Random coefficient growth curve analysis

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

The analyses presented here were carried out using the Mixed procedure from the Ž statistical software package SAS. Instructions for the SAS package can be found in SAS . Language Reference Manual, Version 6, SAS Institute Note that SAS’s Proc Mixed is a very powerful procedure for analyzing many types of mixed linear models, random coefficient growth curves being only one. For more details on mixed models, see e.g., Ž . Ž . Ž . Scheffe 1959 ; Searle et al. 1992 , or Christensen 1987 . Additionally, see Littell et al. Ž . 1996 for a thorough discussion and some excellent examples of mixed model analysis in SAS. 3.1. Preliminary analysis An initial investigation of the data entailed creating a growth curve model for all 4 years and both seasons. The purpose of such a preliminary analysis is to decide on an appropriate model upon which later prediction will be based. Our model was refined as follows. Ž . Ž . After comparing results from using the general Eq. 4 and the no-intercept Eq. 5 , the no-intercept model was deemed most appropriate for modeling shrimp growth. The reasons for this are two-fold. First, the operators did not measure yield until 5 or 6 Ž weeks after shrimp were introduced into the ponds i.e., when the shrimp were large . enough to be captured by the nets being used for measurement . Thus, information on these first few weeks of growth was unavailable and, in practical terms, yield could be considered to be zero at week 0. Secondly, since our goal is prediction at the upper end of the growth curve, the growth behavior at this low end of the curve had very little influence on the forecasts of interest. In other words, using the no-intercept constraint did not significantly alter our prediction results. This preliminary analysis also showed that growth rates for individual ponds were not necessarily similar from year to year. Furthermore, the variability found within growth periods was not uniform. For instance, the growth curves for a set of ponds in summer 1994 were markedly more variable than the growth curves for the same set of ponds in summer 1996. Most importantly, growth rates were shown to be consistent within seasons across different years but not between seasons within the same or different years. In short, summer growth rates differed significantly from winter growth rates. As shown in Fig. 2, although winter growth has a faster linear increase than summer, the growth rate also flattens out earlier. Fig. 3 shows the estimated growth curves for each year for the summer and winter seasons. Note that the variability of growth rates within a season between years is small compared with the variability between seasons. Furthermore, the variables influential in predicting growth differed for the two seasons. While temperature and salinity were found to be statistically significant Fig. 2. Overall estimates of summer and winter shrimp growth. predictors in summer, salinity and turbidity were significant winter growth predictors. Since the predictive factors were not the same for summer and winter, a separate growth rate model was constructed for each season. Stating that a factor is not statistically significant does not imply that it is unimportant in growth. For instance, the operators of the shrimp farm felt that turbidity was influential primarily in the first few weeks of growth. However, at least for the summer season, growth for that period may have been adequately described by other factors already in the model. In other words, if turbidity is highly correlated with one or more factors currently in the model, its inclusion is not necessary for purposes of prediction. The exclusion of temperature in winter illustrates another reason a factor which has an obvious impact on growth may be found insignificant. Specifically, if the factor shows little variation over the growth period, its role in prediction may be minimal. In the winter seasons, temperature levels fluctuated very little over time with an observed standard deviation of only 1.48C, hence its exclusion is not surprising. Thus, a factor may be statistically insignificant in the growth curve model although its impact on growth is unquestioned. 3.2. Summer growth curÕe analysis A detailed description of the growth curve analysis for the summer season follows. The SAS commands and output for this analysis may be found in Appendices A and B, respectively. Recall that for summer, turbidity was found to be an insignificant predictor Fig. 3. Estimated growth rates for 1994–1997. of growth. Since a no-intercept model was chosen for both summer and winter seasons, Ž . the model equation for summer is identical to Eq. 5 except that b and b , the 8 11 coefficients for the effect of turbidity on linear and quadratic growth, respectively, are omitted. The summer model is thus y s b q b t q b s w q b q b t q b s w 2 q ´ . 6 Ž . Ž . Ž . i j 1 6 i j 7 i j i j 2 9 i j 10 i j i j i j Assuming yield data starts at week one and lasts until week n, the matrix form of the growth curve model for the ith pond can be expressed as y s X b q ´ or, more i i explicitly, b 1 1 t s 1 t s y ´ i1 i1 i1 i1 i1 i1 b 6 2 2 t 2 s 4 4 t 4 s y ´ i2 i2 i2 i2 i2 i2 b 7 3 3t 3s 9 9t 9 s y ´ i3 i3 i3 i3 s q 7 Ž . i3 i3 b . . . . . . 2 . . . . . . . . . . . . . . . . . . b 9 2 2 2 y ´ n nt ns n n t n s i n i n i n i n i n i n b 10 Ž . The three components of the historical data necessary for adaptive forecasting are 1 Ž . Ž . the vector of coefficients b, 2 the covariance matrix of b denoted C and 3 the 2 ˆ ˆ 2 overall variance s . We label the estimates of these three quantities b,C and s , ˆ respectively. Note that the b vector obtained from SAS includes coefficients for each year, i.e., X U b s b , b , b , b , . . . , b , b , b , b . Ž . 1 Ž1 994. 1 Ž1995. 1 Ž1996. 1 Ž1997. 10 Ž1994. 10 Ž1995. 10 Ž1996. 10 Ž1997. Ž . The six SAS ESTIMATE statements yield the estimates of b given in Eq. 7 averaged over all 4 years. Likewise, the covariance matrix of b U output by the SAS command COVB is a 24 = 24 matrix. The estimates of C may be calculated by using the matrix equation C s AC U A X where C U is the covariance matrix of b U output by SAS and 1 1 1 w x A s m 0.25 0.25 0.25 0.25 , 1 1 1 with m denoting the Kronecker product operator. The estimates of s 2 , b and C for summer are s 2 s 1.613, ˆ y2.81 = 10 y2 9.79 = 10 y2 1.70 = 10 ˆ b s y1 1.72 = 10 y3 y4.76 = 10 y4 8.26 = 10 and y2 y3 y3 y5 y4 y6 4.6=10 y2.2=10 y1.3=10 6.6=10 y1.3=10 1.2=10 y4 y5 y6 y6 y7 1.2=10 6.7=10 y3.7=10 5.9=10 y4.0=10 y5 y6 y6 y7 4.8=10 y2.4=10 y5.3=10 2.1=10 ˆ C s y7 y7 y9 1.3=10 2.6=10 y9.6=10 y5 y7 1.2=10 y5.7=10 y8 2.9=10 Ž respectively. Note also that the lower diagonal portion of the matrix has been omitted as . it is symmetric.

4. Empirical Bayes analysis