3, which is based on data from a Nitrogen response experiment (Table M6.2) four
treatments are compared for which the net benefits and variable costs are as follows:


Net Benefits Total costs that
($/ha) vary ($/ha)

Treatment I OkgN/ha 400 0
Treatment 2 40kgN/ha 486 30
Treatment 3 80kgN/ha 526 60
Treatment 4 120kgN/ha 535 85


Two lines representing the critical MRR value of 1.0, or 100%, are drawn. The first is drawn from the point
representing the lowest cost/lowest yield treatment point and any treatment point above that line has an
MRR value greater than 1.0 when compared with the lowest treatment The second critical MRR line is








drawn backwards from the maximum yield treatment point and any treatment point in the triangle (A) is
superior to the treatment giving the highest yield since the MRR from such a point to the maximum point
must give an MRR rate less than the critical value of 1.0.

For this set of nitrogen response data there are two intermediate treatments (40kgN/ha and 80kgN/ha)
either of which should be preferred to the maximum treatment. In order to assess, from the diagram, which
of these should be chosen, we draw a third critical MRR line backwards from the higher (80kgN/ha) of the
two alternatives and examine whether the lower treatment point falls above or below this new critical MRR
line. It can be seen immediately that the lower point (40kgN/ha) is below the critical line drawn relative to
the 80kgN/ha and hence we should choose the 80kgN/ha treatment as the best recommendation for farmers.

6. Confidence for Differences

Finally we can develop the ideas of the precision of estimated marginal net benefits and MRR rates to
calculate the confidence that an MRR rate for any particular treatment is greater than the critical MRR rate.
Consider the vertical gaps between the treatment points and the lower critical MRR line. Each gap
represents the best estimate of the net benefit advantage over the minimum acceptable MRR. For the
40kgN/ha treatment the gap is

486 -(400 + 30) = 56.

Using the standard error of a difference of net benefits between two treatments we can calculate the
confidence probability that the gap is genuinely greater than zero.

Suppose the standard error for a difference between the net benefit values for two treatments is 30 $/ha
(based on a large number of degrees of freedom since the mean benefits are derived from experiments at 20
locations). Then the ratio of the advantage over the critical MRR line divided by the standard of the
marginal net benefit is:

for 40 kg N/ha (486 430) / 30 = 1.87

for 80 kg N/ha (526 460) / 30 = 2.20

for 120 kg N/ha (535 485) / 30 = 1.67.

The confidence probabilities can be read from tables of tail probabilities for the Normal distribution
(because the df are large); a short version of the appropriate table is attached as an Appendix, The results
are:

Confidence for advantage with 40 kg N/ha being greater than the critical MRR rate is
0.9693 or almost 97%;
Confidence for advantage with 80 kg N/ha being greater than the critical MRR rate is
0.9861 or nearly 99%;
Confidence for advantage with 120 kg N/ha being greater than the critical Mrr rate is
0.9525 or just over 95%.

We would usually be happy with any of these confidence levels but clearly the greatest confidence attaches
to the advantage for 80 kg N/ha. Because the standard error for marginal net benefit is the same for all
treatment pairs the confidence level depends only on the gap between the treatment net benefit and the
corresponding point on the critical MRR line. Hence any treatment above the upper critical MRR line (in
triangle (A) ) will have a higher level of confidence than that for the maximum treatment. Essentially the
treatment that would be preferred from traditional MRR arguments will also be preferred because it has the
greatest confidence attached to its advantage over the critical MRR rate.








Finally we return to the weed control data to calculate the confidence for the advantages over critical MRR
rates of 0.25, 0.5 or 0.75. The principles of the calculation are exactly as before except that we use tail
probabilities based on the t-distribution because the degrees of freedom on which the standard errors are
estimated are only 15. (Brief tail probabilities are tabulated in the Appendix).


Treatment
2 4


Critical MRR z = (11765-10729)/307 z = (11965-10929)/307
rate = 0.25 = 3.37 = 3.37
Confidence P = 99.8 Confidence P = 99.8

Critical MRR z = (11765-11098)/307 z = (11965-11498)/307
rate = 0.5 = 2.17 = 1.52
Confidence P = 97.7% Confidence P = 91.0%

Critical MRR z = (11765-11466)/307 z = (11965-12066)/307
rate = 0.75 = 0.98 = -0.33
Confidence P = 83.2% Confidence P = 37.1%


For low critical rates both treatments show strong confidence (and if a zero critical MRR rate were
considered treatment 4 would have a minutely stronger confidence). However for higher critical MRR rates
the advantage of treatment becomes increasingly pronounced.

