Published January, 1996
Evaluation of Selected Nonlinear Regression Models in Quantifying Seedling Emergence Rate of Spring Wheat YantaiGan,*ElmerH. Stobbe, and CatherineNjue ABSTRACT
L.). Borowiak (1989) states that a data-fitted model. should be assessed for its sensitivity to the observed set of responses in order to ensure appropriate interpretation. In this study, we determined the relative stability and accuracy of the Gompertz, Logistic, and Weibull regression models in quantifying emergence rate of spring wheat.
Fast and uniform seedling emergenceincreases yield potential of spring wheat(Triticum aestivum L.) in short-season areas. Anaccurate methodof quantifying rate of seedling emergenceis needed. In this study, wecompared the relative effectiveness of the Gompertz,Logistic, and Weibull models in quantifying emergence rate of spring wheat. ’Roblin’ wheat was grown in a growth room under five soil water potential: - 0.002, - 0.165, - 0.41, - 1.00, and - 1.45 MPa. Dailyrecorded emergencedata were fitted to each of the models. The analyses of stability and accuracy functions, residual sumof squares, and variance showedthat the Weibullmodelwas not appropriatein quantifying rate of emergence.The Gompertzand Logistic modelsfunctioned in a similar waywith great stability andaccuracy in most cases. The Gompertzpredictions most closely fitted the observedset of responses with residual points scattered aroundzero. For Iognormallydistributed emergence patterns commonunder field conditions, the Gompertz model provided the most appropriate characterization of emergence.
MATERIALS AND METHODS Roblin wheat was grownin a growth roomwith day/night temperatures of 22/16°C, light level of 20 000 Lux, a13-h photoperiod(0600-1900).A Neuhorstclay loamwas collected from a summer-fallow field, sieved through a 2-mmmesh screen, and dried at roomtemperature. The soil was wetted by sprayingwater onto the soil being mixedin a blender. Five soil water potentials, -0.002, -0.165, -0.41, - 1.00, and -1.45 MPa,were established to generate different seedling emergence curves. Moistenedsoil wasplacedin plastic baskets and allowedto equilibrate before seeding. Plastic containers, 62 cmlong by 17 cmwideby 12 cmdeep, were filled with moistenedsoil to a 50-mm depth. Twenty-six pregerminatedseeds were equidistantly spaced on the soil, and then an additional 25 mmof soil was addedto cover the seeds. Thecontainerswerecoveredwith plastic wrapto prevent evaporationduringgermination.Atotal of 30 containers (five soil water potentials times six replicates) wereplacedon the bench using a completely randomizeddesign. The numberof emergingseedlings was recorded daily, and the percentage of emergencewascalculated by dividing the numberof seedlings emergedon a specific day by 26, the total numberof seeds planted per container. The experiment was repeated. Upon confirmation of homogeneityof variance as determined by Bartlett’s test of homogeneity (Steel and Torrie, 1980), data from the runs were combined. The three nonlinear regression modelsused were:
R
UNIFORM, AND COMPLETEseedling emergence ncreases grain yield potential of spring wheat in short-season areas such as the Canadian prairies (Gan et al., 1992). Rapid emergence provides wheat plants with time and spatial advantages to competewith weeds. Uniform and complete emergence allows" the establishment of optimumcanopy structure, that minimizes interplant competition and maximizescrop yield. Quantifying the seedling emergence rate is a prerequisite for the establishment of managementpractices promoting optimum emergence. Emergencein wheat usually occurs following a sigmoidal-shaped curve, relating the cumulative percent emergence to time (Lafond and Fowler, 1989; Gan et al., 1992). The curve starts with a lag period during which no emergence occurs, followed by an exponential period of rapid emergence, and then levels to a plateau when emergenceis nearly complete. This distribution is frequently skewedto the right (Scott et al., 1984). Nonlinear regression models have been used to describe crop emergence. Blackshaw(1991) used a Logistic equation to describe the emergence response of wheat and other crops to soil temperature and water potential. Days to 50% emergence was used to determine differences in rate of emergence. Gan et al. (1992) found the Gompertzregression model described both rate and percent of emergenceof spring wheat effectively. Similarly, Bahler et al. (1989) found that a Weibull model performed as well as the Logistic model in fitting S-shaped germination curves of alfalfa (Medicagosativa AI.PID
