756 



Fishery Bulletin 88(4), 1990 



Figure 4 



Plot of size at recapture vs. size at tagging for recaptures of Chio- 

 noecetes opilio from Conception Bay. The solid line, which indicates 

 a 17% increase in size while at liberty, seems to separate the data 

 into two clouds of points. Solid symbols represent animals believed 

 to have molted once; open symbols represent animals which molted 

 twice. Bulls-eye symbol represents one animal which was reportedly 

 at liberty for only 29 days and which consequently was e.xcluded from 

 analysis. 



tliat the largest crabs may skip a year between molts. 

 Thus, the information on time at liberty supports the 

 hypothesis that animals in the lower group of f^igure 

 4 molted once while those in the upper group molted 

 twice. 



The data in the lower group in Figure 4 appear to 

 be well described by a linear relationship: 



Recapture size = a + b (size at tagging) + 



(1) 



where e is a random error term. Su[)pose that equa- 

 tion (1) describes the size after one molt. Then the size 

 after two molts would be given by the recursive for- 

 mula (Kurata 19R2): 



Size after two molts = 



n + h ((/ -I- /) (size at tagging)) + e. 



(2) 



Application of equation (2) to the size at tagging should 

 provide a good prediction of the size at recapture for 

 animals in the upper group if the assumption is cor- 

 rect that equation (1) describes the size after one molt. 

 We fitted lines to the two clouds of points in Figin-e 

 4 by ordinary least squares (Table 1). The results are 

 consistent with the hypothesis of one and two molts 

 for the two groups. For example, the fitted e(|uation 

 (I) (with parameters estimated from the lower cloud 



Table 1 



Regressions of length at recapture vs. length at tagging for 

 the two groups of Chionoecetes opilio from Conception Bay 

 shown in Figure 4. Lower cloud animals are presumed to have 

 molted once; upper cloud, twice. Parameter estimates « and 

 b pertain to equations (1) and (2) in the text. 



Attribute 



Lower cloud Upper cloud 



Number of observations 

 Adjusted r- 



Intercept 



(standard error) 

 Slope (standard error) 



Interpretation of 



intercept 

 Interpretation of slojie 



20 

 0.8.5 



17.2.50 (8.93:5) 

 0.941 (0.091) 



a 

 b 



17.250 

 0.941 



72 

 0.92 



12.667 (3.640) 

 1.104 (0.039) 



a(l +b) 

 b- 



6.179 

 1.0.51 



of points) predicts that an animal 80 mm in size will 

 be 92.53 mm after one molt. Inserting this estimate 

 of 92.53 mm into equation (1) gives a predicted size of 

 104.32 mm after another molt. In contrast, a linear 

 regression fitted to the uj)per cloud of points predicts 

 that an animal 80 mm in size will be 100.91 mm after 

 two molts, i.e., 3.41 mm smaller than the estimate from 

 equation (1). Similarly, the size after two molts pre- 

 dicted by equation (1) for a 110-mm CW animal is 

 130.89 mm, whereas the size predicted by the regres- 

 sion for the upper cloud is 134.00 mm, i.e., 3.11 mm 

 larger. Over the range of sizes for which we have data, 

 results fron:i the two equations agree closely. In fact, 

 approximate confidence l)ands for the size after two 

 molts, as determined from the lower regression line, 

 enclose the confidence bands f(.)r the upper regression 

 line over the entire range of the data (Fig. 5; see Ap- 

 pendix A for derivation of approximate confidence 

 hands). We therefore conclude that animals in the lower 

 cloud of points molted once, while those in the upper 

 cloud molted twice. 



The regressions in Table 1 are based on 20 and 72 

 animals. Since the sample sizes are small, it would be 

 useful to use the combined molting data from all 92 

 animals to derive a i)est estimate of the parameters o 

 and /). This can be accomplished by combining equa- 

 tions (1) and (2) into a single equation in the form of 

 a nonlinear regression model. Let Y be the observed 

 size at the time of recovery, X be the size at tagging, 

 and let Z be an indicator variable for whether an animal 

 molted once or twice, i.e.. let 



0, animal molted once 



1, animal molted twice. 



