Technical Memorandum NMFS 

MARCH 1983 

POSSIBLE EFFECTS OF SAMPLING BIASES 

ON REPRODUCTION RATE ESTIMATES FOR 

DOLPHINS IN THE EASTERN TROPICAL PACIFIC 

Tom Polacheck 

NOAA-TM-NMFS-SWFC-26 

U.S. DEPARTMENT OF COMMERCE 

National Oceanic and Atmospheric Administration 

National Marine Fisheries Service 

Southwest Fisheries Center 
NOAA Technical Memorandum NMFS 

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into an agency which establishes national policies an manages and conserves our oceanic, coastal, and 

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In addition to its formal publications, the NMFS uses the NOAA Technical Memorandum series to 

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work and may be referenced in the formal scientific and technical literature. 
NOAA Technical Memorandum NMFS 

Thls TM Series IS used for documentation and timely communication of preliminary results imterim reports, or 

Special purpose InlOrmatlOn and have not received complete formal revlew editorla1 control. or detailed editing 

z-- MARCH 1983 

"*-g 

ORD ~ ,I.."" 

4Qh= 

POSSIBLE EFFECTS OF SAMPLING BIASES 

ON REPRODUCTIVE RATE ESTIMATES FOR 

DOLPHINS IN THE EASTERN TROPICAL PACIFIC 

Tom Polacheck 

Southwest Fisheries Center 

National Marine Fisheries Service, NOAA 

La Jolla, California 92038 

and 

Department of Biology 

University of Oregon 

Eugene, Oregon 97403 

NOAA-TM-NMFS-SWFC-26 

U.S. DEPARTMENT OF COMMERCE 

Malcolm Baldrige. Secretary 

National Oceanic and Atmospheric Administration 

John V. Byrne. Administrator 

National Marine Fisheries Service 

William G. Gordon, Assistant Administrator for Fisheries 
CONTENTS 

Page 

............................................... INTRODUCTION........... 1 

METHODS......................... 2 

RESULTS AND DISCUSSION................................................ 4 

...................................... 

APPENDIX.............................................................. 8 

i 
LIST OF TABLES 

Tab1 e 

1 

2 

3 

f 

9 

4 

Effects of bias bl and b2 on the various reproductive 

estimators for the northern offshore spotted dolphin 

population (S. -attenuata). 

Proporti on of mature females. 

Proportion of nursing calves. 

Proportion female. 

Method 1 pregnancy rate. 

Gross annual reproductive rate (G.A.R.) using Method 1 

pregnancy rates. 

Effects of bias bl and b2 on the various reproductive 

estimators for the southern offshore spotted dolphin 

population (S. -attenuata). 

Proportion of mature females. 

Proporti on of nursing cal ves. 

Proportion female. 

Method 1 pregnancy rate. 

Gross annual reproductive rate (G.A.R.) using method 1 

pregnancy rates. 

Effects of bias bl and b2 on various reproductive 

estimators for the eastern spinner dolphin popul ati on 

(S. -1 ongi rostris). 

Proportion of mature females. 

Proportion of nursing calves. 

Proportion female. 

Method 1 pregnancy rate. 

Gross annual reproductive rate (G.A.R.) using Method 1 

Method 2 pregnancy rate. 

Gross annual reproductive rate (G.A.R.) using Method 2 

pregnancy rates. 

pregnancy rate. 

Effects of bias bl and b2 on the various reproductive 

estimators for the northern whi tebel ly spinner dol phi n 

population (S. -longirostris). 

Proportion of mature females. 

Proportion of nursing calves. 

Proportion female. 

Method 1 pregnancy rate. 

Gross annual reproductive rate (G.A.R.) using Flethod 1 

pregnancy rates. 

ii 
List of Tables - Continued: 

Ta bl e 

5 Effects of bias bi and b;! on the various reproductive 

estimators for the southern whitebelly spinner dolphin 

population (S. - longirostris). 

a Proportion of mature females. 

b Proporti on of nursing cal ves. 

C Proportion female. 

d Method 1 pregnancy rate. 

e Gross annual reproductive rate (G.A.R.) using Method 1 

pregnancy rates. 

iii 
LIST OF FIGURES 

Figure 

1 The sensitivity of the pregnancy rate as estimated by 

Method 1 to bias in the sampling of nursing pairs (bl) 

for the pooled data set (1973-1978) for the eastern 

spinner stock. 

The sensitivity of the proportion mature to biases bl 

and b2 for the pooled data set (1973-1978) for the 

eastern spinner stock. 

The sensitivity of the estimated proportion of immature 

nursing female calves to biases bl and b2 for the pooled 

data set (1973-1978) for the eastern spinner stock. 

The sensitivity of the estimated proportion of females to 

biases bi and b2 for the pooled data set (1973-1978) 

set (1973-1978) for the eastern spinner stock. 

