In the quadratic yield model, the expected yields, E(Y_i), are described by a quadratic function of the total N application rates, N_i. Therefore, the yields, Y_i, i = 1, ∙∙∙, n, were modeled as random variables that depended on N_i in the following way:

N_i denote fixed levels of N application rates, b_j, j = 0, 1, 2, the fixed unknown coefficients of regression, and e_i the error variables, which are assumed to be independent and normally distributed with an expected value 0 and a common unknown variance s² > 0. The unknown coefficients b_j, are estimated using the least-squares estimates b_j. This requires at least three different N-levels. Otherwise the estimates would not be unique.

The economically optimal N application rate, N_opt, is the N-rate where the expected returns above N fertilizer cost, E(RANC) = p×(b₀ + b₁N + b₂N²) – r×N, are maximized. This applies to the following optimum N rate:

N_opt results from the N where the first derivative of the model parabola, E(Y) = b₀ + b₁N + b₂N², equals r×p^–1, which is the ratio between N fertilizer and the product price.

By using the least-squares estimates b_j for estimating the coefficients b_j, the point estimate for N_opt was immediately reached :

Note that this point estimator, is biased¹ because it is not a linear combination of the unbiased least-squares estimators b₁ and b₂. It is in essence a ratio.

According to [10]’s duality with confidence intervals and tests, a confidence interval or, more generally, a confidence set for N_opt consists of all hypothetical values N₀ whose simple null hypothesis,

cannot be rejected. By using (3), the null hypothesis in (5) can be reshaped to

This is a linear hypothesis, so it can be tested in the framework of general linear models using the usual Ftest [11], which is a likelihood ratio test. Therefore, the corresponding confidence set is called a likelihood interval, provided that it is an interval. This usually applies but it does not need to. As mentioned before, it is formed by the set of all N₀ for which H₀ in (5) or (6) cannot be rejected. For those interested in the area of statistical inference, this test and the corresponding derivation of the confidence set, which can be produced by explicit mathematical formulae, is described fully in [12]. The computation of this likelihood-type confidence set for N_opt was implemented in the Fortran program VINO. EXE, which can be downloaded from the internet [13]. It is analogous to that of [14], who calculated it for the ratio of parameters, and to that of [15], who computed it for the maximum of a quadratic regression.

Under the assumptions made (quadratic model, independent and normally distributed homoscedastic errors), tests of linear hypotheses, such as in (6), are considered exact, so exact confidence intervals are obtained, whereas confidence intervals derived according to [11,16,17], which are also called Wald intervals, would only provide approximate confidence intervals. They are symmetric around the point estimate, which is not even unbiased. Therefore, they “may not accurately reflect the actual, often asymmetric, uncertainty in an estimate” [18]. Such an asymmetric situation is given here as seen with rapidly increasing fertilizer application rates. The true yield function decreases more slowly, as indicated by the quadratic model, which overestimates the yield loss due to lodging [19]. Therefore, the Delta method will not be pursued but the likelihood intervals will be used to proceed exactly as mentioned above. These intervals are not symmetric around the point estimate, so neither limit depends on the other and they can adapt better to the data around either limit. The lower limit of the confidence interval is in the focus of ecological interests as it gives the minimum fertilization that cannot be rejected as being optimal.

In the linear-plus-plateau model, the economic optimum, N_opt, equals the transition of the increasing straight line to the horizontal, unless the price ratio rp^–1 is greater than the gradient of the increasing straight line (cf. [20]), which, however, usually not applies. The mentioned transition to the horizontal is a parameter of the linearplus-plateau model. The program PRISM not only estimates the model parameters, but also calculates approximate confidence intervals for them [21]. Therefore, point estimates and confidence intervals for N_opt could be obtained using PRISM. They do not, contrary to the parabola-based confidence sets, depend on prices.

In [27] where the same trial was evaluated to test the precision agriculture hypothesis (no difference between the optimum N-rate in highand low-yielding zone) and to compute confidence intervals for the difference of the N optima, the usual assumptions have been checked. The result was that the data can be assumed to be approximately normally distributed and homoscedastic, and further, a block effect in the randomized complete block design could not be detected, although the N rate trials were very extensive. Therefore, the ordinary regression

analysis and the mentioned methods to compute confidence sets for the N optima, can be applied.

