1. Introduction

Regression is a statistical model that can be used to explain the relationship between exogenous and endogenous variables (Ijomah et al., 2018). Hill model is a non-linear regression model with the four parameters and A technique of finding a non-linear relationship between the dependent variable and a set of (or several) independent variable(s) called Non-linear regression analysis. In this way, non-linear regression is a function which models observational data by a non-linear combination of the model parameters and depends on one or more independent variables. As a result, many situations require non-linear function just like the simple and multiple linear regression functions that seem adequate for modelling a wide variety of relationships between the response variable and independent variables (Seber & Wild, 2003; Roush and Branton, 2005; Ijomah et al., 2018). In particularly, non-linear regression functions have served and will continue to serve as useful models for describing various physical and biological systems e.g. Hill growth model (or Hill model).

Hill model is an S-shaped curve, often referred to as sigmoidal growth model (sigmoidal curve) which has many applications in agriculture, engineering field, signal detection theory also applicable in biochemistry, forestry (height distribution) and most importantly it is used in prediction. Numerous useful families of non-linear regression functions exist such as Richards, Logistics, Weibull, Gomperts, S. Shapes curves etc. These models are referred to as Sigmoidal Growth Models and are all useful in growth analysis. This study only considers Hill Model in growth analysis and its applications.

The data used in this study is an experiment used to determine the amount of transmitted voltage against time collected from Department of Electrical/Electronic Engineering, University of Port-Harcourt. In some cases, it is possible to transform a non-linear regression function to a linear regression function using some appropriate transformations of the exogenous variable Y_i, the parameters and the endogenous variable X_i or any combination of these. If the assumptions for simple or multiple linear regression are satisfied in the transformed variables, then the result can be applied to the transformed problem. By the use of the transformed results, the original problem can obtain its results.

The Hill model of growth with four parameters is expressed as

(1.1)

where y_i is the i^th observation

are the parameters

= represents upper asymptote when the time approaches positive infinite (or the initial parameter at time equal to zero; t = 0)

= represents the shape parameter related to the initial time

= represents growth range

= represents the growth rate (or shape parameter)

x_i = represents time (Rudolf et al., 2012)

Over the years, forecasting a non-linear model has been a major problem. On the course of solving this problem, many statistical models have been formed and transformed; the major tool for solving the problem is regression analysis (simple, multiple, linear or non-linear) for more accurate results, non-linear regression (growth models) has been developed for prediction, which includes: Hill (case study), Gomperts, Richards, Logistic, Weibull, Brody, Robertson etc.

The custom statistical techniques in estimating nonlinear models require initial values to begin the optimization. The non-linear model expression must be written, the parameter names declared an initial parameter value specified. In most cases, the quality of the final solution depends upon the position of this initial value or starting value; after the iteration approach end. The problem of the initial parameter is solved (or reduced) by transformation to linear and OLS estimation before the iteration approach begins.

This paper aims to estimate the Hill growth model coefficients/parameters using the equation of a line (simple linear regression). The objectives are as follows: (1) to derive the Hill Model to an equation of a line using transformation techniques (Logarithm) and its properties. (2) to choose the two initial values for the upper asymptote parameter and growth range parameter increasing by 0.1 and 0.25 rate respectively (3) to identify a suitable Hill Model and its parameters, using OLS estimation.

There is a need to provide an alternative method of choosing the initial values and fitting the Hill growth model. This will provide an alternative way of a solution to this model and enable it to produce the desired level of forecasting. This paper only considers Hill model as a growth model neglecting other growth models such as Richards, Gompertz, Weibull, Brody, Robertson, Bertalanffy etc (Dagogo et al., 2018). Hence, an alternative way of solving Hill Model is explored in this study, where an iterative process, choosing initial values and OLS estimation is used in building the nonlinear model considered.