7. Precision and Variability

The methods in this paper relate to the use of information about the precision of estimation of yields,
benefits, etc. We are not discussing the variability of results across locations, which is an important subject
in its own right and is considered elsewhere. It may be noted, however, that the principles of the methods
developed in this paper could be used with measures of variability in place of precision.









Document IB


REPRESENTATION OF RISK

Assume that we are considering a recommendation to farmers to change from Treatment A to Treatment B
on the basis of results from a substantial number of trials. The Marginal Rate of Return, calculated from the
mean yields derived from the trial data, has been shown to provide an acceptable benefit from the proposed
change. We would now like to examine the information about the variation of the returns over the set of
trials. Essentially we wish to assess the probabilities of inadequate returns and, hopefully, identify those
situations in which the inadequate returns occur.

The approaches described here will be illustrated for two sets of data. Initially we consider alternative
methods for representing risk illustrating the methods for the data in Table 8.2 of the CIMMYT Economics
Training Manual "From Agronomic Data to Farmer Recommendations". The second data set is from
Zimbabwe and is drawn from training notes written by Allan Low. In this data set differential weights are
allowed for the different observations.

1. The Variation in the Initial Data

Consider first the data for the two treatments, OkgN and 80kgN at twenty locations. The Net Benefits are as
shown.


Net Benefits ($/ha) Net Benefits ($/ha)
Location OkgN 80kgN Location OkgN 80kgN


1 441 655 11 542 562
2 511 647 12 512 681
3 383 277 13 285 291
4 391 610 14 387 578
5 250 593 15 375 230
6 322 619 16 494 661
7 490 660 17 485 660
8 458 600 18 295 480
9 180 162 19 485 683
10 250 612 20 463 260


To examine the joint pattern of pairs of benefits by location we plot a graph, shown in Figure 4, of the yield
for 80kgN against the yield for OkgN. The diagonal line through the origin represents equal benefits for the
two treatments, so that points above the line are those for which the 80kgN treatment gives the higher
benefits. This graph shows at once an unusual feature of this data, namely the division into two subsets: the
main group of 15 observations towards the top of the graph and the smaller group of 5 towards the bottom
and generally below the equality line. From this graph we observe also that the highest benefits are
obtained from the 80kgN treatment and that the lowest benefits occur for both treatments with slightly
more for 80kgN.

2. Risk Assessment Ignoring Location Pairing

Traditionally most of the methods of representing risk are based on the separate samples of data for the two
treatments to be compared. The minimum returns analysis, described in the Training Manual, compares the
averages of the 25% lowest benefits for each treatment. Two alternative representations utilising the
complete data samples are the comparison of Cumulative Distributions and the Relative Risk Diagram.








2.1. Cumulative Distributions


The distribution of benefits for each treatment sample can be displayed in a Cumulative Distribution. This
is constructed by first listing the sample values in increasing order. Using the sample for OkgN we get

180,250,250,285,295,322,375,383,387,391,441,458,463,485,490,494,511,512,542.

Now we plot the proportion of values less than each value against that value. Thus there are no values less
than 180. There is one value (out of 20) of 180 and the next value is 250 so that the proportion of values
less than any value between 181 and 250 is 0.05. There are two values at 250 so the proportion less than
any value between 251 and 285 is 0.15. From 286 to 295 the proportion of values less than that value is
0.20, and so on. The cumulative distributions for the OkgN and 80kgN samples are shown in Figure 8.

By superimposing the cumulative distributions for the two treatments, as in Figure 8, we can compare the
ranges and distribution patterns of benefits for the two treatments. The pattern at the lower level of benefits
($150/ha to $300/ha) is very similar for the two treatments. The 80kgN treatment shows a clear advantage
of about $150/ha at each proportion in the upper 70% of the distributions. Ths reflects the results for both
the minimum returns analysis and average benefit increase in the Training Manual, showing little
difference in benefits for the lowest 25% of results for each treatment but at the same time a large overall
average benefit increase of $126/ha.

