COOPERATIVE EXTENSIONUNIVERSITY OF CALIFORNIA

RIVERSIDE. CALIFORNIA 92521

May 8, 1979

TO: Cooperative Extension Staff

FROM:Carol Adams and Jack Hills/ySenior Statistician and Extension Agronomist

RE: SHORT STATISTICAL NOTES #6 RESPONSE SURFACES

"Response surface" sounds like something fancy and complicated, but it isjust a regression of a dependent variable, such as yield, on two independentvariables, such as amount of nitrogen fertilizer and number of applications.The regression may be curvilinear and can include interactions between the twoindependent variables. The usual objective of a response surface experimentis the determination of the combination of the two independent variables thatgives the optimum value of the dependent variable. For example, how muchnitrogen fertilizer in how many applications gives the maximum yield?

When factors A and B (the independent variables) of a two factor experimentare both quantitative (levels of nitrogen fertilizer, different plant spacings,planting dates, etc.), the separate effects of A and B on the dependentvariable, such as yield, are best described by regression lines called responselines or curves. If the interaction of A with B is significant, the importanteffects (main effects and interaction components) are best presented as amultiple regression or response surface showing the combined effects of A andB on yield. A Duncan's multiple range test of all the A X B means cannotdescribe the response of the dependent variable to the changing levels offactor A and factor B nor can it show the nature of the interaction. A res¬

ponse surface function and its graphical presentation "tell the whole story."

Figure 1, a three dimensional graph, shows the story for the yield of sugarbeet roots as affected by the harvest date and nitrogen rate. The responsesurface function, from Little and Hills, is

Y = 5.888 + 2.221H + 4.547N - .096H^ - 1.002N^ + .700NH - .144N^HTo find a particular predicted yield, replace H and N with week of harvest andnitrogen rate respectively and carry out the calculations. For H = 9 and N = 2

Y = 5.888 + 2.221(9) + 4.547(2.5) - .096(9^) - 1.002(2.52) +.700(2.5)(9) - .144(2.52)(9) = 30.9

From the figure you can see that later harvest dates had more yield and theincreases in N from 0.0 to 1.6 resulted in increased yield, whereas increasesin N from 1.6 to 3.2 resulted in a plateau or a possible decrease in yield aftean optimum between N = 1.6 and N = 3.2. The significant interaction resulted

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in the response curves drawn on the surface being non-parallel. For week ofharvest = 0, the increase in yield from N=0toN=1.6is only 4.7, but forweek = 12 the increase is 13.7. The effect of N is much more pronounced atlater harvest dates. Figure 2 shows the isoquant map of the surface in Figure 1.This is like a contour map of mountains. Each curve represents a constantyield. Any combination of N and harvest date that falls in the upper rightcorner of the graph, in the area above and to the right of the isoquant curvefor yield = 30 tons/acre, produces optimum results.

The selection of levels for the two factors to be tested is probably the mostimportant aspect of designing an experiment to develop a response surface. Inestablishing these levels there are three important points to consider: (1) therange that each factor should cover; (2) the number of levels for each factor;and (3) the spacing of the levels. A good deal of experience and knowledge ofthe subject matter is important in making these kinds of decisions. We canillustrate the kinds of decisions to be made by taking a specific example. Saywe want to develop a response surface for the best combination of nitrogenfertilizer rate and date of planting for wheat production. Our experiencemight indicate a range of planting dates from November to February and for Nrates from 0 through 250 lbs/acre. For the second point above, the best ruleis to select as many levels over the desired range as you can and for the thirdpoint, it is usually best to have equally spaced levels of each factor. Thus,for our wheat experiment, we might select planting dates of November, December,January, and February--at evenly spaced weekly intervals and for N rates, 0, 50,100, 150, 200, and 250 lb N/acre. This gives 4 X 6 or 24 treatment combinations.

You may feel that 24 treatments will make an experiment too large to handle forthe number of replications you have planned. In this case it would be betterto reduce the number of replications rather than the treatment combinations.For example, three replications of 24 treatments give 72 plots and thus threereplications should be adequate to give reasonably precise results. Evenreducing the replications to two might give a reasonable response surface.

If 24 treatments are still too many to handle, some compromises might be: 4plant dates X 5 N rates, the latter being 0, 60, 120, 180, and 240 lb N/acreto give 20 treatments; 3 plant dates X 5 N rates to give 15 treatments.

One frequent mistake is to say "No one grows wheat without fertilizer N andtherefore we should omit the 0 N rate." This seriously reduces the ability tocalculate good response curves and this mistake should be avoided.

Once treatment combinations have been decided upon they can be arranged in anysuitable design. The split plot often proves useful, especially when inter¬actions are sure to occur. In our wheat example, plant dates could be mainplots and N rates sub plots.

The calculations for a response surface can be lengthy and involved, but theyare much simplified if the levels of each factor are equally spaced so thatpolynomial coefficients can be used. Little and Hills carry through the calcu¬lations for the response surface in Figure 1 in two main steps:

1. Partitioning the sum of squares for nitrogen, harvest date, andinteraction into linear, quadratic, cubic, linear X linear, etc.,components and testing for significance.

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2. Fitting the response surface function for those effects that were

significant in step 1, then converting the function from usingpolynomial coefficients to using N rate and week of harvest directly.

So far we have talked about polynomial response surfaces where only linear,squared, and interaction terms were used. Polynomial equations are easy to fitand are usually adequate within the observed ranges of the independent variables,but they should not be used for extrapolation. The response of sugar beetyield to N rate in Figure 1 may have been better described by an asymptoticcurve (approaching a plateau with increased N). Functions other than polynomialscan be used for response surfaces. Either factor or the dependent variable maybe transformed, such as l/xi^ or log(x2) or special equations such as asymptoticor growth curve may be used. For non-polynomial response surfaces or just tofacilitate the calculations, you may want to consult your statistician.

Suggested Reading:

Thomas M. Little and F, Jackson Hills Agricultural Experimentation Design andAnalysis. John Wiley and Sons. 1978. pp 116-23, 258-64.

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FIGURE 1

FIGUPE 2

N CWT/fl