280 



Fishery Bulletin 101(2) 



Appendix 



Modeling abundance based on presence absence 

 data and effective sample size 



The negative log-likelihood used to measure goodness of fit 

 for CalCOFI and LAOCSD data in the cowcod assessment 

 model was 



L = -A^[/, ln(/, ) + (!-/, )ln(l-/,)]- A 



I 



where A is a constant; 



A'^ = the number of years with data; A was the effec- 

 tive sample size (tows/yr); and observed /, and 

 predicted /, index values are proportions. 



The constant A has no effect on model estimates but 

 makes the log-likelihood easier to interpret, plot, and un- 

 derstand. Following Methot (1990), it was calculated with 

 the following equation: 



A = ^£),[/, ln(/,)-Kl-/,)ln(l-/,)], 



where the dummy variable D, was one if < /, < 1 and zero 

 otherwise. 



The constant depends only on the data (not the fit) and 

 is the minimum possible log likelihood (if observed and 

 predicted values match exactly). 



Effective sample size calculations were based on the 

 variance of residuals in preliminary model runs (Methot, 

 1990). This manual "iterative re-weighting" approach was 

 repeated several times until assumed and calculated vari- 

 ances were roughly equal. The expected variance of an 

 index value based on standard formulas for proportions 

 and n tows is 



Var(p) = 



p(l-p) 



so that 



p(l-p) 

 VaHp) ' 



with A instead of n for the effective sample size. The vari- 

 ance Var(p) of residuals in one year was calculated by using 

 another standard formula: 



Varip) 



(p-p)-+[(l-p)-(l-p) 



-2(p-pf 



To estimate an effective sample size for the time series as 

 a whole, we calculated the geometric mean of the effective 

 sample sizes for each year 



