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The application of a curve-fitting technique to Brassica napus growth data
Field Crops Research, 2 (1979) 35--43 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
35
THE APPLICATION OF A CURVE-FITTING TECHNIQUE TO BRASSICA NAPUS GROWTH DATA
J.M. C L A R K E *
and G.M. S I M P S O N
Department of Crop Science, University of Saskatchewan, Saskatoon, Sask., S7N O W O (Canada) *Present address: Research Station, Research Branch, Agriculture Canada, Swift Current, Sask., S9H 3X2 (Canada) (Accepted 1 August 1978)
ABSTRACT
Clarke, J.M. and Simpson, G.M., 1979. The application of a curve-fitting technique to Brassica napus growth data. Field Crops Res., 2: 35--43. The use of regression analysis to fit growth data of Brassica napus L. to appropriate polynomial expressions was evaluated. Seven of eight dry weight data sets from material grown in 1975 fitted cubic polynomials, and in 1976 data, six of eight sets fitted cubic polynomials. In the remaining data sets, the cubic terms were non-significant. Crop growth rate and relative growth rate calculated from cubic-fitted data was closer to that calculated from the non-fitted data than that calculated from quadratic-fitted data. Leaf area data fitted quadratic polynomials. Data from high seeding rates gave a poorer fit to polynomials than data from low seeding rates. The lack of fit was greatest in the area of maximum leaf area index.
INTRODUCTION T h e use of regression analysis t o fit g r o w t h d a t a t o a p p r o p r i a t e p o l y n o m i a l e q u a t i o n s has b e e n suggested as a m e a n s o f r e d u c i n g t h e w o r k - l o a d o f p l a n t g r o w t h analysis ( R a d f o r d , 1 9 6 7 ; Nicholls and Calder, 1973). L a n d s b e r g ( 1 9 7 7 ) , h o w e v e r , has criticized t h e use o f p o l y n o m i a l s t o describe p l a n t g r o w t h , since t h e c o n s t a n t s in t h e e q u a t i o n s s e l d o m have biological m e a n i n g . Q u a d r a t i c p o l y n o m i a l s w e r e f o u n d a p p r o p r i a t e f o r fitting d r y weights a n d leaf areas o f Glycine m a x ( B u t t e r y 1 9 6 9 ) . In Callistephus chinensis, d r y weights a n d leaf areas f i t t e d c u b i c p o l y n o m i a l s ( H u g h e s a n d F r e e m a n , 1967). Nicholls a n d Calder ( 1 9 7 3 ) , h o w e v e r , f o u n d little e v i d e n c e t o s u p p o r t t h e use o f c u b i c relationships in A t r i p l e x spp., Poa annua or L o l i u m t e m u l e n t u m . This p a p e r e v a l u a t e s t h e fitting o f d r y w e i g h t and leaf area d a t a o f ~rassica napus L. t o a p p r o p r i a t e p o l y n o m i a l expressions. In a d d i t i o n , the e f f e c t s o f f i t t e d d r y w e i g h t d a t a on t h e derived g r o w t h f u n c t i o n s m e a n relative g r o w t h rate ( R G R ) and m e a n c r o p g r o w t h r a t e ( C G R ) w e r e assessed.