The sensi ti vi ty of the estimated gross annual reproductive 

rate (G.A.R.) using Method 1 pregnancy rates to biases bl and 

b2 for the pooled data set (1973-1978) for the eastern spinner 

stock. 

The sensitivity of Flethod 2 pregnancy rate estimates to 

bias b2 for the pooled data set (1973-1978) for the eastern 

spinner stock. 

The sensitivity of the estimated gross annual reproductive rate 

(G.A.R.) using Method 2 pregnancy rates to biases bl and b2 

for the pooled data set (1973-1978) for the eastern spinner 

stock. 

iv 
POSSIBLE EFFECTS OF SAMPLING BIASES 

ON REPRODUCTION RATE ESTIMATES FOR 

DOLPHIN IN THE EASTERN TROPICAL PACIFIC 

Tom Pol acheck 

Southwest Fisheries Center 

La Jolla, California 92038 

and 

Department of Bi ol ogy 

University of Oregon 

Eugene, Oregon 97403 

INTRODUCTION 

The only avai 1 ab1 e information on the 1 i fe hi story parameters for dol phin 

populations in the eastern tropical Pacific is derived from animals 

incidentally killed during yellowfin tuna purse seine operations. Because of 

the methods by which these samples have been obtained and apparent 

inconsistencies in some of the estimated life history parameters, the question 

of potential biases in these samples has been raised (Powers and Barlow, 

19791). 

Young calves and lactating females may be more vulnerable than other 

animal s during purse seining operations. Analysis of the proportion of 

immature females of the total number of dolphins killed in a set supports the 

suggestion that the overall sample for the spotted dolphin (Stenella 

attenuata) may be biased towards immature animals (Powers and Barlow, 

19/9l). Similar analysis for the data on the eastern spinner and northern 

whi tebel ly spinner dol phi n (Stenel 1 a 1 ongi rostri s did not reveal any 

significant differences, although a trend did exist. In an attempt to correct 

for this bias, the life history parameters for S. attenuata have been 

estimated only from sets in which more than forty doTphins have been killed, 

based on the assumption that large kills should tend to be unselective 

IPowers, J. E. and J . Barlow. 

dolphins in the eastern tropical Pacific, SOPS/79/31. 

SOPS/79/31 Status of Porpoise Stocks Workshop, Southwest Fisheries Center, La 

Jolla, California. 

1979. Biases in the tuna-net sampl ing of 

Working paper, 
2 

(Henderson, et a1 . , 197g2). However, the question of possible biases still 

remains as a number of the estimated life history parameters appears 

inconsistent (e.g., Polacheck, in prep. a and b3). 

The purpose of this report is to present a sensitivity analysis on the 

effects that biased sampling of young calves and lactating females would have 

on the various estimated life history parameters. This sensitivity analysis 

provides one measure of the reliability of the various estimates. Also, where 

i nconsi stenci es in the estimates can be identified, this analysis provides one 

means for evaluating which estimates are likely in error. 

METHODS 

The effects of sampling biases toward nursing pairs (i.e., lactating 

females with their calves) and immature calves were explored by developing a 

simple model for the number of animals in each category which would have been 

in the sample in the absence of the selective factor. These "corrected" 

observations were then used to compute corrected values for the proportion 

mature, the sex ratio, the ratio of lactating to immature females, the 

pregnancy rate and the gross annual reproduction rate. The following symbols 

will be useful in defining how the analysis was conducted. Let: 

bl = the proportional bias or increased vulnerability of nursing 

pairs 

b2 = the proportional bias or increased vulnerability of immature 

calves in addition to bi. 

P 

R 

L 

C 

U 

reproducti ve condi ti on 

I = the number of immature females 

N = the number of nursing calves 

= the number of pregnant females in a sample 

= the number of resting females 

= the number of lactating females 

= the number of lactating and pregnant females 

= the number of mature females unclassified with respect to 

2Henderson, J., W. F. Perrin and R. B. Miller. 1979. Gross annual production 

dol phi n popul ati ons (Stenel 1 a spp. 

SOPS/79/33. 

Southwest Fisheries Center, La Jolla, California. 

in the eastern tropical Paci fic, 

Working paper, SOPS/79/33 Status of Porpoise Stocks Workshop, 

3Polacheck, T. 

rates from reproductive data for dolphin populations in the eastern tropical 

Pacific. NOAA Technical Memorandum, NMFS-SWFC. 

