Pestic. Sci. 1987, 19, 297-307

Experimental Design and Response Surface Analysis of Pesticide Trials*

Derek J. Pike and Anne M. Hasted

Department of Applied Statistics and Statistical Services Centre, University of Reading, PO Box 217, Whiteknights, Reading RG6 2AN, UK

(Received 20 October 1986; accepted 8 November 1986)

ABSTRACT

The search for combinations of pesticides or herbicides which have a synergistic effect is an important area of study and one which presents challenges for the statistician as well as for the experimenter. The aim is to find and to describe the interaction between a number of quantitative factors, and having done so to determine a combination which is economically optimal for a given level of control. The present paper discusses the role of a family of response functions called Inverse Poly- nomials to describe such phenomena. Examples are given of such func- tions and it is shown how it is possible to fit these functions to data and to plot the results on a routine basis. The importance of the experimental design for the estimation of such nonlinear regression models is investig- ated, and examples are given of experimental designs which are ‘optimal‘ with respect to particular aims of the experimenter.

1 INTRODUCTION

The search for combinations of pesticides or herbicides which have a synergistic effect on the pest or weed, whilst having no detrimental effect on the main crop, is an important area of study and one which presents challenges for the statistician as well as for the experimenter. The broad aim is to find and to describe the interaction between a number of quantitative factors, and having done so to determine a combination of levels of them which is economically optimal for a given level of control. *Presented at the symposium ‘Enhancing Pesticide Action through Mixtures’ on 14 October 1986, organized by the Physicochemical & Biophysical Panel of the Pesticides Group of the Society of Chemical Industry.

297 Pestic. Sci. 0031-613)3/87/$03.50 0 Society of Chemical Industry, 1987. Printed in Great Britain

298 D. J. Pike, A. M. Hasted

A convenient illustration is the control of a grass weed, such as blackgrass, in a main crop of wheat. There are generally two main stages to such a study. The first is an efficacy trial to determine the effect of the chemicals on the pest. This trial should be, but is not always, followed through to harvest to determine the effect of the chemicals on the crop yield in the presence of the competing effect of pest control. It is these results which are likely to be of greatest relevance to the farmer. The second stage of the study, governed by registration requirements, is likely to be a selectivity trial to determine the damage caused by the chemicals to a weed-free crop.

The principles underlying the techniques discussed in this paper are applicable to both stages of such a study, but here only the first stage will be used as an illustration. It is reiterated that there are two responses of different types-that of the pest and that of the crop-and that associated with these are different and competing objectives. As control is exerted over the pest the crop yield will increase. However, use of excessive levels of pesticides, while increasing pest control, will reduce crop yield. This paper is constrained to a study of the response to two stimulus variables and these are considered as levels of two active ingre- dients in a pesticide mixture. Consideration could equally well have been given to the levels of oil or wetter added to the active ingredient or to levels of more than two stimulus variables. The problem and its study would essentially be the same although conceptually more complex.

In this paper three facets of the problem are discussed. First, there is the need to find a family of response models capable of describing the behaviour of both the crop and the weed. Second, if a study is to be performed to fit these surfaces it is important that the treatments to be included in the study are chosen so that the surfaces can be fitted as accurately as posible. Third, one must ensure that a suitable means exists for analysis of the data and for presentation of the results.

2 INVERSE POLYNOMIAL RESPONSE SURFACES

It is important to recognise at the outset that this is a problem more suited to techniques of regression than to analysis of variance, which can indicate the presence of such effects as interactions but cannot explain the nature of these effects. Moreover, it will be very rare in an experiment that the best possible combination of factor levels corresponds to a treatment which has actually been used. However, if there is available a flexible approximating function which can represent the responses to the applied treatments, it is possible to use this fitted function to interpolate between them to find the optimal treatment combination. Such a family of functions, called Inverse Polynomials, was first introduced by Nelder.' These have subsequently been used in a range of areas such as enzyme kinetic^,^.^ agriculture,M and nuclear reactor In a problem where y is the response to levels of k stimulus variables x,, . . . xk, the general inverse polynomial function is given by

