6.3. Applications with length covariate

• H₀:
• H_A: .

For each cohort, likelihoods, l₀ and l_A, are computed for the null and alternative hypotheses respectively. Comparing these two likelihoods yields an assessment of the performance of the two models relative to each other. With a likelihood ratio test, the null hypothesis is rejected at the 0.05 level if the ratio is greater than ²_1(0.05) = 3.84. Using Akaike's information criterion (AIC) the null model is rejected if the ratio is greater than 2.0. Using the Bayesian information criterion (BIC), the null model is rejected if the BIC for the length model (BIC_l) is greater than the BIC for the null model (BIC₀), and I report the value BIC_l - BIC₀. I will use these values as a rough measures of the relative performance of the two models.

Three sets of cohorts are analyzed in this section. The first two sets are Snake River spring chinook and steelhead analyzed in previous chapters. The third set is the mid-Columbia fall chinook.

results

The results of the analysis of the 3 data sets are contained in Table 6.1 - Table 6.5. These tables provide averages and standard deviations of length for each release group. Also contained in the tables are parameter estimates for the length model as well as log likelihoods for the null and alternative models, the ratios between the two, and the BIC values.

There is some support for the length model in the Snake River chinook cohorts (Table 6.1 and Table 6.2). Nine out of the 18 cohorts had likelihood ratios greater than 2.0, which is the AIC value at which the null hypothesis is rejected, but the null model is only rejected for five out of 18 cases based on the BIC values. The parameter estimate results are somewhat contrary to what I expected, however. In 14 out of 18 cohorts, is negative, indicating that the model predicts increasing migration rate for decreasing fish lengths. This is also true for 8 out of the 9 cohorts that had likelihoods ratios greater than 2.0.

Table 6.1 Results from the application of the individual covariate travel time model with length covariate to cohorts of Snake River "run-of-the-river" chinook. Note that a negative BIC value lends support to the null model (that is, the model without the length covariate).
cohort #	# of fish	length		parameter estimates			likelihoods
		mean	s.d.	0	1		l0	lA	ratio	BIC
1989
1	55	128.25	9.28	3.61	-0.006	4.42	-177.51	-177.44	0.15	-3.86
2	57	128.18	10.51	4.99	-0.015	7.02	-199.30	-199.07	0.46	-3.58
3	43	128.21	12.47	10.42	-0.056	7.26	-148.90	-147.10	3.60	-0.16
4	64	134.77	12.06	15.49	-0.059	9.97	-159.53	-158.50	2.06	-2.10
5	69	126.26	18.92	12.81	-0.036	8.34	-156.14	-154.79	2.70	-1.53
6	66	124.61	18.06	9.53	-0.016	7.75	-152.75	-152.46	0.59	-3.60
7	64	115.88	17.39	4.16	0.030	11.26	-164.28	-163.75	1.04	-3.12
1990
1	54	115.07	13.02	16.86	-0.069	7.83	-114.29	-112.20	4.17	0.18
2	66	118.95	14.24	15.49	-0.077	10.07	-182.78	-179.99	5.58	1.39
3	52	117.13	15.14	9.54	-0.028	5.90	-122.53	-121.38	2.29	-1.66

Table 6.2 Results from the application of the individual covariate travel time model with length covariate to cohorts of Snake River "run-of-the-river" chinook. Note that a negative BIC value lends support to the null model (that is, the model without the length covariate).
cohort #	# of fish	length		parameter estimates			likelihoods
		mean	s.d.	0	1		l0	lA	ratio	BIC
1991
1	55	124.22	12.06	4.00	-0.009	4.82	-178.31	-178.14	0.33	-3.67
2	66	128.32	9.84	9.03	-0.039	6.02	-197.19	-195.71	2.97	-1.22
3	51	127.88	10.16	5.61	0.004	8.35	-135.23	-135.22	0.01	-3.92
1992
1	50	130.00	9.32	8.14	-0.032	5.51	-147.50	-146.59	1.82	-2.10
1993
1	60	127.08	10.77	1.67	0.015	5.18	-182.64	-182.25	0.79	-3.31
2	46	123.72	11.22	10.29	-0.047	4.54	-124.04	-120.45	7.18	3.35
3	64	120.33	13.07	3.21	0.041	6.92	-135.87	-134.53	2.68	-1.48
4	57	122.02	10.87	12.48	-0.009	7.41	-95.49	-95.47	0.04	-4.00
5	74	121.14	18.26	-4.94	0.148	15.13	-154.11	-148.98	10.27	5.97

The results for the steelhead (Table 6.3 and Table 6.4) are similar to the Snake River chinook results. Five out of the 19 cohorts had likelihood ratios greater than 2.0, and three out 19 has positive BIC values, supporting the null model in most cases. Also, eight out of 19 had negative values for .

