linear and nonlinear-segmented models describing ...

The 11th Conference of Agronomy, Agron. Dept., Fac. Agric., Assiut Univ., Nov. 15-16, 2005

LINEAR AND NONLINEAR-SEGMENTED MODELS DESCRIBING RESPONSE OF MAIZE GRAIN YIELD TO NITROGEN FERTILIZATION Mohammad Hany A. Tageldin and Nasser K. B. El-Gizawy Agronomy Department, Banha University ABSTRACT In nitrogen fertilization trials of maize (Zea mays L.), estimation of both optimum nitrogen fertilizer and maximum grain yield has been of interest to researchers. This requires assessment of statistical models fit to yield data in response to N fertilizer rates before selecting a single model that best fit data. We assessed linear-plateau (L-P), quadratic-plateau (Q-P), quadratic (Q), exponential, and square root models describing four cultivar grain yield responses to five N fertilizer rates (0, 40, 80, 120, and 160 kg N fa -1). This was based upon data from a two-year field experiment conducted in 2001 and 2002. Averaged over cultivars, coefficients of determination, R2 were quite close and considerably high (0.873 to 0.887) for all models, except for the exponential model that yielded fairly low value (0.582). The latter model gave a root mean square error (RMSE) equals 465 whereas the others resulted in values ranged from 248-262. Despite of the closeness of both R2 and RMSE of four models, they resulted in a wide range of predicted optimum N fertilizer rates. The square root model gave absurd values, and the exponential model gave the lowest (58 -76 kg N fa –1). Yet, the linear-plateau calculated lower values than the Q and the Q-P, which both gave equal great values lied outside N data space for three cultivars. For predicted maximum yield, both the exponential and the square root models predicted lower maximum yield than did the others. The quadratic model, however, predicted greater mean maximum grain yield than did the quadratic-plateau (3906.7 kg fa-1 vs. 3898.9 kg fa-1). Residual analysis of the fitted models indicated that the pattern of residuals of the exponential model exhibited a clear systematic bias; none of the other models showed clear systematic bias. INTRODUCTION Describing and quantitatively determining the response of corn yield to applied N fertilizer has been of interest to researchers. This has been done via fitting various mathematical models to yield data in response to applying different N fertilizer rates over several environments (sites and/or years). These models are fitted to help predicting both economic optimum N fertilizer rate and maximum yield or either one. A family of response surface models has been fit to yield data of various crops. Generally, the most common models are the linear (Mamo, Malzer, Mulla, Huggins, and Strock, 2003); splined linear-plus-

plateau (Cerrato and Blackmer, 1990); quadratic, and splined quadratic-plusplateau (Cerrato and Blackmer, 1990; Bullock and Bullock, 1994); square root, and exponential (Cerrato and Blackmer, 1990; and Bélanger, Walsh, Richards, Milburn, and Ziadi, 2000). By comparing several models applied to 12 site-years (six sites and two year) of seed yield data using 10 N fertilizer rates, Cerrato and Blackmer (1990) found that all models fit the data equally using the coefficient of determination (R2 ) statistic. Also the quadratic-plus-plateau was found to describe well yield response in their study.

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Tageldin & El-Gizawy

Predictions of economic optimum N fertilization by applying these models often vary as has been noted by Anderson and Nelson (1975). Reasons for these disagreements among models have not been clearly agreed on (Cerrato and Blackmer, 1990). Since researchers rarely mention reasons of selecting one single model over others; a standardized way of selection, as they further added, has to be established. Two contradicted criteria of selecting a regression equation are involved as described by Draper and Smith (1981): i) model should include as many predictor variables (Xi’s) as possible so that trustworthy fitted values can be determined, ii) Since cost is involved in obtaining and monitoring information on Xi’s , equation should include as few as Xi’s . They have called the compromise between these extremes “selecting the best regression equation”. There is no unique statistical way for this selection as they further added. The most common criteria reported in N response studies are the values of R2 and/or S2 (the residual mean square). It is quite important to check the validity of models by examining the residuals (observed yield-predicted yield) to find out if one or more of model assumptions have been violated. In all of the literature mentioned herein, no one directly or indirectly reported testing statistical significance of model parameters. All models evaluated in Cerrato and Blackmer’s study (1990) gave similar predicted maximum yields(PMY); yet there were clear variations among models to predict economic optimum N rates (EON). The quadratic model neither gave a valid prediction of yield response (Cerrato and Blackmer, 1990; Bullock and Bullock, 1994) nor predicted reasonable optimal N fertilizer rates (Cerrato and Blackmer, 1990; and Bélanger, Walsh, Richards, Milburn, and Ziadi, 2000). The

finding of Cerrato and Blackmer (1990) and Bullock and Bullock (1994) assumed splined functional forms can lead to better, and more realistic information. Aref, Bullock, and Bullock (1997) argued that using various functions to predict mean corn grain yield response to N fertilizer should not ignore stochastic temporal variables eg., rainfall and temperature, which often vary from year to year, for example, in the USA corn belt. They highlighted two shortcomings in Cerrato and Blackmer (1990), and Bullock and Bullock (1994). The first shortcoming of Cerrato and Blackmer (1990) is that the authors use very short-term (one or two years) data sets from agronomic experiments to draw their conclusions. A second major shortcoming of Cerrato and Blackmer (1990), Bullock and Bullock (1994), and many other previous studies is that they only look at the effects of N fertilizer on average (mean) corn yields and do not consider the effect of farmer attitude toward risk when making N fertilizer decisions. Farmers are also concerned about the variance of their yield and profit. The objectives of this study were i) to fit five different models applied to four corn cultivar’s seed yield data, ii) to calculate economic optimum N fertilizer rate of each single model, iii) to calculate predicted maximum yields, and iv) to perform residuals analysis of each model. This evaluation did not aim to select a particular model to be recommended since this certainly requires evaluation based on data collected from more environments (sites and years). MATERIALS AND METHODS A 2-yr field trial was conducted in 2001 and 2002 at the Moshtohor Experiment Center, Kalubiah, Egypt on a clay soil with a pH 7.8 and organic matter content 18 g kg-1 . Before planting, the experimental area had been divided into

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The 11th Conference of Agronomy, Agron. Dept., Fac. Agric., Assiut Univ., Nov. 15-16, 2005

3.0 m x 3.5 m subplots each had five approximately 0.70-cm wide ridges. Seeds of Giza 2, SC 10, TWC 310, and SC 3062 were put in 0.20 m apart hills on 14 June in 2002 and 11 June in 2003. Characteristics of these cultivars are presented in Table 1. In both years, Egyptian clover (Trifolium alexandrinum L.) was the preceding crop. Following planting, a quantity of 23 kg fa-1 P2O5, in the form of calcium mono phosphate (15.5% P2O5) fertilizer, was drilled between ridges. In addition to zero N rate,

two equal split doses of ammonium nitrate (33.5 % N) at a rate of 40, 80, 120, and 160 kg N fa-1 were drilled between ridges just prior to Irrigations 1 and 2 (elongation growth stage). Cultivars and N rates were laid out in a split plot design in four randomized complete blocks, where N rates were the main plots and cultivars were the subplots. At harvest, seed yield was estimated from within the inside three ridges of each subplot. After adjustment to 155 g kg-1 moisture content, seed yield (kg fa-1) was used in the statistical analysis.