the Gompertz, Y = Fexp[-b exp(-kt)], the Logistic, Y = F/[1 + exp(b - kt)], and the Weibull, Y = F/1 - exp[-[(/- a)/b]Cll, whereY is cumulative percent emergenceby time (day) t, is predicted final percent emergence,and a, b, c, and k are constants to be estimated. Emergence began3 d after planting and continueduntil Day13 (i.e., n = 11) wheren is the total numberof days during which emergenceoccurs. For the first two models, m= 3, while the last modelhas tn = 4, wherem is the numberof parameters in the model. Theseparameterswereestimated for each treatmentby fitting equationsto meancumulativeemergence using the least squares estimation methodwith the SAS-NLIN procedure (SASInstitute, 1990).Theseestimates,along withresidual sumof squares and variance, were used to determine modelsstability and accuracy (Borowiak, 1989). Modelstability was based on asymptotic theory of least square estimation. The stability function (SF) was given by:
Y. Gan, Agriculture and Agri-FoodCanada, Semiarid Prairie Agric. Res. Centre, Swift Current, SK, CanadaS9H3X2; E.H. Stobbe, Dep. of Plant Sci., Univ. of Manitoba, Winnipeg, MB, Canada R3T 2N2; C. Njue, Dep. of Statistics, Univ. of Manitoba, Winnipeg, MB,CanadaR3T2N2. Contribution from Dep. of Plant Sci., Univ. of Manitoba, Winnipeg, MB. Received 1 May 1995. *Corresponding author (
[email protected]).
Abbreviations:SF, stability function; CF, criterion function; RSS,residual sum of square.
Published in Crop Sci. 36:165-168 (1996).
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Table 1. Stability function (SF) for the Gompertz, Logistic, grown under five soil water potentials (treatments).
and Weibull regression
equations describing emergence rate of spring wheat
Days after planting Model
3
4
5
6
7
8
9
10
11
12
13
Mean
SF Treatment 1 (-1.45 MPa) Compertz Logistic Weibuli~"
0.15 0.26 5.03
0.48 0.36 8.31
0.69 0.48 1.36
0.68 0.57 2.14
0.66 0.58 3.04 Treatment 2
0.78 1.27 0.56 0.70 3.39 1.55 (- 1.00 MPa)
1.84 0.99 3.44
2.30 1.20 6.70
2.58 1.21 2.74
2.67 1.14 6.94
1.28 0.73 a 4.06 c
Gompertz Logistic Weibull
0.10 0.35 -
0.80 0.64 2.58
0.82 0.78 1.11
0.69 0.75 0.86
0.75 0.66 0.55 0.82 0.61 0.46 0.76 1.22 1.55 Treatment 3 (- 0.41 MPa)
0.49 0.43 1.50
0.48 0.43 1.14
0.48 0.44 0.78
0.49 0.44 0.62
0.57 a 0.56 a 1.10 b
Gompertz Logistic Weibull
0.27 0.42 5.20
0.91 0.80 1.86
0.73 0.79 0.89
0.67 0.74 6.30
0.46 0.38 0.38 0.42 0.37 0.38 2.82 3.22 2.60 Treatment 4 (- 0.165 MPa)
0.40 0.39 1.00
0.41 0.39 0.45
0.42 0.39 0.40
0.42 0.39 0.40
0.49 a 0.49 a 2.28 b
Gompe~z Logistic Weibull
0.22 0.41 8.10
0.93 0.84 1.85
0.76 0.82 0.98
0.66 0.70 0.91
0.43 0.37 0.38 0.39 0.37 0.38 1.13 0.89 0.47 Treatment 5 (- 0.002 MPa)
0.40 0.38 0.41
0.41 0.38 0.41
0.41 0.38 0.41
0.41 0.38 0.41
0.49 a 0.49 a 1.45 b
Gompertz Logistic Weibull
0.26 2.46 3.62
0.89 4.04 2.06
0.71 0.60 0.85
0.68 4.86 3.16
0.41 0.58 0.92
0.42 0.53 0.47
0.43 0.52 0.43
0.43 0.52 0.43
0.50 a 1.75 b 1.64 b
0.48 3.00 2.39
0.38 1.37 1.71
0.38 0.76 2.02
sF values for Weibull modelare in thousands (for treatment 1 only). Means,within a treatment, followed by different letters are significantly different (P < 0.05) based uponthe t-test of ANOVA. = [[S 2 (yi)]/02/1/2, where the biased variance,S 2 (y~) = 2 ¢t~ + o4Ai. Th e o 2is a consistent estimate of variance obtained from repeat data points, ¢t~ is the result obtained from the first order partial derivatives at each emergence day (t~), and Ai is the result obtained from the second order derivatives at t~. Unbiased estimates of o2 were obtained using 12 replicated data points (six from each run) at each emergence day. Small SF values over the range of emergence days indicates stability of the model (Borowiak, 1989).