In prep. a. Estimating current rates of increase and survival 

Polacheck, T. In prep. b. Juvenile survival rates as estimated from the 

proportion of immature females that are nursing for dolphin stocks in the 

eastern tropical Pacific. NOAA Technical Memorandum, NMFS-SWFC. 
3 

M = the number of males 

The subscript c will be used with the above symbols to designate the sample 

values corrected for biases bl and b2, and the subscript o to designate the 

observed sample values. Accounting for biases bl and b2, the corrected number 

in the various reproductive states can be modeled as: 

- 

pc - 

- Rc - Ro 

- 1 

LO 

Lc - - 

l+bl 

- 1 

cc - I+bl co 

uc - uo - uo (k) (LoL: C+ 0 +c;o + Ro 

1) 5bl Lo + co (1, - - (Lo + co + u, ( - 1 -- 

IC l+b2 l+bl Lo + Co + Po + R, 

= Mo - (Io - IC) MC 

(3) 

(4) 

(7) 

Equations 5 and 6 assume that the proportion of undetermined sexually mature 

females that are in the various reproductive states is equal to the proportion 

for which the reproduction states were determined. Equations 6 and 7 assume 

that the sex ratio of nursing calves is .5 and equation 7 assumes that the sex 

ratio of all immature calves is .5. 
4 

Estimates of the proportion mature, the sex ratio, the r a t i o o f lactating 

to immature females, the pregnancy rate as estimated by Method 1 (see 

Henderson et al., 197g2), and the gross annual reproduction rate (G.A.R. were 

calculated using the corrected values for the observed number in each 

reproductive condition for a range of values for bl and b2. The analysis was 

conducted on the five stocks for which the observed numbers in each 

reproductive state are avai 1 ab1 e (Henderson, et a1 . , 19792). 

The sensitivity of Henderson's Method 2 for estimating pregnancy rates 

was a1 so considered. Recalculating the pregnancy rate as estimated by 

Method 2 to account for bias is not straightforward. However, bias bl should 

not introduce a bias into the Method I1 estimates of pregnancy rate. In order 

to assess the effect of bias b2, it was assumed that this bias only applied to 

nursing calves. The number of nursing calves when b2 is corrected for should 

equal the number of lactating females. Therefore, the number of nursing 

calves actually occurring in the sample should equal: 

It was further assumed, as it i s i n Method 2, that the first No individuals in 

the observed cummul ative length-frequency distribution represent nursing 

calves. In order to assess the effect of bias, (b2/l + b2) x No individuals 

were randomly removed from the length distribution of nursing calves and the 

Method 2 pregnancy estimate was recalculated. Since this method of assessing 

the effect of b2 is a stochastic procedure, for each value b2 examined the 

procedure was repeated 10 times. The mean and standard deviations of these 

rep1 ications calculated. The standard deviations of these calculations were 

on the order of 1O1O or less and therefore only the means are reported below. 

RESULTS AND DISCUSSION 

The estimated 1 i fe hi story parameters using the corrected sample values 

for the number of males and for the numbers of females lactating, 

unclassified, and immature and male from equations 3 to 7 are given in 

Appendix Tables 1 to 5 for five populations of dolphin. New estimates are 

given based on a range of values for the proportional biases bl, and b2. For 

the northern and southern offshore spotted dolphin populations and the 

northern and southern whitebelly spinner dolphin populations, values are given 

in Tables 1, 2, 4 and 5, respectively, for the proportion of females that are 

mature, the proportion of calves which are nursing, the proportion of the 

population which is female, the pregnancy rate estimated by Method 1, and the 

estimated gross annual reproduction rate using this pregnancy rate. For the 

eastern spinner porpoise population, the above calculations are given in Table 

3, along with estimates of pregnancy rates and gross annual reproduction rates 

using Method 2. These latter were computed using the simulation technique 

described above. Due to limitations on computer time these simulations were 

not completed for the other populations. 
5 

In Figures 1 to 7, the results for the eastern spinner population 

(Stenel 1 a 1 ongirostri s) have been plotted for each of the reproductive 

estimators, illustrating the general behavior of each of the estimators to 

bias bi and b2. Method 1 pregnancy rates and the proportion of immature 

calves nursing are the most sensitive estimators to bias bl. Method 2 

pregnancy rates, the proportion of mature females and proportion of immature 

calves nursing are the estimators most sensitive to b2. Method 2 pregnancy 

rates tend to be more sensitive to bias b2 than Method 1 pregnancy rates are 

to bias bl of an equal magnitude. Somewhat surprising is the insensitivity to 

bias of the G.A.R. for both Methods 1 and 2 pregnancy rates. A bias of 100% 

in sampling nursing pairs results in a change of about 1% in the estimated 

G.A.R. for the eastern spinner stock for both methods, while a bias of a 

similar magnitude in sampling immature females results in a change of at most 

3%. The reason for this insensitivity is that the G.A.R. is a product of the 

percent female, the percent mature and the pregnancy rate. For both methods, 

the effects of bi and b2 on these parameters either tend to be small and/or 

opposite in magnitude. 

