A Comparative Study of Simple Regression Models to Estimate Fibre Length Growth in Chios Sheep from Common Meteorological Variables

Introduction

Chios sheep is one of the most commonly reared indigenous sheep breeds in Greece¹ with very good performance in milk production.^2,3 This breed has been participated in many genetic improvement programs^2,4,5, showing great potential. One of the products of Chios sheep in which genetic improvement programs have been focused recently is wool⁵ which is irreplaceable for the production of high quality fabrics.^2,6 An important component of the wool quality is the length of fibre⁷ but, up to now, little research has been conducted on this component of Chios sheep. Matsoukis et al.² recently reported a seasonal variation of wool fibre length of this breed in relation to important meteorological variables.

Air temperature (T) is a meteorological variable of primary importance for farm animals.^8,9 It impacts considerably their physiological functions⁸ and probably the fibre length growth (FLG) of Chios sheep taking into account the reported significant correlation between T and FLG in this breed. A similar significant correlation has been reported between the aforementioned variable and sunshine (SUNS)², another common meteorological variable, which has also been reported to affect the postpartum ovarian activity of the Murrah breed of buffalo (Bubalus bubalis L.) in India.¹⁰

Regression analysis is a statistical tool which helps evaluating which variables are capable of predicting a single, continuous dependent variable.^11,12 Regression technique may be divided into two broad categories, that is, linear regression models and nonlinear regression models. Linear regression models are the simplest ones and commonly used in both data analysis and as a part of the learning process.¹³ On the other hand, the more realistic models are often the nonlinear ones. A characteristic example of these models are many growth models.¹⁴

Literature, to our knowledge, presents a gap regarding the prediction of FLG of Chios sheep from T and SUNS. Thus, a need arises to predict Chios sheep FLG from these meteorological variables, using various regression models, linear and nonlinear ones, and compare them for their effectiveness. The basic hypothesis for our current work is that the examined nonlinear models, that is, quadratic, cubic, logarithmic and inverse ones, better predict FLG from T or SUNS than the linear one, taking into consideration that nonlinear models seem to approach better reality.¹⁴

Materials and Methods

Research was carried out in the farm of Artificial Insemination Center of Karditsa Municipality (39⁰21΄18΄΄N, 21⁰54΄19΄΄E), Periphery of Thessaly, Greece, for a two-year period. Six rams (Ovis aries L.) of Chios sheep breed, all about six years old, living in the aforementioned farm, were the experimental material. The examined variables were FLG and two meteorological ones, T and SUNS, concerning the experimental environment of the animals. Details regarding the measuring instruments, their good operation, time of measurements and recording, ways of measurement, estimation and processing of  FLG, T and SUNS were reported in Matsoukis et al.²

Briefly, fibre length measurements were conducted manually once a month with the aid of a ruler. From these measurements, FLG was estimated as the difference between adjacent months. Air temperature was recorded continually every 6 min by appropriate sensors (Hobo type Pro, H08-032-08, USA, accuracy ± 0.2^oC at 25^oC) connected to data loggers and sunshine data were kindly provided by the meteorological station of the Tobacco Station of Agricultural Research, adjacent to the farm.² For our study, mean values of all variables were estimated for each month, for the whole studied period, and were used in the statistical analysis.

Specifically, in order to detect the possible response function of FLG in Chios sheep to each one of the examined meteorological variables, the first step of our statistical analysis was the test of easily available simple regression models. Thus, we applied five models, one linear and four nonlinear, that is, the quadratic, cubic, logarithmic and inverse ones^15,16, defined by the following equations:

y = a + b₁·x (linear)                                                                                                    (1)

y = a + b₁·x + b₂·x² (quadratic)                                                                                  (2)

y= a + b₁·x + b₂·x² + b₃·x³ (cubic)                                                                             (3)

y=a + (b₁·lnx) (logarithmic)                                                                                      (4)

y=a + (b₁/x) (inverse)                                                                                                (5)

where y: dependent variable (FLG), x: independent variable (separately T and SUNS), a: Y-axis intercept, b₁,b₂, b₃: regression coefficients and ln: natural logarithm.

Then, to evaluate the performance of the examined models, specifically, their goodness of fit (quality of prediction), leading to their comparison, a careful visual examination of the fitted regression lines and two well acceptable statistical measures were used; the adjusted coefficient of determination (R²_adj)^17,18 and the root mean square error (RMSE).^17,19 The R²_adj, an adjustment of the coefficient of determination, explains how well a model fits a data. It indicates the proportion of total variation in the response (dependent) variable that is explained by the predictor (independent) variables. The bigger the R²_adj the better the model.¹⁹ The RMSE, a sort of generalized standard deviation²⁰, is used to measure the differences between the values predicted by a model and the observed values.¹⁹ It is considered as one of the most important criteria to compare the suitability of regression models. The smaller its value, the better the model fits the data.²⁰ The statistics was performed using MS Excel 2007^21,22 and IBM SPSS Statistics 23 with results to be considered significant at P≤0.05.