36
MATERIALS AND METHODS
Field trials using the B. n a p u s cultivar Tower were conducted at Saskatoon in 1975 and 1976. The experiment was laid out as a split-plot in randomized complete block, with four replications in 1975 and six in 1976. The main plots consisted of three water regimes -- rainfed, low irrigation and high irrigation. The sub-plot treatments were seeding rates of 2.5, 5, 10 and 20 kg/ha. The sub-plots were 4.9 × 4.9 m, with 0.25 m: quadrats designated for growth analysis sampling. Each sampling site was bordered by a minimum of 0.5 m. Growth analysis samples were taken at 5-day intervals during 1975, and at 7-day intervals during 1976. At each sampling time, all plants in a randomly selected 0.25 m 2 quadrat were cut off at ground level. Dry weights were determined by oven drying for 24 h at 100 ° C. Leaf area was measured with an electronic planimeter (Model LI 3000, Lambda Instruments Corp., Lincoln, Nebr.). Dry weights and leaf areas were determined on all plants of each quadrat during the early part of the season or on representative sub-samples later in the season. The leaf area/time and dry weight/time relationships were fitted in logarithmic form using a least squares regression program (Statistical Research Service, Agriculture Canada, Ottawa, Ont.). Observations for each replicate within each time were entered separately. The time terms were entered step-wise in the program. The affect of the addition of each term to the equations was assessed by an F-test comparing the regression and residual mean squares. In addition, an analysis of variance of the residuals (actual values - predicted values} was performed to determine if the lack of fit was greater at any position on the curves. The growth functions RGR and CGR were calculated using equations given by Radford (1967} for both the original and fitted data. Results are presented for the rainfed and high irrigation treatments only. The low number of observations on the low irrigation treatment precluded the calculation of meaningful regressions. RESULTS
Dry weight data fitted cubic polynomials of the form loge Dry Weight = a + b T + c T 2 + d T 3, where a, b, c, and d are constants, and T is time from
planting in days. Dry weight data obtained in 1975 fitted cubic polynomials in seven of eight cases (Table I). The R 2 values for the cubic equations were greater than those for quadratic polynomials, which ranged from 0.90 to 0.92 in the low irrigation and from 0.87 to 0.91 in the high irrigation. Data from the 5.0 kg/ha seeding rate in the high irrigation treatment fitted a quadratic equation; addition of the cubic term did not significantly reduce the residual sum of squares. The distribution of the residuals was similar for the quadratic and cubic eauations. The 1976 dry weight data fitted cubic polynomials in six of eight cases; in the remaining two sets the addition of the cubic term was not justified {Table H}.
Constant
5
Constant Time Time 2 Time 3
20
(a) (b) (c) (d)
t T h e c u b i c t e r m was non-significant.
Constant Time Time ~ Time 3
10
Time Time 2 Time 3
Constant Time Time 2 Time 3
2.5
Seeding rate (kg/ha)
a + bT + cT 2 + dT ~,1975
-7.9292 0.5160 -0.0068 0.000030
-11.6239 0.6818 -0.0094 0.000043
0.7167 -0.0094 0.000041
-13.4275
-19.4990 1.0047 0.0139 0.000064
Coefficient
Rainfed
1.1458 0.0688 0.0013 0.0000074
1.1252 0.0676 0.0013 0.0000072
0.0903 0.0017 0.0000097
1.5048
1.98483 0.1170 0.0022 0.000013
S.E.
0.94
0.96
0.95
0.94
R2
-7.7330 0.4980 -0.0064 0.000028
-9.8633 0.5640 -0.0070 0.000030
0.4884 -0.0050 0.000016
-9.9433
-16.3663 0.7946 -0.0097 0.000040
Coefficient
High irrigation
o.88t 2.0965
0.96
0.94
0.9526 0.0539 0.0009 0.000005 0.9291 0.0526 0.0009 0.000005
0.1187 0.0021 0.000011
0.94
1.5786 0.0894 0.0015 0.0000083
S.E.
R2
Regression c o e f f i c i e n t s and s t a n d a r d errors f o r dry w e i g h t f i t t e d to cubic p o l y n o m i a l s o f t h e f o r m loge Dry Weight =
TABLE I
e..O
Constant Time Time: Time 3
Constant Time Time s Time 3
Constant Time Time: Time 3
5
10
20
-12.7180 0.6359 -0.0078 0.000032
-21.374 0.9934 - 0.0127 0.000054
-15.0166 0.6346 -0.0068 0.000024
-42.9088 1.8609 -0.0244 0.00011
Coefficient
Rainfed
t T h e cubic t e r m was non-significant.
Constant Time Time: Time 3
2.5
Seeding rate (kg/ha)
a + bT+
2.5066 0.1266 0.0021 0.000011
3.4934 0.1764 0.0029 0.000015
3.5838 0.1810 0.0029 0.000015
6.3187 0.3191 0.0052 0.00003
S.E.
