Comparison of Three Statistical Models Describing Potato Yield Response
to Nitrogen Fertilizer
Gilles Bélanger,* John R. Walsh, John E. Richards, Paul H. Milburn, and Noura Ziadi
ABSTRACT
Quadratic models have been very popular for describing the crop response to fertilization, but they tend to
overestimate the response if the maximum point on the
curves is taken as the best fertilization rate (Neeteson
and Wadman, 1987; Cerrato and Blackmer, 1990; Colwell, 1994). Exponential functions have also been used
to describe the crop response to fertilizer with agronomic crops and vegetables (Neeteson and Wadman,
1987). The square root model can also be a reasonable
choice in many situations (Nelson et al., 1985; Colwell, 1994).
Most studies based on statistical models for predicting
optimal N rates have been conducted on corn (Zea mays
L.) (Cerrato and Blackmer, 1990; Bullock and Bullock,
1994; Isfan et al., 1995), and only a few studies have
been reported on potato (Neeteson and Wadman, 1987).
Our objective was to compare and evaluate three statistical models (quadratic, exponential, and square root)
describing the response of potato to N fertilizer application. More specifically, we focused on fitting each model
to data collected from 12 field trials, each having six N
rates and comparing (i) calculated economic optimum
N rates, (ii) coefficients of determination (R2) and standard errors of the estimate (SE), (iii) trends in differences between measured and calculated data, and (iv)
potential economic losses associated with making incorrect decisions when a response model was selected from
among others. This is part of a larger study aimed at
developing field-specific recommendations based on the
relationship between optimal N rates and spring soil nitrates.
Estimation of optimum fertilizer rates is of interest because of
growing economic and environmental concerns. Optimum fertilizer
rates can be determined by fitting statistical models to yield data
collected from N fertilizer experiments. We evaluated quadratic, exponential, and square root models describing the yield response of potato
(Solanum tuberosum L.) to six rates of N fertilization (0–250 kg N
ha⫺1) with and without supplemental irrigation at four on-farm sites
in each of three years (1995 to 1997) in New Brunswick, Canada.
Economic optimum N rates (Nop) varied among sites and models. The
proportion of variability (R2) explained by the three models was
similar. The quadratic model, however, calculated a greater Nop value
(175 kg N ha⫺1) averaged over all sites than those calculated by the
square root (123 kg N ha⫺1) and exponential (80 kg N ha⫺1) models.
Regression residues of the quadratic model were closer to a normal
distribution than those of the other two models, indicating a less
systematic bias. Economic losses were greatest when the quadratic
model was the most appropriate model, but the data were fitted to
the exponential (loss of $204–240 ha⫺1; all values in Canadian dollars)
or square root model (loss of $58–201 ha⫺1). We conclude that the
quadratic model is the most appropriate for describing the potato
yield response to N fertilizer and predicting Nop for areas with a ratio
of the cost of N fertilizer to the price of potatoes similar to that in
Atlantic Canada.
N
itrogen fertilization recommendations must optimize crop yield and quality, maximize profitability, and reduce the risk of environmental pollution. Fertilizer recommendations are usually based on field trials
that determine the crop response to various rates of
fertilizer application. Data from fertilizer studies can be
fitted to several statistical models to determine optimum
fertilizer rates. The selection of the most appropriate
model for a particular cropping situation is not obvious
(Bock and Sikora, 1990; Angus et al., 1993; Bullock
and Bullock, 1994). In addition, model selection has
considerable effects on estimating optimal fertilizer
rates. For example, different models fitting one data set
can give comparable coefficients of determination (R2)
but different optimal fertilizer rates (Cerrato and Blackmer, 1990; Isfan et al., 1995). Although several statistical
models are commonly used to describe the crop yield
response to fertilizer rates, the choice of one model over
another is rarely explained.
