# A simple allometric model for estimating blueberry fruit weight from diameter measurements

Emilio Jorquera-Fontena, Michel Génard, Alejandra Ribera-Fonseca, Nicolas Franck

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Emilio Jorquera-Fontena, Michel Génard, Alejandra Ribera-Fonseca, Nicolas Franck. A simple allometric model for estimating blueberry fruit weight from diameter measurements. Scientia Horticulturae, 2017, 219, pp.131 - 134. 10.1016/j.scienta.2017.03.009. hal-01521538

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1 **Short Communication**

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3 **A simple allometric model for estimating blueberry fruit weight from diameter measurements**

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5 Emilio Jorquera-Fontena¹, Michel Génard², Alejandra Ribera-Fonseca³, 4*, Nicolás Franck⁵

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7 ¹Facultad de Ciencias Agropecuarias y Forestales, Universidad de La Frontera, Avenida Francisco

8 Salazar 01145, Casilla 54-D, Temuco, Chile. [emilio.jorquera@ufrontera.cl](mailto:emilio.jorquera@ufrontera.cl)

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10 ²UR 1115 Plantes et Systèmes de culture Horticoles, Institut National de la Recherche

11 Agronomique, Avignon, France. [michel.genard@avignon.inra.fr](mailto:michel.genard@avignon.inra.fr)

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13 ³Departamento de Producción Agropecuaria, Facultad de Ciencias Agropecuarias y Forestales,

14 Universidad de La Frontera, Casilla 54-D, Temuco, Chile. [alejandra.ribera@ufrontera.cl](mailto:alejandra.ribera@ufrontera.cl)

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16 ⁴Center of Plant-Soil Interaction and Natural Resources Biotechnology, Scientific and

17 Technological Bioresources Nucleus (BIOREN-UFRO), Universidad de La Frontera, Casilla 54-D,

18 Temuco, Chile.

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20 ⁵Centro de Estudios de Zonas Áridas, Facultad de Ciencias Agronómicas, Universidad de Chile,

21 Casilla 129, Coquimbo, Chile. [nfranck@uchile.cl](mailto:nfranck@uchile.cl)

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23 *Corresponding author:

24 Alejandra Ribera Fonseca

25 Phone: +56 45 2734123/2325669/2325458

26 E-mail address: alejandra.ribera@ufrontera.cl
**Abstract.** Monitoring of fruit growth is a measurement widely used in physiology and agronomy studies. This is normally done from detached fruits, which can lead to erroneous results when fruits grow asynchronously in clusters as occurs in blueberry plants (*Vaccinium* spp.). The aim of this communication is to develop a simple allometric model for estimating blueberry weight from diameter measurements. In three growth seasons, weight and diameter of a total of 416 fruits were measured from four field-grown blueberry cultivars. The obtained data set was then randomly divided resulting in 200 data for calibrating model and 216 for validating. Prior to calibration procedure, the use of cultivar-specific models or a common model for four cultivars was evaluated by Analysis of Covariance between logarithm transformed diameter and cultivar/season variables. From analysis, no interactive effect was observed, so a common allometric model was constructed, giving a power functional relationship between berry diameter and weight. The allometric model was validated with the independent data, giving errors ranging from 4.7% to 6.9%. Residuals showed a random dispersion pattern. Our results showed that a common model was adequate for providing a reliable means of estimating blueberry fruit weight of different cultivars.

**Key words:** Regression, validation, non-destructive methods

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## 1. Introduction

Monitoring of fruit growth is a measurement widely used in studies of physiology and evaluation of management practices (e.g. Godoy et al., 2008; Minchin et al., 2010). Thus, its knowledge has been an important parameter in understanding source-sink relationships, water and nutrient use efficiency, crop quality and yield potential (Génard and Lescourret, 2004, Grossman and DeJong, 1995, Spreer et al., 2009). Habitually, fruit growth curves are developed from detached samples; however, this method can increase variance and lead to erroneous results for species in which fruits grow in clusters (Coombe, 1976). This is because several development stages of the fruit can be observed on a plant at a given time, as occur typically in several fruits species including blueberry (e.g. Tuccio et al., 2011; Godoy et al., 2008). A common method for non-destructively estimating biomass of plant organs is to fit simple equations to data relating biomass to some dimensional characteristic of the plant such as diameter, height, length and width (e.g. Antunes et al., 2008; Roxburgh et al., 2015). In this line, estimating fruit weight from equations using simple measurements of fruit diameter can represent an inexpensive and rapid alternative for accurately assessing fruit weight. In blueberry industry, fruit segregation on the basis of weight and size, is crucial to obtain higher export market value. The aim of this communication is to develop a simple allometric model for estimating blueberry weight from diameter measurements.