Table 1. Characteristics of cultivars used in 2001 and 2002. Developed by Grain Color Type Ag. Res. Centre White Synthetic cultivar Ag. Res. Centre White Single cross hybrid Ag. Res. Centre White Three-way cross hybrid Pioneer Co. Yellow Single-cross hybrid Response Curve Models Commonly used in N response curve analyses, five response curve models(splined linear-plateau, splined quadratic--plateau, quadratic, negative exponential, and square root) were fit to each cultivar’s seed yield data from each year, and to each averaged-over-year cultivar’s seed yield. This was performed using both SAS NLIN (Ihnen and Goodnight, 1985), and GLM procedures (Spector, Goodnight, Sall, and Sarle, 1985). The general regression model takes the form in Eq. [1]

Yi = β 0 + ∑ β i X i + ε i

[1] where Yi is the response variable, here is the grain yield (kg fa-1) and X i is the set of the predictor variables, here is the N fertilizer rate (kg fa-1), β 0 (the intercept), β i (coefficient of respective X i ), and ε i is a random error term. The splined linearplateau model is defined by Eq. [2]  β 0 + β 1 X i + ε i if X < X m  Yi =  if X ≥ X m  Ym + ε i [2]

Cultivar Giza 2 SC 10 TWC 310 SC 3062

where X m is the critical N fertilizer rate, which occurs at the intersection of the linear response and the plateau line, and Ym is the maximum yield, also referred to as the plateau yield. To ensure continuity at the threshold, maximum yield is defined as in Eq. [3] Ym = β 0 + β1 X m , [3] thus we can define ( X m , Ym ) as the knot point at which the response and plateau portions are splined. The splined quadratic-plateau model is defined by Eq. [4]  β + β 1 X i + β 2 X i 2 + ε i if X < X m  Yi =  0  if X ≥ X m  Ym + ε i [4] where Yi is the response variable, here is the grain yield (kg fa-1) and X i is the predictor variable, here is the N fertilizer rate (kg fa-1), β 0 , β1 as before, and β 2 the

X

m is quadratic regression coefficient. the critical N fertilizer rate, which occurs at the intersection of the quadratic

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Tageldin & El-Gizawy

response and the plateau line. Ym is the maximum yield, also referred to as the plateau yield. To ensure continuity at the threshold, maximum yield is defined as in Eq. [3]. Thus we can define ( X m , Ym ) as the knot point at which the response and plateau portions are splined. The quadratic model is defined by Eq. [5] 2 Yi = β 0 + β 1 X i + β 2 X i + ε i [5] where Yi is the response variable, here is the grain yield (kg fa-1) and X i is the predictor variable, here is the N fertilizer rate (kg fa-1). The exponential model(the Mitscherlich model (Cerrato and Blackmer, 1990) is defined by Eq. [6] Yi = β 0 (1 − exp(− β 1 ( X i + β 2 ) ) [6] where Yi is the response variable, here is the grain yield (kg fa-1) and X i is the predictor variable, here is the N fertilizer rate (kg fa-1). The square root model is defined by Eq.[7] 12 Yi = β 0 + β 1 X i + β 2 X i + ε i [7] where Yi is the response variable, here is the grain yield (kg fa-1) and X i is the predictor variable, here is the N fertilizer rate (kg fa-1). Predicted Maximum Yields Calculation of predicted maximum yields (Ym)(kg fa-1) depend on the model used in the analysis. For the quadratic and square root models as in Eq. [5&7], Ym’s were obtained by taking the first derivative of each response equation then equating to zero and solving for x and substituting the value of x into the response equation and solving for y . For the quadratic model,

− bˆ dy ˆ = b1 + 2bˆ2 x = 0 → x = 1 dx 2bˆ2 [8]

b2 2 dy 1 −1 2 = b1 + b2 x =0 →x= dx 2 4b12 [9] For the exponential model (Eq.[6]), Ym approximately equals the estimated value of the intercept ( bˆ0 ) of the model. This means that it predicts maximum yield when fertilizer application at infinity (Cerrato and Blackmer, 1990). For the linear-plateau and quadratic-plateau models (Eq. [2], [4]), the plateau yields were considered the maximum yields. Economic Optimum N Fertilizer Rates (EON) For the linear-plateau model (Eq. [2]), EON values were identified by determining the intersection of the two lines (Ihnen and Goodnight, 1985). For the rest of the models, values were calculated by equating the first derivatives of the response equations to a certain fertilizer-to-corn price ratio and solving for X (Cerrato and Blackmer, 1990). The first derivatives for the quadratic and quadratic-plateau models as in Eq. 8 and for the square root models as in Eq. 9; for the exponential model,

dy = b0 * b2 * e −b2 ( X +b1 ) dx [10] Values of coefficient of multiple determination, R 2 = (1 − ( RSS TSS ) ) , and of residual error mean square(RMS), of the fitted response equations of the used models, were estimated using regression analysis. In addition, the SAS UNIVARIATE procedure was applied to check for the normality of the residuals resulted from different models using the Shapiro-Wilk (W) test (Delong, 1985).