2.2. The Relative Risk Diagram

A related technique displaying the advantage of one treatment against the other is the Relative Risk
diagram (Mead et al; 1986). For this we require the two samples to be ordered together, but maintaining the
identification of each value by its treatment.


OkgN 180 250 250 285 295 322 375 383
80kgN 162 230 260 277 291

OkgN 387 391 441 458 463 485 485 490 494 511 512 542
80kgN 480

OkgN
80kgN 562 578 593 600 610 612 619 647 655 660 660 661 681

OkgN
80kgN 683


The Relative Risk Diagram, shown in Figure 9, is constructed by moving through the joint ordering,
counting along the horizontal axis of the graph the occurrences of the OkgN sample, and up the vertical axis
for the occurrences of the 80kgN sample. The levels of benefit are shown on the diagonal line (with a non-
linear scale reflecting the benefit values which actually occurred in the sample). The diagonal line
represents equal risks. For all critical levels of benefits in the range of interest this diagram shows the
comparative probabilities, for each treatment, of benefits lower than any critical level

Thus, the first (lowest) benefit is 162, which is for the 80kgN treatment so that between benefit levels of
162 and 179 there is a 1/20 (=0.05) chance of a low yield for 80khN and a 0/20 chance for OkgN. After the
first five values we have reached the benefit level of 250 and the chances are 2/20 (=0.10) for 80kgN and
3/20 (=0.15) for OkgN. After ten values we have reached a benefit level of 295 and the chances are 0.25 for






both OkgN and 80kgN. After twenty values we have reached a benefit level of 485 and the chances are 0.30
for 80kgN and 0.70 for OkgN.

The main advantage of this diagram is that it emphasises the direct comparison of risks. The deviation of
the relative risk curve from the diagonal shows the size of the difference in risk. Up to a benefit level of
about S300/ha the risks are similar for the two treatments, with slightly greater risks with 80kgN (the graph
tending to be marginally above the diagonal). From benefits of $300/ha up to about $500/ha the risk of
inadequate benefits increases steadily for OkgN but hardly changes for 80kgN. At the critical benefit level
of $550/ha the relative risks are read off the graph by moving perpendicularly from the diagonal line and
are 100% and 30%. The equivalent information can be extracted from Figure 8 making vertical
comparisons between the two cumulative distributions, but that diagram does not display the magnitude of
the risk differential so clearly.

Various patterns are possible in the relative risk diagram. A linear section of the relative risk curve through
the origin, lying between one of the axes and the diagonal line represents a constant ratio of relative risk,
the ratio of the risks being measured by the slope of the line. A section of the relative risk curve parallel to,
but displaced from, the diagonal line represents a constant difference between the two risks. A horizontal,
or vertical, section indicates rapid increase in one risk with increasing benefit level with no change in the
other risk.

3. Risk Assessment Using Location Pairing

A disadvantage of the methods of the previous section is that they ignore the pairings of the results for each
location. The methods could be applied equally well to data collected at two different sets of locations for
the two different treatments. Not only do we fail to use the fact that the two treatments were both observed
at each location but we formally assume that there is no relationship between the results for the two
treatments. This is usually a most unrealistic assumption and, although the relationship shown in Figure 4
is, as has been noted previously, unusual, it does not suggest that the two sets of benefits should be
assumed to provide independent information.

3.1. The relationship of Change to Current Practice

When the principal interest is in the change in benefit between two treatments, it is often beneficial to use
an alternative form of Figure 4 in which the difference between the two benefits(80kgN OkgN) is plotted
against the average benefit at each location. However from the point of view of the farmer deciding
whether to change about from OkgN to 80kgN the average of the two benefits is not a very appropriate
measure. Instead we should plot the change in benefit against the net benefit for the OkgN treatment at each
location. This graph, which is a skewed rotation of Figure 4, is shown in Figure 7.

This graph displays how the advantages and risks of a change from OkgN to 80kgN are distributed across
the range of present practice, as represented by the benefits for each location of the OkgN treatment. We
can count directly from the graph the proportion of locations for which 80kgN gives a higher benefit than
OkgN,(16 out of 20), and the corresponding risk of 4/20 of failing to achieve an increase in benefit. If a
critical MRR of 100% is assumed, as in the discussion of Table 8.2 in the Training Manual, then the
minimum acceptable increase in net benefits would be $60/ha and again we can count the proportion of
locations giving increases of at least this level (14 out of 20). It may be helpful to draw the line of minimal
acceptable increase on the graph, as shown in Figure 10.

