856 



Properties of the residuals from 

 two tag-recovery models* 



Robert J. Latour 



John M. Hoenig 



Department of Fisheries Science 

 Virginia Institute of Manne Science 

 College of William and Mary 

 Gloucester Point, Virginia 23062 

 E-mail address (for R, J, Latour): latour@vims edu 



Kenneth H. Pollock 



Biomatfiematics Graduate Program 

 Department of Statistics 

 Nortfi Carolina State University 

 Raleigti, Nonh Carolina 27695 



Researchers often use multiyear tag- 

 recovery studies to assess fish popu- 

 lations: yet deriving useful stock as- 

 sessment parameter estimates from 

 the resulting data can be difficult. The 

 reliability of those parameter esti- 

 mates generally depends on data qual- 

 ity and meeting the assumptions in- 

 herent to the models used for analysis. 

 As a result, practical application of 

 multiyear tag-recovery models gener- 

 ally requires that a large portion of 

 the data analysis involve investigation 

 and evaluation of biases due to poten- 

 tial assumption violation. 



Brownie et al. (1985) developed a 

 class of models that has become widely 

 used for the analysis of multiyear tag- 

 recovery data. These models constitute 

 a generalization of the class of models 

 developed by Seber (1970), which have 

 recently been resurrected as an impor- 

 tant tool for the analysis of multiyear 

 tag-recovery data by the development 

 of the software progi'am MARK I White 

 and Burnham, 1999). Although these 

 models are fairly simple and robust, in 

 practical situations at least one of the 

 assumptions is often not supported by 

 the data. 



Approaches that are commonly used 

 to assess the fit of multiyear tag-recov- 

 ery models include the formal good- 

 ness-of-fit test, Akaike's information 

 criterion (AIC) (Akaike, 1973; Burn- 

 ham and Anderson, 1992; Burnham et 



al., 1995) and other related measures 

 such as quasilikelihood AIC (Akaike, 

 1985). Although these measures are 

 informative about overall model fit, 

 they do not provide any information 

 about why a model fit is poor or which 

 assumption(s) is (are) possibly in viola- 

 tion. To remedy this problem, Latour et 

 al. (2001a) conducted a series of simu- 

 lations and demonstrated that distinct 

 patterns in model residuals will be 

 evident if particular assumptions are 

 violated. They discussed in detail the 

 residuals associated with the time- 

 specific parameterizations of the Se- 

 ber (1970) and Brownie et al. (1985) 

 models, as well as the time-specific 

 instantaneous rates model developed 

 by Hoenig etal.( 1998). 



The genesis of the work by Latour 

 et al. (2001a) can be traced to two par- 

 ticular applications of multiyear tag- 

 recovery models. Specifically, Latour et 

 al. (2001b) analyzed tag- recovery data 

 of red drum iSciaenops ocellatits) in 

 South Carolina and found systematic 

 patterns along the diagonals in the 

 upper right corner of the residuals 

 matrix. Frusher and Hoenig (2001) ap- 

 plied a series of tag-recovery models to 

 Australian rock lobster {Jasus edward- 

 sii) data and found consistent patterns 

 in the columns of the residuals matrix. 

 In both instances, the researchers 

 could only speculate as to the cause of 

 these patterns in residuals. Although 



the simulations conducted by Latour 

 et al. (2001a) have since provided 

 reasonable explanations for the ob- 

 served patterns, the development of 

 those diagnostic procedures led to the 

 discovery that the residuals associated 

 with the time-specific Seber ( 1970) and 

 Brownie et al. (1985) models are sub- 

 ject to several constraints. 



This note contains a series of simple 

 mathematical arguments that verify 

 the assertions made by Latour et al. 

 (2001a) about the residuals of the 

 time-specific parameterizations of the 

 Seber (1970) and Brownie et al. (1985) 

 models. Unfortunately, the constraints 

 inherent to the residuals of those 

 models partially cloud a researcher's 

 ability to assess the existence of a pat- 

 tern. As such, knowledge of the inher- 

 ent properties of the residuals of these 

 models is of particular importance, 

 especially because the time-specific 

 parameterizations are commonly used 

 for the analysis of tag-recovery data. 



Materials and methods 



Multiyear tag-recovery models 



Multiyear tagging data are generally 

 represented by an upper triangular 

 matrix of tag recoveries. For example, 

 the matrix for a study with / years of 

 tagging and J years of tag-recovery 

 would be, when / = J. 



'll'l9- 



(1) 



where r = the number of tags recov- 

 ered in year j that were 

 released in year ; (note, / = 

 1,... J\j=i J). 



Application of multiyear tag-recov- 

 ery models generally involves con- 

 structing a matrix of expected values 

 and comparing them to the observed 



* Contribution 2490 of the Virginia Insti- 

 tute of Marine Science, Gloucester Point, 

 VA 23062. 



Manuscript accepted 10 July 2002. 

 Fish. Bull. 100:856-860 (2002). 



