plied to a set of winter wheat experiments in the Paris Basin of France. The proposed methods are quite genModels of response to applied N can be useful for deriving imeral and can be used with different response functions. proved N dose recommendations. Here we show how response model parameters can be estimated and how model predictions and model Parameter estimation in N response models is generN dose recommendations can be evaluated. For parameter estimation, ally based on ordinary least squares (e.g., Mombiela et we use a statistical approach based on random parameter models. al., 1981; Sain and Jauregui, 1993). This method is easily Two methods for evaluating models are applied. The first method implemented. However, the underlying assumption that is to calculate mean squared error of prediction (MSEP) by cross model errors are independent is unrealistic for data like validation, and the second is to perform nonparametric regressions to N response data where several measurements (correevaluate the profitability of calculated optimal N rates. The proposed sponding to different N doses) are made for the same methods are used with a data set consisting of 37 winter wheat (Tritisite-year. In this paper, we propose an alternative cum aestivum L.) experiments. Different functions taking into account method for parameter estimation. Our approach conend-of-winter mineral soil N are evaluated. The results show that the sists of using a statistical model called a random paramedifferent functions all have similar MSEP values for predictions of yield and grain protein content and lead to N recommendations of ter model, which is a specific type of mixed-effects similar profitability. However, there are substantial differences in model. A random parameter model assumes that the MSEP for residual mineral N at harvest. One of these models is form of the modeled response is common to all sitethen compared with a model that does not include any site-year years but that parameter values vary between site-years. characteristic and with a model that does not have random parameters. The parameters are thus treated as having some random We find that using the model without a site-year characteristic leads variation around their mean values. This approach is to predictions that are less accurate and optimal N rates that are less consistent with the data that are usually observed, and profitable by F 17 to F 105 ha21. Another result is that the gross it automatically introduces correlations among errors margin obtained with the optimal N rates calculated using the model for the same site-year. This method was applied by Walwithout random parameters is lower by F 438 to F 550 ha21. lach (1995a, 1995b) to a simple quadratic (Q) model for yield as a function of applied fertilizer. We show in this paper how random parameter models M predicting the responses of winter wheat can be used with the more complex models of Makowski to applied N can be useful for making better N et al. (1999). Then we show how these models can be dose recommendations to farmers. Numerous functions used for predicting the response to applied N and calcuhave been proposed for describing yield response to lating optimal N rates. The possibility of calculating applied N (Anderson and Nelson, 1975; Cerrato and optimal N rates that depend explicitly on prices, yield, Blackmer, 1990; Bullock and Bullock, 1994; Colwell, grain protein content, residual mineral N, and field char1994), but few have been proposed for describing the acteristics is one of the major attractions of the models responses of grain protein content (Murray and Nunn, of Makowski et al. (1999). This is not possible with 1987; Fowler et al., 1989) and residual mineral soil N the balance-sheet method currently used in France and at harvest (Jauregui and Paris, 1985). In a recent study, other countries to derive N fertilizer recommendations Makowski et al. (1999) defined several sets of functions (Stanford, 1973; Rémy and Hébert, 1977; Neeteson, that relate yield, grain protein content, and residual 1990; Meynard et al., 1997). mineral N at harvest to N applied to winter wheat. The Another problem treated in this paper is that of model authors showed that their functions give satisfactory evaluation. Many different response models can be defits to the data when fitted site-year by site-year. The veloped by using different functions. To select a particupurpose of this paper is to show how the functions of lar model for practical use, it is necessary to define Makowski et al. (1999) can be used for predicting the criteria for evaluating the different possible models. In responses to applied N and calculating optimal N rates numerous studies, response models are evaluated by as a function of site-year characteristics. Solving this calculating R2 values (Anderson and Nelson, 1975; Cerproblem supposes the definition of appropriate statistirato and Blackmer, 1990; Sain and Jauregui, 1993). Ancal methods for estimating model parameters and evaluother approach is to study graphically the distribution ating model quality using standard N fertilizer trials. of the model residuals (Cerrato and Blackmer, 1990; Such methods are presented in this paper and are apBullock and Bullock, 1994). These approaches evaluate the fit of the proposed functions to past data. Often, D. Makowski and D. Wallach, unité d’agronomie, INRA, B.P. 27, however, the goal of developing the models is either to 31326 Castanet-Tolosan Cedex, France; and J.-M. Meynard, Laboratoire d’agronomie, INRA INA P-G, BP 01 78850 Thiverval-Grignon, Abbreviations: E, expectation; F, French franc; LP, linear-plus-plaFrance. Received 15 Feb. 2000. *Corresponding author (makowski@ teau; MSEP, mean squared error of prediction; PL, plateau-plustoulouse.inra.fr). linear; PQ, plateau-plus-quadratic; Q, quadratic; QP, quadratic-plusplateau. Published in Agron. J. 93:531–539 (2001).
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