We can also assess whether the risk/advantage is evenly spread over the range of benefits achieved from
the OkgN treatment. The six locations where negative changes or small positive changes occur are fairly
evenly spread. The positive changes in benefit greater than the critical MRR do show a trend, however,
with the largest advantages tending to occur at the lower OkgN performance level.






Because Figure 7 contains all the points for the sample of individual locations the scatter of observations
makes the pattern less easy to perceive. This is a situation where the smoothing techniques, developed in
time series analysis of sequences of economic and other data, may be helpful. There is a wide range of
smoothing methods available and the one used here is a relatively simple one chosen to illustrate the
general philosophy rather because it provides the best results. A smoothed version of Figure 7 is
constructed by considering a series of intervals of the horizontal axis and for each interval calculating the
average change in that interval. Thus for the interval (300 400) there are five points


OkgN Benefit Change in Benefit


322 +297
375 -145
383 -106
387 +191
391 +219


The average change in benefit over this interval is

(297 -145 -106 +191 +219)/5 = 91.2.

By calculating the average change for intervals of width 100 centred on 175, 200, 225,525, 550, we
produce, in Figure 8, the desired smoothed version of Figure 7.

The result shows a pronounced peak (caused mainly by two locations) at a OkgN benefit level about
$250/ha followed by quite a strong dip around OkgN benefit levels around $350 to $400. This, in turn, leads
to a more steady plateau of $100 change in benefit for the higher levels of OkgN benefit. At both ends the
information is, of course, based on very few values.

There are three main influences on this form of diagram. The fact that we are plotting (y-x) against (x) will
tend to produce a negative slope arising from the negative correlation of random errors in the absence of
systematic patterns. At very low yield environments neither treatment can produce good yields and the
yield difference must be small, with the change in benefit being negative. Similarly it might be expected
that at very high yield levels the change in benefit would tend to decrease. In the middle range of the curve
a horizontal section indicates that the improved treatment is maintaining its advantage over the standard, in
defiance of the natural tendency to negative slope. If the slope tends to be positive then the improved
treatment shows an even greater advantage against the natural negative correlation.

Thus the surprising aspect of Figure 8 is not the initial slightly negative value when both treatments
produce poor yields nor the early peak, but the dip between the peak and the final plateau. The three
locations which cause this are seen more clearly in Figure 7 and represent a proportion of medium to good
environments where the improved treatment simply does much worse than elsewhere. There is no trend but
a clear split into good and bad results for the improved treatment.

3.2 Cumulative Expressions of Advantage and Risk

It is sometimes more meaningful to discuss the average performance.for all locations below a particular
threshold than to consider each location separately. The idea of the cumulative distribution in section 2.1
exemplifies this concept (in contrast to a histogram of benefits). We therefore consider another
modification to Figure 7 to consider the cumulative average benefit increase instead of the individual
values.








To construct the cumulative mean advantage graph, shown in Figure 9, we order the locations in increasing
value of OkgN benefit For each location we then calculate the average change in benefit for all those
locations with the same or lower value of benefits from OkgN. The resulting cumulative average
advantages show how the advantage of the improved treatment develops as the yield environment is
gradually improved. The first few steps of the calculation are shown.


Locations in
order of OkgN benefit Changes to be averaged Mean


9 180 -18 -18
5 &10 250 -18,+362,+343 +229
13 285 -18,+362, +343, +6 +173
18 295 -18, +362, +343, +6, +185 +176


After the initial small negative change there is a sharp rise to a value over 200 and then the cumulative
mean advantage settles down, quite quickly to its final level around 130.

Finally, as well as calculating the cumulative mean advantage in Figure 9, we can examine the cumulative
risk, calculated in a parallel manner. As before the locations are listed in order of ascending benefit at
Okgn. For each level of OkgN net benefit we count the proportion of those locations with the same or lower
value of benefits from OkgN which give a positive advantage. A second expression of this form of risk is to
count the proportion of such locations giving a change greater than the 60S/ha required to surpass the
critical MRR. The calculations are again shown for the first steps of the graph.


Locations
in order Changes to be counted Proportions
of Okg N benefit above zero above 60
Okg N benefit

9 180 -18 0/1 0/1
5 & 10 250 -18,+362,+343 2/3 2/3
13 285 -18, +362, +343, +6 3/4 2/4
18 295 -18,+362, +343, +6, +185 4/5 3/5


The resulting graph, with both cumulative proportions is shown in Figure 10.