0.95
0.94
0.95"}"
0.93
R2
-12.3747 0.6221 -0.00757 0.000031
-18.0948 0.8183 - 0.0098 0.000040
-13.6323 0.5601 -0.0056 0.000018
-31.3932 1.2532 -0.01459 0.000057
Coefficient
High irrigation
0.96
0.95
1.9425 0.0939 0.00145 0.0000072
0.97t
0.94
2.2799 0.1102 0.0017 0.0000084
2.2889 0.1106 0.0017 0.0000085
4.4500 0.2151 0.0033 0.000016
S.E.
R:
Regression c o e f f i c i e n t s a n d s t a n d a r d errors for dry w e i g h t f i t t e d t o cubic p o l y n o m i a l s o f t h e f o r m loge Dry Weight = c T : + d T 3, 1976
T A B L E II
C.O {30
39
R 2 values for quadratic p o l y n o m i a l s ranged from 0 . 9 0 t o 0 . 9 5 in the l o w irrigation treatment and from 0 . 9 3 t o 0 . 9 6 in the high irrigation treatment. E x a m i n a t i o n o f one o f the cases where the cubic term was non-significant revealed that the curves p r o d u c e d by quadratic and cubic expressions differed late in the season ( F i g . l ) . Comparison o f CGR calculated from the original, ¢q
E
u~ cq
160
j,'
120 "ICD W
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16/6
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617
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1
16/7
2617
518
15/8
DATE
Fig.1. Comparison of B. napus dry weight fittings to cubic and quadratic polynomials in a case where the cubic term was non-significant; 5 kg/ha seeding rate, high irrigation, 1976 (o original,---quadratic,, cubic). 6.0 "o ~0
\\\
/J
•
t.~ O,,l
'
\\\
d £Z: ~.D ,...'.
\\
20 f
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~
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\X \ % I% X
L,d 20
\
i
~
:~
0.20 I
i
i
i
I
i
-
i
i
16/6
26/6
i
6/7
16/7
L
26/7
i
i
5/8
15/8
DATE Fig.2. Crop growth rate and relative growth rate ofB. napus calculated from original dry weight data and from dry weights fitted to quadratic and cubic polynomials; 5 kg/ha seeding rate, high irrigation, 1976 (o original,---quadratic,--cubic).
40
quadratic and cubic showed that CGR calculated from the cubic data was closest to that calculated from the original data (Fig.2, top). CGR calculated from the quadratic-fitted data gave uncharacteristically negative values late in the season. Similarly R G R calculated from the cubic-fitted data was closest to that calculated from the original data (Fig.2, bottom). R G R calculated from quadratic-fitted data declined linearly with time. Leaf area data fitted quadratic polynomials of the form loge Area = a + b T + cT:, where a, b and c are constants and T is time from planting in days. The R : values ranged from 0.92 to 0.99 in both 1975 and 1976 (Tables III and IV). The high seeding rate treatments gave a poorer fit with high standard errors. This trend was especially evident in the 1976 results (Table IV). The lack of fit was most noticeable in the area of maximum leaf area index (Fig.3). Maximum LAI was both overestimated (Fig.3, 1975) and underestimated (Fig.3, 1976) in the fitted data. T A B L E III Regression c o e f f i c i e n t s and s t a n d a r d errors for leaf area f i t t e d t o a q u a d r a t i c p o l y n o m i a l o f t h e f o r m log c Area = a + b T + c T : , 1975 Treatment Rainfed
High irrigation
Seeding rate
Coefficient
S.E.
R:
2.5
Constant Time Time 2
-10.7348 0.7258 -0.0067
1.0498 0.0462 0.00048
0.98
5
Constant Time Time 2
-8.1321 0.6542 -0.0063
0.6790 0.0299 0.00031
0.99
10
Constant Time Time 2
-4.1380 0.5100 -0.0049
1.2647 0.0557 0.00057
0.93
20
Constant Time Time:
-1.3036 0.4231 -0.0042
0.6195 0.0273 0.00028
0.97
2.5
Constant Time Time:
-4.9375 0.4309 -0.0032
0.9501 0.0351 0.00030
0.95
5
Constant Time Time 2
-2.7606 0.3782 -0.0030
0.7285 0.0269 0.00023
0.96
10
Constant Time Time:
-1.0000 0.3448 -0.0029
0.8254 0.0305 0.00026
0.93
20
Constant Time Time:
-0.6161 0.3695 -0.0033
1.0240 0.0379 0.00032
0.92
41
T A B L E IV
Regression c o e f f i c i e n t s and standard errors for leaf area fitted to a quadratic p o l y n o m i a l o f the form loge A r e a = a + b T + c T 2, 1 9 7 6 Treatment Rainfed
Seeding rate 2.5
Constant Time Time 2
S.E.
