734 



Fishery Bulletin 88(4), 1990 



sets of 2x2 contingency tables were analyzed using 

 the X- statistic, which is approximately distril)uted as 

 the chi-squared distribution, and which is equivalent 

 to the G-statistic for 2 x 2 contingency tables (Sokal and 

 Rohlf 1981). Only two-way interactions can be iden- 

 tified with this procedure, so factors tested were vital- 

 ity vs. injuries, vitality vs. mortality, and injuries vs. 

 mortality. The null hypothesis tested was that effects 

 of both factors were independent of each other, i.e., 

 that no interactions occurred. 



Loglinear analysis was accomplished to determine 

 the odds of moi'tality associated with vitality level and 

 injury presence. The SPSS LOGLINEAR (SPSS Inc. 

 1986) procedure was used to fit a logit model, in which 

 one dichotomous factor— delayed mottality (DMORT)— 

 was designated as a dependent variable, and the re- 

 maining factors— vitality (VIT) and injuries (INJ)— 

 were the independent variables. Only live crabs were 

 used, i.e., those with vitality codes of 1 or 2; injury 

 codes were 1 (uninjured) or 2 (injured), and DMORT 

 was coded as 1 (survived) or 2 (died). The data fit by 

 the model were the log-transformed survival odds (ratio 

 of survivors to mortalities), often referred to as logits, 

 in each category of the independent variables (vitality 

 and injuries): 



Logits = In(ODDS) = ln(f,/f,„) 



(4) 



where 4 and f,„ are the observed frequencies of sur- 

 vivors and mortalities, respectively. The logit model 

 used for fitting entails expressing the logits as a linear 

 model of several parameters: 



ln(tyf,„) = 2(l:, + l„.v + l„.|) (5) 



where f^, f,,, = observed number of survivors or 

 mortalities. 

 Id = parameter estimate for DMORT 

 (mean odds of mortality), 

 l[vv = parameter for interaction of 



DMORT and vitality, 



li,-i = parameter for interaction of 



DMORT and injuries. 



The proportion of crabs in any cell of the model which 

 survive is related to the survival odds by the formula: 



S = (1-fODDS-I)- 



(6) 



This model assumes that a mean level of mortality 

 would occur regardless of other factors, and that each 

 confounding factor increases mortality in a logarithmic 

 fashion, or that the overall odds of survival are the sum 

 of the logs of the parameter estimates for each con- 

 tributing factor. In this model, there are two alterna- 



five outcomes for each category, which have equal and 

 opposite parameter values which must sum to 0. For 

 instance, parameter 1 represents the alternatives of 

 survival vs. mortality; parameter 2 represents the 

 positive contribution of a 'good' vitality code of 1 vs. 

 the negative influence of a 'poor' vitality code of 2; 

 parameter 3 is similar to parameter 2 with regard to 

 the influence of the presence vs. absence of injuries. 

 Since the model represents the log of a ratio, the ef- 

 fect attributable to any single factor is calculated as 

 the difference between parameter estimates for alter- 

 native outcomes, e.g., the total effect of DMORT is 

 Id - ( -'u). <"' 2(lr))- Thus the entire equation is multi- 

 plied by a factor of 2. 



Each cell of the contingency table was coded as 1 or 

 2 for each factor. Crabs which were active, uninjured, 

 and survived, for instance, were coded as 1,1,1. 

 Similarly, crabs which were inactive, injured, and died 

 were coded 2,2,2. The parameter value for the effect 

 of vitality on mortality (Id-v in the above model) was 

 calculated for cases with equal codes for DMORT and 

 VIT (i.e., both e(jual to 1 or both equal to 2); the param- 

 eter estimate for cases with unequal codes was equal 

 but opposite in sign. Individual parameters were con- 

 verted to odds by calculating the antilog. Odds for any 

 combination of categories is equivalent to the product 

 of antilogs of parameters (or the sum of all parameters 

 times 2) for all factors in the model. 



The interaction of immediate mortality (IMORT) with 

 body injuries (BODY) and leg injuries (LEGS) was in- 

 vestigated in an identical manner. Values of the vari- 

 ables BODY and LEGS were set to 1 if nn injury was 

 present, or 2 if an injury was present. 



Contingency table analysis using the X- statistic 

 was performed to test the null hypothesis that imme- 

 diate mortality was independent of three types of leg 

 injury, defined as autotomization, injuries distal to the 

 autotomy or breakage plane, and injuries proximal to 

 the breakage plane. The X- statistic was also used to 

 test the independence of immediate and delayed mor- 

 tality from shell condition, and the number of legs with 

 injuries. 



