Ordinal regression model for pea seed mass

The research was carried out in the Experimental Research Station in Modzurów, Silesian province, Poland (50°09′N 18°07′E, 274 m. a.s.l.). The experimental field soil was Haplic Phaeozem. A randomised complete block design with four replications was used in the field experiment. The following factors were considered: year – 2010 and 2011 (seeds were sown on 9^th of April in 2010 and 4^th of April in 2011); cultivar – edible ‘Tarchalska’ and used as a fodder ‘Klif’; and inoculant – of powder and gel form (Nitragina – powder form inoculant, produced by the BIOFOOD company (Poland) and non-commercial, gel inoculant (IUNG) produced by the Polish Institute of Soil Science and Plant Cultivation). Each plot had an area of 8.4 m². The following pre-sowing fertiliser doses were applied: phosphorus, 48 kg ha⁻¹ (P₂O₅ ); potassium, 72 kg ha⁻¹ (K₂O). Ammonium nitrate was applied as a starting dose with 20 kg ha⁻¹ N. One hundred and twenty seeds of the cultivar ‘Tarchalska’ and 100 seeds of cultivar ‘Klif’ were sown per square meter. Row distance was 15 cm. Before harvest, 15 mature plants per plot (7.67 pods on average from each plant) were randomly selected and collected in order to conduct the biometrical analysis. The crops were harvested on 6^th of August 2010 and on 3^rd of August 2011. In the biometrical analysis, we focused on seeds per pod, pods per plant and seeds per plant. All collected seeds were divided into five classes of weight (<200, 200–240, 240–270, 270–300 and >300 mg) in order to assess how qualitative factors (year, cultivar and inoculant) and quantitative variables (seed mass in pod, the seeds per plant, seeds per pod) influence seed mass.

The pea experiment data was evaluated with an ordinal regression model. The reason of using this model is the prediction of the ordinal outcome. The outcome variable was pea seed’s weight with five ordinal levels: <200, 200–240, 240–270, 270–300 and >300 mg. Ordinary logistic regression models apply with binary dependent variables, but generalisations of logistic regression apply with multicategorical responses. The ordinal regression enables to build models aiming to evaluate the importance of predictor variables, where the dependent variable is ordinal in nature. However, continuous variables could be considered as ordinal because of threshold effects.

Generally, linear regression models do not work very well in prediction of ordinal response variables. Because linear regression models are drawing up to model the outcome variable measured on an interval scale the simplification of linear regression assumption may result in not accurately reflecting the relationship in data. A way to model ordinal variables is to use a generalisation of linear regression – generalised linear model to access cumulative probabilities for outcome variable categories. With this approach, the finding is a predicted probability of being in the corresponding category. The basic form of a generalised linear model is link(γ<!-- ? -->j)=θ<!-- ? -->j−<!-- - -->[β<!-- ? -->1x1+β<!-- ? -->2x2+⋯<!-- ? -->+β<!-- ? -->kxk]$$\begin{array}{} \displaystyle link({\it\gamma}_j)=\theta_j-[\beta_1x_1+\beta_2x_2+\cdots+\beta_kx_k] \end{array} $$(1)

where

γ_j – the cumulative probability for the j^th category of the dependent variable

x₁ … x_k – the predictor variables

k – the number of predictor variables

β₁ … β_k – the regression coefficients

Instead of predicting the cumulative probabilities, the ordinal regression model provides a function of those values. This function is called link function. Generally, an ordinal regression model consists of three major components:

Location component – it is the part of equation (1) that includes predictor variables and related coefficients [β₁x₁ + … + β_kx_k]. It is used to access probabilities of membership of each observation in the categories considered.

The process of selecting predictors for the location component takes both theoretical and empirical sides into consideration. Because the discrimination of potential predictor variables that are expected to influence seed weight was the aim of the study, the analysis started off by including all the variables involved and considered as might be influential. Stepwise procedure was used: variables not helpful in the model were removed from the analysis and the model was re-estimated. Qualitative variables (year, cultivar and inoculant) were entered as the factors in the model, and, on the other hand, quantitative variables (seeds per pod, pods per plant and seeds per plant) were entered as covariates in the ordinal regression model. The Wald test was used to determine whether a certain predictor variable is significant or not. It rejects the null hypothesis of the corresponding coefficient being zero.

A general appraisement of the goodness of fit of the final model enables pseudo-R-square coefficients. These indicators are based on the likelihood ratio. There are three pseudo-R-square measures of determination used to evaluate the ordinal regression model. Cox and Snell’s indicator is a generalisation of the usual R-square designed to evaluate estimates obtained by the maximum likelihood method (Cox and Snell, 1989).

The total set of observations used to construct the regression model for the pea seed mass amounted to n = 5,527 (year – 2010: 2,505, 2011: 3,022; cultivar – Klif: 3,112, Tarchalska: 2,415; inoculant – Control: 1,765, IUNG: 1,981, Nitragina: 1,781).

Weather conditions differed in two years of vegetation (Table 1). In 2010, intensive precipitation was observed in the seedling stage (May) and during the flowering phase, which contributed to the growth and development retardation of vegetative and generative stages. This reduced plant density, number of seeds and pods and consequently resulted in the low yield. The July 2010 was characterised by high temperature conditions, which contributed to the decrease in the yield parameters. In 2011, the precipitation and average temperature were optimal for the plant growth.