R2
-13.1184 0.6716 -0.0052
0.9877 0.0325 0.00025
0.99
5
Constant Time Time 2
-7.8684 0.5156 -0.0040
1.0838 0.0356 0.00028
0.98
10
Constant Time Time 2
-12.2376 0.7029 -0.0058
2.5083 0.0825 0.00065
0.95
20
Constant Time
-7.8586 0.5766 -0.0049
2.7146 0.0893 0.0007
0.93
-7.9330 0.4560 -0.0031
1.3602 0.0426 0.00032
0.98
Time Time 2 5
Constant Time Time 2
-4.2164 0.3656 -0.0025
1.2626 0.0396 0.00029
0.95
10
Constant Time Time 2
-3.6763 0.3702 -0.0026
1.4289 0.0448 0.00033
0.92
20
Constant Time Time 2
0.2772 0.2684 0.0020
1.0137 0.0318 0.00024
0.92
Time s High irrigation
Coefficient
2.5
Constant
DISCUSSION
Cubic polynomials seem to be the most appropriate expressions for fitting B. napus dry weight data. The quadratic polynomials predicted dry weights
to drop just prior to maturity, while the actual dry weights either continued to increase or plateaued during this period. In the cases where the cubic term was non-significant, the quadratic-fitted data produced CGR and RGR curves different from those produced by the original data. In the present case leaf area data of B. napus could be fitted to quadratic polynomials, although environmental conditions had an effect on the goodness of fit. The fit was poorest at high seeding rates. This is probably due to the effect of plant density on leaf area development; at high densities the leaf area/ time relationship was less symmetrical than at low densities due to more rapid leaf senescence after flowering.
42
5.0
1975
.
40
3.0
20 x bJ z
1.0
rr ,cI l.d
.~
0
4.0
i
i
i
i
i
i
I
3.0 2.0' 1.0 0
6/6
I
i
I
i
i
i
,
16/6
26/6
6/7
16/7
26/7
5/8
15/8
DATE
Fig. 3. Comparison of original and quadratic-fitted leaf area index data of B. napus, high
irrigation, seeded at 10 kg/ha (1975) and 20 kg/ha (1976).
Although Nicholls and Calder (1973) have found little support for cubic polynomials for fitting growth data, it is obvious that cubic polynomials are appropriate in some species. The effect observed here of plant density on polynomial fitting is also of some importance, since it would make comparison of data from differing growth conditions difficult. Curve fitting may be useful for comparative purposes, since it can remove fluctuations in data caused by sampling error. However, as Buttery (1969) and Buttery and Buzzell (1974) have noted, curve fitting can conceal genuine environmental or treatment effects. This makes the regression approach to growth analysis using small samples (Radford, 1967; NichoUs and Calder, 1973) less effective if precise results are desired. The only approach to detailed analysis of growth entails accurate, well-replicated sampling at frequent intervals throughout the growth period. This places rather severe limitations on the number of treatments which can be handled, due to the high labour requirement. If it is necessary to compare a larger number of treatments, the work-
43
load could be reduced by measuring dry weight changes only, from which CGR can then be calculated.
REFERENCES Buttery, B.R., 1969. Analysis of the growth of soybeans as affected by plant population and fertilizer. Can. J. Plant Sci., 49: 675--684. Buttery, B.R. and Buzzell, R.I., 1974. Evaluation of methods used in computing net assimilation rates of soybeans (Glycine max (L.)Merrill). Crop Sci., 14: 41--44. Hughes, A.P. and Freeman, P.R., 1967. Growth analysis using frequent small harvests. J. Appl. Ecol., 4: 553--560. Landsberg, J.J., 1977. Some useful equations for biological studies. Exp. Agric., 13: 273--286. Nicholls, A.O. and Calder, D.M., 1973. Comments on the use of regression analysis for the study of plant growth. New Phytol., 72: 571--581. Radford, P.J., 1967. Growth analysis formulae -- their use and abuse. Crop Sci., 7: 171-175.
