Frequentist and Bayesian inference for gas exchanges of drip-irrigated bell pepper<sup>1</sup>

Santos, Fernando André Silva; Rezende, Roberto; Wenneck, Gustavo Soares; Santi, Danilo César; Saath, Reni

doi:10.1590/1983-40632021v5166435

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Research Article • Pesqui. Agropecu. Trop. 51 • 2021 • https://doi.org/10.1590/1983-40632021v5166435 copy

Frequentist and Bayesian inference for gas exchanges of drip-irrigated bell pepper¹

Inferência frequentista e bayesiana para trocas gasosas de pimentão irrigado por gotejamento

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

The water deficit or excess may cause undesirable changes in yield and physiological aspects of irrigated crops. Considering the analysis of experimental data, the use of classical statistical methods may not be sufficient to detect morphological and physiological effects resulting from the conditions employed, being interesting the use of new procedures, such as the Bayesian inference. This study aimed to evaluate the gas exchanges in the bell pepper crop under irrigation depths and different irrigation times, in a protected environment, by applying the parametric and Bayesian methods. The experiment was conducted in a completely randomized design, arranged in a 5 x 2 factorial scheme, with five irrigation depths (60, 80, 100, 120 and 140 % of the crop evapotranspiration) and two irrigation times (8 a.m. and 2 p.m.), with five replications. The stomatal conductance, transpiration rate, photosynthetic rate, intrinsic water-use efficiency, ratio between the photosynthetic rate and the transpiration rate, and total dry matter were evaluated. The gas exchanges were affected by water replacement depths, with little interference from the irrigation times. The linear regression models, according to the irrigation depths, for transpiration rate, stomatal conductance and total dry matter were significant for both the statistical methods, suggesting that the results are similar and even coincident, especially when a priori information are not provided in the Bayesian analysis. Differences in the photosynthetic rate were observed only with the Bayesian method, adjusting linear models according to the irrigation depths, in both the irrigation times.

KEYWORDS:
Capsicum annum L.; protected cultivation; micro irrigation

Variable	Irrigation time	Irrigation depth (mm)
Variable	Irrigation time	60	80	100	120	140
Photosynthetic rate (µmol CO₂ m^-2 s^-1)	Morning	8.970	10.010	9.440	10.410	11.230
	Afternoon	8.670	8.500	9.880	9.920	10.310
	p-value¹ 1 Probability for the F-test.	0.845	0.327	0.775	0.751	0.550
Transpiration rate (mmol m^-2 s^-1)	Morning	1.650	1.860	1.930	2.200	2.130
	Afternoon	1.710	1.510	1.910	1.890	2.220
	p-value¹ 1 Probability for the F-test.	0.769	0.084	0.887	0.122	0.656
Stomatal conductance (mol m^-2 s^-1)	Morning	0.210	0.210	0.230	0.310	0.320
	Afternoon	0.180	0.180	0.280	0.250	0.290
	p-value¹ 1 Probability for the F-test.	0.358	0.276	0.230	0.133	0.389
Intrinsic water-use efficiency [(µmol CO₂ m^-2 s^-1)/(mmol m^-2 s^-1)]	Morning	5.460	5.640	4.880	4.750	5.420
	Afternoon	5.160	5.490	5.140	5.190	4.670
	p-value¹ 1 Probability for the F-test.	0.728	0.865	0.768	0.620	0.399
Total dry matter (g plant^-1)	Morning	36.930	38.560	44.740	39.770	66.610
	Afternoon	41.820	44.000	45.930	50.980	53.770
	p-value¹ 1 Probability for the F-test.	0.553	0.510	0.885	0.179	0.124

