5.4. Predicting model parameters and travel times

My procedure for applying the regression models is as follows. First, using migration rate (r) as a response variable, I construct a sequence of regression equations. The first four are a sequence that increases in complexity; the last two result from dropping a coefficient from the most complex of the first four equations. I then apply these regression equations to the estimated migration rates on a yearly basis. For both of the data sets, there is variability in the number of cohorts for each year, and I chose to analyze years that have at least 20 cohorts. For the Snake River groups, 1989 and 1993 have 20 or more cohorts; for the Clearwater Trap groups, 1991 and 1992 have 20 or more cohorts.

regression equations for migration rate

Migration rate (r) will be predicted using the following six regression models:

model 1) The null model assumes that r is unaffected by the two factors and has average value ₀:

. (5.14)

Variation about the average rate is expressed by _i.

model 2) This model assumes a linear relationship between migration rate and flow:

. (5.15)

River velocity is assumed to be proportional to river flow. The intercept term (₀) is a combination of directed movement independent of flow and a potential non-zero intercept from the river velocity/river flow relationship.

model 3) The linear flow and date model assumes that fish migrate more actively later in the season, by migrating in the higher flow regions of the river and/or by spending a greater proportion of the day in the river flow versus holding up along the shore The model assumes a linear increase in migration tendency with date as expressed by the coefficient ₂:

. (5.16)

model 4) A more realistic model of migration tendency would have fish migrating at a minimum rate early in the season and reach a a maximum rate later in the season. Although a number of models can produce this behavior, I have chosen to use

. (5.17)

The term in the brackets is the CDF of the logistic distribution. Early in the season fish migrate at a rate of r_min, and later migrate at a threshold rate of r_min + r_max. T₀ determines when the migrate changes from low to high, and determines the rate of this change. A sample plot of equation (5.17) is provided in figure 5.7. Thus, the regression equation can be formulated as a regression model

. (5.18)

model 5) This model eliminates ₁ from model 4:

. (5.19)

model 6) This model is created by removing ₀ from model 4:

. (5.20)

regression equations for

To predict values of I used the following two models.

model 1) The null hypothesis assumes that is a constant plus error:

. (5.21)

model 2) This assumes that the "rate of spreading" is linearly related to migration rate. In this formulation, I use determined from the previous regressions, which is based solely on river flow and date of release. The equation is:

. (5.22)

I will select one of the six regression equations to determine in this equation.

Least-squares regression was used to fit each of the six models for r and two models for to the data from individual years. For all regressions, the parameter estimates and standard errors, deviance, and coefficient of multiple determination are reported. Since the residuals are not necessarily normally distributed, I will not conduct F-tests for levels of significance.

results of the regression analyses

The results of this regression analysis for migration rate of the Snake River chinook are contained in Table 5.8 and for the Clearwater Trap chinook in Table 5.9.

Table 5.8 Regression results for the Snake River spring chinook. For models 4, 5, and 6 1 = rmin, and 2 = rmax.
model	parameter estimates (standard error)					resid. ss	mult. R2
	0	1	2		T0
1989 n = 23
model 1	6.90 (0.46)	-	-	-	-	109.30	-
model 2	-13.69 (4.39)	0.22 (0.046)	-	-	-	53.17	0.514
model 3	-5.51 (2.60)	-0.085 (0.048)	0.0020 (0.00027)	-	-	14.50	0.867
model 4	-4.86 (2.94)	0.052 (0.16)	0.11 (0.15)	0.11 (0.12)	101.6 (21.0)	10.95	0.900
model 5	-4.49 (2.29)	-	0.16 (0.029)	0.089 (0.030)	95.0 (4.1)	10.99	0.900
model 6	-	-4.07 (8342.00)	4.23 (8346.00)	0.015 (1.26)	-143.6 (152400.0)	26.84	0.754
1993 n = 25
model 1	7.91 (0.62)	-	-	-	-	231.10	-
model 2	-3.50 (1.89)	0.14 (0.023)	-	-	-	86.95	0.624
model 3	11.26 (5.10)	-0.56 (0.23)	0.0044 (0.0014)	-	-	61.08	0.736
model 4	21.80 (14.54)	-0.28 (0.23)	0.18 (0.10)	0.34 (0.11)	116.3 (1.9)	23.08	0.900
model 5	3.89 (0.55)	-	0.069 (0.0071)	0.50 (0.29)	112.3 (1.5)	33.60	0.855
model 6	-	0.057 (0.011)	0.052 (0.012)	0.58 (0.69)	110.3 (2.4)	44.09	0.809