Both cumulative proportion curves, after the settling-down process of the first two points, show very steady
proportions of about 0.7 and 0.6 respectively, slowly tending upwards to their final values of 0.8 and 0.7.
Thus, like the cumulative mean advantage graph the pattern is of a consistent average level after very few
initial variations and with a slight late upward trend.

In both Figures 9 and 10 we would be pleased to find that the cumulative values settled down quickly. The
cumulative advantage curve in Figure 9 is subject to the same tendency to a downward trend caused by the
negative correlation between (y-x) and (x) as was expected in Figure 8. The interpretation of patterns in
Figure 9 are, generally, very much the same as in Figure 8 except that the cumulative plotting should
reduce the scope for meandering. In fact we do get the negative trend in Figure 9 suggesting that there is a
generally consistent level of advantage, the more extreme peak in Figure 8 being caused by the
exceptionally high advantage of the improved treatment in those locations where the OkgN benefit was low.







The cumulative proportion graph in Figure 10 should also be expected to settle to a steady level and there
should not be the same expectation of a negative trend because the correlation does not apply to the same
extent for counts. Unlike the cumulative advantage curve, the cumulative proportion curve must show, at
least, some jerkiness because each successive observation must go up or down from the previous one.

4. The Zimbabwe Data (Using Weighting)

The second example set of data is from a worked example attached to teaching notes on "Application of
Risk Decision Theory to Net Benefit and MRR Analysis" by Allan Low. There are eighteen sample
locations regarded as representative of a potential domain population. Within the total population it is
estimated that the proportions of "Good", "Medium" and "Poor" locations are 30%, 50% and 20%
respectively. The eighteen sample locations are classified as being six from the "Good" locations, six from
the "Medium" locations and six from the "Poor" locations. It is therefore decided to weight the
observations in the three groups, according to these estimated overall proportions.

Thus the six good sample locations should predict for 30% of the total population; the six medium sample
locations for 50% of the total population; the six poor sample locations for 20% of the total population. The
practical effect of this differential importance is achieved by allocating weights of 3,5 and 2 to the
observations from the three groups to indicate their relative importance ...

This use of weighting is unusual but, provided there are clear grounds for assessing the weights, it offers a
method of using knowledge about the degree of typicalness of sample values. Most commonly we have a
sample which is "randomly" chosen, if randomness is interpreted in the context of various practical
restrictions. We then have to treat each sample as equally informative. However we may know that our
sampling proportion has been different in different areas, deliberately or by the accidents of loss of results.
Weighting is an attempt to correct unbalanced sample proportions. Not weighting would imply acceptance
of the sample proportions as representative of the population proportions. Weighting may also be
appropriate as a mechanism for using subjective judgements about the extremeness or unusualness of
different years.






The data for benefits and benefit changes in the 18 locations are shown.


Class Net Benefit Change in
OkgN 60kgN Benefit Weight


Good 219 432 +213 3
359 460 +101 3
84 243 +159 3
275 498 +223 3
407 508 +101 3
161 479 +318 3

Medium 269 251 -18 5
221 262 +41 5
434 640 +206 5
246 339 +93 5
172 240 +68 5
255 324 +69 5

Poor 14 -75 -89 2
101 164 +63 2
29 9 -20 2
74 102 +28 2
191 154 -37 2
184 94 -90 2


The same seven forms of graphs (numbered 11, 12, 13, 14, 15, 16 and 17) are presented for this second set
of data and additional explanation will be given only where the weighting requires modification of the
construction procedure. Discursive comments will also be restricted to points of contrast with the previous
example.

Figure 11: Joint Variation of the Two Treatments.

The weights for each location point are represented by numbers (3,5,2). The pattern is much more
homogeneous than in the previous example with points around the diagonal of equal benefit for low benefit
levels and tending increasingly above the diagonal for higher benefit levels.

Figure 12: Cumulative Distributions

The counting process includes the weights so that the plot shows the proportion of the total weight (60)
associated with the locations giving benefits less than each value. The first few calculations are shown.