For the square root model, RESULTS AND DISCUSSION Except for the exponential model, the

other four models seemed to fit quite well corn grain yield data of each cultivar in

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The 11th Conference of Agronomy, Agron. Dept., Fac. Agric., Assiut Univ., Nov. 15-16, 2005

both years as indicated by the values of the coefficients of determination, R 2 , as in Table 2. All four models explained as minimum as 80% of the variation in mean corn grain yield in both years, whereas the exponential model explained variation in the range of 0.401 to 0.741 over cultivars in both years (Table 2). In addition, the four models did not seem to fit each cultivar’s grain yield differently. Another criterion used in assessing models are the values of the residual mean square , S 2 , or the root mean square error (RMSE) which is the square root of S 2 as presented in (Table 3). This was equivalently reported as ‘SE’ by Bélanger, Walsh, Richards, Milburn, and Ziadi, (2000). Data in Table 3 indicated that the exponential model resulted in exceptionally high RMSE compared to the other four models, where the latter resulted in similar values in both years. When models were fit to meanover-year grain yield, both R 2 and RMSE values had the same magnitude (Table 4) as for all models in each individual year. As reported by Mead and Pike (1975) there has not been so far enough biological basis for selecting one model over a set of other candidate models. Most often values of R 2 and/or S 2 are used to justify the selection of a particular model. However, to claim a model that best fit a data set, its estimated parameters (bˆi ) have to be different from zero at a certain level of probability ( p < α ) to effectively contribute to the total variation in mean grain yield. In the literature of the N fertilizer response curve analyses, this point has rarely been mentioned directly or indirectly. In the procedures of backward, forward, and stepwise regression analyses, however, Draper and Smith (1981) reported choosing certain α to test the entered or removed fitted parameters . -

High R 2 value does not necessarily imply significance of all estimated model parameters (bˆi ) . In Tables 5 & 6, for example, it is apparent that the quadratic regression coefficients (bˆ2 ) for either the quadratic or the quadratic-plateau fitted equations were non significant at p = 0.05 for certain cultivars in individual years (Table 5), or for mean-over year (Table6) grain yield despite the reasonably 2

high R values for these particular fitted regression equations. Yet, the fitted equations for the exponential model were the only ones that resulted in significant regression coefficients (Tables 5 & 6) despite of the relatively quite low

R 2 values as indicated earlier. 2 The values of R can easily attain 1.0 (or 100%) when all the X values are different, but when repeat runs (replications) exist in the data as in this study; it never reaches 1.0 no matter how the model fits and how many terms are used in the model (Draper and Smith, 1981). Cerrato and Blackmer (1990) 2

proved that greater R values were obtained when models were fitted to four N rates compared to the original 10 rates they have used. Since most N fertilizer response studies usually involve four or 2

five N rates, relying on R as a criterion to choose among different N response models is not reliable enough. Moreover, based on single N fertilizerto-corn price ratio, as Cerrato and Blackmer (1990) stated there should only be one correct economic optimum N fertilizer rates (EON); however, EON values varied much among the fitted models based on mean-over

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Tageldin & El-Gizawy

Table 2. Coefficients of determination (R2 values) for fitted models that relate corn cultivar grain yield to N fertilizer rate in 2001 and 2002. R2 values Cultivar Giza 2 SC 10 TWC 310 SC 3062 Mean Giza 2 SC 10 TWC 310 SC 3062 Mean Over all mean

LinearPlateau

QuadraticPlateau Quadratic 2001 0.806 0.845 0.845 0.886 0.891 0.891 0.850 0.849 0.849 0.854 †-----0.856 0.849 0.862 0.860 2002 0.841 0.869 0.869 0.857 0.882 0.882 0.853 0.872 0.872 0.816 0.858 0.858 0.842 0.870 0.870 0.845 0.866 0.865

Exponential

Square root

0.532 0.682 0.565 0.401

0.829 0.902 0.855 0.856 0.545 0.861

0.518 0.741 0.559 0.546

0.855 0.907 0.872 0.843 0.591 0.869 0.568 0.865

† Value could not be obtained because the model failed to fit yield data.

Table 3. Root mean square error (RMSE) for fitted models that relate corn cultivar grain yield to N fertilizer rate in 2001 and 2002. Root Mean Square Error (RMSE) Cultivar Giza 2 SC 10 TWC 310 SC 3062 Mean Giza 2 SC 10 TWC 310 SC 3062 Mean Over all mean

LinearPlateau

QuadraticPlateau Quadratic 2001 321.96 295.73 295.73 281.77 275.56 275.56 268.02 268.56 268.56 280.73 †-----287.03 288.12 279.95 281.72 2002 305.73 285.72 285.72 316.87 288.40 288.40 274.68 263.34 263.34 281.90 254.83 254.83 294.79 273.07 273.07 291.45 276.02 277.39

† Value could not be obtained because the model failed to fit yield data.

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Exponential

Square root

500.53 458.41 443.97 570.28 493.29

310.79 261.81 263.68 287.37 280.91

533.32 415.00 476.07 443.19 466.89 480.09

300.14 255.35 263.54 268.20 271.81 276.36

The 11th Conference of Agronomy, Agron. Dept., Fac. Agric., Assiut Univ., Nov. 15-16, 2005

year grain yield (Table 7). Fitted models that have very close values of

R 2 differed much in EON values; they further

added

that

this

is

another 2

shortcoming of relying on R values. Ahmed (1994) compared models based on minimum mean square error; he based calculating optimum rate of N fertilizer on the model that had been chosen to fit data. By analogy and based on Cerrato and Blackmer’s argument(1990), models which have the least MSE do not necessarily produce a reasonable optimum N fertilizer rate. Averaged over cultivars, the exponential model resulted in value lower than of each of the linear-plateau (LP), the quadratic-plateau (Q-P) and the Q models. Also, Bélanger, Walsh, Richards, Milburn, and Ziadi (2000) obtained lower value for the exponential compared to the Q model. Based on data in Table 7 and averaged over cultivars, the L-P had lower value than that of either the Q-P and Q models (156 vs. 218 kg N fa-1). The latter trend was also obtained by Cerrato and Blackmer (1990). The L-P model performed and explained crop response to fertilizer quite as well or better than polynomial (parabolic, cubic, etc.) models as reported by Tembo, Brosen, and Epplin (2005). Polynomial specifications that have been used, they further added, do not display a plateau and generally result in nitrogen recommendations higher than a plateau model. In other studies, the Q fitted model predicted higher EON value than that of the Q-P (Cerrato and Blackmer, 1990; Bullock and Bullock, 1994). By comparing the Q fitted model to both the segmented L-P and Q-P fitted models, it appears that i) xm point --the rate of N fertilizer at which the linear/quadratic and plateau portions of the L-P/Q-P functions are splined –-were extremely great whether in each year or