k

Experimental design and response surface analysis of pesticide triak 299

0 c ,O

x' x ? c

0

300 D. J . Pike, A . M. Hasted

Experimental design and response surface analysis of pesticide trials 301

302 D. J . Pike, A . M. Hasted

where P(x) is a polynomial in x , , . . . , x k . To be more specific, using two stimulus variables, pest control, which rises to an asymptote with increasing pesticide levels, could be represented by a two-factor linear inverse polynomial:

( 2 ) X l X 2

PuO+PlOXl + P O l X 2 + P l l X I X 2 Y =

Such a response is shown in Fig. 1. Crop response, on the other hand, which rises to a maximum and then falls,

could be represented as in Fig. 2 by a two-factor quadratic inverse polynomial:

The family is clearly a very flexible one as can be seen from the different forms of function in Figs 1 and 2. The quadratic function in Fig. 2 is a symmetric one in which the crop responses are similar for changes in levels of each of the two stimulus variables. If the crop yield was much more sensitive to levels of x2 than of xl, the response might be as in Fig. 3-identical in form to Fig. 2, but with different parameter values p,,.

3 PRINCIPLES OF EXPERIMENTAL DESIGN

It is not generally recognised that experimental design is important for studies which, at the analysis stage, will involve techniques of regression. Careful choice of the levels of the stimulus variables can have a considerable effect on the precision which attaches to parameters of fitted response functions. In particular, a poor choice of treatment combinations can make it very difficult to fit an approximating function to the resulting data. Factorial experiments are widely used for studies with quantitative stimulus variables but the results are rarely analysed other than by simple analysis of variance. If results are to be analysed using, for example, inverse polynomials, then it should be possible to design an experiment specially tailored to the approximating function.

There is much relevant research work in the theoretical statistical l i terat~re,~JO but probably the most widely applicable technique for designing experiments for fitting multifactor nonlinear models is that of minimisation of the generalised variance of the parameter estimates." Technically, this ensures that the volume of the confidence ellipsoid surrounding the parameter estimates is minimised. Infor- mally, this means that overall the parameters in the model are estimated as precisely as possible, bearing in mind that if the estimation of one parameter is improved by rearrangement of the treatment combinations then estimation of other parameters is likely to be adversely affected. Thus, the mathematical technique performs a balancing act between the estimation of all the parameters of the model. For technical reasons it is necessary to have some prior knowledge of the behaviour of the response functions to be fitted before the experiment can

Experimental design and response surface analysis of pesticide trials 303

be designed. However, it will become apparent in the results which follow that this knowledge may not have a dramatic effect on the resulting design.

3.1 Design for crop yield

The inverse quadratic given in Fig. 2 may represent a typical response of crop yield to application of combinations of two pesticides. The technique of minimisa- tion of the generalised variance has been used to obtain the optimal nine-point design, D,, for fitting such a surface and the results are given in Fig. 4.

From this several facts are clear. The design points are similar to those of a 32 factorial, but with a geometric type of spacing of the levels to keep more observa- tions in the region where the shape of the surface is changing most rapidly. The design levels have not spread to the edges of the region of interest and indeed if the upper limits of x1 andx, are increased the design points do not change. It is also interesting to note that the optimal design points lie in two sets of four around contours of the true surface. The theoretical reasons for this are not known and are subjects for ongoing research.

Notice that the most extreme combinations of factor levels are using amounts of pesticide four or five times that which gives the maximum crop yield. Experimen-

Fig. 4 Optimal design Dy, for crop yield surface.

304 D. J . Pike, A . M. Hasted

ters may resist the use of such levels as being unrealistic in practice, but from the point of view of studies such as the present, the use of such extreme levels, provided they do not kill the crop, is to be encouraged. At an experimental stage it is sensible to do apparently foolish things to discover the effects of extreme treatment combinations. Furthermore, if the form of the response surface does not break down, it will be fitted much more precisely because these extreme conditions have been used in the study.