Ordered values
for OkgN
Weights

Cumulative
Proportion
of Weight


14 29 74 84 101 161

2 2 2 3 2 3

2/60 4/60 6/60 9/60 11/60 14/60


172

5

19/60







Figure 13: Relative Risk


Modified in exactly the same way as Figure 12. The result, though, is more clearly different with a fairly
consistent differential of about 0.25 between the risks for 60kgN (lower risk) and OkgN (higher risk) once
the level of benefits is over 200.

Figure 14: Change v. Current Practice

Modified as for Figure 11. Comments as for Figure 11.

Figure 15: Smoothed Changes

Instead of using simple averages in each interval we use weighted averages. The resulting graph shows less
variation of trend than was achieved for the previous example, though the scatter in Figure 10 has not been
fully smoothed out. There is a clear initial negative benefit change, which is converted into a positive
benefit change by the stage of a current net benefit level of 100; the change then maintains its level with a
very slight upward trend. As mentioned earlier the negative change for the lowest environmental conditions
is quite common. There is no evidence here of the negative correlation, the change in benefit showing a
steady slight increase.

Figure 16: Cumulative Mean Advantage

As for Figure 15 weighted averages are used instead of simple averages. Also a much simpler pattern
showing an initial loss at low OkgN benefit values changing steadily to a gain of about 75 maintained at
that value.

Figure 17: Cumulative Proportion

Modification exactly as for Figure 12. Again a smoother and simpler pattern of change.







, Document IC


DISCRETE AND CONTINUOUS ANALYSIS

Introduction

One important group of experimental research investigations, with consequent recommendations, is
concerned with the effects of differing levels of a quantitative input factor. The most common example is
the investigation of the effect of nitrogen fertilisers and the discussion here will be written in the context of
a nitrogen response relationship. An example of a detailed analysis of nitrogen responses is contained in
document 2B (Mead, 1990c).

A major decision for the experimenter and analyst is whether to develop the research in terms of a discrete
or continuous model. Obviously the underlying model is a continuous one. Any amount of nitrogen could
be applied. But in an experiment or for a recommendation a finite set of alternatives will inevitably be
considered. I think there are three separate stages of experimentation and analysis, for each of which we
have to ask the question "Discrete or Continuous?",

(i) the choice of experimental levels of nitrogen,

(ii) the analysis of the experimental data,

(iii) the calculation of net benefits, marginal rates of return, and the framing of recommendations.

I believe that these questions involve different principles and, although linked, should be considered
independently. Thus it is not, for example, necessary that the sets of alternative levels for the choice of
experimental treatments and for possible recommendations should be identical.

1. The Choice of Experimental Levels

Obviously the experimental levels must be a discrete set. Statistical theory is quite clear that to investigate
the response to a quantitative factor the maximum information is achieved by using the minimum number
of levels compatible with the requirement of being able to estimate the appropriate response function. In
practice most response functions include three parameters and to allow estimation three levels of N are
needed; to also be able to assess whether the response model provides an acceptable fit we need a fourth
level. More levels dissipates the information and gives less efficient estimation of the response function.

The second choice is which levels and again statistical theory indicates clearly that the levels should be
chosen to cover as wide a range as possible, subject to the proviso that the form of the response curve
should be credible over the entire range of levels. In practice this usually means that we take four equally-
spaced levels starting with zero with the third level about the expected (economic) optimum N level.

2. The Analysis and Summary of the Experimental Data

Since the response being investigated is truly continuous and the experimental levels are simply a
representative sample the summary should be in terms of a fitted response function. Statistical analysis
through comparison of pairs of treatments with assessment of significance of treatment differences makes
no logical sense since if there is any pattern of response at all no Null Hypothesis of a zero difference is
credible.

The most suitable response curves are those which allow for non-symmetry of the response. If a quadratic
response is preferred then a quadratic in the square-root of N will probably provide a better fit (measured in
terms of the residual mean square or (l-R2), rather than R2) than the ordinary quadratic. Inverse
polynomials have been found to give generally better fits than ordinary polynomials. From the fitted







response, with estimates of precision, we can predict the response at any level, including any of the
experimental levels, more accurately than from the individual experimental treatment mean yields (again
see document 2B, section 2).

3. Calculation of Net Benefits, etc.

The assessment of possible recommendations in terms of a set of discrete alternatives has obvious intuitive
appeal. It can be presented as a sequence of decisions each of which involves a sufficient change for the
precision of the marginal rate of return to be reasonably precisely estimated (see document IA). However
the set of possible alternatives is not necessarily the same as the set of experimental levels. We might well
wish to consider more alternatives for recommendation than we should wish to use as experimental levels.

