multiple regression

Powerful and practical statistical packages have simplified the analysis and thus developed the application of data science in all research fields. Accordingly, regression has been applied to almost all aspects of the life sciences. However, misuse of this model has been reported in the past decades. This article aims to examine modeling with this important statistical method and introduce readers to the correct use of this method.

This review article uses real data, and the supplementary materials provide the method for performing the regression analysis in SAS and R statistical software and their related codes.

In the required assumptions of the regression model, the residuals of the model must be normally distributed, but performing the normality test for the actual values ​​of the response variable or any of the explanatory variables is not mandatory. Therefore, researchers should not obsess more than necessary about the normal distribution of real data. On the other hand, almost all normality test methods, such as Kolmogorov-Smirnov, are designed for large numbers of data, typically more than a thousand samples. This suggests that using such methods to test the normality of model residuals estimated from a small number of data, mostly less than a hundred cases, would be inaccurate. Another issue regarding applying the regression model is related to the co-linearity of the explanatory variables. There are still signs of correlation in a data set where all variables are generated separately and randomly in a statistical package. This means that it is very hard to find a correlation coefficient equal to zero (r = 0) even between any pair of separate, random variables. Therefore, in all regression models, there are some kinds of correlation between explanatory variables, but the important issue here is that only high correlation causes severe problems in the model. For collinearity test it would be better to use specialized methods such as Variance Inflation Factor (VIF) or Principal Component Analysis (PCA). The linearity of the model is one other assumption of regression model. Data transformation might be helpful under the situation of non-linearity of the model. However, transformation changes the variables unit, altering the array direction in a geometric space. Researchers should be careful regarding the use of modeling a large number of data affects the probability values ​​in variance analysis due to increasing the value of the degree of freedom of the model.

As the number of data points increases, the degree of freedom of the error term increases rapidly. Therefore, the final error mean squared significantly reduces. In contrast, the scatter of data points around the regression line may be too wide. For this reason, using the coefficient of determination, usually called (R-Squared), is a suitable criterion for testing the model's fit. High coefficient values indicate a suitable model for the data set used. It should be noted that in a multiple regression model, the higher the number of explanatory variables used in the model, the higher the value of this coefficient increases. For such conditions, when the number of explanatory variables is large, another form of this coefficient, called the adjusted coefficient of determination (adjusted R2), has been introduced. The use of this coefficient in the approximations creates a limit on the number of variables used in the regression model. Accordingly, the number of variables in the model as explanatory variables should not exceed the number of samples (or the number of tens) in a set, and researchers should avoid using more variables than the number of samples.

Characterization of physical and chemical soil criteria is a key step in understanding the source of spatial variability in the productivity across agricultural fields (21). Crop yield variability can be caused by many factors, including spatial variability of soil texture, crop management, soil physical and chemical properties and nutrient availability (45). Understanding the spatial variability of soil physical and chemical characteristics is essential for crop management, as it is directly contributing to variability in growth and yield of crop (38 & 14). Hence, understanding their spatial variability across agricultural fields is essential in optimizing the application of agricultural inputs and crop yield and it could help significantly in managing the spatial variability in the productivity of soil agroecosystems (30 & 14). Therefore, the objectives of this study were: (i) evaluate the effect of soil physical and chemical criteria on yield indices of wheat and (ii) to investigate the correlation between physical and chmical soil properties and wheat yield. Samplings were performed based on random-systematic method from 30 fields in Khorasan-e Razavi province during 2017 and 2018. Studied characteristics were texture, organic matter (OM), organic carbon (OC), total nitrogen (TN), available P, available K, pH and C:N ratio of soil and seed yield, biological yield, straw yield, 1000-seed weight and harvest index (HI) of wheat. Multiple regression model was used to identify the relationship between soil variables (independent variables) and wheat yield indices (dependent variables). In addition, determining the most important factors of soil physical and chemical properties which have on wheat yield criteria was done by stepwise regression analysis. The results revealed showed that the mean values of seed yield, straw yield, biological yield, 1000-seed weight and HI of wheat were observed with 3588.47 kg.ha-1, 7362.80 kg.ha-1, 10951.27 kg.ha-1, 35.40 g and 48.56%, respectively. The highest and the lowest standard errors were computed for biological yield (198.40) and 1000-seed yield (0.74), respectively. Also, The effect of soil textures was significant (p≤0.05) on soil chemical criteria and wheat yield. The maximum OM, OC, TN, available P, available K and pH were observed for sandy clay with 1.86%, 1.09%, 0.18%, 166.20 ppm, 0.05 ppm and 7.37, respectively. The maximum seed yield and biological yield were related for clay soil (with 4313.83 and 11924.86 kg.ha-1, respectively). The highest correlation coefficients were computed for OM (r=0.935**) and OC (r=0.933**) with 1000-seed weight. The most important factors influencing wheat yield by using step by step regression were OM, available P, TN and available K, respectively. Longterm sustainability of agroecosystems depends on soil quality and its fertility. Poor soil management practices can lead to degraded soil and environmental quality and reduction in crop yields. Results suggest that novel management approaches are needed to maintain the longterm sustainability of soil resources and crop yields without seriously degrading the environment that this will help in reducing the cost of fertilization and improving soil and environmental quality without altering crop yields.

Plant breeders to select their breeding objectives through the physiological and morphological characteristics, require classification of the limitations and capabilities which exists in plants; this issue leads to the concept of the ideotype. Designing plant ideotypes entails appropriate statistical methods. The objective of this study was to introduce a method based on multiple regressions to find ideotype with a case study of sunflower in Gonbad.
Sunflower was chosen as an example. Data were produced using 12 sunflower genotypes in a randomized block design in south of Gonbad city. Using multiple regressions in order to determine the important traits and to show the contribution of each trait in formation of yield. The method identified the relation between yield and all variables in a quantify matter. Also according to the positive or negative correlation between the variables affecting the yield, to designing ideotype, various hypotheses put forward and various aspects of them was examined.
Five important traits in determining sunflower yield were recognized in this study. They were total dry mass production, maximum leaf number, grain weight, the percentage of empty achene and the percentage of oil. Then, the optimal values of each trait were determined by the method. These five variables explained 57% of yield. The results indicate that if the correlation between some traits would be changed, it can be used for the benefit of yield. Regarding negative correlation between oil percentage and total dry mass, several hypotheses were evaluated. If the negative correlation between max. dry matter and oil content is not breakable, the yield of ideotype would have an increasing of 812 (from 2080 to 2892) kg ha-1 in comparison with sunflower hybrids. If with increasing max. dry mater, oil content stay at moderate level, it would be an increasing of 873 (from 2080 to 2953) kg ha-1 in ideotype, and if correlation between max. dry matter and oil content is breakable, it would be an increasing of 999 (from 2080 to 3079) kg ha-1 in ideotype.
The designed ideotype would increase grain yield from an average of 2080 kg ha-1 to 2892-3079 kg ha-1 in the ideotype. It was concluded that the method used in this study, because of concerning the genetic differences between varieties, can be used in determining plant ideotypes in conjunction with other methods and it can guide plant breeders to move through ideotype crops.