246 
Fishery Bulletin 108(2) 
In Equation A3, E is a square measurement error ma- 
trix with L rows and columns that distributes numbers 
at true shell height into observed shell heights bins that 
are larger and smaller than the true shell height. For 
example, the first row of E sums to one and gives the 
probability of observed shell heights for sea scallops in 
the first true shell height bin. The last row of E sums 
to one and gives the probabilities that sea scallops in 
each shell height bin would be assigned to the “plus 
group” because of measurement error. As described in 
the text, we estimated E for sea scallops using results 
from experiment 2. 
Appendix 2 
Equation A3 in Appendix 1 indicates the possibility of 
correcting shell-height data measurement algebraically, 
without resorting to an approach like the CASA model. 
In particular, if the matrix E is invertible, then it may 
be possible to estimate the true sample proportions n 
by multiplying both sides of Equation A3 by the inverse 
matrix E~ u . 
n - pE~ x . (A4) 
However, the inverse calculation in Equation A4 will be 
unreliable if the estimated error matrix E is poorly con- 
ditioned. If the error matrix is poorly conditioned, then 
small inaccuracies in the estimate of E will propagate 
into larger errors in the inverse E~ l and the predicted 
proportions k. 
As described by Horn and Johnson (1985), the condi- 
tion factor for an invertible matrix E is 
k=\e ||||e' 1 ||, (A5) 
where ||Z?|| = the matrix norm of E. 
The condition factor k is always at least one and 
is an upper bound measure of the extent to which 
errors in the original error matrix E (ignoring 
errors in p ) will propagate to its inverse. If k is 
slightly larger than one, then uncertainty in E 1 
and n from Equation A4 will be at most slightly 
greater then uncertainty in E. If k is large, then 
uncertainty in E -1 and n may be much larger 
than uncertainty in E. 
The measurement-error matrices that included 
both bias and imprecision are the most realistic 
according to results from experiment 2. The con- 
dition factors for these error matrices were 2638 
for video and 2.3 for measuring boards (Table 4). 
These condition factors indicate that uncertainty 
in E - 1 and “corrected” shell-height composition 
data could be much higher than uncertainty in 
the original error matrix E for video and at most 
2.3 times higher for measuring boards. 
Bootstrap analyses show the practical signifi- 
cance of condition factors for video and measur- 
ing board data in our study. For example, for the 
video shell-height measurements in experiment 
2, the first step was to resample n data records 
(including one video measurement and the corre- 
sponding caliper measurement) with replacement 
from the data in experiment 2. 
Sample sizes (77 = 670 for video and 77 = 344 for 
measuring boards) were the same as the number 
of experimental measurements and constituted 
an upper bound on the true effective sample size 
because they ignore repeated measurements on 
the same specimens (Table 2). The effect of us- 
ing an upper bound estimate for effective sample 
size was to understate effects of uncertainty in 
error matrices. Our interest was, however, in a 
“best case” scenario with relatively large sample 
sizes. Next, the measurement errors (e.g. video 
or measuring board minus caliper measure- 
0 . 10 - 
0.15 
0.00 
Measurement 
boards 
\ 
S- 
' VA 
Video 
100 
50 - 
-50 - 
-100 
B— — — 
& 
1 1 1 1 1 1 
30 50 
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 
70 90 110 130 150 
1 1 1 1 
185 
B 
• . 
o § 8 
° 
§ 1 ? 
i i T : i 
I — : R_l|— 
if s 8 r i i * ° 
Rath ■ ^ „ o a . - „ » „ .« 
1 - : V 
8 tig 
“ 8 § 
Borers . _ t ; * 8 
lift s ° T T T o 
? § s 
o § 
° ° o 
• 
i i i i i i i 
30 50 
70 
150 185 
90 110 130 
Shell height (mm) 
Appendix Figure 1 
Boxplots showing bootstrap distributions (1000 iterations) of 
estimated true shell-height (SH) composition for Atlantic sea 
scallops ( Placopecten magellanicus ) in experiment 2, based on 
measurement boards (A) and video (B) shell-height data. True 
shell-height compositions were estimated by using bootstrap 
estimates of the inverse of the measurement error matrix E 
and Equation A4. The solid line in (A) shows the actual caliper- 
derived shell-height data in the experiment. The solid line is 
not visible in (B) because of the scale of the y-axis. 
