6.4. Multiple covariate model

In addition to the length covariate, I also incorporate the covariates date of release, river temperature at release, and average river flow during the individual's migration period. For this analysis I add the covariates one at a time in sequential linear models. I chose to do this for the sake of simplicity, but the covariates could be incorporated in nonlinear models based on salmon behavior. Since the covariates are being added one at a time, the importance of adding the new covariate to the previous model is observed. I add the covariates as a multiple linear model in the following nested sequence:

H₀:

H₁:

H₂:

H₃:

H₄: ,

where

X₁ = fish length (in mm),

X₂ = average river flow during the migration period (kcfs),

X₃ = Julian date of release, and

X₄ = river temperature (degrees centigrade) at time of release.

Other sequences could also have been used.

I apply this sequence of models to each year of data from both data sets. I estimate parameters ('s and ) and report likelihoods for each model. The effect of added covariates can be assessed by computing likelihood ratios between successive models and by comparing BIC values. Note that in this case, I report BIC values for the individual models, so that any of the two models can be compared directly, with the simpler model (that is, the one with fewer parameters) being rejected if it has a lower BIC value.

results

The results for the Snake River fall chinook are contained in Table 6.6. The covariate date of release is extremely important in all three years, with likelihood ratio values ranging from 22.86 to 104.28 larger than the next smaller model nested within (i.e., comparing the model with length flow and date to the one with length and flow). On the other hand, the temperature covariate is never important, with likelihood ratio values ranging from 0.0 to 0.79 larger than those of the model nested within. Length and flow both appear to be important covariates, but the results are not as strong as with the date covariate, particularly in the 1992 data. For all three years, it appears that the best model is the one with length, flow, and date (model 3).

Table 6.6 Results of the application of the individual covariate model to Snake River fall chinook. Note that the BIC values are reported for each of the hypotheses. When two hypotheses are compared, the simpler model is not rejected if it has a larger BIC value than the more complex model.
hypothesis	parameter estimates						likelihoods
	0 (int.)	1 (len.)	2 (flow)	3 (date)	4 (temp)		lik.	ratio	BICi
1991 n = 32
0	1.41	-	-	-	-	4.91	-142.04	-	-291.01
1	-1.56	0.044	-	-	-	4.18	-136.90	10.27	-284.20
2	-3.41	0.050	0.028	-	-	3.74	-133.31	17.45	-280.48
3	-17.07	-0.019	0.072	0.099	-	2.61	-121.88	40.31	-261.09
4	-16.79	-0.018	0.072	0.097	-0.0137	2.61	-121.86	40.34	-264.51
1992 n = 40
0	3.03	-	-	-	-	11.27	-164.59		-336.56
1	-2.25	0.069	-	-	-	10.58	-162.07	5.05	-335.20
2	-5.66	0.074	0.074	-	-	10.20	-160.61	7.96	-335.98
3	-43.91	0.004	0.343	0.234	-	5.95	-139.04	51.10	-296.52
4	-42.78	0.010	0.331	0.215	0.1082	5.89	-138.65	51.89	-299.43
1993 n = 251
0	1.42	-	-	-	-	6.89	-1174.27	-	-2359.59
1	-1.07	0.034	-	-	-	6.05	-1156.74	35.07	-2330.06
2	-3.39	0.043	0.023	-	-	4.90	-1119.92	108.70	-2217.74
3	-15.43	0.014	0.053	0.076	-	3.73	-1067.78	212.98	-2163.19
4	-15.43	0.014	0.053	0.075	0.0002	3.73	-1067.78	212.98	-2168.71

The results for the sockeye are contained in Table 6.7. The length covariate appears to be the most important with large increases in the likelihoods relative to the null model. Flow is also important, with large increases in the likelihoods associated with adding this covariate to the model. Also, date appears to be an important covariate but not as important as the previous two. In both years, temperature had little effect on the model. The order of inclusion may have some importance on the relative importance of the covariates, but it appears that the best model should incorporate length, flow, and date, as with the fall chinook.

Table 6.7 Results of the application of the individual covariate model to mid-Columbia sockeye. Note that the BIC values are reported for each of the hypotheses. When two hypotheses are compared, the simpler model is not rejected if it has a larger BIC value than the more complex model.
hypothesis	parameter estimates						likelihoods
	0 (int.)	1 (len.)	2 (flow)	3 (date)	4 (temp)		lik.	ratio	BICi
1992 n = 148
0	16.37	-	-	-	-	35.11	-495.68	-	-1001.35
1	-14.50	0.265	-	-	-	29.63	-470.42	50.53	-955.83
2	-46.60	0.292	0.504	-	-	26.58	-454.35	82.67	-928.69
3	-63.34	0.266	0.459	0.192	-	25.95	-451.00	89.37	-926.99
4	-63.62	0.267	0.461	0.193	0.0015	25.99	-451.00	89.37	-931.98
1993 n = 521
0	21.24	-	-	-	-	40.83	-1627.24	-	-3266.99
1	-33.85	0.612	-	-	-	31.24	-1499.07	256.35	-3016.91
2	-34.55	0.497	0.099	-	-	30.59	-1477.03	300.43	-2979.08
3	-22.91	0.609	0.142	-0.213	-	30.07	-1470.02	314.45	-2971.31
4	-23.15	0.610	0.143	-0.212	0.0002	30.17	-1470.00	314.47	-2977.54

log likelihood versus log sigma

An interesting result is observed by plotting log likelihood versus log sigma for the 5 alternative models in each of the three years (Figure 6.1). In each year the relationship between these two variables is almost perfectly linear. The inverse relationship indicates that some of the variability in arrival times that was attributed to random movement in the null model is actually the result of population heterogeneity. Thus the more relevant information about the individuals available, the more precise the predictions about arrival times will be. This analysis found that the covariates fish length, river flow, and date of release are important; other covariates may also be determined to be important, further increasing the precision.