3.2 Design for pest control

Figure 1 showed a typical pest-control response to increasing levels of two pesticides. The challenge here is to ask what type of design is needed to fit such a response and how similar is the design to that recommended for estimating crop yield. For this model, one knows that the best thing to do is to use some of the design points at the maximum levels of the stimulus variables we can imagine. However, it is also known that to constrain the region of estimation need not result in much loss of efficiency of the design. The design has therefore been constrained to lie in the same region as that for crop yield, and the resulting nine- point design, D,, is given in Fig. 5 .

11

x " !

1 1 I I I 2 4 6 8

X1

Fig. 5 Optimal design Dp, for pest control surface.

Experimental design and response surface analysis of pesticide trials 305

N X '

0 2 4 6 8 10 x1

Fig. 6 Uncertainty regions for the compromise design.

It is clear that, although there are differences between the two designs, there are surprising similarities in the general spacing of the observations. Figure 6 shows the two designs together with corresponding uncertainty regions for the design points.

3.3 Compromise design For dual objectives

It would be ideal in practical research of this type if a single design could be used to meet a dual purpose. Accordingly, uncertainty regions for the points shown in Fig. 6 have been taken and an experimental design, D, chosen with points in the middle of the region. This design can be compared with D, and D, by using D to fit the surfaces for crop yield and pest control and by defining the efficiencies of D relative to D, and D, as

(4)

( 5 )

Generalised variance of D, E , = { Generalised variance of D

Generalised variance of D, Generalised variance of D

} x100%=78%

} x100%=72% E,= { where r is the number of parameters in the fitted model.

306 D. J. Pike, A . M . Hasted

TABLE 1 Efficiencies (%) for a Range of Designs for Pest Control and Crop Yield Surfaces

Optimal Optimal Compromise Equally Geometrically pest crop design spaced 32 spaced 32

control yield (D) factorial factorial design design (DPl fDyI

Pest control 100 53 72 52 59 Crop yield 48 100 78 32 94

These figures need to be seen in the light of a range of possible designs that would seem sensible. Table 1 shows the efficiencies for estimation of the pest- control and crop-yield surfaces of five designs. These are the designs D,, D, and D already discussed; and, in addition, 32 factorial designs equally spaced and geometrically spaced over the region used for design D. It is clear that D is a very successful compromise design for this problem.

4 DISCUSSION

This paper has shown the possibilities for using a single family of response functions for the joint evaluation of pest control and crop yield in response to levels of two pesticides. In addition it has been shown in principle how a single experimental design can be produced and used to study both problems. The fitting and presentation of results from such studies is readily performed using such packages as SAS and SAS/GRAPH.”

The implications for the future could be considerable. Extension to more factors is complex, but the principles remain the same, in that theoretical evalua- tions should lead to the development of sensible rules of thumb. A major implication for research is the need for interplay between the experimenter and the statistician. The statistician is in a position to provide useful experimental designs for such complex problems. However, this cannot be done unless the experimenter knows something about the nature of his problem and of his objectives. The marrying of these ideas could result in more precise conclusions drawn from a more efficient use of available resources.

The topic this paper has not addressed is that of economics. Pesticides cost the manufacturer money to produce and need to be sold at a profit. Use of the pesticides potentially brings the farmer increased yield and hence increased profit. The subtle task of the manufacturer is to set the price of the product at a level such that his profit is substantial, but which still promises the farmer sufficient return from increased crop yield to encourage him to spend money on the product. It is not difficult to use the response surfaces discussed in this paper to formulate the requisite economic equations. The point of interest is that one encounters again the two competing objectives of maximising profit for both the manufacturer and the farmer. The solution to the problem has to be seen as an acceptable compromise between the two.

Experimental design and response surface analysis of pesticide trials 307

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