Table 6.3 Results from the application of the individual covariate travel time model with length covariate to cohorts of Snake River steelhead. Note that a negative BIC value lends support to the null model (that is, the model without the length covariate).
cohort #	# of fish	length		parameter estimates			likelihoods
		mean	s.d.	0	1		l0	lA	ratio	BIC
1989
1	64	185.89	29.98	27.94	-0.042	12.62	-84.61	-83.51	2.20	-1.96
2	79	182.48	20.62	8.13	0.058	17.19	-126.17	-125.68	0.97	-3.40
3	47	168.94	17.28	18.66	-0.019	11.69	-90.22	-90.16	0.11	-3.74
1990
1	61	182.57	23.54		-0.011	9.93	-101.38	-101.30	0.17	-3.94
2	95	176.74	14.80	0.67	0.067	6.84	-158.91	-154.53	8.76	4.21
3	146	171.97	16.02	8.33	0.021	12.01	-287.36	-287.11	0.50	-4.49
4	68	173.94	16.79	6.42	0.015	13.98	-168.08	-168.02	0.11	-4.10
5	61	169.43	17.40	11.38	-0.014	8.22	-128.11	-127.93	0.35	-3.76

Table 6.4 Results from the application of the individual covariate travel time model with length covariate to cohorts of Snake River steelhead. Note that a negative BIC value lends support to the null model (that is, the model without the length covariate).
cohort #	# of fish	length		parameter estimates			likelihoods
		mean	s.d.	0	1		l0	lA	ratio	BIC
1991
1	50	181.12	16.12	6.17	0.023	7.15	-89.39	-89.07	0.64	-3.27
2	126	178.67	15.29	24.78	-0.055	10.62	-201.54	-199.96	3.15	-1.69
3	56	173.95	16.23	13.99	-0.005	9.45	-94.36	-94.35	0.02	-4.01
4	51	165.82	17.33	11.55	0.050	12.75	-68.50	-68.23	0.54	-3.39
1992
1	67	181.40	17.63	8.21	0.011	6.40	-114.00	-113.84	0.31	-3.89
2	154	176.60	17.70	17.05	-0.014	10.23	-245.01	-244.83	0.36	-4.68
3	90	171.41	14.71	15.14	-0.038	7.34	-214.97	-213.39	3.15	-1.35
1993
1	50	178.28	21.30	7.47	0.032	10.85	-88.79	-88.35	0.88	-3.03
2	87	177.00	20.10	-3.21	0.114	9.44	-123.06	-115.73	14.66	10.19
3	59	173.93	19.03	-13.20	0.183	17.88	-100.69	-97.86	5.66	1.59
4	40	175.38	16.32	10.32	0.032	12.18	-64.90	-64.77	0.26	-3.43

Although the length covariate appears to have some importance in the travel time model for these two groups, it would be difficult to implement the length model based on these data because of the variability in parameter estimates. More information will be required to understand why the relationship between migration rate and fish length is sometimes positive and sometimes negative.

The results for the mid-Columbia fall chinook (Table 6.5) strongly support the inclusion of the length covariate in the travel time model. All the BIC values are positive, with 4 out 5 values greater than 10.0. Also there is consistency in the values of and , with most estimates of in the 3.0 - 5.0 range and most estimates of in the 0.10 to 0.14 range. Thus, including length information in the travel time model for these fish would be quite useful.

Table 6.5 Results from the application of the individual covariate travel time model with length covariate to cohorts of mid Columbia fall chinook. Note that a negative BIC value lends support to the null model (that is, the model without the length covariate).
cohort #	# of fish	length		parameter estimates			likelihoods
		mean	s.d.	0	1		l0	lA	ratio	BIC
1991
2	97	63.32	4.38	-0.60	0.062	6.86	-393.05	-390.38	5.33	0.76
1992
1	75	71.37	7.16	-5.11	0.125	5.58	-288.71	-264.16	49.11	44.79
4	63	69.00	7.21	-3.64	0.102	4.60	-239.05	-219.76	38.57	34.43
1993
1	61	66.80	6.49	-5.36	0.145	3.99	-222.15	-195.54	53.21	49.10
3	115	66.20	5.46	-4.96	0.134	5.92	-425.60	-404.48	42.24	37.50