Precision Agriculture, 2, 265�279, 2000� 2001 Kluwer Academic Publishers. Manufactured in The Netherlands.

The ‘‘Null Hypothesis’’ of PrecisionAgriculture ManagementB. M. WHELAN AND A. B. McBRATNEY b.whelan@acss.usyd.edu.au

Australian Centre for Precision Agriculture, McMillan Building A05, Uni�ersity of Sydney,NSW 2006, Australia

Abstract. As precision agriculture strives to improve the management of agricultural industries, theimportance of scientific validation must not be forgotten. Eventually, the improvement that is impartedby precision agriculture management must be considered in terms of profitability and environmental

Ž .impact both short and long term . As one form of precision agriculture, we consider site-specific cropmanagement to be defined as: ‘‘Matching resource application and agronomic practices with soil andcrop requirements as they vary in space and time within a field.’’ While the technological toolsassociated with precision agriculture may be most obvious, the fundamental concept will stand or fall onthe basis of scientific experimentation and assessment. Crucial then to scientifically validating theconcept of site-specific crop management is the proposal and testing of the null hypothesis of precisionagriculture, i.e. ‘‘Given the large temporal variation evident in crop yield relative to the scale of a singlefield, then the optimal risk aversion strategy is uniform management.’’ The spatial and temporalvariability of important crop and soil parameters is considered and their quantification for a crop field isshown to be important to subsequent experimentation and agronomic management. The philosophy ofprecision agriculture is explored and experimental designs for Precision agriculture are presented thatcan be employed in attempts to refute the proposed null hypothesis.

Keywords: experimentation, spatial variability, management zone

Introduction

Ž .Precision agriculture PA should be considered as a philosophical shift in themanagement of variability within agricultural industries. It must be aimed at

Ž .improving profitability and�or environmental impact both short and long term .As with all such endeavours to further knowledge in science-based disciplines, theconcepts and acceptance of the PA philosophy will ultimately rest on the successfulcompletion of scientific experimentation and assessment.

Ž .As one form of PA, Site-Specific Crop Management SSCM can be defined as:‘‘Matching resource application and agronomic practices with soil and crop re-quirements as they vary in space and time within a field.’’ This definition impliesthat assessment should have both an economic and environmental component, andthat considerable effort or risk is associated with management of the often complexcropping system. It also requires that the variability in soil and crop requirements,across space and time, be correctly characterized.

Figure 1 provides an example of the PA decision process that could then beemployed following a study of field variability. This model begins with the premisethat variability in crop yield is the initial signal that variable-rate treatment mightbe warranted. Another model might begin with the observation of soil variability.

WHELAN AND McBRATNEY266

Figure 1. Management decision-tree for SSCM. A simple model based on crop yield and the economicimperative.

However, until the environmental cost of fertiliser wastage is imposed to produc-tion, the economic imperative of optimising crop yield will no doubt guide manage-ment decisions.

In this model, differential treatment is then examined as an option based on:

� the degree of variation� the cause�s of variation� suitability for management intervention

Continuously variable treatment or division of a field into management sub-unitsis determined based on the spatial dependency observed. This decision marks the

‘‘NULL HYPOTHESIS’’ OF PRECISION AGRICULTURE MANAGEMENT 267

point of a conceptual schism. If variability and treatment can be observed andcontrolled at a fine scale then the question becomes:

Should fields be treated as continuously �ariable in yield potential or can someclassification into management units of ‘‘homogeneous’’ yield potential be accepted?

If the latter is chosen then another question arises:

Should these units be treated with uniform rates of ameliorants if the controllingfactor for application was not used to define the management unit?

The answers to such questions are most likely complex, site-specific and as yetunknown. Options at this point in the model are more than likely governed bylimiting factors such as technology, economics and lack of research.

Finally, some form of predictive model must be employed to enable a scientificand agronomically sensible examination of the implications of differential asopposed to uniform treatment, and the interpretation of the results in the form ofa spatial management plan. Research relevant to this realm of site-specific man-agement requires well-formed hypotheses and experiments.