The results of this sensitivity analysis can be used to suggest where 

likely biases may exist in the reproductive samples when inconsistencies in 

these estimates are identified. This will be illustrated using the results 

for the eastern spi nner dol phi n popul ation , where the fol 1 owi ng three 

inconsistencies have been identified in the reproductive estimators: 

1. 

2. 

3. 

The pregnancy rate as estimated by Method 1 should be greater .than 

or equal to the rate estimated by Method 2 (Goodman, unpublished 

manuscript; Polacheck, unpublished manuscript). 

The observed proportion of females that are mature, given current 

estimates of the age of maturity and either Method 1 or 2 estimates 

of pregnancy rates, is likely only to be observed in a rapidly 

declining population. In a viable population, given the present 

estimates of age of maturity and pregnancy rates, this proportion 

would be significantly greater (Polacheck, in prep a3). 

For a nondeclining population, given the current range of estimates 

for the age of weaning based on Method 1 and the age of maturation, 

the observed proportion of immatures that are nursing is also too 

low (Polacheck, in prep b3). 

Inconsistency 1 can be reconciled by either bl or b2 although the 

necessary magnitude of bl is over a third greater than for b2 (i.e. .95 

compared to .60). A bias of b2 sufficient to reconcile the two pregnancy 

rate estimates would be sufficient to reconcile inconsistencies 2 and 3. In 

attempting to evaluate whether biases bl or b2 can reconcile inconsistencies 2 

and 3, it is not sufficient to look only at their effects on the estimated 

proportion mature or estimated proportion nursing since these inconsistencies 

are also sensitive to other estimated parameters which could be affected by 

these biases. Thus, both an increase in the estimated proportion mature or in 

the pregnancy rate would tend to reconcile inconsistency 2 (Polacheck, in prep 

a3). However, accounting for a bias bl would increase the pregnancy rate 

(Method 1) and decrease the proportion mature while bias b2 would have just 

the opposite effect. It appears that for the values of the observed estimates 
6 

and their sensitivity to bias, increasing the proportion mature would be the 

dominant factor. Therefore only bias b2 is likely to account for this 

i nconsi stency . Simil arly , i nconsi stency 3 can be reconci 1 ed by a decrease in 

tQe age of weaning or an increase in the proportion mature (Polacheck, in prep 

b 1. While not considered in this paper, biases bl and b2 could both affect 

the estimated age of weaning. The age of weaning or the length of lactation 

have been estimated in two ways. Accounting for bias of type b2 would 

increase the estimate of the proportion nursing and either not affect or 

decrease the age of weaning (Polacheck, in prep b3). Therefore, bias b2 could 

explain inconsistency 2. Accounting for bias bl could also result in a 

decrease in the age of weaning. However, accounting for bias bl would also 

result in a decrease in the proportion nursing. The potential for a decrease 

in the age of weaning due to bl is probably insufficient to counterbalance the 

decrease in the proportion nursing. Thus, bias bl is unlikely to be able to 

account for thi s inconsistency . The resul ts of thi s sensitivity analysi s 

combined with the three inconsistencies identified above would suggest that 

the observed samples for the eastern spinner stock are likely to be biased in 

the overrepresentation of immature animals. Given the large magnitude of b2 

necessary to reconcile the two pregnancy rate estimates, if this is the only 

factor contributing to the inconsistency, it is possible that a combination of 

both bl and b2 may be occurring in the sample, although in order to reconcile 

the other two biases, b2 must be the dominant bias. 

It is possible that other biases than those considered here could be 

operating. For instance, it is possible that pregnant females are underrepresented in the observed samples. A1 ternatively, the seasonal nature of 

the samples combined with the known marked seasonality in breeding (Barlow, 

197g4) may be introducing bias. Barlow (19794) explored the possible effect 

that this factor might have on Method 1 pregnancy rate estimates and concluded 

that it would be small given the long length of gestation. The effect of this 

factor on Method 2 pregnancy rate estimates is not obvious, but it could be 

that little sampling occurs when a high proportion of the calves are 

approaching the age of weaning. 

The above apparent inconsistencies in the data for the eastern spinner 

porpoise popul ation are either non-exi stent or not as strong for the other 

porpoise populations in the eastern tropical Pacific for which reproductive 

data are available. This fact, combined with the results of Powers and Barlow 

(1979l) on possible bias in these samples, suggests that the sampling process 

is nonuniform across stocks. In particular, the question of what factors 

could explain the large bias suggested by this paper that are unique in the 

4Barl ow , 3. 

eastern tropical Pacific, SOPS/79/26. Working paper, SOPS/79/26 Status of 

Porpoise Stocks Workshop, Southwest Fisheries Center, La Jolla, California. 

1979. Reproductive seasonal j ty i n pel agi c dol phi ns of the 
7 

capture and sampling of eastern spinner needs to be explored. Stuntz (19805) 

suggested that a possible factor could be that most eastern spinners are 

captured in mixed aggregations with spotted dolphin and apparently occupy a 

subordinate position within the purse seine. However, there is no direct 

evidence that this results in biased samples. Questions such as this need to 

be considered before the representativeness of the kill samples can be fully 

assessed. 

