Growth Model:
Fitting to parametric as well as non-parametric growth models were exempted in order to get the estimate of the biological parameters. Specific methodological approaches are required by the analysis of longitudinal growth data. To establish individual growth patterns and to estimate, so-called, biological parameters of the growth curve are one of the main interests of longitudinal growth studies.
For example, the timing and intensity of the adolescent growth spurt. These appearances are introducing us with information about the shape of the growth curve, rather than informing about the size, which is attained at a particular age.
Many researchers tried to fit parametric and also non-parametric growth models for estimating the biological parameters and they were also dealing with different models to get more revising with great accuracy.
Some of the models are mentioned below: (According to year published)
1.Gompertz and the logistic growth model.
2.Jenss model.
3.Count model.
4.Double logistic model.
5.PB models.
6.ICP model.
7.Reed models.
8.SSC model.
9.JPPS model.
10.JPA-1 and JPA-2 model.
11.Modified ICP model.
12.BTT model.
13.Kernel’s (non-parametric) model.
Gompertz and Logistic Growth Model:
In the analysis of growth process Deming(1957) and Merrell(1931) discussed in detail the properties of the Gompertz and the logistic function. The Gompertz curve equation can be written as:
------(1)
Where, Depending variable, i.e., stature.
Independent variable, i.e. age.
Lower asymptote, i.e. stature at the start of the adolescent growth cycle.
Adolescent gain, i.e. stature gain during the adolescent growth cycle.
Constant of integration, i.e. depending on the position of the origin.
Rate of constant, i.e. .
This curve equation may be considered as the individual’s constant rate of maturation through the adolescent growth cycle according to Deming(1957). It can be seen that the velocity curve is asymmetrical with a point of inflection at by differentiating equation (1). This can be considered as the abscissa of the maximum for its first derivative of equation (1) and the ordinates are:
------(2)
------(3)
Equation (2) expresses that about 37% of the adolescent growth is attained at age .
The logistic curve equation, on the other hand is:------(4)
Here the parameter meanings are the same as stated above in the Gompertz function. From equation (4), the velocity curve is symmetrical about its point of inflection at . Marubiniet.al. in 1971 clearly showed the asymmetry of the Gompertz and the symmetry of the logistic curve.
Jenss Model:
Jenss and Bayley(1937)(where credit is given to Dr.LowellReed for its development) first described a four-parameter nonlinear Jenss model. This is a negatively accelerated exponential and approaches a linear asymptote having positive slope. This model can be written as:
Where, is age (years), is observed stature (cm) or body weight (kg), and is random error. , and are positive parameters, and is negative.
The be the growth or acceleration constant, is independent of scale and measures the ratio of the acceleration of growth at any given age, to the acceleration at the preceding age, be noted by Jenss and Barley(1937).
Thus to compare the growth of different characteristics within the child, or to study the growth of the same characteristic in different children can be needed by . The acceleration’s magnitude of the constant is what largely determines the shape of an individual curve. Since then the model has been used by others (Berkey, 1982; Deming and Washburn, 1963; Manwani and Agarwal, 1973).
Count Model:
Count(1943) first applied the Count model in human stature of Chinese population (children) has been applied this model within the age range three month to seven years. Many other researchers (Tanner et. al.,1956; Israelsohn, 1960, Wingerd, 1970; and Mata, 1978) has been applied this model.
The linear Count model can be written as:
Where, is age (years), is random error and is physical measurement (stature or body weight), and , and are the parameters of the model.
The location of zero age is an implicit fourth parameter in the model. Some authors have used conception, or other points that make the interpretation of parameters especially convenient, for age zero.
Double Logistic Model:
In 1973 Bock, Wainer,Peterson, Thissen, Murrayand Roche described that the double logistic model fit the growth data from childhood to adulthood. The double logistic model can be written as:
------(1)
Where be the stature (in cm), be the age (in years), and , , , and be the five parameters. Mature size has to be inserted in the function in this model.
A component of prepubertal growth, which continues in reduced degree until maturity is forecasted by the first term of the right hand side of the above equation and the second term describes the contribution of the adolescent spurt. The explanations of the parameters are as follows (units are shown in the parentheses):
Upper limit of the prepuberal component (cm).
Determines the initial slope of the prepuberal component , and is implicit in , the maximum velocity of growth of the prepuberal component .
Determines the location in time of the prepuberal component (years).
Adult stature (cm).
Contribution of the adolescent component to adult stature (cm).
Determines the slope of the adolescent component, and is implicit in , the maximum velocity of growth of the adolescent component .
Age at maximum velocity of the adolescent component (years).
PB Model:
In 1978 Preece and Baines developed a procedure for fitting individual serial record of stature from age 2 to adulthood and describes the properties of biological parameters of the proposed growth model. The model was as follows:
------(1)
Where, is the stature (in cm.) at age and is a growth parameter vector . is the equation parameter, which is the estimated adult stature. The parameters and are rate constants, and and are related to the stature and age at take-off of the adolescent growth spurt.
Several authors (Billiwicz and Mc Gregor, 1982; Bogin et al., 1990; Bogin et al., 1992; Brown and Townsend, 1982; Byard et al., 1993; Cameron et al., 1982; Guo et al., 1992; Hauspie, 1980; Hauspie et al., 1980a; Hauspie et al., 1980b; Jolicoeur et al., 1988; Jolicoeur et al., 1991; Jolicoeur et al., 1992; Ledford and Cole, 1998; Mirwald et al., 1981; Qin et al., 1996; Tanner et al., 1982; Zemel and Johnston, 1994) were also used the above model.
