TWO COVARIATES MODEL Y = A + B + e Analysis of per capita growth response r to ln(N_t) + ln(N_t+1), declaring terms: t + tplus1 Ln density data for Figure 11(a): t tplus1 r 10 10 0.1 10 15 5.2 10 20 9.7 15 15 0.0 15 20 4.8 15 25 10.1 20 20 0.0 20 25 5.0 20 30 10.0 t and tplus1 are both covariates of the response: Source DF Seq SS Adj SS Adj MS F P 1 t 1 0.00 73.51 73.51 2536.36 <0.001 2 tplus1 1 147.02 147.02 147.02 5072.72 <0.001 3 Error 6 0.17 0.17 0.03 Total 8 147.19 COMMENT: These covariates are not orthogonal, and information about one increases the explanatory power of the other. Covariate t contributes nothing to explaining r without the presence also of covariate tplus1 (it has Seq SS = 0.00). Ln density data for Figure 11(b): t tplus1 r 12.0 17.5 5 15.5 18.8 4 19.0 22.2 3 21.5 24.0 2 24.0 25.1 1 26.0 26.7 0 t and tplus1 are both covariates of the response: Source DF Seq SS Adj SS Adj MS F P 1 t 1 17.29 0.28 0.28 4.02 0.139 2 tplus1 1 0.00 0.00 0.00 0.01 0.923 3 Error 3 0.21 0.21 0.07 Total 5 17.50 COMMENT: These covariates are not orthogonal, and information about one reduces the explanatory power of the other. Covariate tplus1 contributes negligible variation beyond that contributed by covariate t, which loses explanatory power when adjusted for tplus1 (it has reduced adjusted SS). More information is obtained from analysis with sequential SS, or the inclusion of an interaction term. __________________________________________________________________ Doncaster, C. P. & Davey, A. J. H. (2007) Analysis of Variance and Covariance: How to Choose and Construct Models for the Life Sciences. Cambridge: Cambridge University Press. http://www.southampton.ac.uk/~cpd/anovas/datasets/