Variable	Irrigation time	Irrigation depth (mm)
Variable	Irrigation time	60	80	100	120	140
Photosynthetic rate (µmol CO₂ m^-2 s^-1)	Morning	7.39 (0.91)⁽¹⁾ (1) Mean (standard deviation); CI: credible interval of 95 %; * significant at 5 % of probability; ns: not significant at 5 % of probability.	9.41 (1.33)	9.88 (1.02)	11.80 (1.34)	11.23 (0.54)
	Afternoon	7.01 (1.04)	9.10 (1.32)	9.44 (1.01)	11.10 (1.15)	10.69 (0.61)
	CI	ns	ns	ns	ns	ns
Transpiration rate (mmol m^-2 s^-1)	Morning	1.65 (0.15)	1.86 (0.14)	1.93 (0.12)	2.20 (0.19)	2.13 (0.17)
	Afternoon	1.71 (0.15)	1.51 (0.14)	1.91 (0.13)	1.89 (0.19)	2.22 (0.17)
	CI	ns	ns	ns	ns	ns
Stomatal conductance (mol m^-2 s^-1)	Morning	0.21 (0.02)	0.21 (0.01)	0.23 (0.03)	0.31 (0.03)	0.32 (0.03)
	Afternoon	0.18 (0.02)	0.18 (0.01)	0.28 (0.03)	0.25 (0.03)	0.29 (0.03)
	CI	ns	ns	ns	ns	ns
Intrinsic water-use efficiency [(µmol CO₂ m^-2 s^-1)/(mmol m^-2 s^-1)]	Morning	4.49 (0.28)	5.95 (0.57)	4.88 (0.37)	5.19 (0.67)	4.58 (0.61)
	Afternoon	4.45 (0.28)	4.16 (0.65)	5.14 (0.37)	5.19 (0.52)	4.89 (0.52)
	CI	ns	*	ns	ns	ns
Total dry matter (g plant^-1)	Morning	33.28 (5.25)	38.58 (7.09)	44.76 (6.33)	39.79 (4.57)	58.44 (7.63)
	Afternoon	39.32 (5.21)	44.00 (7.11)	45.93 (6.35)	50.98 (4.59)	53.70 (6.90)
	CI	ns	ns	ns	ns	ns

Variable	Irrigation time	Coefficients	Estimate	Standard error	p-value¹ 1 Probability for the student t-test; β0: constant of the model; β1: linear coefficient of the model.
Photosynthetic rate (µmol CO₂ m^-2 s^-1)	Morning	β0	7.556	1.768	0.000^** * and **: significant at 5 % and 1 % of probability, respectively; ns non-significant at 5 % of probability.
	Morning	β1	0.025	0.017	0.156^ns
	Afternoon	β0	7.109	1.768	0.000^** * and **: significant at 5 % and 1 % of probability, respectively; ns non-significant at 5 % of probability.
	Afternoon	β1	0.024	0.017	0.175^ns
Transpiration rate (mmol m^-2 s^-1)	Morning	β0	1.311	0.228	0.000^** * and **: significant at 5 % and 1 % of probability, respectively; ns non-significant at 5 % of probability.
	Morning	β1	0.006	0.002	0.005^** * and **: significant at 5 % and 1 % of probability, respectively; ns non-significant at 5 % of probability.
	Afternoon	β0	1.154	0.228	0.000^** * and **: significant at 5 % and 1 % of probability, respectively; ns non-significant at 5 % of probability.
	Afternoon	β1	0.007	0.002	0.003^** * and **: significant at 5 % and 1 % of probability, respectively; ns non-significant at 5 % of probability.
Stomatal conductance (mol m^-2 s^-1)	Morning	β0	0.109	0.040	0.010^* * and **: significant at 5 % and 1 % of probability, respectively; ns non-significant at 5 % of probability.
	Morning	β1	0.002	0.000	0.000^** * and **: significant at 5 % and 1 % of probability, respectively; ns non-significant at 5 % of probability.
	Afternoon	β0	0.092	0.040	0.027^* * and **: significant at 5 % and 1 % of probability, respectively; ns non-significant at 5 % of probability.
	Afternoon	β1	0.001	0.000	0.001^** * and **: significant at 5 % and 1 % of probability, respectively; ns non-significant at 5 % of probability.
Intrinsic water-use efficiency [(µmol CO₂ m^-2 s^-1)/(mmol m^-2 s^-1)]	Morning	β0	5.727	1.011	0.000^** * and **: significant at 5 % and 1 % of probability, respectively; ns non-significant at 5 % of probability.
	Morning	β1	-0.005	0.010	0.616^ns
	Afternoon	β0	5.773	1.011	0.000^** * and **: significant at 5 % and 1 % of probability, respectively; ns non-significant at 5 % of probability.
	Afternoon	β1	-0.006	0.010	0.515^ns
Total dry matter (g plant^-1)	Morning	β0	20.904	9.507	0.034^* * and **: significant at 5 % and 1 % of probability, respectively; ns non-significant at 5 % of probability.
	Morning	β1	0.254	0.091	0.008^** * and **: significant at 5 % and 1 % of probability, respectively; ns non-significant at 5 % of probability.
	Afternoon	β0	25.999	9.507	0.009^** * and **: significant at 5 % and 1 % of probability, respectively; ns non-significant at 5 % of probability.
	Afternoon	β1	0.203	0.092	0.032^* * and **: significant at 5 % and 1 % of probability, respectively; ns non-significant at 5 % of probability.