Hypotheses and scientific assessment

Early scientific endeavour employed Baconian principles in experimental designswhich involved the construction of scenarios and the collection of responseobservations in the hope of distilling an answer. This is governed by what MedawarŽ .1979 describes as the principle of induction. Alternatively, Galilean principles ofexperimentation attempt to provide an assessment between possible outcomes ofexperimental treatments. It is this framework that most modern scientists use inconstructing specific experiments to test hypotheses. It is the hypothesis�a suppo-sition or theory that is to be critically examined�that directs and eventuallydefines all good experiments.

There are two avenues of philosophical thought on the criteria for assessing aŽ .good hypothesis. The Fisherian Fisher, 1959, 1970 perspective applies mathemati-

cal probability to ascertain the probability or the likelihood that experimentalobservations are consistent with an hypothesis. In this approach, the hypothesisbeing tested is usually a ‘‘null hypothesis’’�that the treatments have no effect�andthe probability refers to the chance of this being proven correct based on theobserved data. In the Fisherian view, the ‘‘alternative hypothesis’’ is framed as theopposite to the null hypothesis. A good alternative hypothesis is therefore judgedon the basis that the probability of the null hypothesis being proven correct is small

Ž .based on the observed data. In the view of Popper 1963 a good scientifichypothesis should contain a high degree of empirical content. As more information

Ž .is included within an hypothesis i.e. the more it attempts to explain it becomesless probable of being true than a hypothesis that attempts to explain less of thesame subject. Popper argues that if scientific experimentation is aimed at the

WHELAN AND McBRATNEY268

advancement of knowledge then we should be seeking hypotheses with a decreas-ing probability and which are therefore easy to test or refute. Popper considersscience to be directed at solving problems of increasing complexity based on newtheories. Good theories should raise more problems.

In a nutshell then, a good hypothesis can only be judged in the Fisherian viewafter experimentation, based on the low probability that observations conform tothe null hypothesis. The Popperian view can be used to frame good hypothesesbefore testing because it sees only highly testable or improbable hypothesesregarded as suitable for testing. The difference is therefore between probabilisticconfirmation of highly specific theories or refutation of more complex, less proba-ble theories. As we move towards precision management in a naturally variableenvironment, the two views will need to be taken under consideration in order toprogress precision agriculture.

The null hypothesis

The null hypothesis is generally considered to be the hypothesis of no effect inŽ .Fisherian statistics. It may also be regarded in a Popperian sense as the negative

argument to a theory i.e. that the theory under consideration is false. It is commonŽto test the null hypothesis using the probability of an experimental result or a

.more extreme one occurring if the null hypothesis held true.The testing procedures are commonly known as ‘‘tests of significance.’’ In

general terms, the routinely applied procedures standardise an experimental obser-Žvation to a formal observation frequency distribution. The probability value i.e.

.the p-value is calculated as that proportion of the distribution that is as, or moreextreme, than the standardised experimental observation. The commonly used

Ž .significance levels e.g. 0.05, 0.01 are merely benchmark extreme proportions ofthe observation frequency distribution.

The choice of standard test to be applied depends on the particular form of data,with each test employing an appropriate formal observation frequency distribution.

Ž .Proportionality or count data non-continuous data is often analysed using theChi-squared distribution. Continuous variables, where an individual comparisonbetween treatments is required, often involves Student’s t-distribution. Where anumber of treatments are to be compared as samples from a single population then

Ž .the F statistic distribution related to the analysis of variance is used.There are very broad falsification rules for significance based on probability in

Žgeneral use. The levels of significance against which the p-value statistic is.compared are usually chosen with little consideration given to the impact on the

decisions�recommendations arising from the experiment if the null hypothesis isrejected. Of particular importance to PA is the influence of large numbers ofobservations in a data set. Large numbers of observations provide high degrees of

Ž .freedom. In such cases e.g. yield sensor data , small differences may be regardedas significant at a certain probability level but could have little bearing onreal-world management decisions. This must be borne in mind when using proba-bility to falsify an hypothesis in PA.

‘‘NULL HYPOTHESIS’’ OF PRECISION AGRICULTURE MANAGEMENT 269

An alternative to probability testing is the technique of assessing the mathemati-Ž .cal likelihood Fisher, 1959, 1970 of each hypothesis being correct based on the

experimental data. In essence the calculated likelihood of each hypothesis can beused to rank the order in which the hypotheses are supported by the experimentaldata. Specifically, the ratio of likelihood’s for the null and the alternative hypothe-ses can be used to quantify the relative support that each hypothesis is given by thedata. This method could then be used in conjunction with estimates of the relative

Ž .‘‘costs of misinterpretation’’ based on accepting�rejecting the null hypothesis toprovide more practical inferences from the experimental data. This method maywell be suited to the science of PA. An advocate of likelihood analysis, GoodmanŽ .1993 provides a succinct discussion on the global debate regarding the pitfalls andmisinterpretations that accompany the exclusive use of the p-value statistic.