2. Review Works on Hill Growth Model

The Hill model was first introduced by Archobald Vivian Hill who published the model first in its currently known form on 22^th January 1910. This equation possesses three of the most important features of scientific models that is information, understanding and memorizing knowledge; it can be used to find an acceptable solution to practical problems, which we shall see in this review, and it can serve as a starting point for the development of a particular model and more detailed models. The Hill model is the first milestone in quantitative pharmacology research, being the first formula that reacts a reversible association (as an effect) to the variable concentration of one of two associating substances, as long as the other substance is present in a constant and relatively low concentration. Thereby, the Hill growth model was the first exact quantitative model in pharmacology that react to changes (or receptor model). The Hill equation (model) and its properties show that it is a good case study for understanding the development of quantitative models in life sciences.

Carla et al., (2018), applied the Hills growth model to the treatment regimens that is the anti-microbial Pharmacokinetics (PK) and the antimicrobial Pharmacodynamic (PD) against the pathogen causing the disease. Their study aimed to use a model to analyze the relationship between the antimicrobial drug concentration and the pathogen populace. Their experiment captured the relationship between the antimicrobial drug concentration and its effect on the growth rate of the exposed pathogen population (Wen et al., 2018).

Schmitt et al., (2013) researched on, muscle biomechanics, a vital and broad field. Given the research, it was known that in biology microscopic muscle models can predict muscle characteristics and functioning of biological muscles quite well; unfortunately, they require a large number of parameters like the Hill model.

Zoran (2011) studied the relationship between the Hill function (model) and the more complicated Adair-Klotz model. He carried out an experiment using the dose-response curve, the result of the experiment showed that the curve (dose-response) obtained from Hill model agrees well with the dose-response curve obtained from Adair-Klotz Model describe strongly co-operative binding. Hence, the alternative technique suggested in the paper will help to construct a Hill model that its results indicate an improved analysis of growth performance, efficiency and the prediction of future or past growth rate.

3. Materials and Methods

- Materials

A data set was considered in this project for the illustration of the fitted growth model. The data used in this study is an experiment used to determine the amount of transmitted voltage against time collected from the Department of Electrical/Electronic Engineering, University of Port-Harcourt (Appendix). The Microsoft Excel solver is used in the analysis of this study, however, there are others possibility for solving this kind of analysis such as Wolfram Mathematical, MATLAB, Minitab, Gretl statistical software’s etc. In the software’s, this problem (for all four parameters at once) can be fitted with a simple build-in command nonlinear model-fitting algorithm.

- Methods

The method of nonlinear least square estimation was applied by taking the derivatives concerning the parameters with additive and multiplicative error terms. A nonlinear regression model is similar to linear regression model both seek to graphically track a particular response from a set of predictor variables. Non-linear models such as Hill growth model in this work are developed because the function is created a series of approximations (iterations) that may stem from trial and error. Many researchers and statisticians use several established methods such as the Gauss-Newton method and the Levebberg-Marquardt method. This research work used the modified version of the Ordinary Least Square method that is

1) Input the arbitrary value for as the initial guess values for the iteration process.

2) Input the data and initial guess values on the Excel solver; then run the iteration to obtain the results.

- Model Specification

The Hill Growth Model with Multiplicative error term (Rudolf et al., 2012), then Equation (1.1) can be expressed as

(3.1)

where are the error terms.

Re-write Equation (3.1) as

(3.2)

By applied transformation technique to Equation (3.2) (or taking the natural logarithm of both sides); we have

(3.3)

Equation (3.3) is in simple linear regression form, with is the residual sum of squares of the model, where the number of parameters has been reduced from four to two parameters.

Note:

By applying the non-linear least square method, using step 1 and 2 above algorithms and

Let be the initial parameters as follow;

and = (-0.75, -0.50, -0.25, 0, …); where is increase by 0.1 and = is increased by 0.25.

Note: One mathematical property of the Hill growth model is as follow;

Asymptotes, if for

- Model Selection Criteria

(1) R-Square (R²):

The R-Square statistic measures the success of the regression in predicting the values of the dependent variable (Nduka & Ogoke, 2016). It assumes that every independent variable in the model help to explain variation in the dependent (y) and thus gives the percentage of explained variation if all independents in the model affect the dependent variable (y). The R² statistic is defined as

(3.5)

where is the total sum of squares

is the regression sum of squares and are the original and modelled data values.

