NOTE Latour et al.: Properties of the residuals from two tag recovery models 



859 



Q 



' T.,-C., 



[ T, ; 







(T,_,-C,,] 



\ ■'/-2 ; 



{Ri-Ci + R2) + Ri 



+ «4 



+ ■■• + /?, 



+ /?, 



Utilize the definition of T^ = R^ + T", | - C, , to systemati- 

 cally simpliCy and cancel starting with the innermost par- 

 enthetic expression: 



Q 



(R,C, 



T, 



■' 2 Q 



+ iTi-Ci) 



^3^3 



^2 ^1 ^2 ^2 





(T -C ' 

 Systematically factor out terms of the form — ' '' 



Q 



[t, 



_V ■'/-I I 



\T,_,-C,_,+R,] 



T,=C,=r„+r.„+-- + r,^ 



Q = \^^ 



c^in- 



+ (T, -Ci) 



c, 



The expression inside the innermost square brackets is 

 equal to 1 (recall Tj = Cj). Hence, we have 



which demonstrates that the column sum of expected 

 values equals the column sum of observed recoveries, as 

 desired for the /"^ column. Similar arguments hold for the 

 other columns. 



Column sums when l>J 



The proof that the column sums of the residuals matrix 

 equal zero when the recovery matrix is nonsquare is simi- 

 lar to the proof above except for making use of the defini- 



Q = 





+ { 



T -c ) = "^'^^ * •^'•^^ ~ ^'^' = R 



+ r,., +  



which shows the sum of the expected values in row 1 is 

 equal to the sum of the observed data. Similar arguments 

 hold for the other rows. 



Row sums when l> J 



Row sums when I = J 



To show that the row sums (excluding the "never seen 

 again" cell) of the residuals matrix equal zero, we must 

 demonstrate the sum of the observed data equals that of the 

 expected values. Consider the sum of the expected values 

 associated with the first row of the recovery matrix: 



Q = £„ + £,., +  + £;„= Af,/, + iV,S/, +  + A'l'S, ■••S,_i)/,. 



Now substitute for ^ and S, on the right hand side: 



Q = N, 



{R,cA^^{R, iT,-C,) N, 



N,TJ 



•••+7V. 



I Ni T, R, ) 



' R, (Ti-Ci) N., 





v^. 



R, 



(R,_,(T,^,-C,^,)N,]](R,C,] 



[N,_ 



T,^ 



«/ JJ 



I N,T, ) 



Cancel and factor out the term T^ - Cj (recall that T^ 



R,y. 



As with the proof of the column sums when / > J, the defin- 

 tion T,^^ = T^^. J - C,^ jis needed to show the row sum of a 

 nonsquare matrix (excluding the "never seen again" cellsj 



are zero. 



"Never seen again" cells 



The likelihood function for the Brownie-type model is a 

 product multinomial and the parameters for each row are 

 constrained to sum to one. Therefore, the expected values 

 in a row are simply an apportionment of the number 

 tagged to the years of recovery and the "never seen again" 

 category. Hence, the sum of the estimated expected values 

 has to equal the row sum (including the "never seen again" 

 cell), which implies the residuals of the "never seen again" 

 cells are always equal to zero. 



Discussion 



The residuals of multiyear tag-recovery models can be very 

 helpful for evaluating model performance. Unfortunately, 

 examining the residuals matrix for patterns is not a com- 

 monly employed procedure for assessing model fit in practi- 

 cal situations. The work by Latour et al. (2001a) was intended 