ICP Model:
In 1987 Karlberg developed the ICP model and this model divides growth into three distinct phases, such as, Infancy, Childhood, and Puberty. These three distinct phases functional form as follows:
An Infancy component: This component assumed to start during fetal life with a rapidly decelerating course ceasing at 3 – 4 years of age and also it was explained by an exponential function:
A Childhood component: This component starts during the first year of life having a slowly decelerating course and continuing until end of growth. A second degree polynomial function explained this component and this polynomial function could be written could be written as:
A Puberty component: This component representing the additional growth induced by puberty and accelerating up to age at peak velocity (age = ), then decelerating until the end of the growth (age = ). A logistic function represented this component and that function was:
In all the above three functions is stature for the relevant component at time in years from birth, and is the middle of the first one year interval after age at peak velocity where the overall gain becomes less than that in the Childhood component.
Reed Models:
TheReedmodelsare the extension of the Count model. Berkey and Reed(1987) developed these models. The first-order Reed model can be written as:
The first-order Reed model has four parameters and it is more flexible than the Count model since it allows an inflexion point. The second-order Reed model can be written as:
Where the fifth parameters allows a second inflexion point. The first-order version was shown to perform well on height between 3 months and 6 years but few children needed the second-order version, which was proposed by Berkey and Reed(1987).
SSC Model:
Shohoji and Sasaki(1987) described a growth model, which has six parameters. It can be written as:
Where is postnatal age, is stature at age , is adult stature, is a weighting function given by , is a function of stature in infancy given by and is a error.
The weight average of adult stature is the stature at age and stature predicted from an infancy model . The Gompertz function is the weight takes the value 0 at , then switches from 0 to 1 at , with parameter controlling the suddenness of the switch. The function is the Count model for infant stature and body weight. However the Jenss-Bayley function is another infant stature model with one extra parameter, combining an exponential and a linear component, which performs appreciably better (Berkey, 1982; Cole, 1993) suggested modifying the Shohoji-Sasaki model to use the Jenss-Bayley rather than the Count model as its childhood component. Another seven-parameter model (KS7) was described by Kanefuji and Shohoji(1990) extending that of Shohoji and Sasaki(1987), replacing the Count model by . The combines an exponential infancy, linear childhood and logistic puberty component is the SSC (Shohoji-Sasaki modified by Cole) model, and in this sense is similar to Karlberg’sICP model (Karlberg, 1987; Ledford and Cole, 1998).
JPPS Model:
A seven-parameter model was described by Jolicoeur et al. (1988) and written as:
Where is post-conceptual age, is stature at age , is adult stature, , and are positive age scale factors, , and are positive dimensionless exponents, and is the error.
Note that is age post-conception, i.e., assuming a constant gestation of 9 months.
The total age, which is defined as measured from the time of fertilization, and pass through the origin utilize in the above model. It is the practice that, total age can be estimated by adding the average duration of pregnancy (0.75 year in humans) to the age after birth, unless pregnancy is actually known to have been shorter or longer than average. Growth models passing through the origin with respect to total age are based on the fact that, in placental mammals, the fertilized egg is microscopically small, and its dimensions can be considered as practically null in comparison with those of the adults. Models passing through the origin with respect to total age are defined before birth as well as after birth and may be particularly suitable if prenatal data are to be included in the analysis or if prenatal extrapolations so that . Moreover, provided the constraint that some parameters are non-negative is respected, the initial section of growth models passing through the origin with respect to total age remains realistic even when there are few data concerning young ages (Ledford and Cole, 1998).
JPA-1 and JPA-2 Model:
Joulicoeur et al. (1992) have described an extension to the model of Jolicoeur et al. (1988) where the age offset is estimated from the data rather than being constrained at 0.75, to improve the fit in infancy. The extended models were as follows:
------(1)
------(2)
JPA-1 is the nickname of the model in equation (1) and JPA-2 is the nickname of the model in equation (2). The models JPA-1 retains the theoretically desirable quality of passing through the origin with respect to total age while, JPA-2 fits human stature data better than all other asymptotic models proposed till 1991(Jolicoeur et al., 1992).
Modified ICP Model:
To convert the non-stationary time series of growth observations into a stationary time series for the Fourier analysis, the modified ICP(Johnson, 1993) model was used. This model described the combined form of two phases such as Childhood and Puberty.
The modified form was as follows:
BTT Model:
The sum of three generalized logistic terms is the Bock-Thissen-duToit(BTT) model. The form of the logistic term is:
Where is the time (age) variable, is the amount of growth contributed by the term, the quantity in the exponential function is the “logit”, and are its slope and intercept, respectively, and is a fixed shape constant.
In 1994Bock et al. described the triphasic generalized logistic model by summing up three phases of growth; early, middle and adolescent. This triphasic generalized logistic model can be written as:
Where the set of parameters , and refer to the parameters of early, middle and adolescent phases of growth, respectively.
In terms of expressions particular to each model the distance functions can be differentiated to obtain their velocity and acceleration curves, which are described below:
For PB model (Preece and Baines, 1978):
Let
ThenVelocity:
Acceleration:
For JPPS model (Jolicoeur et al., 1988):
Let
ThenVelocity:
Acceleration:
For JPA-1 model (Jolicoeur et al., 1992):
Let
ThenVelocity:
Acceleration:
For JPA-2 model (Jolicoeur et al., 1992):
Let
ThenVelocity:
Acceleration:
For SSC model (Cole, 1993):
Let
ThenVelocity:
Acceleration:
For BTT model (Bock et al., 1994):
Velocity:
Acceleration:
Velocity:Velocity is a directed term; it has a direction and a magnitude and is indicated by the letter . Velocity as directed term is defined as the ratio of the directed displacement to the required time , i.e.
The direction of is the same as the direction of the displacement.
M-POPS: 508 Modeling for Human Growth - 1 -