## 2. Materials and methods

### 2.1. Fruit data

In three growth seasons, a total of 416 healthy berries from three northern highbush blueberry cultivars (V. corymbosum cultivars 'Bluegold', 'Brigitta' and 'Elliot') and one southern blueberry cultivar (V. corymbosum x Vaccinium darrowi Camp. cultivar 'Legacy') were collected from three commercial orchards located in the Región de La Araucanía, Chile (Table 1).
The cultivars used differing in fruiting season; Bluegold and Elliot are early- and late-season cultivars, respectively, whereas Legacy and Brigitta are mid-season cultivars. Plants of the orchards ranged from three to five-year-old and were established at a spacing of 3 × 0.9m, in north-south oriented rows on an Andisol. In each orchard, the average fertilization was 70 kg N ha⁻¹, 40 kg P₂O₅ ha⁻¹ and 75 kg K₂O ha⁻¹, the irrigation was applied as needed, and pests and diseases were controlled according to the locally recommended practices. Winter pruning was annually done in each orchard, removing old wood, damaged canes, unproductive branches, and short shoots that develop from the base of the plant.

Fruits were weekly picked from fruit coloring phase (stage in which berries are changing from green to pink to blue in the cluster) up to beginning of commercial harvest for each season. After picking, samples were placed in a cooler and then rapidly transported to the laboratory for evaluating equatorial diameter and fresh weight by using digital caliper (Mitutoyo, Japan, accuracy ±0.01mm) and technical balance (Gibertini Elettronica S.r.l., Italy), respectively.

The data set was randomly divided to obtain data for both calibrating (n= 200) and validating (n= 216) model (external validation).

## 2.2. Allometric model calibration

Preliminary regression analysis showed that the relationships between fruit weight and diameter followed a power fit typical of allometric models, for all situations in which fruit was obtained (Eq. 1):

$$FW = aFD^b \qquad (\text{Eq. 1})$$

where FW is berry weight, FD is berry diameter, a and b are allometric coefficients.

In order to discriminate the use of cultivar-specific models (including the season effect) or a common model for four cultivars, an Analysis of Covariance was performed on data linearized by natural logarithm transformation (Eq. 2)

$$\ln FW = \ln a + b \times \ln FD \qquad (\text{Eq. 2})$$
Thus, when interaction between logarithm transformed diameter (continuous variable) and cultivar/season (categorical variable) is not significant, the slopes among groups are similar, so a common model for four cultivars can be constructed (Warton and Weber, 2002; Chen et al., 2010).

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## 2.3. Model goodness-of-fit and validation

Model results were compared with independent data through graphical procedures. The Relative Root Mean Squared Error (RRMSE, Eq. 3), a common criterion to quantify the mean difference between estimated data and observed data (Kobayashi and Us Salam, 2000) was calculated:

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$$RRMSE = \left( \sqrt{\frac{\sum n_i(x_i - y_i)^2}{N}} \right) / \bar{y} \qquad (\text{Eq. 3})$$

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where $n_i$, $x_i$, and $y_i$ are the number of observed data, the estimated data, and the observed data at the measurement date $i$, respectively, $N$ is the total number of observed data and $\bar{y}$ the mean of observed values. The smaller the value of RRMSE, the closer to the measurement the estimation is (Grechi et al., 2008).

Model residuals were also calculated and plotted against data estimated by the model to check their normal distribution (i.e. homoscedastic behaviour).

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## 2.4. Data analysis

In order to characterize the data set for calibrating model, results obtained from cultivars/season were subjected to Multiple Comparison of Mean (MCM) by using Scheffé's test ($\alpha$ = 0.05). This test was selected because the sample size among groups was different (Ruxton and Beauchamp, 2008). The MSM, analysis of covariance and model fitting were performed with R software version 2.15.0 (R Development Core Team, 2012).

