2. Growth curve models
Ž .
Wishart 1938 was the first to consider the problem of analyzing growth rates. These models, known as random coefficient models in the statistical literature, have since
Ž received considerable attention see, for example, Rao, 1959; Potthoff and Roy, 1964;
. Grizzle and Allen, 1969; Laird and Ware, 1982; or Littell et al., 1996, chapter .
Suppose we measure the total mass of shrimp in each pond on a weekly basis for a given year and season, say summer 1995. Conceptually, ‘‘pond’’ is the experimental
unit whose growth we measure over time. Let us assume that shrimp growth rates for summer 1995 are similar across ponds due to identical weather conditions, similar feed,
turbidity, salinity, and so on. A simple linear model to describe this average growth is
y s b q b w q b w
2
q ´ 1
Ž .
i j 1
i j 2
i j i j
where y is the yield, w
is the week number, w
2
is the week squared, and ´ is the
i j i j
i j i j
error associated with the ith pond and jth week. The reason for including the square of the weeks is to account for a nonlinear growth rate over time. Explicitly, when the initial
rate of growth is nearly linear but tapers off as the shrimp get closer to full size, the growth curve flattens out. The square term in the model provides for this.
Ž .
The 20 ponds can be thought of as a random sample from a hypothetical population Ž .
of summer 1995 ponds represented by Eq. 1 . Define the growth curve equation for a specific pond i as
y s b q b w q b w
2
q e 2
Ž .
i j 0 i
1 i i j
2 i i j
i j
where e are assumed to be Gaussian errors with mean zero and variance s
2
. The
i j
coefficients b , b and b
are realizations of b , b and b
for the ith pond,
0 i 1 i
2 i 1
2
Ž . respectively. In other words, the random growth curves for each pond given in Eq. 2
are regression lines which deviate about the overall population growth curve given by Ž .
1 . To illustrate, in Fig. 1 the dark line represents a hypothetical population growth curve and the two lighter lines represent individual growth curves from two hypothetical
ponds.
Fig. 1. Random coefficient growth curves for two hypothetical ponds.
Ž . Ž .
From Eq. 2 , we see that the error term ´ for the overall linear growth model 1
i j
consists of deviations from the population parameters plus random error, i.e., ´ s b y b
q b y b w q b y b
w
2
q e .
Ž .
Ž .
Ž .
i j 0 i
1 i 1
i j 2 i
2 i j
i j
Ž . We can rewrite the growth curve Eq. 1 as
y s b q b w q b w
2
q b y b q b y b
w q b y b w
2
q e .
Ž .
Ž .
Ž .
i j 1
i j 2
i j 0 i
1 i 1
i j 2 i
2 i j
i j
For purposes of prediction, the b coefficients and s
2
are estimated from previous years’ data. The b coefficients, however, are peculiar to the specific pond, season and
year. Ž .
To further refine the model in Eq. 1 for the ith pond and jth week, we add terms for the covariates temperature, t , salinity, s , and turbidity, u . The new model is
i j i j
i j
y s b q b w q b w
2
q b t q b s q b u q b t w
i j 1
i j 2
i j 3 i j
4 i j
5 i j
6 i j i j
q b s w q b u w q b t w
2
q b s w
2
q b u w
2
q ´ . 3
Ž .
7 i j i j
8 i j
i j 9 i j
i j 10
i j i j
8 i j
i j i j
Here b is the coefficient for the y-intercept, b is the coefficient for linear growth in
1
weeks and b is the coefficient for growth in squared weeks. The coefficients for
2
temperature, salinity and turbidity are given by b , b and b , respectively. The
3 4
5
remaining coefficients, b through b , are for various interactions of interest. Note that
6 11
Ž .
further interactions e.g., b t s w could have been included in this model.
12 i j i j i j
Ž . It is illustrative to write Eq. 3 as
y s b q b t q b s q b u q b q b t q b s q b u
w
Ž .
Ž .
i j 3 i j
4 i j
5 i j
1 6 i j
7 i j 8
i j i j
q b q b t q b s q b u w
2
q ´ . 4
Ž .
Ž .
2 9 i j
10 i j
11 i j
i j i j
Here b , b , b and b are the coefficients for the intercept of the growth curve, b ,
3 4
5 1
b , b and b are the coefficients for the linear slope of the curve, and b , b , b and
6 7
8 2
9 10
b are the coefficients for the quadratic portion of the model. More specifically, for
11
example, b and b
measure the effect of salinity on linear and quadratic growth,
7 10
respectively. In certain situations it may be reasonable to assume a no-intercept model, in which
Ž . case b , b , b and b from Eq. 4 would be set equal to zero rather than estimated,
3 4
5
yielding y s b q b t q b s q b u
w q b q b t q b s q b u w
2
q ´ .
Ž .
Ž .
i j 1
6 i j 7 i j
8 i j
i j 2
9 i j 10
i j 11
i j i j
i j
5
Ž .
In summary, the yield in a pond can be modeled as a growth curve which increases weekly. The method for forecasting growth to determine when to harvest the shrimp
consists of two components. First, we estimate the growth coefficients and the amount of pond-to-pond variability from historical data. Second, we use the year-to-date data to
estimate adjustments to the historical growth curve coefficients. These corrected coeffi- cients can then be used to forecast growth in the current year.
3. Random coefficient growth curve analysis
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