MATERIALS AND METHODS
Field Experiments
The experimental data used in this study and the conduct
of the experiments were previously reported by Bélanger et
al. (2000). Briefly, the study was conducted at four on-farm
sites in each of three years (1995 to 1997) in the upper StJohn River Valley of New Brunswick, Canada. The sites are
referred as S1 to S4 in 1995, S5 to S8 in 1996, and S9 to S12
in 1997. At each site, soil cropped to two potato cultivars
(Russet Burbank and Shepody) received 0, 50, 100, 150, 200,
and 250 kg N ha⫺1 as ammonium nitrate, which was placed
in a band 2 cm to the side of and 2 cm below the seed piece
at planting. At each site, the experiment consisted of two large
blocks (irrigated and nonirrigated). Within each block, a splitplot arrangement of the experimental treatments was used,
with cultivars as main plots and N fertilization rates as subplots
with four replications. Individual plots consisted of six rows
each 7.6 m in length. There were 1.5 m between plots within
a block and 24.3 m between the irrigated and nonirrigated
blocks. Irrigation applications were scheduled using the Wisdom computer software program (IPM Software, Madison,
G. Bélanger and N. Ziadi, Soils and Crops Research and Development
Centre, Agriculture and Agri-Food Canada, 2560 Hochelaga Blvd.,
Sainte-Foy, Québec, Canada, G1V 2J3; J.R. Walsh, McCain Foods
Limited, Florenceville, New Brunswick, Canada, E7L 3G6; J.E. Richards, Atlantic Cool Climate Crop Research Centre, Agriculture and
Agri-Food Canada, 308 Brookfield Road, P.O. Box 39088, St. John’s,
Newfoundland, Canada A1E 5Y7; and P.H. Milburn, Potato Research
Centre, Agriculture and Agri-Food Canada, P.O. Box 20280, Fredericton, New Brunswick, Canada E3B 4Z7. Contribution 665 Agriculture
and Agri-Food Canada. Received 28 July 1999. *Corresponding author ([email protected]).
Abbreviations: Nop, optimum N rate; SE, standard error of the estimate.
Published in Agron. J. 92:902–908 (2000).
902
903
BÉLANGER ET AL.: THREE STATISTICAL MODELS FOR POTATO YIELD RESPONSE TO N
WI). Water was applied when soil moisture reserves were
reduced to 65% of the soil water holding capacity. Irrigation
was applied at a rate of 0.68 cm h⫺1 with a portable overhead
irrigation system. Among the four sites per year, supplemental
irrigation ranged from 148 to 217 mm in 1995, 50 to 70 mm
in 1996, and 76 to 121 mm in 1997 (Bélanger et al., 2000).
At harvest, total and marketable tuber yields were determined. Marketable tuber yield was determined as total tuber
yield minus small tubers and defects. Defects consisted of
roughs and tubers with hollow heart, brown center, stem-end
discoloration, insect and wireworm damage, sunburn, and rot.
Data from the two cultivars were combined, since the analysis
of variance indicated no significant cultivar ⫻ N fertilization
interaction (Bélanger et al., 2000).
Models
To describe the potato yield response to N fertilizer, three
statistical models (quadratic, exponential, and square root)
were fitted to the data using the NLIN procedure of the SAS
software (SAS Inst., 1990). Economically optimum N fertilizer
rates for the three models were computed for total and marketable tuber yields within each site with and without irrigation.
The Nop (kg N ha⫺1) is defined as the rate of N application
where $1 of additional N fertilizer returned $1 of potatoes,
and it describes the minimum rate of N application required
to maximize economic return (Colwell, 1994). This analysis
assumes that fertilizer N costs are the only variable costs and
that all other costs are fixed. The Nop was calculated by setting
the first derivative of the N response curve equal to the ratio
between the cost of fertilizer and the price of potatoes for the
three tested models. The ratio of the cost of N fertilizer ($0.86
kg⫺1 N; all values presented are in Canadian dollars) to the
price of potatoes ($143 Mg⫺1 tuber) was 0.006 and was referred
to as CP.
For the three statistical models, Y is the tuber yield in Mg
ha⫺1 (total or marketable), N is the N fertilization rate in kg
N ha⫺1, and a, b, and c are parameter estimates using the
NLIN procedure of SAS (SAS Inst., 1990). The quadratic
model is
Y ⫽ a ⫹ bN ⫹ cN 2
[1]
and Nop is calculated as
Nop ⫽ (CP ⫺ b)/2c
[2]
The square root model is
Y ⫽ a ⫹ bN1/2 ⫹ cN
[3]
Nop ⫽ (0.5b/CP ⫺ c)2
[4]
and
The exponential model (i.e., the Mitscherlich model) is
the models were calculated according to the equation
SE ⫽
冤兺(Y n ⫺⫺3Y ) 冥
meas
calc
2 1/2
[7]
where Ymeas is the measured yield, Ycalc is the calculated yield,
and n is the number of observations.
The analysis of residues (measured yields ⫺ calculated
yields) for total and marketable yields with and without irrigation was also used as a criterion to evaluate the three models.