The calculation of net benefits will be considerably more accurate if the predicted values from the response
curve fitted to the experimental mean yields are used. This is essentially a smoothing argument. For
example, it would be generally accepted that the yield response to a sequence of equally-spaced levels of N
will follow a pattern of decreasing increments. Experimental treatment means, because of their associated
standard errors will frequently produce successive yield increments which appear not to adhere to this
expectation. The standard error of the fitted estimate of the yield difference between the two middle levels
of four equally-spaced levels is less than half that of the observed difference between the experimental
treatment mean yields.

4. Summary

We should use

(i) three or four widely spread levels of N for the experimental treatments,

(ii) an appropriate response curve to summarise the experimental data,

(iii) the predictions from the fitted response curve when assessing benefits, and

(iv) a set of discrete alternative levels of N when determining the best recommendation.















Figure 1


Marginal
Net benefits Net benefits

12500-

-2000
MRR 0.99

12000 -

1500


11500 -
M1100 0.71
1000


11000
MRR = 0.43 500


10500

(-0


10000 -


Total costs that vary















Figure 2


Net benefits

12500-




12000 -




11500 -




11000 -




10500




10000


Marginal
Net benefits


2000




1500




1000




-500




0


2500 3000 3500 4000 4500
Total costs that vary










Figure 8


N


Marginal
et benefits Net benefits
540 140
80Kg


520 120




500 -- 100


40 g
480 80


Critical
460 / MRR 1.0 60


OKg

440 Critical MRR 1.0 40



420 20



400 4 0


I I I I
0 20 40 60 80 100
Total costs that vary







80 kg N
Net benefit


Figure 4


Joint variation


800




600




400




200


0 100 200 300 400 500


Cumulative distributions


1.00




0.75




0.5




0.25


0 kg N/ha


OkgN
Net benefit


600


Figure 5

80 kg N/ha


Net benefits




Figure 6


Relative risk


80 kg N
benefits
1.0 ---


0.75




0.5




0.25


0 0.25 0.5 0.75


1.0
0 kgN
Net benefit


Change in
benefit


300 -




200 -




100 -


0




-100 -


-200 -


Figure 7


**

S Marginal
benefit
----------- =60




100 a 200 300 400 500 600


0kgN
Benefit







Figure 8


Change in
benefits


300




200




100




0




-100


Smoothed


300 400 500 600

0kgN
Benefits


Figure 9


Change in
benefit


300




200




100




0


Cumulative mean advantage


300 400 500


600

0kgN
Net benefit


200


-100




















Cumulative proportion < value


1.00




0.75 -




0.5-




0.25 -





0 100 200 300 400 500 600

0kgN
Net benefit


Figure 10












60 kg N
Net benefits


600




500




400




300




200




100




0O




-100 -


300 400 500 600


Net benefits
0kgN


Figure 11


3 5


100


200






Cumulative distribution
S (weighted)


-100


Figure 12


Low Zimbabwe example


Net benefit


I I I I0 I
0 100 200 300 400 500


600
Net benefit


Relative risk diagram
(weighted)


1.00




0.75


60 kg N


0.5




0.25


Figure 13


Low Zimbabwe


1.0




0.8 -




0.6 -




0.4 -




0.2 -


OkgN


60kg N


0 0.25 0.5 0.75
0kgN










Change in
benefits


300 -




200 -




100




0




-100 -


Change against 0 kg N
(weighted)


3


2 100
2 100


1
200
2


I 3
5 300


Change in


300




200


100


-100


Smoothed differences
(weighted)


200


Figure 15





















600

0kgN
Benefit


Figure 14


1
500


0 kgN
Benefit







Figure 16


Net Benefit
advantage
(60 kgN zero N)

100









50









0









-50


-100


Cumulative mean advantage on 0 kg N
(weighted)


100 200


300 400


Low Zimbabwe


-a

















--- I
500


Net benefits
(Zero N)





















Figure 17
Cumulative proportion
0kgN
1.00-




0.75 -




0.5-




0.25 -





0 100 200 300 400 500


Net benefit















































































1104,111,I(II)mll 6. (h' U ll" y
M.11"' 11141 VVIw.11 llllpl)Vi'1111'111




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