NOTE Ldtour et al : Properties of the residuals lioni two tag-recovery models 



857 



data. The matrix of expected values corrospondinK to the 

 time-specific parameterization of Brownie et al. (1985), 

 wliicii is referred to as model 1, takes tlie form 



£, 



- N,f, ■■■N,{S.ySj_,)f:, 



- - - N,f,, 



(2) 



where A^, = the number tagged in year (; 



f\ = the tag recovery rate in year /; and 

 S^ = the survival rate in year /. 



As stated above, the Brownie et al. (1985) models 

 constitute a generalization of those developed by Seber 

 (1970). The only difference lies in the definition of the tag 

 recovery rate. Specifically, Seber (1970) modeled the tag 

 recovery rate in year i as /j = ( 1 - S, )r,, where r, is the rate 

 at which tags are reported from killed fish in year ; re- 

 gardless of the source of mortality. The matrix of expected 

 values associated with time-specific parameterization of 

 the Seber (1970) models, which we will refer to as model 

 1 , takes the form (when 1 = J) 



E = 



N,{l-S,)r, N,St(l-S.,)r.,  iV,(S,-S;_,l(l-S., )o 

 N.^{l-S.,)r.^ ■■■N.,{S.ySj_j)a-Sj)r, 



(3) 



The data in each row of Equation 1 follow a multinomial 

 distribution and maximum likelihood estimation can be 

 used to derive parameter estimates from either model 1 

 or model 1 . Program MARK has emerged as the leading 

 software package for deriving these estimates (White and 

 Burnham, 1999). 



Patterns in residuals 



Latour et al. (2001a) manipulated a hypothetical perfect 

 data set (i.e. the observed number of tag recoveries was 

 equal to the expected number of tag recoveries) to simulate 

 four specific forms of assumption violation for niultiyear 

 tag-recovery models. For each scenario, they analyzed the 

 modified data with model 1, model 1', and a time-specific 

 parameterization of the instantaneous rates (IR) models 

 (Hoenig et al., 1998) and noted any patterns in the residu- 

 als matrix that resulted from each particular assumption 

 violation. Specifically, they found the following: 1) the 

 presence of nonmixing (which violates the assumption 

 that the tagged population is representative of the target 

 population) leads to consistent patterns on the main and 

 super diagonals of the residuals matrix (the main diago- 

 nal contains the ( 1,1),(2,2),...,(/J) cells and the first super 

 diagonal contains the (1,2), (2, 3), ...(/-I,/) cells in a square 

 matrix); 2) permanent emigration from the study area of 

 individuals within a tagged cohort (which violates the 

 assumption that all tagged fish within a cohort are subject 

 to the same annual survival and tag- recovery rates) leads 



to a pattern of negative residuals along the diagonals of 

 the upper right corner of the residuals matrix; 3) tag- 

 induced mortality or immediate loss of tags due to poor 

 tagging (which violates the assumptions that tags are not 

 lost and survival rates are not affected by tagging) leads 

 to row patterns in the residuals matrix (note that these 

 patterns are detectable only in the residuals matrix of the 

 IR model); and 4) a change in the natural mortality rate 

 (which violates the frequently imposed assumption that 

 natural mortality is constant over time) leads to column 

 patterns in the residuals matrix (again, this only applies 

 to the IR model). 



Constraints on residuals of model 1 and model 1' 



Latour et al. (2001a) asserted without proof that the 

 residuals associated with model 1 and model 1 are sub- 

 ject to several constraints. Specifically, they stated that 

 the relationship E/j = r'u always holds, regardless of the 

 number of years of tagging and tag-recovery (note that £„ 

 is the expected number of tags recovered in year j that 

 were released in year /). This implies that the observed 

 data and the expected value associated with the (1,1) cell 

 are always identical and that the residual for that cell is 

 always equal to zero. They also stated that the residuals 

 associated with the implicit "never seen again" category 

 are also always equal to zero (recall that under a multino- 

 mial formulation, one of the possible outcomes is to never 

 recapture a tagged fish). Collectively, these constraints 

 imply that the residuals matrix derived from using model 

 1 or model 1' to analyze data from a study with / years of 

 tagging and J years of tag-recovery takes the form 



resid -■ 



0.00 {)\2-Ey2)  ( >\., ' Eij ) 0.00 



- ( 7-22 - E.,2 )  •  ( r.,j - E.,j ) 0.00 



; ; ■. ; o.oo 



(r,j-E,j) 0.00 



(4) 



where r,^ and £,^ are as defined previously and the last 

 column of the matrix represents the residuals associated 

 with the "never seen again" category. 



In addition to the aforementioned zero residuals, Latour 

 et al. (2001a) stated that the sum of each row and each 

 column of the residuals matrix must equal zero and that 

 for the case when / = J (i.e. the recovery matrix is square), 

 the constraint that Ej, = r,, is also present (i.e. the residual 

 associated with the (/, /) cell is always equal to zero). 



In the context of searching for patterns in residuals, 

 these constraints have the following implications. First, 

 the presence of residuals that are constrained to be zero 

 essentially reduces the total number of values that are 

 available for inspection and ultimately forces conclusions 

 about the existence of a pattern to be based on the signs 

 of fewer residuals. For short-term tagging studies (e.g. 

 3-4 years), the loss of residuals for inspection makes it 

 extremely difficult to evaluate model performance because 

 each row, column, and diagonal of the residuals matrix al- 

 ready contains only a few values. Second, because the sum 

 of each row of the residuals matrix must total zero, and 