when averaged over years (Tables 5 & 6); ii) all three fitted models yielded relatively high EON values for some cultivars (Table 7). In the above both cases, values were located beyond the X data space (0 - 160 kg N fa-1); and iii) both Q and Q-P models resulted in quite identical EON values (Table 7). These results contradicted what has been reached at in the literature (Cerrato and Blackmer, 1990; Bullock and Bullock, 1994). In the traditional L-P and Q-P functions, x m and y m are treated as parameters, and it is implicit that all factors that affect the plateau are fixed and entirely controllable (Tembo, Brosen, and Epplin, 2005); however, these factors mostly tend to vary randomly. They further added that weather conditions vary temporally and soil nutrients in a particular field tend to vary stochastically from place to another. Hence, they concluded that the position of the knot point is greatly conditioned on these random factors. Aref, Bullock and Bullock (1997) criticized ignoring stochastic weather factors when predicting corn grain yield using various functions. Based on Tempo, Brosen, and Epplin’s (2005) argument, shift in the knot position is likely to occur under the nature of the US environmental condition due mainly to weather or soil stochastic factors. Over the two years of field trial of this research, we have not encountered any unusual weather change or disease incident. The position of the critical point xm is likely to be shifted away due to the expected high residual soil N from using Egyptian clover as the previous crop in both years, and applying relatively high doses of N fertilizer. The first factor exerted an upward shift in the estimated intercept bˆ0 ; this is indicated by the relatively great values of the intercepts of the fitted models (Tables 5 & 6). Moreover, the high rate of change in grain

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Tageldin & El-Gizawy

Table 4. Coefficients of determination (R2 values), and root mean square error (RMSE) for fitted models† to mean-over-year data that relate corn cultivar grain yield to N fertilizer rate. L-P Q-P Q Exponential Sq root 2 Cultivar R values Giza 2 0.842 0.876 0.876 0.536 0.860 SC 10 0.885 0.900 0.900 0.723 0.918 TWC 310 0.887 0.888 0.888 0.580 0.891 SC 3062 0.878 0.884 0.884 0.490 0.881 Mean 0.873 0.887 0.887 0.582 0.887 Root Mean Square Error (RMSE) Giza 2 294.790 268.848 268.848 505.316 284.774 SC 10 281.918 252.300 252.300 425.459 237.724 TWC 310 236.180 234.767 234.767 443.536 231.938 SC 3062 237.409 238.467 238.467 486.074 240.902 Mean 262.574 248.595 248.595 465.096 248.834 † L-P=linear-Plateau, Q-P=quadratic-Plateau, Q=quadratic, Sq root= square root.

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The 11th Conference of Agronomy, Agron. Dept., Fac. Agric., Assiut Univ., Nov. 15-16, 2005

Table 5. Estimated parameters of fitted models to data describing relationship between corn cultivar grain yield and N fertilizer rate in 2001 and 2002. Estimated Parameter

b0 b1 Xm † Plateau ‡

b0 b1 b2 Xm† Plateau ‡

b0 b1 b2 b0 b1 b2 b0 b1 b2

Giza 2 2001 2002 Linear-Plateau 1461.59 1292.27 11.02* 177.70

11.82*

SC 10

TWC 310 2001 2002

2001

2002

2101.52

1773.79

1973.58

15.41*

16.40*

11.73*

SC3062 2001

2002

1800.74

2484.50

2297.92

11.11*

11.43*

9.97*

199.70

122.27

112.86

135.89

172.09

187.33

176.90

3420.95 3653.55 Quadratic-Plateau 1297.05 1147.22

3986.78

3625.20

3568.63

3712.21

4625.22

4062.30

2010.79

1677.16

1936.38

1687.59

§ ----

2145.21

22.86*

24.47*

15.01*

16.76*

----

17.60*

19.25*

19.07*

-0.051 ns

-0.045 ns

-0.064*

-0.075 ns

-0.029 ns

-0.035 ns

----

-0.047 ns

187.22

210.41

176.66

161.59

256.07

237.05

----

184.50

3099.31 Quadratic 1297.05

3154.26

4030.36

3654.49

3858.60

3674.65

----

3769.68

1147.22

2010.79

1677.16

1936.38

1687.59

2520.33

2145.21

19.25*

19.07*

22.86*

24.47*

15.01*

16.76*

9.63*

17.61*

-0.051 ns

-0.045 ns

-0.064*

-0.075*

-0.029 ns

-0.035 ns

0.011ns

-0.048*

Exponential 2597.07 2500.62

3537.35

3241.73

3096.34

2943.61

3620.06

3326.62

14.11*

16.85*

15.86*

18.85*

17.82*

0.052*

0.047*

0.056*

0.053*

0.063*

0.059*

Square root 1314.01 1171.03

1952.04

1588.17

1902.55

1669.32

2522.32

2155.74

4.85 ns

6.75 ns

1.39 ns

-1.49 ns

4.98 ns

5.61 ns

13.01*

4.03 ns

82.49 ns

67.77 ns

148.60*

185.19*

71.35 ns

73.46 ns

-21.14 ns

79.48 ns

14.67* 0.048*

13.90* 0.046*

15.46*

* =Significant, and ns = non significant at the 0.05 probability level. † X m = critical N rate (kg fa-1) which occurs at the intersection of linear / quadratic response and the linear lines. ‡ Plateau corn grain yield (kg fa-1). § Values could not be obtained because the model failed to fit yield data.

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Table 6. Estimated parameters of fitted models to mean-over-year data describing relationship between corn cultivar grain yield and N fertilizer rate. Estimated parameter Cultivar

b0

Giza 2 SC 10 TWC 310 SC 3062

1376.94 1942.51 1856.60 2391.21

Giza 2 SC 10 TWC 310 SC 3062

1222.14 1843.98 1811.98 2332.78

Giza 2 SC 10 TWC 310 SC 3062

1222.14 1843.98 1811.98 2332.78

Giza 2 SC 10 TWC 310 SC 3062

2548.85 3389.54 3019.98 3473.35

b1

b2

Linear-Plateau --------------------Quadratic-Plateau 19.16* -0.048* 23.67* -0.070* 15.89* -0.032 ns 13.62* -0.018 ns Quadratic 19.16* -0.048* 23.67* -0.070* 15.89* -0.032 ns 13.62* -0.018 ns Exponential 14.29* 0.047* 14.81* 0.049* 16.37* 0.054* 18.35* 0.061* Square root 5.80 ns 75.137 ns ns -0.054 166.89* 5.29* 72.41* 8.52* 29.17 ns 11.42* 15.73* 12.19* 10.70*

Xm

†

Plateau ‡

188.70 118.14 135.85 182.12

3532.87 3800.12 3512.12 4339.97

198.09 168.54 245.67 372.99

3120.30 3838.47 3763.72 4873.23

-------------

-------------

-------------

-------------

Giza 2 1242.52 ------SC 10 1770.11 ------TWC 310 1785.94 ------SC 3062 2339.04 ------* =Significant, and ns = non significant at the 0.05 probability level. † X m = critical N rate (kg fa-1) which occurs at the intersection of linear / quadratic response and the linear lines. ‡ Plateau corn grain yield (kg fa-1).