35
THE APPLICATION OF A CURVE-FITTING TECHNIQUE TO BRASSICA NAPUS GROWTH DATA
J.M. C L A R K E *
and G.M. S I M P S O N
Department of Crop Science, University of Saskatchewan, Saskatoon, Sask., S7N O W O (Canada) *Present address: Research Station, Research Branch, Agriculture Canada, Swift Current, Sask., S9H 3X2 (Canada) (Accepted 1 August 1978)
ABSTRACT
Clarke, J.M. and Simpson, G.M., 1979. The application of a curve-fitting technique to Brassica napus growth data. Field Crops Res., 2: 35--43. The use of regression analysis to fit growth data of Brassica napus L. to appropriate polynomial expressions was evaluated. Seven of eight dry weight data sets from material grown in 1975 fitted cubic polynomials, and in 1976 data, six of eight sets fitted cubic polynomials. In the remaining data sets, the cubic terms were non-significant. Crop growth rate and relative growth rate calculated from cubic-fitted data was closer to that calculated from the non-fitted data than that calculated from quadratic-fitted data. Leaf area data fitted quadratic polynomials. Data from high seeding rates gave a poorer fit to polynomials than data from low seeding rates. The lack of fit was greatest in the area of maximum leaf area index.
INTRODUCTION T h e use of regression analysis t o fit g r o w t h d a t a t o a p p r o p r i a t e p o l y n o m i a l e q u a t i o n s has b e e n suggested as a m e a n s o f r e d u c i n g t h e w o r k - l o a d o f p l a n t g r o w t h analysis ( R a d f o r d , 1 9 6 7 ; Nicholls and Calder, 1973). L a n d s b e r g ( 1 9 7 7 ) , h o w e v e r , has criticized t h e use o f p o l y n o m i a l s t o describe p l a n t g r o w t h , since t h e c o n s t a n t s in t h e e q u a t i o n s s e l d o m have biological m e a n i n g . Q u a d r a t i c p o l y n o m i a l s w e r e f o u n d a p p r o p r i a t e f o r fitting d r y weights a n d leaf areas o f Glycine m a x ( B u t t e r y 1 9 6 9 ) . In Callistephus chinensis, d r y weights a n d leaf areas f i t t e d c u b i c p o l y n o m i a l s ( H u g h e s a n d F r e e m a n , 1967). Nicholls a n d Calder ( 1 9 7 3 ) , h o w e v e r , f o u n d little e v i d e n c e t o s u p p o r t t h e use o f c u b i c relationships in A t r i p l e x spp., Poa annua or L o l i u m t e m u l e n t u m . This p a p e r e v a l u a t e s t h e fitting o f d r y w e i g h t and leaf area d a t a o f ~rassica napus L. t o a p p r o p r i a t e p o l y n o m i a l expressions. In a d d i t i o n , the e f f e c t s o f f i t t e d d r y w e i g h t d a t a on t h e derived g r o w t h f u n c t i o n s m e a n relative g r o w t h rate ( R G R ) and m e a n c r o p g r o w t h r a t e ( C G R ) w e r e assessed.
