Humans continue to transform the global nitrogen cycle at a record pace, reflecting an increased combustion of fossil fuels, growing demand for nitrogen in agriculture and industry, and pervasive inefficiencies in its use. Much anthropogenic nitrogen is lost to air, water, and land to cause a cascade of environmental and human health problems. Simultaneously, food production in some parts of the world is nitrogen-deficient, highlighting inequities in the distribution of nitrogencontaining fertilizers. Optimizing the need for a key human resource while minimizing its negative consequences requires an integrated interdisciplinary approach and the development of strategies to decrease nitrogen-containing waste.

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This paper deals with statistical inferences based on the varying-coe cient models proposed by Hastie and Tibshirani (1993). Local polynomial regression techniques are used to estimate coe cient functions and the asymptotic normality of the resulting estimators is established. The standard error formulas for estimated coe cients are derived and are empirically tested. A goodness-of-t test technique, based on a nonparametric maximum likelihood ratio type of test, is also proposed to detect whether certain coe cient functions in a varying-coe cient model are constant or whether any covariates are statistically signi cant in the model. The null distribution of the test is estimated by a conditional bootstrap method. Our estimation techniques involve solving hundreds of local likelihood equations. To reduce computational burden, a onestep Newton-Raphson estimator is proposed and implemented. We show that the resulting one-step procedure can save computational cost in an order of tens without deteriorating its performance, both asymptotically and empirically. Both simulated and real data examples are used to illustrate our proposed methodology.

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This article develops a local partial likelihood technique to estimate the timedependent coefficients in Cox's regression model. The basic idea is a simple extension of the local linear fitting technique used in the scatterplot smoothing. The coefficients are estimated locally based on the partial likelihood in a window around each time point. Multiple time-dependent covariates are incorporated in the local partial likelihood procedure. The procedure is useful as a diagnostic tool and can be used in uncovering time-dependencies or departure from the proportional hazards model. The programming involved in the local partial likelihood estimation is relatively simple and it can be modified with few efforts from the existing programs for the proportional hazards model. The asymptotic properties of the resulting estimator are established and compared with those from the local constant fitting. A consistent estimator of the asymptotic variance is also proposed. The approach is illustrated by a real data set from the study of gastric cancer patients and a simulation study is also presented.

This paper deals with statistical inferences based on the varying-coe cient models proposed by Hastie and Tibshirani (1993). Local polynomial regression techniques are used to estimate coe cient functions and the asymptotic normality of the resulting estimators is established. The standard error formulas for estimated coe cients are derived and are empirically tested. A goodness-of-t test technique, based on a nonparametric maximum likelihood ratio type of test, is also proposed to detect whether certain coe cient functions in a varying-coe cient model are constant or whether any covariates are statistically signi cant in the model. The null distribution of the test is estimated by a conditional bootstrap method. Our estimation techniques involve solving hundreds of local likelihood equations. To reduce computational burden, a onestep Newton-Raphson estimator is proposed and implemented. We show that the resulting one-step procedure can save computational cost in an order of tens without deteriorating its performance, both asymptotically and empirically. Both simulated and real data examples are used to illustrate our proposed methodology.