A likelihood ratio confidence interval in the quadratic model need not be symmetric around the point estimate. In Figure 2 for example, where the design of all N levels

including the zero N rate is considered, the confidence interval [204 kg∙ha^–1, 328 kg∙ha^–1] for the high-yielding zone is only 36 kg∙ha^–1 long to the left of the point estimation 204 kg∙ha^–1, whereas to the right it is 88 kg∙ha^–1, which is nearly three times as long. This is, above all, due to the fact that yields from very high N rates do not seem to sink. For the low-yielding zone, a point estimation of 199 kg∙ha^–1 and a confidence interval of [173 kg∙ha^–1, 260 kg∙ha^–1] present a similar situation with regards to the asymmetry. It seems that a concave parabola with a vertex at the far right of the point estimation is easily compatible with the measured data, whereas a concave parabola with a vertex at the far left of the point estimation is not suited for fitting the yields, since those to the right of such a vertex do not sink.

We can see in Figures 2-5 that all confidence intervals or, more generally, confidence sets are very long.

In Figure 2 for example, the 95% confidence interval for N_opt in the low-yielding zone [173 kg∙ha^–1, 260 kg∙ ha^–1], is somewhat shorter than the 95% confidence interval for N_opt in the high-yielding zone [204 kg∙ha^–1, 328 kg∙ha^–1]. This can be explained as follows. In contrast to the high-yielding zone, the higher N doses in the lowyielding zone did not result in further increases in yield while a small yield depression could even be observed at 260 kg∙ha^–1 that forced the regression function to go down. Consequently, the parabola’s maximum can be more clearly determined so that a long confidence interval could also be avoided. Higher N rates in the higheryielding zone have a shortening effect on the confidence interval for N_opt.

When considering the enormous length of both the 95% confidence intervals, it becomes clear that even in such extensive N rate trials, the ex post estimated optimum N rates can only be roughly estimated. It is not possible, in retrospect, to limit the optimal nitrogen quantity to a level of less than 87 kg∙ha^–1 length in the low-yielding zone or 124 kg∙ha^–1 in the high-yielding zone. The length of the confidence intervals results from the fact that N fertilization have a very wide marginal profit area. From the economic point of view, however, the additional N quantity used ruins the economic advantage of the increase in yield. This leads to a very flat function around the optimum for returns above N fertileizer cost, making it compatible with many model parabolae with widely ranging vertices. The set of these vertices is the confidence interval, which is therefore very long.

The length of the confidence intervals gives us a first insight into the uncertainty concerning the true, unknown N_opt. Although their enormous length makes clear that reasonable statements about N_opt can hardly be made, this source of uncertainty is the only one that can be controlled statistically because the confidence sets could be shortened if more data were available.

In order to analyze the influence of the N rate trial design on the point estimate and confidence set for the economically optimum N rate, point estimates and confidence intervals for N_opt with different designs of N levels were calculated. In Figure 2 and Figure 4, all N levels (0, 80, 100, 120, 140, 160, 180, 200, 220, 240 and 260 kg∙ha^–1) are considered, whereas in Figure 3 and Figure 5 the zero N rate is disregarded.

Taking zero fertilization not into account leads to a higher estimation of the economically optimum N rate, N_opt, and to the lengthening of the confidence interval. This can be observed in the quadratic model (compare Figure 3 to Figure 2) as well as in the linear-plus-plateau model (compare Figure 5 to Figure 4). In the lowyielding zone, the 95% likelihood confidence set based on the quadratic model (Figure 3) is no longer an interval. It is made up of all real numbers with the exception of an interval. The reason for such degenerated confidence sets is that the shape of the parabola, concave or convex, around which the data disperse, cannot be clearly recognized when the zero N rate is omitted. The data could be fitted passably by a line with a positive gradient and thus, almost equally, by very wide parabolae whose vertices lie to the far left if they are convex and to the far right if they are concave. In this way, a gap-type confidence set arises, which contains N-coordinates for profit maxima (on the right of the gap) as well as profit minima (on the left of the gap).

