Francis et al.: Quantifying annual vanation in catchability 



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K = 



2(;i-/i„5)/(/i„5-/i„0,5) if/i</i„5 



2(/J - /J„,5 ) / (//o,„5 - jUo.5 ) if /^ > ;io.5 ' 



where fj = the MASK from the assessment data; and 



^^ = the rth quantile of the samphng distribution 

 of/i. 



To calculate the n^ we assumed that the standardized resid- 

 uals follow a Student's ^distribution with n-2 degrees of 

 freedom, where n is the number of indices in the data set. 

 [We assumed n-2 degrees of freedom because, in an assess- 

 ment with only a single series of relative biomass indices, 

 only two parameters can be estimated, e.g. initial biomass 

 and q (Francis, 1992). When there are many data inputs 

 there may be many more than two parameters estimated.) 

 For each value of the sample size n, the jj^ were estimated 

 by simulating 1000 data sets of size n from a /-distribution 

 with n-2 degrees of freedom, calculating the median abso- 

 lute value for each simulated data set, and taking the rth 

 quantile of this set of 1000 medians. 



We used the v statistics in two ways. We tested the null 

 hypothesis that the CVs were, on average, of the correct 

 size by using a simple signs test (under this hypothesis 

 we would expect about 50% of the fc's to be of each sign). 

 If significantly more than half are positive (or negative) 

 this shows a tendency to use CVs that are too large (or 

 too small). This test considers all the CVs at once. We also 

 tested each CV separately; a value of k' greater than 2 (less 

 than -2) is statistically significant. 



Next, we investigated how much, if at all, we should 

 change the assumed CVs to make their size appropriate. 

 This was done by changing the assumed CVs, recalculat- 

 ing the residual statistic and checking to see whether the 

 new values of k" were evenly distributed about zero. We did 

 this separately for the CPUE and trawl survey indices. For 

 the former we simply set them all to a single default value 

 and searched for the default value that produced an even 

 distribution. For the latter, we assumed that the CV asso- 

 ciated with annual variation in catchability was the same 

 for all stocks and "added" this CV to the observation error 

 CVs to obtain assumed CVs for the stock assessments. Note 

 that CVs are "added" as squares, so that when we "add" 

 CVs of 0.2 and 0.3 we get 0.36 [= (0.22+0.32)05]. Here we 

 were searching for the value of the catchability CV that 

 produced an even distribution of k. 



Strictly speaking we should rerun each assessment each 

 time we change a CV. However, it is not practical to do this 

 for so many assessments. Thus we have to assume that 

 changing a CV will not change the model estimates too 

 much. Our experience is that this is true for assessments 

 with only one series of biomass indices. It is least likely 

 when there are more than one series and these show mark- 

 edly different trends. 



Can we detect years of extreme trawl survey 

 catchability? 



First, as an informal procedure to identify possible years of 

 extreme trawl survey catchability, the trawl survey biomass 



indices were standardized (by dividing each time series for a 

 particular species by its mean) and plotted by survey. Next, 

 the following more formal procedure was used to identify 

 extreme years. For each species in a trawl survey data set, 

 the survey years were ranked in order of increasing biomass 

 index, and then these ranks were averaged across species to 

 obtain a mean rank for each year. Then the rank deviations, 

 d = I r^ - 0.5(m+1) I , were calculated, where r, is the mean 

 rank for year I , n = the number of survey years, and 0.5(n+l) 

 is the overall mean of the mean ranks (Table 2). 



The following simulation procedure was used, for each 

 series, to determine which years should be labeled as ex- 

 treme (i.e. how large the d/s need to be to be statistically 

 significant). 



1 The actual biomass indices were replaced by ran- 

 domly generated indices (by using a uniform distri- 

 bution [because our statistic is based on ranks it does 

 not matter what distribution is used to generate the 

 biomass indices] ); 



2 Mean ranks, and rank deviations, were calculated for 

 each survey year by using these simulated biomass 

 indices; 



3 The largest of these rank deviations, d^^^ p was 

 stored; 



4 Steps 1 to 3 were repeated 999 times, generating 

 ^.ax..fo'-7 = 2,...,1000; 



5 Year i was labeled as extreme if d^ was greater than 

 or equal to at least 95% of the d^^^^. 



In other words we asked, for each rank deviation d^, how 

 likely we would be to observe a deviation at least as large 

 as this if there were no between-species correlations. If the 

 probability were less than or equal to 0.05, we would label 

 the year as extreme. 



As a diagnostic tool, to examine possible reasons for 

 these extreme years, we calculated between-year changes 



