Hanrahan and Juanes: Estimating the school size of Thunnus thynnus thynnus 
425 
form a larger group. While schools were remixing during 
such encounters, portions of one group would sometimes 
join another, maintaining the same number of indepen- 
dently acting groups, but changing the number of individ- 
uals within each group. 
Particularly in larger schools, individual fish positions 
were observed to be dynamic, yet the overall shape of 
the school remained relatively constant. The mean size of 
schools observed was 18.88 individuals (n= 74, SD=13.90). 
The smallest schools observed had 2 ( /? =3 ) individuals, 
and the largest had 45 (n=l). The frequency distribution of 
school sizes was not normal (Shapiro- Wilks’ W, P<0.0001) 
and had two prominent modes centered at 5-10 individu- 
als and 35-40 individuals (Fig. 4). 
No statistically significant relationships between envi- 
ronmental variables and N s were observed (Table 1). How- 
ever, low power due to small sample sizes may have re- 
duced our ability to detect significant effects. 
Predicting NFS from surface counts 
The relationship between the number of individuals in 
each depth interval and school size was linear in all 
cases. Although the r 2 values for each N i -N s regression 
were relatively low, their slopes appeared to be similar 
and were within a narrow range (0.14-0.23). However, 
ANCOVA revealed that the slopes were significantly dif- 
ferent (P<0.001). Although the slopes were significantly 
different, the number of individuals in each interval 
remained in the same proportion except at low school sizes 
( < 15 individuals). 
The regression model incorporating the number of fish 
in the surface interval of the school as the independent 
variable and the number of individuals in the remaining 
portion of the school as the dependent variable had an r 2 
of 0.67 (P<0.0001) (Eq. 1, Table 2). However, this regres- 
sion model is likely biased owing to heteroscedasticity in 
the dependent variable. A second least-squares regression 
model, incorporating a weight of 1/variance, achieved an 
r 2 of 0.74 (PcO.0001) (Eq. 2, Table 2). The third regression 
model incorporated the number of fish in the surface in- 
terval (N 1 ) and the second interval (N 2 ) of the school as 
independent variables, the number of fish in the remain- 
ing portion of the school as the dependent variable, and 
1/variance as the weight. This model had an r 2 of 0.70 
(P<0.0001) (Eq. 3, Table 2). Partial P-tests for these models 
could not be executed because the dependent variable N — 
(N t + . . . N t ) changed depending on the number of school 
depth intervals used to predict NFS. 
Three least-squares regression models were used to pre- 
dict school size from school length and width. The model us- 
ing length as the independent variable and N as the depen- 
dent variable had an r 2 of 0.74 (P<0.0001) (Eq. 4, Table 2). 
The second model predicted N s from school width, achiev- 