ŽHowever, if probabilities are to be used and they seem entrenched in modern.scientific assessment then it would be prudent to consider the correlation between

observations in data sets gathered from continuous monitors and fine-scale sam-pling. Knowledge of the correlation is most important when assessing the number

Ž .of independent observations and thereby the degrees of freedom available for anŽ .analysis. Barnes 1988 provides a method for calculating the effective number of

independent samples from a correlated data set. With large, spatially dense datasets gathered by PA technologies, this process should be given careful considera-tion prior to analysis.

Precision agriculture and scientific assessment

On the other hand, these large, spatially dense data sets should also provide theŽ .opportunity to pose more complex in the Popperian sense null hypotheses. It is

imperative that such hypotheses are formulated to test whether the change of farmmanagement focus from whole-of-field to the within-field scale is feasible orwarranted. These changes of scale in crop production processes and treatmentsŽ .the underlying philosophy of PA will also call for new agronomic experiments and

Ž .interpretations McBratney and Whelan, 1995; Kachanoski and Fairchild, 1996 .Ultimately, the assessment as to whether PA can be included in management

decisions must answer the following general question. Can the scale of variabilityin space and time be quantified, an optimal scale identified, and does it present anymanagement opportunities?

Spatial and temporal �ariability

Spatial �ariation. The magnitude of spatial variation in many soil and cropattributes has often been documented using classical statistics at individual sites.

Ž .Table 1 provides a summary of co-efficient of variation CV values based on aŽ .comprehensive review of the literature Whelan, 1998 . While these figures confirm

the presence of variability, at present it would be extremely difficult to develop a

WHELAN AND McBRATNEY270

Ž .Table 1. Median CV values % for a number of soil attributesand crop yield measured over a range of sampling scales

MedianŽ .Attribute CV %

Soil texture Sand 37Silt 18Clay 18

Soil structure Bulk density 5Soil O.M. 18

Ž .Soil moisture w dag�kg 113 3Ž .� m �m 9

Soil nutrients N 38P 38K 23

Soil pH 5Crop yield 14

variable-rate management plan in any field if the pattern of spatial variationdisplayed is random.

Table 2 is again a comprehensive review of the available literature, this timebased on spatial structure analysis of the observed variability. The estimates of

Ž . Žapparent range a suggest that spatial correlation for most attributes with the.possible exception of soil moisture extends to a distance that would be manage-

able with current technology. The moderate to strong spatial structure indicatesthat the variation is not randomly distributed in the field. Together, these at-tributes displayed in a transitive variogram are encouraging from a differentialmanagement viewpoint as they would be representative of an attribute spatiallydistributed in contiguous ‘‘patches.’’ This assessment is also confirmed by the

Ž .analysis of McBratney and Pringle 1997 .

Table 2. Median semivariogram model parameters for a number of soil attributes and crop yieldmeasured over a range of sampling scales

Median variogram parameters

Ž .Attribute C0 C C0 � C a m Spatial structure

2Ž .Soil texture % 2.4 9.3 11.7 63 Strong2Ž .Soil moisture % 0.00049 0.00045 0.00094 22 Moderate

2Ž .Soil nitrogen mg�kg 1.2 2.0 3.2 117 Moderate�strong2Ž .Soil phosphorous mg�kg 26.9 11.0 37.9 180 Moderate�weak

2Ž .Soil potassium mg�kg 887 391 1278 157 Moderate�weak2Ž .Soil pH units 0.021 0.15 0.171 105 Strong

2Ž .Crop yield t�ha 0.37 0.63 1.0 83 Moderate�strong

Ž . ŽC0 � nugget semivariance intercept ; C � spatial structure semivariance semivariance which may be. Žexplained by spatial dependence of samples ; C � C0 � sill semivariance semivariance at which spatial

. Ždependence ceases ; a � apparent range of spatial dependence maximum distance of spatial depen-.dence .