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The 11th Conference of Agronomy, Agron. Dept., Fac. Agric., Assiut Univ., Nov. 15-16, 2005

Table 7. Predicted economic optimum N fertilizer rate† and maximum grain yield by each fitted model‡ to mean-over-year data. L-P Cultivar Giza 2 188.70 SC 10 118.15 TWC 310 135.85 SC 3062 182.12 Giza 2 SC 10 TWC 310 SC 3062 Mean

3532.86 3800.87 3512.12 4339.97 3796.45

Q-P Q Exponential Predicted economic optimum N rate Kg fa-1 179.50 179.50 73.54 155.26 155.26 76.10 216.07 216.07 65.79 321.21 321.21 58.57 Predicted maximum grain yield 3120.30 3134.94 2548.85 3838.47 3844.43 3389.54 3763.72 3765.77 3019.98 4873.23 4881.66 3473.35 3898.93 3906.70 3107.93

†The fertilizer-to-corn price ratio was 1.93, which was calculated based on L.E. 2.0 kg-1 N, and

‡ L-P=linear-Plateau, Q-P=quadratic-Plateau, Q=quadratic, Sq root= square root. § could not be estimated ¶ could not be estimated since estimated b1 was extremely low, and estimated b2 was quite high..

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Sq root

41.97 -----§ 46.78 2.95 1972.18 -----¶ 2528.26 2413.94 2304.79

L.E. 1.036 kg-1 grain corn.

Tageldin & El-Gizawy

yield of each cultivar, represented by the slope bˆ1 due to the equal increment of added N fertilizer dose, shifted away the knot points. The economic optimum N fertilizer rate (EON), except for the L-P model, depends on a predetermined N-to-grain corn price ratio. In the Q-P fitted model, since the critical N fertilize rate lies outside the N rate data set (Table 6) for each cultivars, this means that the original data set, and also up to this critical point, is totally explained by just the quadratic fitted model. This may explain the equal values of EON of both the Q and the Q-P fitted models. By simple calculations, as the price of fertilizer decreases in relation to grain corn price (narrow ratio), variations in EON values become greater within the same fitted model, and among them. This has been explained by Cerrato and Blackmer (1990) on the basis that lower fertilizer prices shifts optimal fertilizer rates into horizontal parts of the curves where minor differences in response occur relative to large differences in fertilizer rates. At moderate price ratios, the economic optimum N rate of the L-P model is independent of the price ratio since EON value depends upon where the critical N rate, xm , lies. Data in Table 7 show that the models predicted different maximum yields averaged over years and cultivars. Both the exponential and the square root fitted models tended to predict lower maximum yield than did the other three models; yet, the square root model predicted extremely low maximum yield. These low values of the fitted square root model were based on illogical quite low N rates for Giza 2, TWC 310, and SC3082 cultivars ( ≅ 42.0, 47.0, 3.0 kg N fa-1). The SC10 cultivar maximum yield could not be estimated since bˆ1 was extremely low

( = - 0.0538) and bˆ2 was quite high (=166.899). As has been indicated by Bullock and Bullock (1994), the quadratic model predicted greater mean maximum grain yield than did the quadratic-plateau model (3906.7 kg fa-1 vs. 3898.9 kg fa-1) (Table 7). The discrepancies in predicting maximum grain yield by the fitted models provides little confidence for using maximum yield as a criterion for choosing one single model or a set of models over others. The prediction process itself varies between models, since both the Q and the square root models depend upon equating the first derivative with respect to N rate (X) to zero and solving for X to get Y. For the exponential model, maximum yield equals the value of predicted intercept ( bˆ0 ), but it equals the plateau yield for both L-P and Q-P models. Figures 1a through 1e represent analysis of the residuals (observed yield, yi – predicted yield, yˆ i ) from each fitted model. Points lie above the horizontal line indicate that the model under predicted yield and points under this horizontal line indicate that the model over predicted yield. The pattern of residuals of the exponential model in Fig. 1d indicates that there is a clear systematic bias. It seems that the fitted model overestimated yield at 40 and 80 kg N fa -1 rates, but it underestimated it at 120 and 160 kg N fa –1 for the four cultivars. The other four fitted models also differed in the way they predicted yields; none exhibited clear systematic bias. Testing normality of the residuals resulted from applying all five models to data from individual years as well as from mean-over-year using the Shapiro-Wilk (W) test (Delong, 1985) indicated failing to reject the hypothesis of normality

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The 11th Conference of Agronomy, Agron. Dept., Fac. Agric., Assiut Univ., Nov. 15-16, 2005

600 Giza 2

400 200 0 - 200

0

50

100

150

200

- 400 - 600

Observed yield - predicted yield (kg/fa)

600 SC 10

400 200 0 - 200

0

50

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- 400 - 600

600 TWC 310 400 200 0 0

50

100

150

200

- 200 - 400

600 SC 3082

400 200 0 0 200 -

50

100

150

200

- 400 - 600

Applied N fertilizer rate (kg/fa)

Fig. 1a. Residuals (observed yield- predicted yield) when fitting a linear + plateau model to averaged-over-year cultivar yields.

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Tageldin & El-Gizawy

600 Giza 2

400 200 0 - 200

0

50

100

150

200

- 400 - 600

Observed yield - predicted yield (kg/fa)

600 SC 10

400 200 0 - 200

0

50

100

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- 400 - 600

600 TWC 310

400 200 0 - 200

0

50

100

150

200

- 400 - 600

600 SC 3082

400 200 0 - 200 0

50

100

150

200

- 400 - 600

Applied N fertilizer rate (kg/fa)

Fig. 1b. Residuals (observed yield- predicted yield) when fitting a quadratic + plateau model to averaged-over-year cultivar yields.

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The 11th Conference of Agronomy, Agron. Dept., Fac. Agric., Assiut Univ., Nov. 15-16, 2005

600 Giza 2

400 200 0 - 200

0

50

100

150

200

- 400 - 600

Observed yield - predicted yield (kg/fa)

600 SC 10

400 200 0 - 200

0

50

100

150

200

- 400 - 600

600 TWC 310

400 200 0 - 200

0

50

100

150

200

- 400 - 600

600 SC 3082

400 200 0 0 200 -

50

100

150

200

- 400 - 600

Applied N fertilizer rate (kg/fa)

Fig. 1c. Residuals (observed yield- predicted yield) when fitting a quadratic model to averaged-over-year cultivar yields.