5Stuntz, W. E. 1980. Variation in age structure of the incidental kill of 

spotted dolphins, Stenella attenuata, in the U.S. tropical purse-seine 

fishery. Southwest t-isheries Center Admin. Report No. LJ-80-06. 
8 

APPEND I X 

Tables of the sensitivity of the various reproductive estimates to biases 

bl and b2 for five stocks of porpoises in the eastern tropical Pacific. The 

observed estimates for each stock are based on the pooled data set from 1973- 

1978 (Henderson, et a1 , 1979). 
9 

Table 1. Effects of bias bl and b2 on the various reproductive 

estimators for the northern offshore spotted dolphin 

population (S. - attenuata). 

la. Proportion of mature females 

Value of B2 

Bi 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.565 0.609 0.645 0.675 0.700 0.722 

0.200 0.556 0.601 0.637 0.667 0.693 0.715 

0.400 0.550 0.594 0.631 0.662 0.687 0.710 

0.600 0.544 0.589 0.626 0.657 0.683 0.705 

0.800 0.540 0.585 0.621 0.652 0.679 0.701 

1.000 0.536 0.581 0.618 0.649 0.675 0.698 

lb. Proportion of nursing calves. 

Value of 82 

81 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.337 0.405 0.472 0.539 0.607 0.674 

0.200 0.298 0.357 0.417 0.476 0.536 0.595 

0.400 0.266 0.320 0.373 0.426 0.480 0.533 

0.600 0.241 0.289 0.338 0.386 0.434 0.482 

0.800 0.220 0.264 0.308 0.352 0.397 0.441 

1.000 0.203 0.243 0.284 0.324 0.365 0.405 

IC. Proportion female. 

Value of 82 

81 0.00 0.20 

0.000 0.556 0.561 

0.200 0.545 0.548 

0.400 0.536 0.539 

0.600 0.529 0.531 

0.800 0.523 0.525 

1.000 0.518 0.519 

Id. Method 1 pregnancy rate. 

Value of 81 

0.40 0.60 0.80 1.00 

0.565 0.568 0.571 0.573 

0.552 0.554 0.557 0.559 

0.541 0.544 0.545 0.547 

0.533 0.535 0.536 0.538 

0.526 0.528 0.529 0.530 

0.521 0.522 0.523 0.523 

0.000 0.375 

0.200 0.403 

0.400 0.426 

0.600 0.446 

0.800 0.462 

1.000 0.477 

le. G.A.R. using Method 1 pregnancy rates. 

Value of BZ 

81 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.118 0.128 0.137 0.144 0.150 0.155 

0.200 0.122 0.133 0.142 0.149 0.155 0.161 

0.400 0.125 0.136 0.145 0.153 0.160 0.165 

0.600 0.128 0.139 0.149 0.156 0.163 0.169 

0.800 0.130 0.142 0.151 0.159 0.166 0.172 

1.000 0.132 0.144 0.153 0.162 0.168 0.174 
10 

Table 2. Effects of bias bl and bz on the various reproductive 

estimators for the southern offshore spotted dolphin 

population (S. -attenuata). 

2a. Proportion of mature females. 

Value of B2 

Bi 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.673 0.711 0.742 0.767 0.787 0.804 

0.200 0.673 0.712 0.742 0.767 0.788 0.805 

0.400 0.674 0.712 0.743 0.767 0.788 0.805 

0.600 0.674 0.713 0.743 0.768 0.788 0.805 

0.800 0.674 0.713 0.743 0.768 0.788 0.805 

1.000 0.674 0.713 0.744 0.768 0.788 0.806 

2b. Proportion of nursing calves. 

Value of B2 

B1 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.464 0.557 0.650 0.743 0.836 0.928 

0.200 0.419 0.503 0.587 0.671 0.755 0.839 

0.400 0.382 0.459 0.535 0.612 0.688 0.765 

0.600 0.351 0.422 0.492 0.562 0.632 0.703 

0.800 0.325 0.390 0.455 0.520 0.585 0.650 

1.000 0.302 0.363 0.423 0.484 0.544 0.605 

2c. Proportion female. 

Value of B2 

Bi 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.594 0.600 0.605 0.610 0.613 0.616 

0.200 0.584 0.589 0.594 0.598 0.601 0.603 

0.400 0.576 0.581 0.585 0.588 0.591 0.593 

0.600 0.569 0.574 0.577 0.580 0.583 0.585 

0.800 0.564 0.568 0.571 0.574 0.576 0.578 

1.000 0.559 0.563 0.566 0.569 0.571 0.572 

2d. Method 1 :pregnancy rate. 