‘‘NULL HYPOTHESIS’’ OF PRECISION AGRICULTURE MANAGEMENT 271

Temporal �ariation. If we consider crop yield, it is evident that yield quantitiesvary within fields and that its magnitude may change between fields. It should alsobe important for farm management to quantify the extent to which crop yieldvaries with time. Annual changes in yield CV values for whole fields can be quitesignificant and the effect can be different for different crops and different soil

Ž .units within the field Whelan, 1998 . Significant temporal variability would in-crease the difficulty of yield goal determination and operations planning.

Table 3 shows farm-wide season summaries for two growing seasons withŽsubstantial differences in mean yields between the two seasons for wheat 1.56

. Ž .t�ha and 4.28 t�ha and the two seasons for sorghum 6.20 t�ha and 3.08 t�ha .The mean standard deviation for wheat increases slightly from 0.75 t�ha to0.88 t�ha with the decrease in mean yield. Sorghum shows an increase in meanstandard deviation from 1.07 t�ha to 1.31 t�ha with the decrease in mean yield. Itwould appear that sorghum yield is more variable across the farms than wheatyield, even with its greater overall mean yield.

The range of standard deviation values for wheat extends from 0.44 t�ha to1.02 t�ha, and from 0.67 t�ha to 1.62 t�ha for sorghum. With 3 standard

Ž .deviations encompassing 99% of the normal distribution Fisher and Yates, 1963 ,these figures intimate that within a single farm block, the wheat yield may possiblyvary between �1.32 t�ha and �3.06 t�ha from the farmwide mean. Sorghumyields may show variability from the farmwide mean of between �2.01 t�ha and�4.86 t�ha over 2 seasons.

However, with the spatial density of yield data obtained from real-time yieldmonitors, it has now become possible to estimate the variability across time in cropyield at the within-field scale. An estimate of the temporal variance should providean indicator of seasonal or climatic influences on crop yield. The temporal variancemay be simply estimated by Equation 1.

n 2Y � YÝ ž /i , j i

j�12� � 1Ž .T , i n � 1

where � 2 � temporal variance at point i, Y � yield value at point i in each yearT , i i, jj, Y � mean yield value at point i for all years.i

This estimate of temporal variance must be a comparison of yield values betweenseasons from fixed points in the field. Such data can only be reliably obtained byspatial prediction onto a single grid. Reliable and robust prediction methodsŽ .Whelan et al., 1996 are vital to this process.

Ž .The estimates of temporal variance over 2 seasons included in Table 4 aresubstantially larger than the yearly spatial variance in all cases. The inference isthat the variation in yield attributable to spatial variability in physico-chemicalattributes of the cropping system is much less than that induced by season toseason climatic variability in this instance.

It may also be postulated that a high temporal variance in conjunction with a lowspatial variance for a number of years suggests a relatively uniform field on which

WHELAN AND McBRATNEY272

Tab

le3.

Yie

ldda

tafo

r2

seas

ons

ona

farm

bloc

kba

sis

Yie

ldM

oist

ure

No.

ofda

taM

ean

Std

dev.

C.V

.M

ean

Std

dev.

C.V

.Ž

.Ž

.Ž

.Ž

.Ž

.Ž

.Ž

.Y

ear

&cr

opF

arm

Are

aha

poin

tst�

hat�

ha%

%v�

v%

v�v

%

1995

whe

atM

arin

ya49

.00

3879

92.

060.

9747

.110

.05

1.33

13.2

Mai

dens

77.2

932

082

0.96

0.78

81.3

8.81

0.44

5.0

1996

whe

atM

arin

ya22

1.86

2578

813.

870.

7820

.210

.36

1.69

16.3

Mai

dens

77.2

911

0942

4.54

0.59

13.0

10.8

80.

666.

1R

omak

a20

2.11

2261

455.

500.

7914

.411

.16

1.51

13.5

Bom

mer

a26

.78

2778

92.

520.

8332

.910

.51

0.64

6.1

Cab

ro20

5.14

2442

333.

690.

8623

.310

.05

1.14

11.3

KW

ee17

7.17

2120

894.

230.

7517

.711

.42

0.80

7.0

1996

sorg

hum

Mar

inya

76.3

510

9682

6.43

0.74

11.5

12.7

31.

169.

1R

omak

a13

8.09

1890

496.