36
MATERIALS AND METHODS
Field trials using the B. n a p u s cultivar Tower were conducted at Saskatoon in 1975 and 1976. The experiment was laid out as a split-plot in randomized complete block, with four replications in 1975 and six in 1976. The main plots consisted of three water regimes -- rainfed, low irrigation and high irrigation. The sub-plot treatments were seeding rates of 2.5, 5, 10 and 20 kg/ha. The sub-plots were 4.9 × 4.9 m, with 0.25 m: quadrats designated for growth analysis sampling. Each sampling site was bordered by a minimum of 0.5 m. Growth analysis samples were taken at 5-day intervals during 1975, and at 7-day intervals during 1976. At each sampling time, all plants in a randomly selected 0.25 m 2 quadrat were cut off at ground level. Dry weights were determined by oven drying for 24 h at 100 ° C. Leaf area was measured with an electronic planimeter (Model LI 3000, Lambda Instruments Corp., Lincoln, Nebr.). Dry weights and leaf areas were determined on all plants of each quadrat during the early part of the season or on representative sub-samples later in the season. The leaf area/time and dry weight/time relationships were fitted in logarithmic form using a least squares regression program (Statistical Research Service, Agriculture Canada, Ottawa, Ont.). Observations for each replicate within each time were entered separately. The time terms were entered step-wise in the program. The affect of the addition of each term to the equations was assessed by an F-test comparing the regression and residual mean squares. In addition, an analysis of variance of the residuals (actual values - predicted values} was performed to determine if the lack of fit was greater at any position on the curves. The growth functions RGR and CGR were calculated using equations given by Radford (1967} for both the original and fitted data. Results are presented for the rainfed and high irrigation treatments only. The low number of observations on the low irrigation treatment precluded the calculation of meaningful regressions. RESULTS
Dry weight data fitted cubic polynomials of the form loge Dry Weight = a + b T + c T 2 + d T 3, where a, b, c, and d are constants, and T is time from
planting in days. Dry weight data obtained in 1975 fitted cubic polynomials in seven of eight cases (Table I). The R 2 values for the cubic equations were greater than those for quadratic polynomials, which ranged from 0.90 to 0.92 in the low irrigation and from 0.87 to 0.91 in the high irrigation. Data from the 5.0 kg/ha seeding rate in the high irrigation treatment fitted a quadratic equation; addition of the cubic term did not significantly reduce the residual sum of squares. The distribution of the residuals was similar for the quadratic and cubic eauations. The 1976 dry weight data fitted cubic polynomials in six of eight cases; in the remaining two sets the addition of the cubic term was not justified {Table H}.
Constant
5
Constant Time Time 2 Time 3
20
(a) (b) (c) (d)
t T h e c u b i c t e r m was non-significant.
Constant Time Time ~ Time 3
10
Time Time 2 Time 3
Constant Time Time 2 Time 3
2.5
Seeding rate (kg/ha)
a + bT + cT 2 + dT ~,1975
-7.9292 0.5160 -0.0068 0.000030
-11.6239 0.6818 -0.0094 0.000043
0.7167 -0.0094 0.000041
-13.4275
-19.4990 1.0047 0.0139 0.000064
Coefficient
Rainfed
1.1458 0.0688 0.0013 0.0000074
1.1252 0.0676 0.0013 0.0000072
0.0903 0.0017 0.0000097
1.5048
1.98483 0.1170 0.0022 0.000013
S.E.
0.94
0.96
0.95
0.94
R2
-7.7330 0.4980 -0.0064 0.000028
-9.8633 0.5640 -0.0070 0.000030
0.4884 -0.0050 0.000016
-9.9433
-16.3663 0.7946 -0.0097 0.000040
Coefficient
High irrigation
o.88t 2.0965
0.96
0.94
0.9526 0.0539 0.0009 0.000005 0.9291 0.0526 0.0009 0.000005
0.1187 0.0021 0.000011
0.94
1.5786 0.0894 0.0015 0.0000083
S.E.
R2
Regression c o e f f i c i e n t s and s t a n d a r d errors f o r dry w e i g h t f i t t e d to cubic p o l y n o m i a l s o f t h e f o r m loge Dry Weight =
TABLE I
e..O
Constant Time Time: Time 3
Constant Time Time s Time 3
Constant Time Time: Time 3
5
10
20
-12.7180 0.6359 -0.0078 0.000032
-21.374 0.9934 - 0.0127 0.000054
-15.0166 0.6346 -0.0068 0.000024
-42.9088 1.8609 -0.0244 0.00011
Coefficient
Rainfed
t T h e cubic t e r m was non-significant.
Constant Time Time: Time 3
2.5
Seeding rate (kg/ha)
a + bT+
2.5066 0.1266 0.0021 0.000011
3.4934 0.1764 0.0029 0.000015
3.5838 0.1810 0.0029 0.000015
6.3187 0.3191 0.0052 0.00003
S.E.
