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

Chios sheep is a promising sheep breed, with wool, one of its products, to be of special interest to genetic improvement programs. Recently, it has been reported significant linear correlation between the fibre length growth (FLG) of Chios sheep, an important component of its wool quality, and each of the meteorological variables air temperature (T) and sunshine (SUNS), but nothing is known about the prediction of FLG from T and SUNS. Thus, this work aims to investigate the effectiveness of five simple regression models (linear, quadratic, cubic, logarithmic and inverse), concerning the aforementioned prediction, using visual examination and two widely accepted statistical measures, the adjusted coefficient of determination (R 2adj ) and the root mean square error (RMSE). Results showed that the applied nonlinear regression models were characterized by higher R 2adj and lower RMSE in comparison to the linear one, irrespective of input variable. The inverse model presented the greatest effectiveness to predict FLG from T and SUNS, separately (maximum R 2adj and minimum RMSE), followed by the logarithmic and the linear ones, under visual examination and applied statistical measures. Air temperature was superior to SUNS in all cases (higher R 2adj and lower RMSE), when comparing the regression models of the same type to check their effectiveness for predicting FLG. The findings of our study could be a decisive step towards a better exploitation of the examined meteorological variables for the sustainable production of Chios sheep.


Introduction
Chios sheep is one of the most commonly reared indigenous sheep breeds in Greece 1 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 wool 5 which is irreplaceable for the production of high quality fabrics. 2,6 An important component of the wool quality is the length of fibre 7 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 functions 8 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 is 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 (39 0 21΄18΄΄N, 21 0 54΄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 their 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 o C at 25 o C) 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: where y: dependent variable (FLG), x: independent variable (separately T and SUNS), a: Y-axis intercept, b 1 ,b 2 , b 3 : 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 (R 2 adj ) 17,18 and the root mean square error (RMSE). 17,19 The R 2 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 R 2 adj the better the model. 19 The RMSE, a sort of generalized standard deviation 20 , 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 o C (January) to 28.6 o C (July) and 3700 h (December) to 19585 h (July), respectively, with their means±SE to take the values 16.2±1.7 o C (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 R 2 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 R 2 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.
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 R 2 adj with minimum RMSE and minimum R 2 adj with maximum RMSE, respectively, followed the descending orders Inverse>Cubic>Quadratic> Lo-garithmic>Line-ar for T as input variable (Table  1) and Inverse>Quadratic>Lo-garithmic>Cubic> Line-ar 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 R 2 adj and lower RMSE. The aforementioned superiority may be attributed, in general, to the noticeable role of T to the performance of domesticated farm animals 23 , 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.

Conclusion
In conclusion, the examined simple nonlinear regression models (inverse, logarithmic, cubic, quadratic) were characterized by higher R 2 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 R 2 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 R 2 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
Thanks are due to State Scholarships Foundation of Greece which partly funded 'the experimental part of this study.

Funding
The author(s) received no additional financial support for the research, authorship, and/or publication of this article.

Conflict of Interest
The authors declare that there is no conflict of  interest.