SF (yi)
Modelaccuracywasdeterminedusing the criterion function (CF): CF(yi) = RSS(yi) - o2(n - m) +
~Ai,
whereRSSis the residual sumof square, 02 is a consistent estimate of varianceobtained fromrepeated data points, ~. is a constant, and EAiis the sumof Ai obtained from the second order derivatives at each emergenceday. It is noted that CF has three components:accuracy as measuredby RSS, linear stability assessed by 02(n - m), and nonlinear stability ac-
Table 2. Criterion function (CF), coefficient of determination (R2), and parameter estimates for the Gompertz, Logistic, and Weibull regression equations describing the seedling emergence rate of spring wheat grown under five soil water potentials (treatments). Parameter estimates CF
2 R
Gompertz Logistic Weibull
0.094 0.091 348.3
0.91 0.92 0.77
95.1(8.21)~" 83.3 (3.83) 77.5 (5.39) 77.3 (4.12) Treatment 2 (- 1.00 MPa)
3.206 (0.536) -6.538 0.669) -67.98 (2.26)
0.390 (0.078) 0.744 (0.086) -41.03 (1.88)
Gompertz Logistic Weibull
0.064 0.073 0.532
0.97 0.97 0.96
4.749 (0.428) - 8.042 (0.968) 3.500 (0.915)
0.891 (0.078) 1.391 (0.169) 2.001 (0.187)
Gompertz Logistic Weibull
0.015 0.017 0.681
0.96 0.97 0.94
85.6 (1.34) 83.9 (1.69) 87.8 (1.53) 3.001 (0.411) Treatment 3 (- 0.41 MPa) 90.2 (0.92) 89.4 (0.74) 2.314 (0.404) 89.3 (0.55) Treatment 4 ( - 0.165 MPa)
4.874 (0.339) - 8.024 (0.446) 2.908 (0.417)
1.091 (0.073) 1.641 (0.091) 2.755 (0.466)
Gompertz Logistic Weibull
0.015 0.013 0.417
0.97 0.97 0.96
89.3 (0.43) 88.5 (0.76) 87.8 (0.88) 2.977 (0.359) Treatment 5 ( - 0.002 MPa)
5.152 (0.175) - 8.429 (0.498) 2.129 (0.384)
1.160 (0.054) 1.747 (0.077) 2.066 (0.229)
Gompertz Logistic Weibull
0.012 0.012 0.201
0.94 0.85 0.87
85.4 (0.71) 84.6 (0.66) 84.7 (0.47)
4.653 (0.256) -7.649 (0.385) 2.834 (0.223)
1.024 (0.054) 1.534 (0.077) 2.449 (0.229)
Model
F
a
b
k or c
Treatment 1 (-1.45 MPa)
The numbersin parentheses are SE.
