LAI ET AL.: AGE DETERMINATION OF PACIFIC COD 



APPENDIX A.— ANALYSIS BY LOG-LINEAR MODEL 



In our study, there are three factors, ageing method (M), age class (A), and re- 

 peatability (R ). Our sampling model is product-multinomial, using the terminology of 

 Fienberg (1981, Sec. 3-2), since the number offish being aged is fixed for each ageing 

 method after deleting unreadable or damaged age-structures. The aged fish were cross- 

 classified into corresponding cells denoted by factors A and R (Table 1). 



Let y,jfi be the observed cell frequency for the j'th ageing method, thejth row of age 

 class, and the /jth column of repeatability, and let m^^ be the expected value ofyijk  The 

 general log-linear model (called the saturated model because it includes the highest 

 three-factor interaction) for our three-way (5x8x2) contingency table is 



Qyk = \0g{m,jk) = JJL + \f^ + \/ + \f + Xj"" + ^fk"" + ^r + ^yk"" (A.l) 



where, as in the usual analysis of variance model, all effects sum up to zero over any 

 subscript. Let be the marginal mean of 8^;^ over the subscript which is replaced by " + " 

 to indicate averaging, then the parameters in (Equation A.l) can be written as 



M- = 6 + + + ^ij = 6y+ ~ 9i++ ~ 9+7+ + 6 + + + 



kf - e,++ - e+++ x^^ - 8,+^ - e,++ - 0++^ + 8+++ 



kf = 0+7+ - 8+ + + kf^ ^ d+jk - Q+J+ -Q++k + Q+++ ( A.2) 



Xf = 0++k - 0+ + + Xjf/« = Q,jk - 8,,+ - 8,+^ + 8,+ + 



~^+jk + 0+J+ + + +* ~ + + + - 



Log-linear models are "hierachical", i.e., higher-order interaction terms can be included 

 only if related lower-order terms are included. For example, X^"*^ is not included unless 

 X^, X^^, and X^'^ are all included. 



Once all expected cell frequencies (m^^) are estimated, the goodness-of-fit for the 

 selected model can be tested using the likelihood ratio test statistic 



G^-2211y..^o,i^) (A.3) 



which has approximately a x^ distribution with degrees of freedom (df) = number of 

 cells - number of parameters (Fienberg 1981, sec. 3-3 and 3-4). 



Using the partition property of G^, we can decide whether an effect or an interaction 

 should be included. In Table 3, for example, Hq: k^^^ = can be tested by examining 

 G2 = 28.08 - 0.00 - 28.08. This is not significant at the 1% level (referred to a x^ 

 distribution with df = 28). Similarly, G^ = 20.79 for Hq: k^^ = 0, which exceeds the 

 upper 1% tail value of a x^ distribution with df = 4, and is rejected. This means that 

 ^MAR ^j]j j^Q^ Yte included in the model but X^"* will. Hence, our best log-linear model 

 is Equation (2). 



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