Value of B1 

0.000 0.572 

0.200 0.597 

0.400 0.617 

0.600 0.634 

0.800 0.649 

1.000 0.661 

2e. G.A.R. using Method 1 pregnancy rates. 

Value of B2 

Bi 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.228 0.244 0.257 0.267 0.276 0.283 

0.200 0.234 0.250 0.263 0.274 0.282 0.290 

0.400 0.239 0.255 0.268 0.279 0.287 0.295 

0.600 0.243 0.259 0.272 0.283 0.291 0.299 

0.800 0.247 0.263 0.275 0.286 0.295 0.302 

1.000 0.249 0.265 0.278 0.289 0.297 0.305 
11 

Table 3. Effects of bias bi and b2 on various reproductive 

estimators for the eastern spinner dolphin population (s. longirostris). 

3a. Proportion of mature females. 

Value of BZ 

Bl 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.437 0.482 0.521 0.554 0.583 0.608 

0.200 0.422 0.467 0.506 0.539 0.568 0.594 

0.400 0.411 0.456 0.494 0.527 0.557 0.582 

0.600 0.402 0.446 0.484 0.518 0.547 0.573 

0.800 0.394 0.438 0.476 0.510 0.539 0.565 

1.000 0.387 0.431 0.470 0.503 0.532 0.558 

3b. Proportion of nursing calves. 

Value of B2 

B1 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.210 0.252 0.294 0.336 0.378 0.420 

0.200 0.181 0.217 0.254 0.290 0.326 0.362 

0.400 0.159 0.191 0.223 0.255 0.287 0.319 

0.600 0.142 0.171 0.199 0.228 0.256 0.285 

0.800 0.129 0.154 0.180 0.206 0.231 0.257 

1.000 0.117 0.141 0.164 0.188 0.211 0.234 

3c. Proportion female. 

Value of B2 

Bi 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.506 0.507 0.508 0.508 0.509 0.509 

0.200 0.496 0.496 0.496 0.495 0.495 0.495 

0.400 0.489 0.487 0.486 0.485 0.485 0.484 

0.600 0.482 0.481 0.479 0.478 0.476 0.475 

1.000 0.473 0.471 0.468 0.466 0.464 0.462 

0.800 0-477 0.475 0.473 0.471 0.470 0.468 

3d. Method 1 pregnancy rate. 

Value of B1 

3e. G.A.R 

Bi 

0.000 0.336 

0.200 0.367 

0.400 0.393 

0.600 0.415 

0.800 0.434 

1.000 0.451 

using Method 1 pregnancy rates 

Value of 82 

0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.074 0.082 0.089 0.095 0.100 0.104 

0.200 0.077 0.085 0.092 0.098 0.103 0.108 

0.400 0.079 0.087 0.094 0.101 0.106 0.111 

0.600 0.080 0.089 0.096 0.103 0.108 0.113 

0.800 0.082 0.090 0.098 0.104 0.110 0.115 

1.000 0.083 0.091 0.099 0.106 0.111 0.116 
Table 3. (continued) 

3f. Method 2 pregnancy rate. 

Value of B2 

BI 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.447 0.420 0.381 0.342 0.318 0.301 

39. G.A.R. using Method 2 pregnancy rates. 

Value of B2 

Bl 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.099 0.103 0.101 0.096 0.094 0.093 

0.200 0.094 0.097 0.096 0.091 0.089 0.089 

0.400 0.090 0.093 0.091 0.087 0.086 0.085 

0.600 0.087 0.090 0.088 0.085 0.083 0.082 

0.800 0.084 0.087 0.086 0.082 0.081 0.080 

1.000 0.082 0.085 0.084 0.080 0.078 0.078 
13 

Table 4. Effects of bias bl and b2 on various reproductive 

estimators for the northern whi tebel ly spi nner dolphin 

population (S. longirostris). - 

4a. Proportion of mature females. 

Value of 82 

Bi 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.527 0.572 0.609 0.641 0.667 0.690 

0.200 0.518 0.563 0.601 0.632 0.659 0.682 

0.400 0.511 0.556 0.594 0.625 0.652 0.676 

0.600 0.505 0.550 0.588 0.620 0.647 0.671 

0.800 0.500 0.545 0.583 0.615 0.642 0.666 

1.000 0.495 0.541 0.579 0.611 0.639 0.662 

4b. Proportion of nursing calves. 

Value of 82 

Bi 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.261 0.314 0.366 0.418 0.470 0.523 

0.200 0.228 0.273 0.319 0.364 0.410 0.455 

0.400 0.202 0.242 0.282 0.323 0.363 0.403 

0.600 0.181 0.217 0.253 0.290 0.326 0.362 

0.800 0.164 0.197 0.230 0.263 0.296 0.328 

1.000 0.150 0.180 0.210 0.240 0.270 0.301 

4c. Proportion female. 