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130 **3. Results**

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132 **3.1. Berry diameter and weight**

133 The mean fruit diameter and weight from different cultivars/seasons ranged from 13.35 to
134 17.75 mm and from 1.19 to 2.58 g, respectively (Table 2). As observed, the highest values were
135 measured for cv. Legacy in the season 2015-2016, while the lowest values for cv. Brigitta. When
136 comparing cultivars, the mean diameter and weight of the fruit was significantly higher in Legacy
137 and Bluegold than in Elliot and Brigitta (Table 2).

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139 **3.2. Evaluation of regression slopes and allometric model**

140 Diameter and weight of the fruits were regressed on logarithm transformed data in order to evaluate
141 whether the interactive effect of berry diameter (continuous variable) and cultivar/season
142 (categorical variable) is significant for estimating berry weight (Table 3). Regression slopes of
143 logarithm transformed data were no statistically different among cultivars/season, thereafter whole
144 data set for calibration procedure was used for constructing a common model for four cultivars.
145 Then, fruit weight data were plotted against fruit diameter data, giving a power functional
146 relationship as shown in the Fig. 1. The relation between these variables was highly significant with
147 *p*-values of the coefficients lower than 2 x10<sup>-16</sup>.

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149 **3.3. Model goodness-of-fit and validation**

150 For model validation, estimated versus actual berry weights were plotted (Fig. 2). As observed, the
151 values were closely distributed alongside the 1:1 line, although there was some tendency to
152 overestimate the weight, which can be noticed on regression slope, being slightly higher than one
153 (Fig. 2A). The external validation of the model showed that the model goodness-of-fit (RRMSE)
154 decreased from cv. Bluegold (4.7%) to cv. Elliot (6.9%), with cv. Legacy and Brigitta showing
intermediate values (Fig. 2A). Residual were randomly distributed along estimated data, which indicates absence of heteroscedasticity (Fig. 2B).

## 4. Discussion

Empirical models are required for the non-destructive estimation of biomass of a plant organ of interest from easily obtainable measurements. These models, which are based on allometric theory, use the statistic relations between a variable of interest and one or several variables used as predictor (Niklas, 1994).

Many abiotic and biotic factors may potentially affect allometric relationships, including genetics, ontogeny, size, age, structure, site, climate, and their interactions (Misle et al., 2013; Niklas, 1994), however our results indicated that the relationship between fresh weight and diameter was sufficiently stable irrespectively to the genotype and growth seasons (Table 3). From this, a common allometric model was calibrated, giving a power functional relationship as showed in the Fig. 1. On the other hand, according to the principle of parsimony and the objective of proposing easy-to-use models, this finding was convenient. In the model, as a fruit grows, its dimensions change in ways that maintain their functional balance, but not in the ways that maintain constant ratios between dimensions (Castellan-Estrada et al., 2002). Similar allometric relationships have been reported in studies of fruit physiology. In most of these, fruit diameter was included as predictor of fresh and dry weight (e.g. Gautier et al., 2001; Gilbert et al., 2005; Díaz-Pérez et al., 2007; Grechi et al., 2008). Our allometric model was validated with independent data, giving higher goodness-of-fit for evaluated cultivars (Fig. 2A). This showed that a common model was adequate for providing a reliable means of estimating blueberry fruit weight of different cultivars. On the other hand, residuals follow a normal distribution, with data randomly distributed, which indicates absence of heteroscedasticity (Fig. 2B). When the dispersion pattern of residuals follows a
180 heteroscedastic behavior, the estimated standard errors of the regression coefficients are often
181 underestimated, giving an incorrect perception of accuracy (Chatterjee and Hadi, 2006).

182 In conclusion, blueberry fruit weight was satisfactorily estimated on the basis of diameter
183 measurements via allometric approach. Despite that model showed to be reliable for estimating
184 berry weight of four blueberry cultivars, it must be validated when applied to other genetic
185 materials. Further studies must be done in blueberry fruits to establish allometric equations for
186 relating other fruit quality traits to easily obtainable non-destructive measurements

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188 **5. Acknowledgments**

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190 The first author acknowledges to the Convenio Marco 2016, Ministerio de Educación, Chile –
191 Universidad de La Frontera, Chile.

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193 **6. References**

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195 Antunes W.C., Pompelli M.F., Carretero D.M., DaMatta F.M. 2008. Allometric models for non-destructive leaf area estimation in coffee (*Coffea arabica* and *C.canephora*). Annals of Applied Biology, 153: 33–40.