221.

3321

.413

.15

1.28

9.7

Cab

ro90

.92

1360

735.

881.

1118

.912

.72

1.05

8.3

1997

sorg

hum

Mar

inya

102.

8687

002

3.38

1.50

44.4

14.6

66.

2142

.4R

omak

a65

.95

6052

92.

661.

1141

.712

.75

1.61

12.6

‘‘NULL HYPOTHESIS’’ OF PRECISION AGRICULTURE MANAGEMENT 273

Table 4. Descriptive statistics for the yield variability in 4 wheat fields over space and timeŽ . Ž .2 seasons after Whelan, 1998

Area Mean yield Variance2Ž . Ž . Ž .Field name ha Year t�ha t�ha

Field 1 16.14 Spatial 1995 1.31 0.66Spatial 1996 3.44 0.50Temporal 2.70

Field 2 8.12 Spatial 1995 1.89 0.44Spatial 1996 4.03 0.34Temporal 2.58

Field 3 10.85 Spatial 1995 2.66 0.34Spatial 1996 4.65 0.14Temporal 2.20

Field 4 77.29 Spatial 1995 0.90 0.49Spatial 1996 4.53 0.22Temporal 6.98

crop yield is overwhelmingly governed by climatic conditions. Just what is high andlow variance in both space and time has yet to be determined.

Figure 2 displays the yield maps for a single field over 3 consecutive seasons.Table 5 shows the spatial and temporal statistics. Examining 3 years of data for thesame field and crop reveals a further dilemma in the assessment of temporalvariability and its application in PA management. Over 3 seasons, the temporalvariability is at least twice the largest spatial variability. Between two seasons, thetemporal variability may be at least 3 times the spatial variability or of the samemagnitude as the spatial variability. Figure 3 shows how the temporal variabilitychanges over the field in each comparison.

These maps may eventually prove to be useful for confirming whether patternsin yield maps are temporally stable and assessing the suitability of a field forexperimenting with differential treatment. Low, uniform, temporal variability overa number of years would suggest a field which produced a similar spatial yieldpattern over time. This could be a uniform, or a consistently spatially variable, yieldpattern. When used in conjunction with the crop yield maps, the type of pattern

Ž Žcan be confirmed and further analysis such as cluster analysis Lark and Stafford,..1997 can be considered to determine if significantly different yield potentials

exist.At present, a field with uniformly high temporal variability would be unsuitable

for PA. Traditional uniform treatment would be advised. Fields displaying amixture of low and high temporal variability could also be considered for furtherexamination. However an estimate of yield potential for the regions of hightemporal variability would prove difficult to obtain.

Figure 3 shows that the situation may not be always clear. Figures 3a and 3b areexamples where the magnitude of temporal variance changes in space. Figure 3ashowing more spatially contiguous ‘‘zones’’ of temporal variance. Figure 3c shows agenerally uniform temporal variance map. With all years in the analysis, Figure 3d

WHELAN AND McBRATNEY274

Ž . Ž . Ž .Figure 2. Wheat yield maps for Field 5 a season 1996, b season 1997, c season 1998.

falls somewhere in between. Obviously, the spatial pattern and magnitude of yieldŽ . Ž . Ž .in 1996 Figure 2a and 1998 Figure 2c were very similar, while 1997 Figure 2b

displayed a significant departure from the magnitude and, to a degree, spatialpattern of wheat yield.

If this field is assessed using Figure 3c, then the expected pattern and potentialof the next wheat yield would not be difficult to determine and could be used tolayout experiments to test the effects of PA treatments. It would, however, neglect

Ž .the possibility of encountering a year such as 1997 Figure 2b . Using Figure 3d

‘‘NULL HYPOTHESIS’’ OF PRECISION AGRICULTURE MANAGEMENT 275

Ž .Table 5. Descriptive statistics for the yield variability in 1 wheat field over space and time 3 seasons

Area Mean yield Variance2Ž . Ž . Ž .Field name ha Year t�ha t�ha

Field 5 100 1996 5.41 0.261997 3.64 0.611998 5.53 0.37

Ž .Temporal 1996�1997 1.79Ž .Temporal 1996�1998 0.22Ž .Temporal 1997�1998 2.00Ž .Temporal all years 1.34

would reduce the risk that such neglect could have on the outcome of any furtherexperimentation.