Value of B2 

Bi 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.509 0.510 0.510 0.511 0.511 0.512 

0.200 0.498 0.498 0.498 0.498 0.498 0.498 

0.400 0.490 0.489 0.489 0.488 0.488 0.487 

0.600 0.484 0.482 0.481 0.480 0.480 0.479 

0.800 0.479 0.477 0.475 0.474 0.473 0.472 

1.000 0.474 0.472 0.470 0.469 0.467 0.466 

4d. Method 1 pregnancy rate. 

Value of 81 

4e. G.A.R. 

B1 

0.000 0.353 

0.200 0.378 

0.400 0.398 

0.600 0.416 

0.800 0.430 

1.000 0.443 

using Method 1 pregnancy rates. 

Value of 82 

0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.095 0.103 0.110 0.116 0.120 0.125 

0.200 0.098 0.106 0.113 0.119 0.124 0.128 

0.400 0.100 0.108 0.116 0.122 0.127 0.131 

0.600 0.101 0.110 0.118 0.124 0.129 0.133 

0.800 0.103 0.112 0.119 0.125 0.131 0.135 

1.000 0.104 0.113 0.121 0.127 0.132 0.137 
14 

Table 5. Effects of bias bl and b2 on various reproductive 

estimators for the southern whitebel ly spinner dolphin 

population (2. longirostri s). 

5a. Proportion of mature females. 

Value of 52 

Bi 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.639 0.680 0.712 0.739 0.761 0.780 

0.200 0.637 0.678 0.711 0.737 0.760 0.778 

0.400 0.636 0.677 0.710 0.736 0.759 0.777 

0.600 0.635 0.676 0.709 0.735 0.758 0.776 

0.800 0.634 0.675 0.708 0.735 0.757 0.776 

1.000 0.633 0.674 0.707 0.734 0.756 0.775 

5b. Proportion of nursing calves. 

Value of 82 

Bi 0.00 0.20 0.40 0.60 0.80 1.00 

-0.000 0.316 0.380 0.443 0.506 0.569 0.633 

0.200 0.278 0.334 0.390 0.445 0.501 0.556 

0.400 0.248 0.298 0.348 0.397 0.447 0.497 

0.600 0.224 0.269 0.314 0.359 0.404 0.449 

0.800 0.204 0.245 0.286 0.327 0.368 0.409 

1.000 0.188 0.225 0.263 0.301 0.338 0.376 

5c. Proportion female. 

Value of BZ 

81 0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.522 0.523 0.524 0.525 0.526 0.527 

0.200 0.512 0.513 0.514 0.514 0.515 0.515 

0.400 0.505 0.505 0.506 0.506 0.506 0.506 

0.600 0.499 0.499 0.499 0.499 0.499 0.499 

0.800 0.495 0.494 0.494 0.494 0.494 0.494 

1.000 0.491 0.490 0.490 0.490 0.489 0.489 

5d. Method 1 pregnancy rate. 

Value of 81 

5e. G.A.R. 

Bi 

0.000 0.295 

0.200 0.310 

0.400 0.322 

0.600 0.332 

0.800 0.340 

1.000 0.347 

using Method 1 pregnancy rates. 

Value of 82 

0.00 0.20 0.40 0.60 0.80 1.00 

0.000 0.098 0.105 0.110 0.114 0.118 0.121 

0.200 0.101 0.108 0.113 0.118 0.121 0.124 

0.400 0.103 0.110 0.116 0.120 0.124 0.127 

0.600 0.105 0.112 0.118 0.122 0.126 0.129 

0.800 0.107 0.114 0.119 0.124 0.127 0.130 

1.000 0.108 0.115 0.120 0.125 0.129 0.132 
15 

0.50 

0.45 

0.40 

0.35 

0.30- ~ 

0.00 0.20 0.40 0.60 0.80 1 .oo 

PROPORTIONAL BIAS OR INCREASED 

VULNERABILITY OF NURSING PAIRS (b 1) 

Figure 1. The sensitivity of the pregnancy rate as estimated by Method 1 to 

bias in the sampling of nursing pairs (bl) for the pooled data 

set (1973-1978) for the eastern spinner stock. 
16 

3 

Ir 

u) 

W 

a 

a 

r 

W 

L 

L 

0 

z 

0 

IU 

0 

n 

0 

a 

n 

I 

0.65 

0.60 

0.55 

0.50 

0.45 

0.40 F' 

b2= 1.0 

b2 = 0.8 

b2 = 0.6 

b2 = 0.4 

b2 = 0.2 

b2 = 0.0 

n QC I I I ~~ I I V.VU 

0.00 0.20 0.40 0.60 0.80 1 .oo 

PROPORTIONAL BIAS OR INCREASED 

VULNERABILITY OF NURSING PAIRS (bl) 