198 Castelan-Estrada M., Vivin P. Gaudillere J.P. 2002. Allometric relationships to estimate seasonal
199 above ground vegetative and reproductive biomass of *Vitis vinifera* L. Annals of Botany, 89:
200 401-408.

201 Chatterjee S., Hadi A.S. 2006. Regression analysis by example. Hoboken, NJ, USA, Wiley.

202 Chen H., Felker S., Sun S. 2010. Allometry of within-fruit reproductive allocation in subtropical
dicot woody species. American Journal of Botany, 97: 611–619.

204 Coombe B.G. 1976. The development of fleshy fruits. Annual Review of Plant Physiology, 27:
205 207–228.
206 Díaz-Pérez J.C., Muy-Rangel M.D., Mascorro A.G. 2007. Fruit size and stage of ripeness affect postharvest water loss in bell pepper fruit (*Capsicum annuum* L.). Journal of Science and Food Agriculture, 87:68-73.

209 Kobayashi K., Us Salam M.2000. Comparing simulated and measured values using mean squared deviation and its components. Agronomy Journal, 92:345-352.

211 Gautier H., Guichard S., Tchamitchian M. 2001. Modulation of competition between fruits and leaves by flower pruning and water fogging, and consequences on tomato leaf and fruit growth. Annals of Botany, 88: 645-652.

214 Gilbert C., Lescourret F., Génard M., Vercambre G., Pérez Pastor A. 2005. Modelling the effect of fruit growth on surface conductance to water vapour diffusion. Annals of Botany 95: 673-683.

216 Godoy C., Monterubbianesi G., Tognetti J. 2008. Analysis of highbush blueberry (*Vaccinium corymbosum* L.) fruit growth with exponential mixed models. Scientia Horticulturae, 115: 368-376.

219 Grechi I., Hilgerts N., Génard M., Lescourret F. 2008. Assessing the peach fruit refractometric index at harvest with a simple model based on fruit growth. Journal of the American Society of Horticultural Science 133:178-187.

222 Grossman Y.L., DeJong T.M. 1995. Maximum fruit growth potential and seasonal patterns of resource dynamic during peach growth. Annals of Botany, 75, 553-560

224 Minchin P.E.H., Snelgar W.P., Blattmann P., Hall A.J. 2010. Competition between fruit and vegetative growth in Hayward kiwifruit. New Zealand Journal Crop and Horticulture, 38: 101-112.

227 Misle E., Kahlaoui B., Hachicha M., Alvarado P. 2013. Leaf area estimation in muskmelon by allometry. Photosynthetica, 51: 613-620.

229 Niklas K.J. 1994. Plant Allometry: The scaling of form and process; University Chicago Press: Chicago, IL, USA.
231 R Development Core Team, 2012. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.

233 Roxburgh SH, Paul KI, Clifford D, England, JR, Raison RJ (2015) Guidelines for constructing allometric models for the prediction of woody biomass: How many individuals to harvest? Ecosphere 6: 38 doi: 10.1890/ES14-00251.1.

236 Ruxton G.D., Beauchamp G. 2008. Time for some a priori thinking about post hoc testing. Behavioral Ecology, 19: 690-693.

238 Spreer W., Ongprasert S., Hegele M., Wünsche J.N., Muller J. 2009. Yield and fruit development in mango (*Mangifera indica* L. cv. Chok Anan) under different irrigation regimes. Agricultural Water Management, 96: 574-584.

241 Tuccio L., Remorini D., Pinelli P., Fierini E., Tonutti P., Scalabrelli G., Agati G. 2011. Rapid and non-destructive method to assess in the vineyard grape berry anthocyanins under different seasonal and water conditions. Australian Journal of Grape and Wine Research, 17: 181-189.

244 Warton D. I., Weber N.C. 2002. Common slope tests for bivariate errors-in-variables models. Biometrical Journal, 44: 161-174.

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<table><caption>257 Table 1. Cultivars, season and orchard locations</caption><thead><tr><td>Cultivar</td><td>Season</td><td>Orchard location</td></tr></thead><tbody><tr><td>Bluegold</td><td>2015-2016</td><td>38°84' S 72°69' W</td></tr><tr><td>Brigitta</td><td>2010-2011</td><td>38°29' S 72° 23'</td></tr><tr><td>Elliot</td><td>2008-2009</td><td>39°19' S 72°25' W</td></tr><tr><td>Elliot</td><td>2015-2016</td><td>38°84' S 72°69' W</td></tr><tr><td>Legacy</td><td>2010-2011</td><td>38°29' S 72° 23'</td></tr><tr><td>Legacy</td><td>2015-2016</td><td>38°84' S 72°69' W</td></tr></tbody></table>

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**Table 2.** Mean fruit diameter and weight for four field-grown blueberry cultivars on data set used for calibrating model. Standard deviation (in brackets) and letters indicating statistic differences by Scheffe's test ($\alpha$ = 0.05) are shown.