2.509 (0.214)
GANET AL.: NONLINEAR REGRESSION MODELS QUANTIFYING WHEAT EMERGENCE RATE
167
lOO A ~ ~o ¯ ,-. 6o
.~_ ~ 20
o
4O B
~/ inflection time Ai MPa
m 30 ~ -1.00 MPa
i ~ 20
2
4
6
8
10
12
14
Days after initial emergence Fig. 1. Observed cumulative percentemergence andcurvespredictedby Gompertz (solid lines) andLogistic(dottedlines) models(A), Gompertz-predicted emergence rate withtime(B), for springwheatgrownundersoil waterpotentialof -0.165, -1.00, and-1.45 MPa. counted for by ~4 EAi. Betweentwo data-fitted models, the one with a smaller CF value is more accurate (Borowiak, 1989). Analysis of variance was performedon meanSF values to determinethe significant differencesin stability betweenmodels within each treatment. Modelwas considered to be a fixed factor. Individual SF values for each emergenceday wereused to estimate error variance. A logarithmic transformationwas applied to the SFvalues, since the meansand standarddeviations werenot independent. RESULTS
AND DISCUSSION
Analysis of stability and criterion functions revealed that the Weibull model was neither stable nor accurate relative to the other two modelswhenfitted to cumulative emergence. This model had the highest SF values at most emergence days (Table 1) and had the highest
values in all treatments (Table 2). At the lowest soil water potential (-1.45 MPa), the Weibull model produced exceptionally large SF and CF values, and the lowest R2 value (Tables 1 and 2). The large SF value was attributable to its asymptoticvariance of the residuals at individual emergence day, while the large CF and the low R2 values were due to the large residual sum of squares in the model-fitting (data not shown). It can concluded that the Weibull model is not appropriate in describing seedling emergence in spring wheat. Both Gompertzand Logistic models performed equally well whenfitted to the sameset of emergencedata (Tables 1 and 2). These two models had similarly small SF values at most emergence days (Table 1) and small values in all treatments (Table 2). Exceptions were that when soil water potential was extremely low (-1.45 MPa), the Gompertz model had a higher mean SF value
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CROP SCIENCE, VOL. 36, JANUARY-FEBRUARY 1996
than the Logistic (Table 1), because the Gompertz model produced large SF values from emergence Day 9 to 13. The reverse also was true: When soil water potential was the highest (-0.002 MPa), the Logistic model had a significantly higher mean SF value than the Gompertz, because the Logistic model produced large SF values from emergence Day 1 to 6. Dispersed data-fitting points at any emergence days, either early or late, resulted in an unstable model, and the extremely fast and slow seedling emergence appeared to favor different models. Between the Gompertz and Logistic models, the decision as to which model would most appropriately fit emergence data was easier in some cases than others. Data-fitted curves (Fig. 1A) showed that the two curves generally fit the observed set of responses well. However, when water potential was —1.00 MPa, the Logistic predictions underestimated final percent emergence but overestimated emergence from Day 7 to 10. Residual points for estimates provided from the Gompertz model scattered from 0.03 to —0.03, while residual points for estimates generated from the Logistic spread between 0.06 and -0.06. Emergence of spring wheat followed a sigmoid pattern, with the distribution of emergence skewed to the right in most cases (Fig. IB). Similar patterns of emergence have been observed in winter or spring wheat (Lafond and Fowler, 1989; Gan et al., 1992). These patterns of seedling emergence can be best described by the Gompertz regression model to determine (i) inflection time (t = \nb k ~ l ) , (ii) maximum emergence rate (ymax = kF e~l ), and (iii) median response time, where F, b, and k are constants estimated from the model, and e is the natural log base. Inflection time reflects the days required to produce a significant increase in emergence rate after initial emergence. The maximum emergence rate indexes the magnitude of the increased emergence rate. The median response time or time to 50% emergence, which can be calculated by substituting the constant parameters in the equation with estimated values, provides a measure of location for nonnormally distributed data that is analogous to the mean of a normal distribution. These values provide a complementary information package for interpretation to emergence data. In this study, we used the parameters generated by the Gompertz model to describe the effect of soil water potential on the rate of seedling emergence. As soil water potential decreased, Gompertz-predicted median response time (Fig. 1A) and inflection time (Fig. IB) increased but the maximum emergence rate (Fig. IB) decreased. Compared with the -0.165 MPa treatment, the -1.45 MPa treatment delayed median response time
by 4.4 d (Fig. 1A), increased inflection time by 4.2 d (Fig. IB), but decreased maximum emergence rate by 18.4% d~' (Fig. IB). Soil water potential of -0.41 MPa resulted in a similar emergence pattern to the —0.165 MPa treatment, whereas the extremely high water potential (—0.002 MPa) slightly delayed inflection and median response time, but decreased maximum emergence rate as compared with the —0.165 MPa treatment (data not shown). The highest water potential may have caused an anaerobic condition around roots, decreasing the water uptake rate of roots. Taylor (1983) states that optimum water uptake and subsequent growth of roots occurs only when the soil is well-aerated so that the supply of O2 is sufficient to meet the demands of both plant and soil organisms. The Logistic model makes the simplifying assumptions that the rate of emergence is normally distributed and the response is symmetry, which are not always true. For example, when the seedbed has adequate soil moisture, the majority of seedlings will emerge during the first few days and emergence distribution will be skewed to the right. In addition, the Logistic model does not provide ancillary information about the distribution of emergence response, and hardly measures the dispersion of the response over time. However, when the sole objective is to determine effect of treatment on time to first or/and median emergence, the Logistic model provides a simple and useful description of emergence, as documented by other researchers (Lafond and Fowler, 1989; Blackshaw, 1991).