Variable	Irrigation time	Coefficients	Average (SD)	CI
Photosynthetic rate (µmol CO₂ m^-2 s^-1)	Morning	β0	4.606 (1.204)	(2.214; 6.989)^* * significant at 5 % of probability;
	Morning	β1	0.049 (0.011)	(0.027; 0.072)^* * significant at 5 % of probability;
	Afternoon	β0	5.429 (1.764)	(1.946; 8.943)^* * significant at 5 % of probability;
	Afternoon	β1	0.043 (0.017)	(0.009; 0.077)^* * significant at 5 % of probability;
Transpiration rate (mmol m^-2 s^-1)	Morning	β0	1.314 (0.264)	(0.787; 1.830)^* * significant at 5 % of probability;
	Morning	β1	0.006 (0.002)	(0.001; 0.011)^* * significant at 5 % of probability;
	Afternoon	β0	1.156 (0.209)	(0.007; 0.002)^* * significant at 5 % of probability;
	Afternoon	β1	0.007 (0.002)	(0.003; 0.011)^* * significant at 5 % of probability;
Stomatal conductance (mol m^-2 s^-1)	Morning	β0	0.109 (0.041)	(0.027; 0.189)^* * significant at 5 % of probability;
	Morning	β1	0.001 (0.0004)	(0.0007; 0.0023)^* * significant at 5 % of probability;
	Afternoon	β0	0.092 (0.045)	(0.003; 0.180)^* * significant at 5 % of probability;
	Afternoon	β1	0.0015 (0.0004)	(0.0006; 0.0023)^* * significant at 5 % of probability;
Intrinsic water-use efficiency [(µmol CO₂ m^-2 s^-1)/(mmol m^-2 s^-1)]	Morning	β0	3.880 (0.308)	(-0.629; 0.547)^ns ns non-significant at 5 % of probability.
	Morning	β1	0.008 (0.004)	(0.0003; 0.0169)^* * significant at 5 % of probability;
	Afternoon	β0	4.545 (0.908)	(2.738; 6.344)^* * significant at 5 % of probability;
	Afternoon	β1	0.006 (0.009)	(-0.011; 0.023)^ns
Total dry matter (g plant^-1)	Morning	β0	24.860 (8.233)	(8.491; 41.110)^* * significant at 5 % of probability;
	Morning	β1	0.189 (0.079)	(0.033; 0.346)^* * significant at 5 % of probability;
	Afternoon	β0	22.570 (10.420)	(2.059; 43.240)^* * significant at 5 % of probability;
	Afternoon	β1	0.232 (0.099)	(0.034; 0.426)^* * significant at 5 % of probability;

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Probability for the F-test.

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Frequentist and Bayesian inference for gas exchanges of drip-irrigated bell pepper1

Inferência frequentista e bayesiana para trocas gasosas de pimentão irrigado por gotejamento

ABSTRACT

Frequentist and Bayesian inference for gas exchanges of drip-irrigated bell pepper¹