<table><thead><tr><th>Cultivar</th><th>Season</th><th>Berry diameter (mm)</th><th>Berry weight (g)</th><th>N</th></tr></thead><tbody><tr><td>Bluegold</td><td>2015-2016</td><td>16.00 (±2.25) b</td><td>1.95 (±0.74) b</td><td>50</td></tr><tr><td>Brigitta</td><td>2010-2011</td><td>13.35 (±3.24) d</td><td>1.19 (±0.85) c</td><td>50</td></tr><tr><td>Elliot</td><td>2008-2009</td><td>15.52 (±1.30) bc</td><td>1.65 (±0.36) bc</td><td>21</td></tr><tr><td>Elliot</td><td>2015-2016</td><td>15.83 (±1.31) cd</td><td>1.14 (±0.31) c</td><td>27</td></tr><tr><td>Legacy</td><td>2010-2011</td><td>14.97 (±1.81) bcd</td><td>1.55 (±0.57) bc</td><td>16</td></tr><tr><td>Legacy</td><td>2015-2016</td><td>17.75 (±2.09) a</td><td>2.58 (±0.79) a</td><td>33</td></tr><tr><td colspan="5">Comparison among cultivars</td></tr><tr><td>Bluegold</td><td></td><td>16.00 (±2.25) A</td><td>1.95 (±0.74) A</td><td>53</td></tr><tr><td>Brigitta</td><td></td><td>13.35 (±3.24) B</td><td>1.19 (±0.85) B</td><td>50</td></tr><tr><td>Elliot</td><td></td><td>14.57 (±1.54) B</td><td>1.36 (±0.41) B</td><td>48</td></tr><tr><td>Legacy</td><td></td><td>16.84 (±2.39) A</td><td>2.24 (±0.86) A</td><td>49</td></tr></tbody></table>

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**Table 3.** Linear regression between log-transformed weight and log-transformed diameter data, and ANCOVA for log-transformed diameter and cultivar/season variables. Determination coefficient (R²) and significance values are shown. Abbreviations: ln = natural logarithm, FW = fresh weight, FD = fruit diameter.

<table><thead><tr><th>Cultivar / season</th><th>Linear regressions on<br>logarithmic transformed data</th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td>Legacy / 2015-16</td><td>ln(FW) = 2.68 ln(FD) - 6.79</td><td>0.98</td></tr><tr><td>Legacy / 2008-09</td><td>ln(FW) = 2.77 ln(FD) - 7.10</td><td>0.97</td></tr><tr><td>Bluegold / 2015-16</td><td>ln(FW) = 2.74 ln(FD) - 6.97</td><td>0.98</td></tr><tr><td>Elliot / 2015-16</td><td>ln(FW) = 2.77 ln(FD) - 7.17</td><td>0.97</td></tr><tr><td>Elliot / 2008-09</td><td>ln(FW) = 2.97 ln(FD) - 7.68</td><td>0.98</td></tr><tr><td>Brigitta / 2010-11</td><td>ln(FW) = 2.87 ln(FD) - 7.43</td><td>0.99</td></tr></tbody></table>

ANCOVA

<table><thead><tr><th>Variable</th><th>p-value</th></tr></thead><tbody><tr><td>ln (FD)</td><td>2.20 x10<sup>-16</sup></td></tr><tr><td>Cultivar / season</td><td>2.67 x10<sup>-15</sup></td></tr><tr><td>ln (FD) x cultivar / season</td><td>0.07</td></tr></tbody></table>

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<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Allometric relationship between berry diameter (FD) and berry weight (FW). Fitted model, *p*-value and *n* are shown.</figcaption></figure>
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<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Estimated versus observed berry weight (A), and residuals versus estimated fruit weight (B) for four field-grown blueberry cultivars. In A, Relative Root Mean Squared Error (RRMSE), line 1:1 (dotted line) and regression between plotted data are showed.</figcaption></figure>