(2) Adjusted R-Square (R²adj):

In the least-squares regression, increasing the number of regressors in the model leads to an increase in R-Square. Hence R-Square alone cannot be employed as a meaningful comparison of the model. The adjusted R-Square (R²adj) tells us the percentage of variation explained by only those independent variables that do not belong to the model (Nduka & Ogoke, 2016).

The adjusted R-square is defined as

(3.6)

where n is the sample size, p is the model parameter, SST and SSE are defined in Equation (3.5) above.

(3) Bayesian information criterion (BIC):

Bayesian information criterion (BIC) is a criterion for model selection among a finite set of models: the model with the lowest BIC is preferred (Henry, 2010; Arimie et al., 2018). The BIC is expressed as

(3.7)

where

RSS (or SSE) is the residual sum of squares of the model.

n is the number of data points (or the number of observations)

p is the number of free parameters to be estimated:

ln is the natural Logarithm

(4) Akaike Information Criterion (AIC)

The residual sum of squares from the regression model was also used to calculate AIC, Arimie et al., (2018):

(3.8)

where all parameters are defined as in Equation (3.7).

Similarly, the model with the lowest AIC is preferred. Note:

4. Analysis and Results

The data in Appendix was used to build a suitable Hill growth model and its parameter estimates.

Firstly, figure 4.1 scatter plot of the amount of transmitted voltage against time with a non-linear model called exponential model with its R² value of 86.7%. This plot showed a slightly S-shaped curve and non-linear relationship between the amounts of transmitted voltage against time. Hence, a suitable Hill growth model can be identified.

Figure 4.1. Scatter plot of the amount of transmitted voltage against time

The results of the ordinary least square technique after logarithm transformation and the steps below are in Appendices are summarized in Table 4.1, 4.2 and 4.3.

Step 1: suggest the initial arbitrary value for as the initial guess values for the iteration process to start.

Step 2: Then, input the data sets and the initial guess values on the Excel solver; then run the iteration to obtain the results OLS.

Step 3: Continue the process (iterations) by increasing and decreasing arbitrary value for until the model selection criterion convergence; therefore stop the process.

Table 4.1 below shows the values of the intercept and other parameters estimates with R², Mean Square Error, AIC and BIC for various iterations. When increase by 0.1 and increase by 0.25.

Table 4.1. Hill model parameters estimates when       increase by 0.1);       increase by 0.25)

We observe that the R-Square values obtained increased slightly after thirteen (13) iterations. Therefore, we increase from 0.1 to 0.3 and increase from 0.25 to 0.75 in Table 4.2.

Similarity, Table 4.2 below shows the values of the intercept and other parameters estimates with R², Mean Square Error, AIC and BIC for various iterations. When increase by 0.3 and increase by 0.75.

Table 4.2. Hill parameters estimates when       increase 0.3);       increase 0.75)

We observe that the R-Square values are closed to converge and close to 1 after seven (7) iterations. Therefore, we decrease from 0.3 to 0.25 and decrease from 0.75 to 0.10 in Table 4.3.

Likewise, Table 4.3 below shows the values of the intercept and other parameters estimates with R², Mean Square Error, AIC and BIC for various iterations. When increase by 0.25 and increase by 0.1.

Table 4.3. Hill parameters estimates when       increase 0.1);       increase 0.25)

We observe the convergence of R² (86.5%) and R² (adj) (86.0%) at the 26^th and 27^th iteration. Note that R² of 86.5% is similarly (or the same) to the exponential model with 86.7%.

The estimated Hill Growth model parameters are β₀ = 3.6, β₂ = 9.75, β₁ = 1.0×10¹⁸ and β₃ = -18.29

By substitution into Equation (3.4):

Hence, the identified Hill growth model is

5. Conclusions

The Hills Growth Model is nonlinear, then logarithmic transformation was applied to make the linear model paved way for its estimation of parameters. The study was able to show an alternative method of estimating the Hill Growth Model parameters using a real data set. A suitable Hill growth model was determined using Mean Square Error, R-Squared, Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). The results of the sum of squared residual, R-Squared, AIC and BIC showed that the identified Hill Growth Model is adequate and can be used for forecasting of the amount of transmitted voltage against time.

Appendices

Table 1. Voltage (volts) and Time (minute)

Hill Growth Model Parameter Estimation