Precision agriculture and the null hypothesis

Given this information, it is possible to frame a null hypothesis for SSCM. Thehypothesis can be stated as: ‘‘Given the large temporal variation evident in cropyield relative to the scale of a single field, then the optimal risk aversion strategy isuniform management.’’

The null hypothesis is framed this way because uniform management is thecurrent practice and the strategy which is to be replaced if a better alternative canbe found. This may not be a null hypothesis in the true Fisherian sense, but morein line with Popper’s view on what is a desirable hypothesis to test. It includesinformation that is essential to examining PA operations and is not merely a‘‘negative’’ hypothesis. Testing the hypothesis will require clear well-thought out

Ždefinitions of ‘‘optimal’’ with respect to economic, environmental and other. Ž .outcomes and ‘‘uniform’’ uniform is clearly a question of scale .

One alternative hypothesis that can be put forward then is: ‘‘Management ofvariability at a finer spatial resolution than is currently undertaken would be animprovement on uniform management.’’

Framing and testing such hypotheses should be considered vital to PA becausethe adoption of SSCM practices without reasonable testing may well lead to lowerprofitability and poorer environmental outcomes. If this should occur with anumber of ‘‘advocated’’ products or practices then it is easy to imagine thatlong-term harm may be inflicted on the concept of SSCM.

Testing the hypothesis. As more information becomes available on the variability inspace and time of the most influential soil and crop attributes, these hypothesesmay be tested under a general range of experimental conditions, that is to sayacross broad space and time ranges. At present, the testing should be restricted tosite-specific conditions, which makes the task simpler but may limit informationrelevance to the specific field or farm.

WHELAN AND McBRATNEY276

Ž . Ž .Figure 3. Temporal variability maps for Field 5 a between season 1996�1997, b between seasonŽ . Ž .1997�1998, c between season 1996�1998, d over 3 seasons.

A good test will be required to probe the treatment effects of differentialmanagement and uniform management over space and time. In the publishedliterature on PA, very few response experiments propose a formal hypothesis andanalyse the data in an effective method to test the hypothesis. Most provide onlyan economic comparison of net returns from uniform versus variable treatmentover space. Mostly, the hypothesis as proposed here is probably being informallyassumed. The results are predominantly from sites where there is lots of informa-

‘‘NULL HYPOTHESIS’’ OF PRECISION AGRICULTURE MANAGEMENT 277

tion gathered on variability and also significant control over application timing.Intuitively, under these conditions the null hypothesis as proposed here will oftenbe rejected.

In situations where there is greater natural variation and�or little informationŽ .or reduced density of information on the variability, then the null hypothesis maybe accepted. Such conditions may be found when much larger areas are consideredŽ .as in Australian farm fields .

To test a well framed null hypothesis, a suitable experimental design must beimplemented. Here it is assumed that prior response information has been gath-

Ž .ered using methods such as those outlined in Pringle et al. 1999 or by some othersuitable process. A knowledge of response variation is vital for formulating treat-ment rates.

With knowledge of response to the variable of interest, there are two basic stepsthat may be considered. Firstly, a management zone approach whereby areas of afield are deemed homogeneous in response and are treated similarly within, butdifferentially between, the zones. Secondly, management based on the assumptionthat the attribute of interest is continuously variable and will be treated accord-ingly.

Figure 4 outlines diagrammatically the necessary treatments required to test thenull hypothesis under the two different management schemes. For the uniform

Ž .treatments, U is the field mean calculated once and applied in each year of thes, tŽ .experiment while U is the field mean calculated each year taking into accounts

Ž .some timely measurement or environmental prediction. U is spatially ands, tŽ .temporally uniform and U is only spatially uniform. The estimated treatment fors

Ž .U would require prior information gathered over a number of years.s, tŽ .The differential treatment V is varied based on the spatial location ofs

Ž .observations and applied in each year of the experiment while V is calculateds, tŽ .taking into account some seasonal measurement or environmental prediction. Vs

Figure 4. Diagrammatic representation of an experimental design sequence for testing the nullhypothesis.

