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This article in AJ

  1. Vol. 71 No. 2, p. 352-354
    Received: June 26, 1978

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Fitting the Law of the Minimum to Fertilizer Applications and Crop Yields1

  1. P. E. Waggoner and
  2. W. A. Norvell2



Liebig's Law of the Minimum is a simple, logical, and successful predictor of crop response to fertilization. A statistical method is reported for fitting the Law by least squares regression to yield responses to two essential factors, with optional inclusion of a limit to maximum yield.

The Law maybe written: y = MIN(a+bq, c+dz2 m); where y is yield, z1 and z2 are two essential nutrients, and m is a maximum imposed by another factor. The expression MIN( ) defies y as the minimum of the specified values; and a,b,c and d are the intercepts and slopes relating y to z1 and z2. The parameters were estimated from crop responses to fertilizer by searching for the minimum residual sum of squares: where for each treatment with yield y1, X1 = z2 if c+dz2 ≤ a+bz1 and m, X1 = (a−c+bz1/d if a+bz1 < c+dz2 and m, X1 = (m−c)/d If m < a+bz1 and c+dz2.

This logical law fit yields of corn that received N and P fertilizer more closely than three empirical functions, and the standard error of estimate for the law was within 14% of that of the most closely fitting function for two legumes that received K and P. Where the Law of the Minimum is logical, it can now be fitted statistically to observations, identifying nutrients limiting each yield and estimating the return from added nutrient.

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