The gap could disappear if, in addition, the error probability were so “little” that every vertex can be considered as being a “little” compatible with the data. In practice, all these confidence sets are, of course, worthless. Based on the research findings, feasible fertilization rules cannot be given at a high level of confidence. The gap type or the extreme length of these confidence sets, without zero fertilization, is an indication of the fact that the function for the returns above N fertilizer cost, which is to be maximized, must be relatively flat in the area of its maximum, making it very difficult to locate.

If, however, the zero N rate is taken into account (Figure 2), the parabola is forced to sink as a result of the low yield under no fertilization, but not just to the left of the maximum: The symmetry of the parabola means that it would sink to the right as well. In this way, the region around the maximum is less flatly modeled as it appears to be in reality. The shorter length of the confidence interval that results from this may also possibly be attributed to a weakness in the quadratic model in the area of N optimum.

However, by excluding zero fertilization, the model is limited to a really small area of interest and the weakness is then of little importance. The smaller the region, the better it can be modeled by a simple function. On the one hand, the confidence interval is extended when omitting the zero fertilization, but on the other hand, it is also made more trustworthy since it is less influenced by the model assumptions.

In the introduction five function types used for modeling the yield response are mentioned (quadratic, linearplus-plateau, Mitscherlich, quadratic-plus-plateau, squareroot). The evaluation of N rate trials based on these models is related to the point estimates for N_opt. Usually, confidence intervals are only computed in the linearplus-plateau model, where N_opt is one of its parameter. The confidence intervals of [2], however, are also based on the quadratic model, but they were not compared with those based on the linear-plus-plateau model. Yet we shall see that such a comparison, which shall be made in the following, reveals large differences.

As far as the goodness of fit is concerned, it can be stated that both models, the quadratic (Figure 2 and Figure 3) and the linear-plus-plateau model (Figure 4 and Figure 5), fitted equally good. Reference [6] also made the same conclusion, but this does not validate the finding completely because, despite this similarity, [6] and many other authors found marked discrepancies with respect to the N_opt (cf. [28-31]).

Such discrepancies also apply to the confidence intervals for N_opt and their length. The linear-plus-plateau model provides smaller confidence sets. When considering all N levels, as in Figure 4, the increasing straight line’s slope is 0.026 (Mg∙ha^–1) (kg∙ha^–1)^–1 = 26 for the high-yielding zone and 0.033 (Mg∙ha^–1) (kg∙ha^–1)^–1 = 33 for the low-yielding zone. Figure 5 depicts the design without the zero N rate, where the slope of the increasing straight line of the linear-plus-plateau model is 19 for the highand 15 for the low-yielding zone. All these slopes are greater than the realistic price ratios rp^–1. Thus, the economic optimum, N_opt, equals the transition of the increasing straight line to the horizontal, where confidence intervals for N_opt could be obtained using PRISM [17].

The shorter length of the confidence intervals can be attributed to the fact that the transition from the increaseing straight line to the horizontal is clearly easier to identify than the position where a parabola has a rather flat positive gradient (Figure 2 and Figure 3). It is not surprising that the inferences from the two models, the linear-plus-plateau and the quadratic one, which are based on such very different biological assumptions, are incompatible. Nevertheless, it may be surprising that the confidence intervals for N_opt differed so much in both models. When considering only the design with all N levels, these confidence intervals did not even overlap. In the high-yielding area, the linear-plus-plateau model resulted in a 95 % confidence interval of [158 kg∙ha^–1, 200 kg∙ha^–1] (Figure 4), whereas the quadratic model yielded a confidence interval of [204 kg∙ha^–1, 328 kg∙ha^–1] (Figure 2). The same pattern was seen in the low-yielding area: If the linear-plus-plateau Model is assumed to be true, then the N_opt estimate lies between 106 and 148 kg∙ ha^–1 (Figure 4), whereas the quadratic model considers fertilization between 173 and 260 kg∙ha^–1 (Figure 2) as optimal.

The latter clearly shows that a confidence interval does not sufficiently enough reflect the uncertainty of the true optimum N rate. The fact that the true exact model is unknown and that no simple model reflects the reality in an appropriate way makes the uncertainty much greater.