WHELAN AND McBRATNEY278

Ž .is varied in space, V is varied in space and time. In the management zones, tŽ .approach, the differential treatments V and V would in effect be uniforms s, t

treatments calculated independently for each zone. For management of continuousŽ .variability, the differential treatments V and V would be individually calculateds s, t

for the desired space and time co-ordinates.It is the comparative analysis of the response to these treatments that is required

to test the null hypothesis of PA. Ultimately it should be an indicator of theeconomic and environmental response that should be measured. The response willalso need to be observed over time for satisfactory assessment and eventualacceptance or rejection of the null hypothesis.

Summary

It has been the author’s intention to prompt reflection on the scientific evaluationof PA. Much data remains to be gathered to test the philosophy of PA. What does

Žappear definitive is that temporal variability in crop production indicators i.e..yield at the within-field scale is often larger in magnitude than spatial variability.

This will increase the risk of economically and environmentally inappropriateactions if differential treatments are solely based on spatial information. Scientificanalysis of treatments that consider both the spatial and temporal variability arerequired. To this end, a null hypothesis and an alternative hypothesis have beenproposed for PA. It is imperative that experiments are framed to rigorously testsuch theories. Acceptance of precision agriculture as a new system of managingvariability in agricultural production will undoubtedly hinge on repeatable evidencefor the rejection of the null hypothesis.

Acknowledgments

The authors wish to thank Craig and Judy Boydell for invaluable support to thiswork, and the referees for insightful, constructive suggestions.

References

R. J. Barnes, Bounding the required sample size for geologic site characterisation. MathematicalŽ .Geology 20, 477�490 1988 .

Ž .R. A. Fisher, Statistical Methods and Scientific Inference, Oliver and Boyd, Edinburgh, UK, 1959 ,175 pp.

Ž . ŽR. A. Fisher, 1970 Statistical Methods for Research Workers, 14th ed. Edinburgh, UK: Oliver and Boyd,362 pp.

R. A. Fisher and F. Yates, Statistical Tables for Biological, Agricultural and Medical Research, 6th ed.Ž .Hafner Publishing Company, New York, USA, 1963 , 146 pp.

S. N. Goodman, p-values, hypothesis tests, and likelihood: implications for epidemiology of a neglectedŽ .historical debate, American Journal of Epidemiology 137, 485�496 1993 .

‘‘NULL HYPOTHESIS’’ OF PRECISION AGRICULTURE MANAGEMENT 279

R. G. Kachanoski and G. L. Fairchild, Field scale fertiliser recommendations: the spatial scalingŽ .problem, Canadian Journal of Soil Science 76, 1�6 1996 .

R. M. Lark and J. V. Stafford, Classification as a first step in the interpretation of temporal and spatialŽ .variation of crop yield, Annals of Applied Biology 130, 111�121 1997 .

A. B. McBratney and M. J. Pringle, Spatial variability in soil�implications for precision agriculture. In:Precision Agriculture ’97: Proceedings of the 1st European Conference on Precision Agriculture, edited by

Ž .J. V. Stafford BIOS, Oxford, UK, 1997 , pp. 3�32.A. B. McBratney and B. M. Whelan, The Potential for Site-Specific Management of Cotton Farming

Ž .Systems CRC for Sustainable Cotton Production, Narrabri, Australia, 1995 , 46 pp.Ž .P. B. Medawar, Ad�ice to a Young Scientist Harper & Row, New York, USA, 1979 , 109 pp.

ŽK. R. Popper, Conjectures and Refutations: The Growth of Scientific Knowledge Routledge, London,.1963 , 431 pp.

M. J. Pringle, A. B. McBratney, and S. E. Cook, Some methods of estimating yield response to aspatially variable input. In: Precision Agriculture ’99: Proceedings of the 2nd European Conference on

Ž .Precision Agriculture, edited by J. V. Stafford Sheffield Academic Press, Sheffield, UK, 1999 , pp.309�318.

B. M. Whelan, Reconciling Continuous Soil Variation and Crop Yield�a study of some implications ofŽwithin-field variability for site-specific crop management. Unpublished Ph.D. thesis, University of

.Sydney, Australia, 1998 .B. M. Whelan, A. B. McBratney, and R. A. Viscarra Rossel, Spatial prediction for precision agriculture.

In: Precision Agriculture: Proceedings of the 3rd International Conference on Precision Agriculture,Ž .edited by P. C. Robert, R. H. Rust, and W. E. Larson ASA�CSSA�SSSA, Madison, WI, 1996 , pp.

331�342.