0.95
0.94
0.95"}"
0.93
R2
-12.3747 0.6221 -0.00757 0.000031
-18.0948 0.8183 - 0.0098 0.000040
-13.6323 0.5601 -0.0056 0.000018
-31.3932 1.2532 -0.01459 0.000057
Coefficient
High irrigation
0.96
0.95
1.9425 0.0939 0.00145 0.0000072
0.97t
0.94
2.2799 0.1102 0.0017 0.0000084
2.2889 0.1106 0.0017 0.0000085
4.4500 0.2151 0.0033 0.000016
S.E.
R:
Regression c o e f f i c i e n t s a n d s t a n d a r d errors for dry w e i g h t f i t t e d t o cubic p o l y n o m i a l s o f t h e f o r m loge Dry Weight = c T : + d T 3, 1976
T A B L E II
C.O {30
39
R 2 values for quadratic p o l y n o m i a l s ranged from 0 . 9 0 t o 0 . 9 5 in the l o w irrigation treatment and from 0 . 9 3 t o 0 . 9 6 in the high irrigation treatment. E x a m i n a t i o n o f one o f the cases where the cubic term was non-significant revealed that the curves p r o d u c e d by quadratic and cubic expressions differed late in the season ( F i g . l ) . Comparison o f CGR calculated from the original, ¢q
E
u~ cq
160
j,'
120 "ICD W
•
-
'\\~
8O
>.rr Cb J I-C)
i///
•
z
4O
0
16/6
26•6
617
i
i
i
1
16/7
2617
518
15/8
DATE
Fig.1. Comparison of B. napus dry weight fittings to cubic and quadratic polynomials in a case where the cubic term was non-significant; 5 kg/ha seeding rate, high irrigation, 1976 (o original,---quadratic,, cubic). 6.0 "o ~0
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16/6
26/6
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6/7
16/7
L
26/7
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5/8
15/8
DATE Fig.2. Crop growth rate and relative growth rate ofB. napus calculated from original dry weight data and from dry weights fitted to quadratic and cubic polynomials; 5 kg/ha seeding rate, high irrigation, 1976 (o original,---quadratic,--cubic).
40
quadratic and cubic showed that CGR calculated from the cubic data was closest to that calculated from the original data (Fig.2, top). CGR calculated from the quadratic-fitted data gave uncharacteristically negative values late in the season. Similarly R G R calculated from the cubic-fitted data was closest to that calculated from the original data (Fig.2, bottom). R G R calculated from quadratic-fitted data declined linearly with time. Leaf area data fitted quadratic polynomials of the form loge Area = a + b T + cT:, where a, b and c are constants and T is time from planting in days. The R : values ranged from 0.92 to 0.99 in both 1975 and 1976 (Tables III and IV). The high seeding rate treatments gave a poorer fit with high standard errors. This trend was especially evident in the 1976 results (Table IV). The lack of fit was most noticeable in the area of maximum leaf area index (Fig.3). Maximum LAI was both overestimated (Fig.3, 1975) and underestimated (Fig.3, 1976) in the fitted data. T A B L E III Regression c o e f f i c i e n t s and s t a n d a r d errors for leaf area f i t t e d t o a q u a d r a t i c p o l y n o m i a l o f t h e f o r m log c Area = a + b T + c T : , 1975 Treatment Rainfed
High irrigation
Seeding rate
Coefficient
S.E.
R:
2.5
Constant Time Time 2
-10.7348 0.7258 -0.0067
1.0498 0.0462 0.00048
0.98
5
Constant Time Time 2
-8.1321 0.6542 -0.0063
0.6790 0.0299 0.00031
0.99
10
Constant Time Time 2
-4.1380 0.5100 -0.0049
1.2647 0.0557 0.00057
0.93
20
Constant Time Time:
-1.3036 0.4231 -0.0042
0.6195 0.0273 0.00028
0.97
2.5
Constant Time Time:
-4.9375 0.4309 -0.0032
0.9501 0.0351 0.00030
0.95
5
Constant Time Time 2
-2.7606 0.3782 -0.0030
0.7285 0.0269 0.00023
0.96
10
Constant Time Time:
-1.0000 0.3448 -0.0029
0.8254 0.0305 0.00026
0.93
20
Constant Time Time:
-0.6161 0.3695 -0.0033
1.0240 0.0379 0.00032
0.92
41
T A B L E IV
Regression c o e f f i c i e n t s and standard errors for leaf area fitted to a quadratic p o l y n o m i a l o f the form loge A r e a = a + b T + c T 2, 1 9 7 6 Treatment Rainfed
Seeding rate 2.5
Constant Time Time 2
S.E.