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Tageldin & El-Gizawy

800 600 400

Giza 2

200 0 0 200 400 -

50

100

150

200

100

150

200

100

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- 600 - 800 - 1000

Observed yield - predicted yield (kg/fa)

800 600

SC 10

400 200 0 - 200 0

50

- 400 - 600 - 800

1000 800 600 400 200 0 200 0 - 400 - 600 - 800 - 1000

800 600 400 200 0 - 200 0 - 400 - 600 - 800 - 1000

TWC 310

50

SC 3082

50

Applied N fertilizer rate (kg/fa)

Fig. 1d. Residuals (observed yield- predicted yield) when fitting an exponential model to averaged-over-year cultivar yields.

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The 11th Conference of Agronomy, Agron. Dept., Fac. Agric., Assiut Univ., Nov. 15-16, 2005

600 Giza 2

400 200 0 - 200

0

50

100

150

200

- 400 - 600

Observed yield - predicted yield (kg/fa)

600 SC 10

400 200 0 - 200

0

50

100

150

200

- 400 - 600

600 TWC 310

400 200 0 - 200

0

50

100

150

200

- 400 - 600

600 SC 3082

400 200 0 - 200 0

50

100

150

200

- 400 - 600

Applied N fertilizer rate (kg/fa)

Fig. 1e. Residuals (observed yield- predicted yield) when fitting a square root model to averaged-over-year cultivar yields.

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Tageldin & El-Gizawy

(P < W); therefore these models are valid to fit the data. Based on mean-over-year data, both the Q and Q-P fitted models had higher W (W = 0.957), on the average, compared to those of both the L-P and the square root models, which had equal values (W = 0.948). The exponential model had the least value(W = 0.941). Probabilities of failing to reject the hypothesis of normality ranged from as low as P = 0.088 to as high as P = 0.822 over all models and cultivars. Cerrato and Blackmer (1990) found that both the exponential and square root fit least well. Bélanger, Walsh, Richards, Milburn, and Ziadi (2000) found that the quadratic model had a higher W than the exponential, but the square root model did not give a valid description of the yield response. They emphasized how residual analysis aids considerably in the total process of model selection. CONCLUSIONS In general, the discrepancies among models concerning predicted optimum N fertilizer rate or predicted maximum grain yield as concluded in this study or in the literature have been resulted from shortterm experiments. Although long-term fertilizer experiments are too difficult to carry out, response curve fit needs trials to be conducted over many sites on quite variant soils for many years. These longterm trials will certainly help measuring how, for example, predicted optimum N fertilizer by a particular model would vary over many site-year data. Aref, Bullock, and Bullock (1997) suggested that the variance, rather than just the mean value, may be of more concern. Stemming from their variance issue, we suggest that minimum variance may be applied as a model selection criterion.

REFERENCES Ahmed, F. A. 1994. Multivariate and response curve analyses for important yield factors in maize. Ph.D thesis, Zagazeeg Univ. Anderson, R.L., and L.A. Nelson. 1975. A family of models involving intersecting straight lines and concomitant experimental designs useful in evaluating response to fertilizer nutrients. Biometrics 31:303-318. Aref, S., D. Bullock, and D. Bullock. 1997. Consideration of farmer perception of risk, rainfall probability and realistic production function in the calculation of economic optimum nitrogen rates for corn. Illinois Fertilizer Conference Proceedings, January 27-29. Bélanger, G., J.R. Walsh, J.E. Richards, P.H. Milburn, and N. Ziadi. 2000. Comparison of three statistical models describing potato yield response to nitrogen fertilizer. Agron. J. 92:902-908. Bullock, D.G., and D.S. Bullock. 1994. Quadratic and quadratic-plus-plateau models for predicting optimal nitrogen rate of corn: A comparison. Agron. J. 86:191-195. Cerrato, M.E., and A.M. Blackmer. 1990. Comparison of models for describing corn yield response to nitrogen fertilizer. Agron. J. 82:138-143. Delong, D.M. 1985. The univariate procedure. p. 1181-1191. In SAS user’s guide: Basics, 1985 ed. SAS Institute Inc., Cary, NC. Draper, N.R., and H. Smith. 1981. Applied regression analysis. 2nd ed. John Wiley & Sons. New York, USA.

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‫‪Biometrics 31:803:851.‬‬ ‫‪Spector, P.C., J.H. Goodnight, J.P. Sall,‬‬ ‫‪and W.S. Sarle. 1985. The GLM‬‬ ‫‪procedure. p. 433-506. In SAS‬‬ ‫‪user’s guide: Statistics, 1985 ed.‬‬ ‫‪SAS Institute Inc., Cary, NC.‬‬ ‫‪Tembo, G., B.W. Brosen, and F. M.‬‬ ‫‪Epplin. 2005. Stochastic linear‬‬ ‫‪response plateau (LRP) functions.‬‬ ‫‪ last consulted‬‬ ‫‪7/7/2005.‬‬

‫‪Ihnen, L.A., and J.H. Goodnight. 1985.‬‬ ‫‪The NLIN procedure. p. 575-606. In‬‬ ‫‪SAS user’s guide: Statistics, 1985 ed.‬‬ ‫‪SAS Institute Inc., Cary, NC.‬‬ ‫‪Mamo, M., G. L. Malzer, D. J. Mulla, D.‬‬ ‫‪R. Huggins, and J. Strock. 2003.‬‬ ‫‪Spatial and Temporal Variation in‬‬ ‫‪Economically Optimum Nitrogen‬‬ ‫–‪Rate for Corn. Agron. J. 95:958‬‬ ‫‪964.‬‬ ‫‪Mead, R., and D.J. Pike. 1975. A review‬‬ ‫‪of response surface methodology‬‬ ‫‪from a biometric viewpoint.‬‬

