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/