Table 5.9 Regression results for the Clearwater Trap spring chinook. For models 4, 5, and 6 1 = rmin, and 2 = rmax.
model	parameter estimates (standard error)					resid. ss	mult. R2
	0	1	2		T0
1991 n = 25
model 1	3.92 (0.32)	-	-	-	-	60.54	-
model 2	-8.06 (1.37)	0.19 (0.021)	-	-	-	13.83	0.772
model 3	6.49 (2.01)	-0.30 (0.064)	0.0024 (0.00031)	-	-	3.70	0.939
model 4	3.55 (4.81)	-0.026 (0.11)	0.078 (0.065)	0.14 (0.15)	112.2 (1.8)	3.28	0.946
model 5	2.26 (0.30)	-	0.065 (0.0067)	0.18 (0.048)	112.0 (1.3)	3.32	0.945
model 6	-	0.042 (0.0037)	0.048 (0.0054)	0.25 (0.077)	111.5 (1.1)	3.55	0.941
1992 n=35
model 1	4.14 (0.32)	-	-	-	-	120.00	-
model 2	1.02 (1.16)	0.067 0.024	-	-	-	97.31	0.189
model 3	5.21 (0.46)	-0.28 (0.023)	0.0023 (0.00014)	-	-	10.34	0.914
model 4	4.02 (3.54)	-2.15 (600.95)	2.33 (601.00)	0.0080 (0.41)	-200.9 (50000.3)	10.64	0.911
model 5	2.84 (0.20)	-	0.13 (0.027)	0.10 (0.03)	131.7 (5.6)	10.64	0.911
model 6	-	0.072 (0.0031)	0.096 (0.011)	0.26 (0.12)	133.8 (3.6)	15.21	0.873

These results show that some of the variability in migration rate can be related to the factors river flow and date of release. For all 4 years of data analyzed, the multiple R² values are greater than .736 for model 3 through 6. The linear equation (model 3) works well; in three of the four cases, its R² values are close to those of the nonlinear models. Although model 4 yields consistently high R² values, the standard errors are high, diminishing its predictive capabilities, and in one case (Clearwater trap, 1992) the regression results are unrealistic. Model 5 offers an improvement over model 4. The R² values are the same as or close to those of model 4, and the standard errors are small. Model 6 does not work as well as model 5, and in one case (Snake trap, 1993), it yields unrealistic results. Models 3 and 5 are the best candidates for predicting migration rates. The advantage of model 3 is that is has one fewer parameter, but the threshold time relationship contained in model 5 might be more realistic and can more reasonably handle dates outside those observed in this analysis. A plot of regression model 5 for r is contained in Figure 5.8. Note that if Julian date is held constant, there is a linear relationship between r and date. Also, if flow is held constant, the nonlinear relationship between r and date is apparent.

Based on the results of the previous regressions, I used model 5 to determine for the regressions. The results of these regressions are contained in Table 5.10. In all four cases there is a positive linear relationship between and (R² = .589 - .845).

Table 5.10 Regression results using estimates of as the response variable.
model	parameter estimates (stand. error)		resid. ss	mult. R2
	0	1
Snake trap 1989
model 1	9.23 (0.48)	-	115.60	-
model 2	3.03 (0.95)	0.90 (0.13)	36.12	0.688
Snake trap 1993
model 1	7.33 (0.38)	-	87.75	-
model 2	3.29 (0.75)	0.51 (0.089)	36.06	0.589
Clearwater trap 1991
model 1	6.27 (0.35)	-	73.38	-
model 2	2.67 (0.58)	0.92 (0.14)	25.16	0.657
Clearwater trap 1992
model 1	7.17 (0.51)	-	307.00	-
model 2	0.80 (0.52)	1.54 (0.12)	47.70	0.845

predicting travel times

The goal of the regression analysis is to determine model parameters based on predicting factors. These in turn will be used to predict the arrival distribution of fish at a downstream site based on the passage distribution at an upstream site. In this section, I demonstrate this procedure by using the results from regressions to try to predict the arrival times of the 1993 Snake trap chinook cohorts and the 1992 Clearwater trap chinook (since the 1993 sample size is small) at Lower Granite Dam. In this analysis, I apply the two parameter travel time model with parameters predicted for each cohort to determine predicted arrival time distributions at Lower Granite Dam. I then pool together the predicted arrival distributions for the cohorts to yield an arrival distribution for all the fish through the year. This distribution is compared to the data and the sum of the squared deviations is reported.