The residues were reported as a function of either the rate of
N applied or the deviations from the economic optimum N
rates. In addition, a statistical test, based on the values of two
parameters W and P (Shapiro-Wilk test, Delong, 1985), was
used to determine whether the residues of each of the models
were normally distributed.
The potential economic losses or gains from the incorrect
selection of the model were calculated to determine which of
the three models was the most suitable. The potential economic losses or gains were calculated as the difference between
the gains or losses associated with the use of less or more N
fertilizer, and the gains or losses associated with reduced or
increased yield. For example, using the exponential model
when the quadratic model is correct results in an economic
gain associated with the use of less N fertilizer but in an
economic loss associated with a reduced yield. For each site,
and for total and marketable yield with and without irrigation,
we calculated the difference in yield and N rate when an
incorrect decision was made in selecting one model from
among others. The economic losses or gains were then estimated using a cost of $0.86 kg⫺1 fertilizer N and a price of
$143 Mg⫺1 potato tubers. For example, if the quadratic model
is correct but the exponential model was used to estimate Nop,
the yield was calculated using both exponential and quadratic
models with the Nop calculated by the exponential model.
The difference in calculated yield between the two models
represents the yield loss or gain. The difference in N rate is
calculated as the difference between Nop values estimated by
the two models. Average losses or gains were then calculated
by adding losses and gains of all sites and dividing by the
number of sites used. A positive value is considered a gain
and a negative value represents a loss.
RESULTS AND DISCUSSION
The response to N fertilizer was reported by Bélanger
et al. (2000). Nitrogen fertilization significantly (P ⬍
0.05) increased both total and marketable yields at all
sites except at S4. No interaction between N fertilization
and cultivars occurred, and therefore data for Russet
Burbank and Shepody were combined.
Economic Optimum Rates of N Fertilization
Y ⫽ a ⫹ b exp (cN )
[5]
Nop ⫽ 1/c log (CP/bc)
[6]
and
The Nop values were not estimated when c had a positive
value both for quadratic and exponential models, because the
Nop estimation is based on relationships with a continuous
diminishing form; and when b had a negative value for the
square root model, to avoid having Nop estimates as the square
of a negative value (Colwell, 1994).
The coefficients of determination (R2) were computed from
the analysis of variance routine provided on the SAS listing:
R2 ⫽ 1 ⫺ (residual SS/corrected total SS), where SS is the
sum of squares. The SEs of total and marketable yields for
Economic optimum N rates varied greatly among the
tested models and sites (Table 1). The Nop values fell
outside the range of tested rates of N fertilizer at more
sites with the square root model than with the exponential and quadratic models (Table 1). These values were
omitted when the average Nop was calculated because
they were derived by extrapolation, which may considerably decrease their reliability (Neeteson and Wadman,
1987). The Nop calculated by the quadratic model was
greater than that of the exponential model at all sites.
The square root model calculated Nop values that were
between those calculated by the quadratic and exponential models at most sites. In general, when data for mar-
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AGRONOMY JOURNAL, VOL. 92, SEPTEMBER–OCTOBER 2000
Table 1. Economic optimum rates of N fertilization (Nop) calculated by three models for total and marketable yield with and without
irrigation at each site.
Nop
Nonirrigated
Sites
Quadratic
Exponential
Irrigated
Square root
kg N
Total yield
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
Average
Marketable yield
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
Average
Quadratic
Exponential
Square root
ha⫺1
105
170
110
144
250
173
208
207
–‡
250
135
135
172
8
39
9
40
–†
–†
159
157
–‡
156
87
9
74
78
82
80
110
42
–†
–§
–†
127
–†
–†
85
75
217
151
205
–‡
225
203
155
181
169
183
183
148
184
109
41
92
–‡
137
116
61
134
86
72
63
10
82
–†
106
–†
–§
–†
–†
164
–†
174
–†
212
136
112
155
143
150
144
–‡
173
200
–†
–‡
185
230
155
171
7
66
45
43
–‡
99
146
–†
–‡
78
132
9
69
56
192
126
110
–§
–†
–§
29
–§
–†
–†
89
100
198
195
144
108
–†
210
150
220
154
173
–‡
166
172
128
96
74
41
218
116
53
130
71
90
–‡
46
97
–†
236
150
25
–†
–†
149
–†
177
–†
–§
153
148
† Values ⬎250 kg N ha⫺1 are not presented.
‡ The Nop value was not calculated when the fitted parameter c had a positive value.
