TABLE 3.— Midpoint (h) and duration (h) of stage of walleye pollock embryos for Stages I- VII at experimental temperatures 2° 



6°, 8°, and 1 1 C. See Table 1 for description of stages. 



2°c 



5°C 



6°C 



8°c 



n°c 



FIGURE 1. — Age-stage relation of walleye pollock embryos at 2°, 5°, 

 6°, 8°, and 11°C. Solid lines are regression lines of embryo age on 

 developmental stage with experimental data adjusted for pre- 

 experimental time and temperature; x's represent experimental 

 data. 



and the data were fitted to a third- degree polynomial 

 model using stage as the predictor variable (Table 4) . 

 The regression equations in Table 4 were then used 

 to derive the temperature- dependent coefficients, 

 a, ,(7), . . . , a 3 (7), (Table 5), for the generalized pre- 

 dictive equation for embryo age, 



y = a (T) + ai (T)X + a., (TjX 2 + a^T)^ (1) 



where y = age (in hours), X = stage, and T = water 

 temperature (°C). Equations for estimating the tem- 

 perature coefficients are least square fits of the 

 regression coefficients in Table 4 to a second degree 

 polynomial model of temperature (Table 5). Over the 

 temperature range of 2°-ll°C, the standard error 

 (obtained by comparing predicted with observed 

 values) was normally distributed and equaled 3.03 h. 

 The temperature- dependent coefficients in our 

 predictive model represent the rate (units: hours/ 

 stage) of embryogenesis in walleye pollock and are 

 only meaningful when used in context with the 

 embryo- staging classification in Table 1. 



Table 4. — Regression equations for estimating age (in hours) of 

 walleye pollock embryos at constant temperatures of 2°, 5°, 6°, 

 8°, and 1 1°C, where /= predicted age andX= Stages I- VII. See 

 Table 1 for description of stages. 



Temperature 



(°C) Regression equations 



2 

 5 

 6 

 8 



11 



y = 50.0- 73.4365* + 28 8571 X 2 - 1.2778X 3 

 y = 16.2857 - 27.5436X + 1 2.351 2X 2 -0.3194X 3 

 y= 14.5714-25.9524X+ 12.3036X 2 -0.4583X 3 

 y = 6.0286 - 1 2.3960X + 6.3286X 2 - 0.061 1 1 X 3 

 y= 0.6000- 5.01 11X + 3.91 19X 2 -0.0056X 3 



Table 5 . — The temperature- dependent co- 

 efficients, a ....a 3 , estimated from the least- 

 square fits of the regression coefficients in 

 Table 4. 7"= mean incubation temperature 

 (°C). See Table 1 for description of 

 stages. 



To verify that the curves in Figure 1 were different 

 for each temperature, we statistically compared the 

 curves for embryonic development at the two closest 

 temperatures, 5° and 6°C. Cumulative measure- 

 ments of development are dependent and violate 

 assumptions underlying usual statistical com- 

 parisons among treatment groups. To avoid this prob- 

 lem of dependency, we transformed the development 

 curves of embryonic development into time in- 

 crements between midpoints of adjacent stages. 

 Time increments are less correlated than cumulative 

 measurements yet contain the same information 

 (Box 1950). The time increments were then used as 

 the dependent variable in the analysis of variance. 

 The analysis of variance showed a significant tem- 

 perature effect {P< 0.01) and significant interaction 

 (P<0.05), which indicates a real difference in 

 development rates at 5° and 6°C. 



To further substantiate that the generalized predic- 

 tive Equation (1) is valid for estimating age of walleye 

 pollock embryos at any stage over the temperature 

 range of 2°-l 1°C, we regressed age (In) as a function 



892 