Figure 2. The sensitivity of the proportion mature to biases bl and b2 for 

the pooled data set (1973-1978) for the eastern spinner stock. 
17 

(3 

z 

v) 

U 

3 

z 

W 

I 

a 

a 

I 

0 I 

r 

3 

v) 

W 

> 

J 

0 

LL 

0 

z 

0 

Ia 

0 

n 

0 

a 

n 

a 

I 

PROPORTIONAL BIAS OR INCREASED 

VULNERABILITY OF NURSING PAIRS (b 1) 

Figure 3. The sensitivity of the estimated proportion of immature nursing female calves to biases bl and b2 for the pooled data set (1973- 

1978) for the eastern spinner stock. 
18 

W 

4 

a 

L 

W 

LL 

z 

0 

k 

U 

0 

0 

U 

I 

n 

n 

0.5 1 

0.50 

0.49 

0.48 

0.47 

0.46 

0.45 

0.00 0.20 0.40 0.60 0.80 1 .oo 

PROPORTIONAL BIAS OR INCREASED 

VULNERABILITY OF NURSING PAIRS (b 1) 

Figure 4. The sensitivity of the estimated proportion of females to biases 

bl and b2 for the pooled data set (1973-1978) set (1973-1978) for 

the eastern spinner stock. 
19 

K 

a 

6 

0.12 

0.1 1 

0.10 

0.09 

0.08 

0.07 

0.00 0.20 0.40 0.60 0.80 1 .oo 

PROPORTIONALY BIAS OR INCREASED 

VULNERABILTY OF NURSING PAIRS (bi 1 IFigure 5. The sensitivity of the estimated gross annual reproductive rate 

(G.A.R.) using Method 1 pregnancy rates to biases bl and b2 for 

the pooled data set (1973-1978) for the eastern spinner stock. 
20 

0.45 

0.40 

0.35 

0.30 

0.25 

0.00 0.20 0.40 0.60 0.80 1 .oo 

PROPORTIONAL BIAS OR INCREASED 

VULNERABILITY OF IMMATURE CALVES (bp) 

Figure 6. The sensitivity of Method 2 pregnancy rate estimates to bias b2 

for the pooled data set (1973-1978) for the eastern spinner 

stock. 
21 

0.1 1 

0.1 0 

a 

a 0.09 

6 

0.08 

OD7 

0.0 

0.2 

0.4 

0.6 

0.8 

1 .o 

0.00 0.20 0.40 0.60 0.80 1 .oo 

PROPORTIONAL BIAS OR INCREASED 

VULNERABILITY OF NURSING PAIRS (bl) 

Figure 7. The sensitivity of the estimated gross annual reproductive rate 

(G.A.R.) using Method 2 pregnancy rates to biases bl and b2 for 

the pooled data set (1973-1978) for the eastern spinner stock. 
RECENT TECHNICAL MEMORANDUMS 

Copies of this and other NOAATechnical Memorandums are available from the National Technical 

Information Service, 5285 Port Royal Road, Springfield, VA 221 67. Paper copies vary in price. Microfishe copies cost $3.50. Recent issues of NOAATechnical Memorandumsfrom the NMFS Southwest 

Fisheries Center are listed below: 

NOAA TM-NMFS SWFC 16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

An evaluation of tagging, marking, and tattooing techniques for 

M. J. WHITE, JR., J. G. JENNINGS, W. F. GANDY and L. H. 

CORNELL 

(November 1981) 

Local stability in maximum net productivity levels for a simple 

T. POLACHECK 

(April 1981) 

Environmental data contouring program EDMAP2. 

L. EBER 

(April 1982) 

The relationship between changes in gross reproductive rate and 

the current rate of increase for some simple age structred models. 

T. POLACHECK 

(May 1982) 

Testing methods of estimating range and bearing to cetaceans 

T. D. SMITH 

(1 982) 

"An annotated bibliography of the ecology of co-occurring tunas 

(Katsuwonus pelamis, Thunnus albacares) and dolphins (Stenella 

attenuata, Stenella longirostris and Delphinus delphis in the 

eastern tropical Pacific" 

small delphinids. 

model of porpoise population sizes. 

aboard the R/V D.S. Jordan. 

S. D. HAWES 

(November 1982) 

Structured flotsam as fish aggregating devices. 

R. S. SHOMURA and W. M. MATSUMOTO 

(November 1982) 

Abundance estimation of dolphin stocks involved in the eastern 

tropical Pacific yellowfin tuna fishery determined from aerial and 

ship surveys to 1979. 

R. S. HOLT and J. E. POWERS 

(November 1982) 

Revised update and retrieval system for the CalCOFl oceanographic data file. 

L. EBER and N. WILEY 

(December 1982) 

A preliminary study of dolphin release procedures using model 

D. B. HOLTS and J. M. COE 

(December 1982) 

purse seines.