‫ﺍﻟﻤﻠﺨﺹ ﺍﻟﻌﺭﺒﻰ‬ ‫ﺍﻟﻨﻤﺎﺫﺝ ﺍﻟﺨﻁﻴﺔ ﻭﺍﻟﻨﻤﺎﺫﺝ ﺍﻟﻼﺨﻁﻴﺔ ﺍﻟﻘﻁﻌﻴﺔ ﺍﻟﺘﻲ ﺘﻨﺎﻗﺵ ﺍﺴﺘﺠﺎﺒﺔ ﻤﺤﺼﻭل ﺤﺒﻭﺏ ﺍﻟﺫﺭﺓ ﺍﻟﺸﺎﻤﻴﺔ‬ ‫ﻟﻠﺘﺴﻤﻴﺩ ﺍﻟﻨﻴﺘﺭﻭﺠﻴﻨﻲ‬ ‫ﻤﺤﻤﺩ ﻫﺎﻨﺊ ﺃﺤﻤﺩ ﺘﺎﺝ ﺍﻟﺩﻴﻥ ﻭ ﻨﺎﺼﺭ ﺨﻤﻴﺱ ﺒﺭﻜﺎﺕ ﺍﻟﺠﻴﺯﺍﻭﻯ‬ ‫ﻗﺴﻡ ﺍﻟﻤﺤﺎﺼﻴل —ﺠﺎﻤﻌﺔ ﺒﻨﻬﺎ‬ ‫ﻓـﻲ ﺘﺠـﺎﺭﺏ ﺍﻟﺘﺴﻤﻴﺩ ﺍﻟﻨﻴﺘﺭﻭﺠﻴﻨﻲ ﻟﻠﺫﺭﺓ ﺍﻟﺸﺎﻤﻴﺔ‪ ،‬ﻓﺘﻘﺩﻴﺭ ﻜل ﻤﻥ ﻤﻌﺩل ﺍﻟﺘﺴﻤﻴﺩ ﺍﻷﻤﺜل ﻭ ﺃﻋﻠﻲ ﻤﺤﺼﻭل‬ ‫ﺤـﺒﻭﺏ ﻴﻌﺩﺍ ﻤﻥ ﺃﻫﺩﺍﻑ ﺍﻟﺒﺎﺤﺜﻴﻥ ‪.‬ﻫﺫﺍ ﻴﺘﻁﻠﺏ ﺘﻘﻴﻴﻡ ﻨﻤﺎﺫﺝ ﺍﺤﺼﺎﺌﻴﺔ ﺘﻁﺒﻕ ﻋﻠﻲ ﺍﻟﺒﻴﺎﻨﺎﺕ ﻭ ﺫﻟﻙ ﺍﺴﺘﺠﺎﺒﺔ ﻟﻤﻌﺩﻻﺕ‬ ‫ﺘﺴﻤﻴﺩ ﻭ ﺫﻟﻙ ﻗﺒل ﺍﺨﺘﻴﺎﺭ ﺍﻟﻨﻤﻭﺫﺝ ﺍﻟﺫﻱ ﻴﻁﺎﺒﻕ ﺍﻟﺒﻴﺎﻨﺎﺕ ﺒﻁﺭﻴﻘﺔ ﺃﻓﻀل ‪.‬ﻟﻘﺩ ﻗﻤﻨﺎ ﺒﺩﺭﺍﺴﺔ ﺍﻟﻨﻤﻭﺫﺝ ﺍﻟﺨﻁﻲ ﺫﻭ ﺍﻟﺠﺯﺀ‬ ‫ﺍﻟﻤﻭﺍﺯﻱ ﻟﻤﺤﻭﺭ ﺍﻟﺴﻴﻨﺎﺕ‪ ،‬ﻭﺍﻟﻨﻤﻭﺫﺝ ﻤﻥ ﺍﻟﺩﺭﺠﺔ ﺍﻟﺜﺎﻨﻴﺔ ﺫﻭ ﺍﻟﺠﺯﺀ ﺍﻟﻤﻭﺍﺯﻱ ﻟﻤﺤﻭﺭ ﺍﻟﺴﻴﻨﺎﺕ‪ ،‬ﻭﺍﻟﻨﻤﻭﺫﺝ ﻤﻥ ﺍﻟﺩﺭﺠﺔ‬ ‫ﺍﻟﺜﺎﻨـﻴﺔ‪ ،‬ﻭ ﻭﺍﻟﻨﻤﻭﺫﺝ ﺍﻷﺴﻲ‪ ،‬ﻭ ﻭﺍﻟﻨﻤﻭﺫﺝ ﺫﻭ ﺍﻟﺠﺫﺭ ﺍﻟﺘﺭﺒﻴﻌﻲ ‪.‬ﻭﺫﻟﻙ ﺒﺎﺴﺘﺨﺩﺍﻡ ﺒﻴﺎﻨﺎﺕ ﻤﺤﺼﻭل ﺃﺭﺒﻊ ﺃﺼﻨﺎﻑ ﻤﻥ‬ ‫ﺍﻟﺫﺭﺓ ﺍﻟﺸﺎﻤﻴﺔ ﺍﻟﻨﺎﺘﺠﺔ ﻋﻥ ﺍﺴﺘﺨﺩﺍﻡ ﺨﻤﺱ ﻤﻌﺩﻻﺕ ﻤﻥ ﺍﻟﺴﻤﺎﺩ ﺍﻟﻨﻴﺘﺭﻭﺠﻴﻨﻲ ﻭﻫﻲ ﺼﻔﺭ‪160 ،120 ،80،40 ،‬ﻜﺠﻡ‬ ‫ﻥ ﻟﻠﻔـﺩﺍﻥ ﻓﻲ ﺘﺠﺭﺒﺔ ﺤﻘﻠﻴﺔ ﻋﺎﻤﻲ ‪2001‬ﻭ ‪2002.‬ﻜﺎﻥ ﻤﻌﺎﻤل ﺍﻟﺘﺤﺩﻴﺩ ﻤﺘﻘﺎﺭﺒﺎ ﻭﻤﺭﺘﻔﻌﺎ )‪(0.887--0.873‬ﻓﻴﻤﺎ‬ ‫ﻋﺩﺍ ﺍﻟﻨﻤﻭﺫﺝ ﺍﻷﺴﻲ ﺍﻟﺫﻱ ﺃﻋﻁﻲ ﻗﻴﻤﺔ ﻤﻨﺨﻔﻀﺔ ﻨﺴﺒﻴﺎ‬

