Stephen R. Carpenter. 2002. Ecological futures: building an ecology of the long now. Ecology 83:2069-2083.

Appendix A. Calculation of probability distributions.

The distributions shown in Fig. 5 were derived from data shown in Figs. 2-4 using standard normal theory for Bayesian analysis of linear regression models (Gelman et al. 1995).  The posterior based on P load only (distribution 1 in Fig. 5) was calculated from the model

 log(A) = b0 + b1 log(L) + (A.1)

Here A is chlorophyll, L is P load, bi are regression coefficients, and is a normally-distributed error with mean zero and variance 2.  The regression coefficients and 2 are assumed to be unknown and must be estimated from data.  The prior distribution for analysis of Eq. 1 was uniform for b0, b1, and log()  (Gelman et al. 1995).

A second prior (distribution 2 of Fig. 5) was needed for analysis of the model including both P load and crustacean mean length Z:

 log(A) = c0 + c1 log(L) + c2 Z + (A.2)

The ci are regression coefficients, and is a normally-distributed error with mean zero and variance 2. As before, the regression coefficients and 2 are to be estimated from the data.  The joint posterior distribution obtained for b0 and b1 was used as the joint prior for c0 and c1.  The prior mean regression coefficient for the effect of Z was obtained by linear regression on the data in Fig. 3B fitting the model

 log(A) = a0 + a2 (Z - Ž) + (A.3)

where Ž is the mean of Z.  The estimate of a2 was used as the prior mean for c2.  The prior variance of a2 from this regression was 0.09.  However, this is an underestimate of the variance of c2, because we cannot disentangle the role of P input (vs. P concentration) in the data of Fig. 3.  Therefore I calculated the prior variance according to Raftery et al. (1997) which was 0.9 in this case, reflecting greater uncertainty.  The prior covariances of c2 with c0 and c1 also followed Raftery et al. (1997). This leads to the prior shown as distribution 2 of Fig. 5.   This prior was used to calculate the posterior based on P load and crustacean length (distribution 3 in Fig. 5) using Eq. 2 and the data of Fig. 4, following Gelman et al. (1995).

Literature Cited

Gelman, A., J. B. Carlin, H. S. Stern, and D. B. Rubin.  1995.  Bayesian data analysis.  Chapman and Hall, London, UK.

Raftery, A. E., D. Madigan, and J. A. Hoeting.  1997.  Bayesian model averaging for linear regression models.  Journal of the American Statistical Association 92:179-191.

[Back to E083-037]