For comparison purposes, I use three approaches to determine model parameters. In the first approach, I pool together the cohorts from the four other years, apply regression model 5 for r and model 2 for , and determine regression coefficients for these data. The regression coefficients along with the covariates date of release and river flow are then used to determine model parameters for the fifth year's cohorts. This approach uses independent data from four years to predict arrival time distributions for the fifth and is the standard method for using the travel time model predictively.

In the second approach, instead of using independent data to determine regression coefficients, I use the "in-year" regression coefficients to predict model parameters. In other words, I take the regression coefficients from the 1993 analysis (reported in Table 5.8) and use the 1993 covariates to determine model parameters for the 1993 cohorts. Again, I use model 5 for r and model 2 for . This is a bit circular but represents the best that these regression equations can do if we have perfect knowledge of the regression coefficients.

The third approach uses the maximum likelihood estimates (mle's) of the model parameters for each of the cohorts. This represents the best that travel the model can do to predict arrival times if we have perfect knowledge of the individual cohorts.

The regression results for the four years pooled data are contained in Table 5.11. These are the coefficients to be applied to the fifth year's data with the first method outlined above. Even with the pooled data, the regressions for r are reasonable (R² = .681 and .840). For , the Clearwater trap regressions had a very low R² value, indicating that this model offers little improvement over the null model of constant .

Table 5.11 Results from the application of regression model 5 for r and model 2 for to the Snake trap and Clearwater trap spring chinook four year composite data.
model	parameter estimates (standard error)					resid. ss	mult. R2
	0	1	2		T0
Snake trap chinook 1989-1992
model 5 for r	1.53 (0.69)	-	0.099 (0.019)	0.099 (0.036)	105.8 (3.9)	81.20	0.681
model 2 for	2.50 (0.31)	0.95 (0.12)	-	-	-	145.90	0.519
Clearwater trap chinook 1989-1991, 1993
model 5 for r	2.61 (0.39)	-	0.14 (0.030)	0.096 (0.028)	129.3 (4.8)	61.97	0.840
model 2 for	6.04 (0.54)	0.21 (0.11)	-	-	-	222.6	0.060

Plots of the predicted arrival distributions and the actual observations are shown in Figure 5.9 (Snake trap) and Figure 5.10 (Clearwater trap). For the Snake trap spring chinook, the predicted arrival distribution based on independent data (top plot) captures the general shape of the data but misses some of the details. The predicted arrival distribution based on the "in-year" regression (middle plot) reduces the sum-of-squares by 18 per cent and begins to capture some of the bimodality of the data. The predicted curve based on the mle's is sharply two-peaked (bottom plot) and confers an additional 46 per cent reduction in the sum-of-squares, a substantial improvement over the previous two. It is interesting to note that while the "in-year" regression for migration rate had an R² of .855 and .589 for , using the actual mle's reduced the sum-of-squares of the arrival distribution by over 50 per cent (that is, comparing the middle and bottom plots).

For the Clearwater trap fish (Figure 5.10), the "in-year" regression based arrival distribution reduces the sum-of-squares by 10 per cent, and the mle based distribution by an additional 15 per cent over the arrival distribution based on independent data. In this case, the arrival distribution based on independent data performs well when compared to the "in-year" arrival distribution.

For both these two data sets, the arrival distribution based on independent data captures the general shape of the observations. While comparisons to the plots based on "in-year" regressions and on mle's indicate that improvements could be made, these improvements may not necessarily enhance the utility of the model. The types of management actions based on these plots (such as increased spill or augmented flows) would probably not be tuned to fine scale variability but would be based on the gross features captured by the top plots.