§ The Nop value was not calculated when the fitted parameter b had a negative value.
ketable and total yields with and without irrigation are
considered, the Nop calculated by the quadratic model
was the highest (175 kg N ha⫺1), followed by the square
root model (123 kg N ha⫺1) and by the exponential
model (80 kg N ha⫺1).
When all models were considered, the average Nop
across sites with irrigation varied between 82 and 184 kg
N ha⫺1 for total yield, and between 97 and 172 kg N ha⫺1
Fig. 1. Example of potato yield response to N fertilization, indicating
how each model fits the data. The data correspond to total yield
under irrigation at S2. Arrows indicate the Nop values for each
model.
for marketable yield (Table 1). With no supplemental
irrigation, the average Nop across sites varied between
74 and 172 kg N ha⫺1 for total yield, and between 69
and 171 kg N ha⫺1 for marketable yield. These results
corroborate other studies (Nelson et al., 1985; Bullock
and Bullock, 1994) where the large variation in Nop was
related to inappropriate model selection. Cerrato and
Blackmer (1990) pointed out the potential economic
and environmental importance of selecting the best response model when making fertilizer recommendations.
The higher Nop obtained with the quadratic model is
in agreement with results reported in the literature
(Bock and Sikora, 1990; Cerrato and Blackmer, 1990).
The quadratic curve must be symmetrical around its
maximum (Fig. 1), which may lead to higher optima
(Neeteson and Wadman, 1987). The exponential response curve, however, may have its optimum at low
fertilizer N rates (Nelson et al., 1985; Neeteson and
Wadman, 1987). In a study conducted in 99 sites in the
Netherlands with potato and sugar beet, Neeteson and
Wadman (1987) reported that Nop varied largely with the
two tested models and that higher values were obtained
with a quadratic than a modified exponential model.
The Nop was 28 kg N ha⫺1 greater with irrigation
than without irrigation for marketable yield when the
exponential model was used (Table 1), but the difference was only 8 kg N ha⫺1 for total yield. The largest
difference in Nop with and without irrigation was, however, obtained with the square root model for both total
(72 kg N ha⫺1) and marketable (48 kg N ha⫺1) yields.
BÉLANGER ET AL.: THREE STATISTICAL MODELS FOR POTATO YIELD RESPONSE TO N
905
Table 2. Coefficients of determination (R2) for three models describing the relationship between potato yield and N rate with and
without irrigation at each site.
R2
Nonirrigated
Sites
Total yield
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
Marketable yield
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
Irrigated
Quadratic
Exponential
Square root
Quadratic
Exponential
Square root
0.49
0.91
0.74
0.63
0.95
0.98
0.76
0.99
–†
0.85
0.84
0.21
0.35
0.76
0.70
0.76
–†
–†
0.23
–†
–†
0.87
0.77
0.62
0.40
0.81
0.82
0.74
0.89
–†
–†
–†
0.66
–†
–†
0.54
0.98
0.88
0.96
–†
0.97
0.98
0.95
0.98
0.86
0.88
0.77
0.67
0.99
0.90
0.98
–†
0.98
0.99
0.99
0.95
0.95
0.90
0.81
0.95
–†
0.94
–†
–†
–†
–†
0.98
–†
0.92
–†
0.83
0.90
0.61
0.75
0.74
0.73
–†
0.95
0.33
–†
–†
0.73
0.73
0.27
0.30
0.70
0.77
0.84
–†
0.97
0.32
–†
–†
0.80
0.71
0.68
0.48
0.84
0.78
0.82
–†
–†
–†
0.83
–†
–†
–†
0.43
0.97
0.93
0.99
0.44
–†
0.94
0.90
0.91
0.84
0.92
–†
0.75
0.94
0.96
0.90
0.34
0.98
0.97
0.89
0.91
0.90
0.91
–†
0.94
–†
–†
0.89
0.27
–†
–†
0.87
–†
0.89
–†
–†
0.92
† R2 values were not determined when the data could not be fitted by the model (see Table 1) or when Nop values were ⬎250 kg N ha⫺1.
With the quadratic model, the average Nop for total and
marketable yield was similar for potato grown with and
without irrigation. In this study, however, 60% of the
sites had no significant interaction between irrigation
and N fertilization and there was no yield response to
irrigation at 4 of the 12 sites (Bélanger et al., 2000).
When considering only the sites (S1, S2, and S9) where
the yield response to irrigation was ⬎9 Mg ha⫺1 (Bélanger et al., 2000) and for which Nop was estimated, the
Nop for both marketable and total yield was 35 kg N
ha⫺1 greater with irrigation than without irrigation when
the quadratic model was used. Under the conditions of
Table 3. The standard error of the estimate (SE) for three models describing the relationship between potato yield and N rate with and
without irrigation at each site.
