This F ratio has (s-1) and (n-p) degrees of freedom where n is the 

 number of observations used to build the time series model and p is the 

 number of terms estimated by the model. 



To determine if the relative proportions of selected important 

 shore-zone species differed in the current year from previous years, an 

 expansion of the Krumbein and Tukey (1956) analysis of proportions was 

 applied to the relative proportions of the five most abundant shore-zone 

 taxonomic groups. A similar approach was used by Briggs and O'Connor 

 (1971) . The analysis consisted of using the arcsine square root trans- 

 formation of the relative proportions and distributing the sums of 

 squares as in the following analysis of variance model: 



X = X + P + Y + S k + T + (TY) + (TS) + (13) 

 Rl + (RP)ii + (RYl ±jl + (RS) ijkl J + (TR) lm J + 



(TRP),, + (TRY").., + (TRS) ' 



klm ij lm ij klm 



where the model components are defined in Table 2. 



There is no error term in this model because proportion data are 

 multinomial. The theoretical variance (BV) is approximately 821/n where 

 n is the average number of individuals in each sample (Bartlett 1947). 

 The total sum of squares was apportioned following Krumbein and Tukey 

 (1956) and the expected mean squares determined. Because changes in the 

 relationship among taxa were of interest, F values for significance 

 tests were calculated only for terms that included taxon crossed with 

 some other factor. The F ratios for testing the significance of the 

 taxon x period and taxon x year interactions were "manufactured" by 

 taking the ratio of the sum of two mean squares. The appropriate degrees 

 of freedom were calculated using the method given by Snedecor and Cochran 

 (1967). Table 3 summarized the effects of interest and composition of 



46 



