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

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, b_i 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 c_i 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.