SE
Nonirrigated
Sites
Total yield
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
Marketable yield
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
Irrigated
Quadratic
Exponential
Square root
Quadratic
Exponential
Square root
2.29
1.46
2.45
1.12
1.39
1.83
1.17
–†
–†
2.33
2.09
2.46
1.92
0.75
1.53
0.90
–†
–†
1.21
1.02
–†
14.68
1.46
1.71
1.73
0.69
1.22
0.94
6.63
–†
–†
–†
13.57
–†
–†
1.96
1.83
1.81
1.90
–†
2.22
0.90
0.52
2.02
3.06
1.47
0.92
3.17
0.58
1.57
0.85
–†
6.20
0.37
0.18
1.98
2.17
1.20
0.82
1.14
–†
1.18
–†
–†
–†
–†
0.28
–†
2.47
–†
0.81
1.70
2.16
0.75
1.68
0.89
–†
2.10
2.10
1.88
–†
3.74
1.94
2.60
1.39
0.80
1.54
0.68
–†
0.99
10.47
–†
–†
2.70
1.88
1.71
1.21
6.23
1.51
0.73
–†
–†
–†
1.91
–†
–†
–†
1.98
1.18
1.74
1.89
0.90
–†
1.74
0.52
1.72
3.65
1.22
–†
2.80
1.58
1.28
1.50
0.96
1.28
1.05
0.54
1.69
2.40
1.19
–†
1.62
–†
–†
1.61
1.02
–†
–†
0.60
–†
2.53
–†
–†
1.59
† SE values were not determined when the data could not be fitted by the model (see Table 1) or when Nop values were ⬎250 kg N ha⫺1.
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AGRONOMY JOURNAL, VOL. 92, SEPTEMBER–OCTOBER 2000
Fig. 2. Regression residues (measured yield – calculated yield) when models were fitted to data from individual sites for total yield without
irrigation with the (a ) quadratic model, (b ) exponential model, and (c ) square root model. Each point represents one N rate at one site.
this study, N requirements were greater when there was
a strong positive response to irrigation. Furthermore,
both the quadratic and exponential models revealed
only minor differences in N requirements for total or
marketable yield.
Coefficient of Determination
The three models explained a large proportion of the
variability as indicated by coefficients of determination
⬎0.70 in most cases. Indeed, 79% of R2 values were
⬎0.70 (Table 2). In general, there was little difference
between R2 values obtained by the three tested models.
With similar R2 values, however, a large variation in
calculated Nop was obtained with the three models. This
is illustrated by the data obtained at S2, for which the
calculated Nop varied between 41 and 151 kg N ha⫺1
(Fig. 1). The coefficient of determination is, therefore, a
poor criterion for selecting a model to identify economic
optimum rates of N fertilization; this agrees with other
studies (Cerrato and Blackmer, 1990; Colwell, 1994).
Greater R2 values were observed with irrigation than
without irrigation, indicating the large variability of
yield data when potato is grown under water stress.
Standard Error of the Estimate
The standard error of the estimate varied greatly
among sites and models (Table 3). The quadratic model
consistently had the lowest SE values for both total
BÉLANGER ET AL.: THREE STATISTICAL MODELS FOR POTATO YIELD RESPONSE TO N
907
Fig. 3. Regression residues (measured yield – calculated yield) when models were fitted to data from individual sites for total yield without
irrigation with the (a ) quadratic model, (b ) exponential model, and (c ) square root model. Points from each site and N rate are positioned
relative to calculated economic optimum rates of fertilization, which are located in the centers of the figures.
and marketable yields with and without irrigation. The
exponential and square root models had some SE values
⬎3, especially when no irrigation was applied (Table 3).
The SE varied between 0.65 and 3.24 for the quadratic
model, 0.24 and 11.74 for the square root model, and
0.32 and 12.70 for the exponential model.