Table 1

Weather conditions in the two experimental years

Tabelle 1. Wetterbedingungen in den zwei Versuchsjahren

All three factors (year, cultivar and inoculant) had significant effects on seed yield parameters. Seed mass, seed per pod, pod per plant and seeds per plant were higher in 2011 than those in 2010. Fodder cultivar ‘Klif’ had more seeds per pod, pods per plant and seeds per plant, whilst ‘Tarchalska’ had a higher seed mass. The use of inoculants had no effect on the seed mass but significantly influenced the seeds per pod, pods per plant and seeds per plant with highest numbers IUNG application.

Tables 3 and 4 show the descriptive statistics of quantitative and qualitative variables used in the model. The process of the ordinal model construction involved specification of different arrangements of predictor variables based on their theoretical justification. The ultimate model was built on the premise of the significance of predictor variables in dependent variable categories description. Factors retained in the model were considered as important in explanation of seed mass. Additional variables that were initially considered got rejected. These factors, as not relevant for the seed mass modelling, are excluded from the selected model.

Table 2

Means and analysis of variance of seed mass and quantitative variables (seeds per pod, pods per plant and seeds per plant) as affected by the qualitative variables (year, cultivar and inoculant)

Tabelle 2. Mittelwerte und Varianzanalyse des Samengewichts und der quantitativen Variablen (Samen pro Hülse, Hülsen pro Pflanze und Samen pro Pflanze) in Abhängigkeit der qualitativen Variablen (Jahr, Sorte und Inokulum)

Table 3

Model fitting information

Tabelle 3. Informationen zur Modellanpassung

Table 4

Statistical evaluation of considered factors (ordinal regression model) for seed mass categories

Tabelle 4. Statistische Evaluierung der betrachteten Faktoren (ordinales Regressionsmodelles) hinsichtlich der Kategorien des Samengewichts

Model fitting information (Table 3) shows that using predictor variables significantly improves the model’s ability of adequate seed weight predictions. McCullagh and Nelder (1989) indicate that the difference between −2 times the log-likelihood for the intercept-only model and for the final model can be interpreted as chi-square statistics. The significant chi-square statistic indicates that the final ordinal regression model gives better predictions than those based on marginal probabilities for the outcome variable categories.

Table 4, which contains the parameters’ estimates, shows the final ordinal regression model. The parameters of the ordinal regression model were estimated by the iterative algorithm. Options set for the algorithm were link function (log–log), maximum number of iterations (100), maximum step-halving (5), parameter convergence (e⁻⁶), delta (0 – no value was added to zero cell frequencies) and singularity tolerance (e⁻⁸). The criterion of log-likelihood was not applied (the algorithm stops if the change in the log-likelihood is less than this criterion).

The overall assessment of the goodness of fit of the ordinal regression model was done according to the Nagelkerke (1991) pseudo-R-square for the reason that the coefficient attains values in the zero-to-one range. The estimate amounted value of 0.461, indicating the meaningful proportion of variance in the dependent variable associated with the predictor variables. From the theoretical standpoint, the threshold parameters are not as important as location estimates. The location parameters relate the predictor variables to the cumulative probabilities of dependent variable.

According to the statistical tests’ results, the variables seeds per pod, pods per plant, seeds per plant year and cultivar are meaningful predictors for the outcome variable (pea seed mass categories). The variable inoculant is marginally significant. However, direct interpretation of the coefficients in the model in Table 4 is difficult because of the character of the link function. On the other hand, the sign of a particular parameter coefficient provides meaningful insight into the effect of a predictor variable in the model. The signs of the estimates in principle indicate the direction of a particular predictor effect. Positive signs of coefficients of seeds per pod and pods per plant suggest a positive relationship between these predictors and dependent variable. As such variable increases, so does the probability of being in one of the higher categories of the seed weight category. Conversely, negative estimates indicate inverse relationship. An increase in the number of seeds per plant variable predisposes a pea to be classified into the lower outcome variable categories.

Our results correspond with the observations of other authors. Halsted and Byron (1918) analysed the seed position in a pod of soybean. The author noticed that the number of seeds and the weight of seeds in a pod are variable. The seed mass mainly depends on the number of seeds and their position in a pod. Additionally, the seed mass is inversely proportional to seeds per pod. The author proved that the largest seed mass is gained in single seed pods. In double seed pods, an earlier developed seed is always the largest. However, in the triple seed pods, the largest seed mass is gained in the central part of the pod. Furthermore, node position on the main stem (Atta et al., 2004), cultivar and weather conditions (Zając et al., 2013; Dacko et al., 2016) influence the seed dry weight. Zając et al. (2013) and Dacko et al. (2016) supported the above study, claiming that cultivars and weather conditions have the largest impact on seed weight. Zając et al. (2013) proved that biometrical parameters of pods revealed considerable differences between the pea cultivars. The fodder pea cultivar ‘Klif’ was more productive because of the higher number of seeds and the higher mass of seeds in pods. The edible pea cultivar ‘Tarchalska’ produced a large amount of heavier seeds in the first two reproductive nodes. Application of IUNG inoculant resulted in a rapid increase in the number of pods and seed weight on the bottom part of reproductive shoots but the weight of seeds and pods decreased at consecutive nodes. Seed mass significantly depended on weather conditions. Plants were more productive in optimal weather conditions. Under high temperature conditions, as prevailing in July 2010, both seed position on the plants and time of pod set appeared to contribute considerably to the variance in seed mass. Seeds produced under high temperature conditions were lighter and smaller than those grown under normal temperature conditions.