R2
-13.1184 0.6716 -0.0052
0.9877 0.0325 0.00025
0.99
5
Constant Time Time 2
-7.8684 0.5156 -0.0040
1.0838 0.0356 0.00028
0.98
10
Constant Time Time 2
-12.2376 0.7029 -0.0058
2.5083 0.0825 0.00065
0.95
20
Constant Time
-7.8586 0.5766 -0.0049
2.7146 0.0893 0.0007
0.93
-7.9330 0.4560 -0.0031
1.3602 0.0426 0.00032
0.98
Time Time 2 5
Constant Time Time 2
-4.2164 0.3656 -0.0025
1.2626 0.0396 0.00029
0.95
10
Constant Time Time 2
-3.6763 0.3702 -0.0026
1.4289 0.0448 0.00033
0.92
20
Constant Time Time 2
0.2772 0.2684 0.0020
1.0137 0.0318 0.00024
0.92
Time s High irrigation
Coefficient
2.5
Constant
DISCUSSION
Cubic polynomials seem to be the most appropriate expressions for fitting B. napus dry weight data. The quadratic polynomials predicted dry weights
to drop just prior to maturity, while the actual dry weights either continued to increase or plateaued during this period. In the cases where the cubic term was non-significant, the quadratic-fitted data produced CGR and RGR curves different from those produced by the original data. In the present case leaf area data of B. napus could be fitted to quadratic polynomials, although environmental conditions had an effect on the goodness of fit. The fit was poorest at high seeding rates. This is probably due to the effect of plant density on leaf area development; at high densities the leaf area/ time relationship was less symmetrical than at low densities due to more rapid leaf senescence after flowering.
42
5.0
1975
.
40
3.0
20 x bJ z
1.0
rr ,cI l.d
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0
4.0
i
i
i
i
i
i
I
3.0 2.0' 1.0 0
6/6
I
i
I
i
i
i
,
16/6
26/6
6/7
16/7
26/7
5/8
15/8
DATE
Fig. 3. Comparison of original and quadratic-fitted leaf area index data of B. napus, high
irrigation, seeded at 10 kg/ha (1975) and 20 kg/ha (1976).
Although Nicholls and Calder (1973) have found little support for cubic polynomials for fitting growth data, it is obvious that cubic polynomials are appropriate in some species. The effect observed here of plant density on polynomial fitting is also of some importance, since it would make comparison of data from differing growth conditions difficult. Curve fitting may be useful for comparative purposes, since it can remove fluctuations in data caused by sampling error. However, as Buttery (1969) and Buttery and Buzzell (1974) have noted, curve fitting can conceal genuine environmental or treatment effects. This makes the regression approach to growth analysis using small samples (Radford, 1967; NichoUs and Calder, 1973) less effective if precise results are desired. The only approach to detailed analysis of growth entails accurate, well-replicated sampling at frequent intervals throughout the growth period. This places rather severe limitations on the number of treatments which can be handled, due to the high labour requirement. If it is necessary to compare a larger number of treatments, the work-
43
load could be reduced by measuring dry weight changes only, from which CGR can then be calculated.
REFERENCES Buttery, B.R., 1969. Analysis of the growth of soybeans as affected by plant population and fertilizer. Can. J. Plant Sci., 49: 675--684. Buttery, B.R. and Buzzell, R.I., 1974. Evaluation of methods used in computing net assimilation rates of soybeans (Glycine max (L.)Merrill). Crop Sci., 14: 41--44. Hughes, A.P. and Freeman, P.R., 1967. Growth analysis using frequent small harvests. J. Appl. Ecol., 4: 553--560. Landsberg, J.J., 1977. Some useful equations for biological studies. Exp. Agric., 13: 273--286. Nicholls, A.O. and Calder, D.M., 1973. Comments on the use of regression analysis for the study of plant growth. New Phytol., 72: 571--581. Radford, P.J., 1967. Growth analysis formulae -- their use and abuse. Crop Sci., 7: 171-175.