Distribution of Residues
To be reliable, models should not have any systematic
bias; therefore, the regression residues should have a
normal distribution. An example of the analysis of regression residues is shown for total yield without irrigation (Fig. 2a, 2b, and 2c). Points above the horizontal
line (residue ⫽ 0) indicate measured values ⬎ calculated
values, and points below the horizontal line indicate the
inverse case. According to a statistical test (ShapiroWilk test, Delong, 1985), the residues from the square
root model did not have a standard normal distribution
(W ⫽ 0.68; P ⬍ 0.001). The quadratic model had a
higher W (W ⫽ 0.98; P ⫽ 0.82) than the exponential
model (W ⫽ 0.90; P ⫽ 0.49). Hence, the distribution of
residues of the quadratic model was closer to a normal
distribution than that of the exponential model, and the
square root model did not give a valid description of
the yield response. This analysis of residues highlights
the importance of model selection to describe the yield
response to N fertilizer and illustrates that the quadratic
model was generally more reliable than the two other
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AGRONOMY JOURNAL, VOL. 92, SEPTEMBER–OCTOBER 2000
models at describing the yield response to N fertilizer.
These results confirm those reported by Cerrato and
Blackmer (1990), indicating that some models fit the
yield response to N fertilizer with less systematic bias
than others. The results of the analysis of regression
residues obtained with total yield under irrigation were
similar to those presented above (data not shown). We
conclude that, on the basis of the analysis of residues,
the quadratic model best describes the data reported in
this study.
The residues for each site-year are also presented as
a function of the deviations from the economic optimum
N rates (Fig. 3a, 3b, and 3c). The economic optimum
rates of fertilization are located in the centers of the
figures. The quadratic model tended to overestimate
yield at the rates of fertilization it identified as optimum
and underestimate yield at the rates of fertilization
greater than Nop. A similar observation was reported by
Cerrato and Blackmer (1990). The exponential model
tended to underestimate yield at the rates of fertilization
it identified as optimum, whereas the square root model
either underestimated or overestimated yield at rates
of N fertilization greater than Nop.
Economic Analysis
Potential economic losses resulting from an incorrect
selection of a response model ranged from $6 to 441
ha⫺1 (Table 4). In all situations, losses were observed
when the quadratic model was the most appropriate
model but the data were fitted to the exponential or
the square root model. As an example, for total yield
without irrigation, less N fertilizer and lower yields are
calculated by the exponential model when the quadratic
model is correct. Consequently, the economic loss due
to lower yield is $203 ha⫺1 greater than the economic
gain due to fertilizer saving. In 75% of cases, a gain was
obtained when data were fitted to the quadratic model
despite the two other models being the correct choice.
Thus, the quadratic model minimizes the risks of potential economic losses when predicting Nop.
It is noteworthy that the average calculated Nop across
sites by the quadratic model (175 kg N ha⫺1) is close
to the existing recommendation for potato in Atlantic
Canada (Bernard et al., 1993). Even though it might be
appropriate for a general recommendation, the use of
a fixed rate such as 175 kg N ha⫺1 might result in underor overfertilization in many situations because of the
range in Nop values (Table 1). The calculated Nop and
the economic analysis presented in this paper are both
a function of the cost of fertilizer and the price of potatoes, and we chose representative values for potato production in Atlantic Canada. The superiority of the quadratic model to describe the potato yield response to N
fertilizer might not apply to other crops or to potatoes
grown in areas where the ratio of the cost of N fertilizer
to the price of potatoes is significantly different than
that in Atlantic Canada.
CONCLUSION
The three statistical models explained a similar proportion of the variability in the yield response to N
Table 4. Mean economic losses or gains resulting from incorrect
model selection.
Mean economic losses or gains†
Model used to identify Nop
If quadratic If exponential If square root
is correct
is correct
is correct
$ ha⫺1
Nonirrigated total yield
Quadratic
Exponential
Square root
Nonirrigated marketable yield
Quadratic
Exponential
Square root
Irrigated total yield
Quadratic
Exponential
Square root
Irrigated marketable yield
Quadratic
Exponential
Square root
0‡
⫺203
⫺229
35
0
95
113
⫺158
0
0
⫺204
⫺201
25
0
⫺13
⫺111
⫺396
0
0
⫺441
⫺79
62
0
180
⫺6
⫺246
0
0
⫺240
⫺58
77
0
107
15
⫺106
0
† Calculations are based on a cost of $CDN 0.86 kg⫺1 for N fertilizer and
a price of $CDN 143 Mg⫺1 of potato tuber.
‡ Positive values indicate a gain and negative values indicate a loss.
fertilization, but the Nop values calculated by the three
models varied greatly. The quadratic model fitted the
data with less bias than the other two models, and calculated Nop values that minimize the risks of potential
economic losses. We conclude that the quadratic model
is best suited to describe the yield response of potato
to N fertilizer and to predict the economic optimum N
rates for areas with a ratio of the cost of N fertilizer to
the price of potatoes similar to that in Atlantic Canada.
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