‫‪(0.582).‬ﻭﻗﺩ ﺃﻋﻁﻲ ﺍﻷﺨﻴﺭ ﻗﻴﻤﺔ ﻤﺭﺘﻔﻌﺔ ﻟﻠﺠﺫﺭ ﺍﻟﺘﺭﺒﻴﻌﻲ‬

‫ﻟﺘﺒﺎﻴﻥ ﺍﻟﺨﻁﺄ )‪(465‬ﺒﻴﻨﻤﺎ ﺘﺭﺍﻭﺤﺕ ﺍﻟﻘﻴﻡ ﻟﺒﻘﻴﺔ ﺍﻟﻨﻤﺎﺫﺝ ﻤﻥ ‪248‬ﺍﻟﻲ ‪262.‬ﻭ ﺒﺎﻟﺭﻏﻡ ﻤﻥ ﺘﻘﺎﺭﺏ ﻗﻴﻡ ﻜل ﻤﻥ ﻤﻌﺎﻤل‬ ‫ﺍﻟﺘﺤﺩﻴﺩ ﻭﺍﻟﺠﺫﺭ ﺍﻟﺘﺭﺒﻴﻌﻲ ﻟﺘﺒﺎﻴﻥ ﺍﻟﺨﻁﺄ ﻟﻠﻨﻤﺎﺫﺝ ﺍﻷﺭﺒﻌﺔ‪ ،‬ﻓﻘﻴﻡ ﻤﻌﺩل ﺍﻟﺘﺴﻤﻴﺩ ﺍﻷﻤﺜل ﺍﻟﻨﺎﺘﺠﺔ ﻟﻡ ﺘﻜﻥ ﻤﺘﻘﺎﺭﺒﺔ ﺒﺎﻟﻤﺭﺓ ‪.‬‬ ‫ﻓﺄﻋﻁﻲ ﺍﻟﻨﻤﻭﺫﺝ ﺍﻟﺠﺫﺭ ﺍﻟﺘﺭﺒﻴﻌﻲ ﻗﻴﻤﺎ ﻏﺭﻴﺒﺔ ﻭ ﻏﻴﺭ ﻤﻨﻁﻘﻴﺔ‪ ،‬ﻭﺃﻋﻁﻲ ﺍﻷﺴﻲ ﺃﻗل ﻗﻴﻡ ‪(76- 58‬ﻜﺠﻡ ﻥ ﻟﻠﻔﺩﺍﻥ ‪).‬ﺃﻤﺎ‬ ‫ﺍﻟﻨﻤﻭﺫﺝ ﺍﻟﺨﻁﻲ ﺫﻭ ﺍﻟﺠﺯﺀ ﺍﻟﻤﻭﺍﺯﻱ ﻟﻤﺤﻭﺭ ﺍﻟﺴﻴﻨﺎﺕ ﻓﻘﺩ ﺃﻋﻁﻲ ﻗﻴﻤﺎ ﺃﻗل ﻤﻥ ﻜل ﻤﻥ ﺍﻟﻨﻤﻭﺫﺝ ﻤﻥ ﺍﻟﺩﺭﺠﺔ ﺍﻟﺜﺎﻨﻴﺔ ﺫﻭ‬ ‫ﺍﻟﺠﺯﺀ ﺍﻟﻤﻭﺍﺯﻱ ﻟﻤﺤﻭﺭ ﺍﻟﺴﻴﻨﺎﺕ‪ ،‬ﻭﺍﻟﻨﻤﻭﺫﺝ ﻤﻥ ﺍﻟﺩﺭﺠﺔ ﺍﻟﺜﺎﻨﻴﺔ ﺍﻟﻠﺫﺍﻥ ﺃﻋﻁﻴﺎ ﻗﻴﻤﺎ ﻋﺎﻟﻴﺔ ﻤﺘﺴﺎﻭﻴﺔ ﻭﻗﻌﺕ ﺨﺎﺭﺝ ﻨﻁﺎﻕ‬ ‫ﻤﻌﺩﻻﺕ ﺍﻟﻨﻴﺘﺭﻭﺠﻴﻥ ﺍﻟﻤﺴﺘﺨﺩﻤﺔ ‪.‬ﺃﻤﺎ ﺒﺎﻟﻨﺴﺒﺔ ﺍﻟﻲ ﺃﻋﻠﻲ ﻤﺤﺼﻭل ﺤﺒﻭﺏ‪ ،‬ﻓﻘﺩ ﺘﻨﺒﺄ ﻜل ﻤﻥ ﺍﻟﻨﻤﻭﺫﺝ ﺍﻷﺴﻲ ﻭﻨﻤﻭﺫﺝ‬ ‫ﺍﻟﺠﺫﺭ ﺍﻟﺘﺭﺒﻴﻌﻲ ﺒﺄﻗل ﻗﻴﻡ ﻤﻘﺎﺭﻨﺔ ﺒﺎﻟﻨﻤﺎﺫﺝ ﺍﻷﺨﺭﻱ ‪.‬ﻓﻲ ﺤﻴﻥ ﺘﻨﺒﺄ ﺍﻟﻨﻤﻭﺫﺝ ﻤﻥ ﺍﻟﺩﺭﺠﺔ ﺍﻟﺜﺎﻨﻴﺔ ﺒﻤﺘﻭﺴﻁ ﻗﻴﻡ ﺃﻋﻠﻲ ﻤﻥ‬ ‫ﻤﺜﻴﻠـﺘﻬﺎ ﺍﻟـﻨﺎﺘﺠﺔ ﻤـﻥ ﺍﻟﻨﻤﻭﺫﺝ ﺍﻟﺨﻁﻲ ﺫﻭ ﺍﻟﺠﺯﺀ ﺍﻟﻤﻭﺍﺯﻱ ﻟﻤﺤﻭﺭ ﺍﻟﺴﻴﻨﺎﺕ ‪(3906‬ﻤﻘﺎﺒل ‪3898‬ﻜﺠﻡ ﻟﻠﻔﺩﺍﻥ ‪).‬ﻭ‬ ‫ﺒﺘﺤﻠﻴل ﺍﻷﺜﺭ ﺍﻟﻤﺘﺒﻘﻲ ﻟﻠﻨﻤﺎﺫﺝ ﺍﻟﺨﻤﺴﺔ‪ ،‬ﻓﻘﺩ ﺃﻅﻬﺭ ﺍﻟﻨﻤﻭﺫﺝ ﺍﻷﺴﻲ ﺘﺤﻴﺯﺍ ﻤﻨﺘﻅﻤﺎ ﻭﺍﻀﺤﺎ ﻓﻲ ﺤﻴﻥ ﻟﻡ ﻴﻅﻬﺭ ﺃﻱ ﻤﻥ‬ ‫ﺍﻵﺨﺭﻴﻥ ﺃﻱ ﺘﺤﻴﺯ ﻭﺍﻀﺢ‬

‫‪.‬‬

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