Results and Discussion

The monthly averages concerning FLG of Chios sheep ranged from 1.1 cm (October) to 2.5 cm (January) with 1.68±0.07 to be their mean±standard error of the mean (SE). As regards T and SUNS, their monthly averages ranged from 3.3 ^oC (January) to 28.6 ^oC (July) and 3700 h (December) to 19585 h (July), respectively, with their means±SE to take the values 16.2±1.7 ^oC (T) and 11016.1±993.2 h (SUNS).

The statistical results of the tested regression models with FLG regressed on T and SUNS, separately, are shown in Tables 1 and 2, respectively. For the examined linear and nonlinear models, significant R²_adj values were confirmed. These values ranged from 0.298 to 0.544 with regard to T and 0.200 to 0.292 with regard to SUNS. The higher and the lower R²_adj values were estimated for the inverse and linear models, respectively, for both T and SUNS.  The RMSE ranged from 0.226 to 0.281 and 0.282 to 0.300, with regard to the model inputs T and SUNS, respectively. The lower and the higher values of RMSE were confirmed in the cases of inverse regression model and linear one, respectively, irrespective of the aforementioned variables.

Table 1: Regression Parameters of the Tested Regression Models Concerning Air Temperature (Independent Variable) and Fibre Length Growth (Dependent Variable) in Chios Sheep.

a: Y-axis intercept, b₁, b₂, b₃: Regression coefficients, ***, **: Significance at P≤0.001 and P≤0.01, respectively, RMSE: Root mean square error.

Table 2: Regression Parameters of the Tested Regression Models Concerning Sunshine (Independent Variable) and Fibre Length Growth (Dependent Variable) in Chios Sheep.

a: Y-axis intercept, b₁, b₂, b₃: Regression coefficients, **, *: Significance at P≤0.01 and P≤0.05, respectively, RMSE: Root mean square error.

The goodness of fit of the tested regression models, from a purely statistical point of view, taking into account that the best and the worst models were characterized by the combinations of maximum R²_adj with minimum RMSE and minimum R²_adj with maximum RMSE, respectively, followed the descending orders Inverse>Cubic>Quadratic>Logarithmic>Linear for T as input variable (Table 1) and Inverse>Quadratic>Logarithmic>Cubic>Linear for SUNS input (Table 2). When comparing the regresssion models of the same type to investigate the effectiveness of the independent variables (T and SUNS) for predicting FLG, T was superior to SUNS in all cases, due to its coincidence with higher R²_adj and lower RMSE. The aforementioned superiority may be attributed, in general, to the noticeable role of T to the performance of domesticated farm animals²³, like sheep.²⁴

Although the cubic and quadratic models appeared a high rank, in general, regarding their effectiveness to predict FLG, taking into account the applied statistical criteria (Tables 1 and 2), the visual examination of the respective fitted regression lines (data not shown) led us to the ascertainment that  these models are unsuitable to predict sufficiently the FLG of Chios sheep. The quadratic and cubic models were characterized by one hump (an actual shape of asymmetric U) and two humps (one facing upward and the other down), respectively.²⁵ These humps had no physical meaning, taking also into consideration the growth pattern of sheep fibre length in relation to the examined meteorological variables.^2,26,27,28 Thus, finally, under the given data and time period, the more suitable and consequently recommended regression model to predict FLG was the inverse model, followed by the logarithmic and the linear one (last in the rank order list), independently of the studied predictors (T and SUNS). Nevertheless, a greater time period is required for a more precise estimation of FLG, based on T and SUNS, for Chios sheep.

In conclusion, the examined simple nonlinear regression models (inverse, logarithmic, cubic, quadratic) were characterized by higher R²_adj and lower RMSE in comparison to the linear model. The inverse model showed the greatest effectiveness in predicting FLG from T and SUNS, separately (maximum R²_adj and minimum RMSE), followed by the logarithmic and the linear ones, under visual examination and applied statistical criteria. The comparison of the regression models of the same type to check the effectiveness of T and SUNS for predicting FLG of Chios sheep, showed that T was superior to SUNS in all cases (higher R²_adj and lower RMSE). The findings of our study could be helpful in establishing sophisticated plans towards a better exploitation of the meteorological variables for the sustainable production of this interesting sheep breed.

Acknowledgements and Funding Sources

Thanks are due to State Scholarships Foundation of Greece which partly funded this work.

Conflict of Interest               

The authors declare that there is no conflict of interest.

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