Estimation of Disease Incidence in Fish Populations

National Marine Fisheries Service, Norfhw9est and Alaska Fisheries CePrntt.~, 2725 Moratdake Boe4Ceva1.d East, Sea tile, W.4 981 12, USA

WORLUND, D. D., AND G . TAYLOR. 1983. Estimation of disease incidence in fish popu- iatiosms. Can. %. Fish. Aquat. Sci. 40: 2194-2197.

A method is described to estimate the disease incidence in large populations of fish where the material analyzed is composed of a nunmber of pooled individuals. A procedure for calculating a confidence interval estimate is presented, and the bias of the estimate discussed. The methods were developed for estimating disease incidence in hatchery populations sf juvenile Pacific salmon (OnccsrhyqrncBzus spp.), but are applicable to any situation in which the sampling plan can be assumed to follow the binomial distribution.

WORLUND, D. D., AND G. TAYLOR. 1983. Estimation of disease incidence in fish popu- lations. Can. J. Fish. Aquat. Sci. 40: 2194-2197.

L'article qui suit contienk une description de mkthodes pekrnettant d'estirner l'accunence de maladies dans de grandes populations de poissons pour lesquelles le materiel analysk est constitbad par plusieurs individus group&. On y presente une rnCthode d'eskimation des intervalles de confiance, avec analyse du biais d'estimation. Ces nlCtRodes ont ttt miscs au point en vue d'estimer l'occumence de maladies ckcz des populations d'CIsvage de sau- moneaux du Pacifique (Onco8-baynchus spp.), mais elks ssnt applicables egalement a tokate situation dans laquells on peut supposer que le plan de I'echantillonnrtge suit une distribution binomiale .

Received March 8, 1983 Accepted August 10, 1983

OSSI~ZNDER and Wedemeyer (1973) described a method for determining the sample size needed to detect a given inci- dence of diseased fish in a population. As an example they presented sample sizes needed for various incidence levels and large population sizes such as a hatchery production of juvenile Pacific salmon (Oncorkmpnshus spp.). Sample sizes were. determined so that the probability is no less than 0.95 to have at least one diseased fish in the sample if the disease incidence is at least a specified value. In their example, saran- gle sizes of 60 and HSO individual fish were found adequate for detecting disease incidences as low as 5 and 296, respec- tively, when sampling from populations as large as 10b.

Because of thc cost and time associated with disease deter- minations, individual fish are not usually examined. Instead, material from say five fish are combined and the resulting pool is analyzed; for example, one diseased individual in the sample will cause one of a number of groups of five fish to be disease positive. If all groups are disease negative, the inci- dence is unlikely (probability of 0.05) to be as great as that originally specified. On the other hand, if some of the groups are disease positive it is of interest to determine the incidence level. Here we develop a method for estimating disease inci- dence when samples are drawn from a large population and

Meihods - Let R be the proportion of a large population of fish that is disease psitive. Suppose that n samples, each a group of & fish, are drawn from the population; then U = I - R is the probability of obtaining a disease negative individual, L'' is the probability that all fish in a group sf k are disease negative, and P = 1 - uk is the probability that at least one fish in a group is disease positive; Le., the group of k fish is disease positive.

Let x be the number of groups found to be disease positive in the sample of n groups. Assuming that x is a binomial variate with parameters m and P , the probability of observing exactly x disease positive groups is:

The maximum likelihood estimate of R is given by:

(2) i = ( 1 - Ll) = 1 - [I - ( . u / f ? ) ] ' i A

To construct a confidence interval estimate for W, where n , .u. and k are fixed, consider the following probabilities:

are pooled and analyzed in this manner. n ! (3) PR(m 2 XI = 2 ( 1 - uh)jl1 ( l ~ k l ( n nr)

n, fn! (n - n1) ! 'Present address: Bureau of Sport Fislaerics and Wildlife, Olympia

Fish Health Labc~ratory, BLympia, WA 98502, USA. n !

(4) PR(m 6 .x) = ( 1 - ~ 7 h 1 ~ ~ 7 , ~ ~ h ) { i r m)

Printed in Canada (57285) ,,, = rn! 4, - rn)! HrnprimC au Canada (%7%85)

2194

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y T

exas

A&

M U

nive

rsity

on

11/1

4/14

For

pers

onal

use

onl

y.

NOTES % 195

In equation (3) let R , be the largest value of R = i - h?' for which PR(nz 2 x) S a: and in equation (4) let R,, be the smallest value of R for which PW(rn 6 .a-) S (1. R1 and R,, are then the respective lower and upper limits of a IBBO(1 - 2cr)% confidence interval for R . In practice, a finite number of decimal places for W , and R,, must be predetermined and equations (3) and (4) solved by trial and emor.

Table B gives solutions for equations (21, (31, and (4) in situations where samples of 12 and 30 groups of five fish each are obtained. This table is in current use by the Olympia Fish Health Laboratory for sampling hatchery populations of juve- nile salmon. The value of a in equations (3) and (4) was set at 0.05 to produce a 90% confidence interval. Copies of computer prsgranms for obtaining the estimates are available from the authors.

In 'Fable 1 the upper limit of the 90% confidence interval is 4.9% for the event of observing zero disease positive groups of five fish from a total of I2 groups. An assertion can be nude that for any particular value greater than or equal eo 4.9%, the probability of observing zero disease positive groups is less than or equal to 0.05. This agrees well with the disease incidence detected for Iarge populations as determined by Ossiander and Wedemeyer ( 1973). A similar result holds for the sainple of 30 groups of five fish each.

A When n (i.e. number of groups) is Large and 0 < R < 1 confidence intervals can also k obtained by making use of the asymptotic normality of maximum likelihood esti- mates (Wetherill 1981). A l0O( I -2a)% confidence interval for R is:

where 2, is the point of the standard normal distribution and A

- 1 .O/V(R) is the expectation of the second-order derivative of the log of PR(.r) with respect to Cf. The variance of k , V ( k j. can be shown to $6 given by:

and is estimated by substituting (1 - R ) fur U. Equality holds A

if R is an unbiased estimate csf R. Using (5) and (61, 90% confidence intervals corresponding

to values of 100(i) of 3.6, 7.8 , 12.9, 19.7, and 30.1 and for ia = 12 and k = 5 arc (0-81, (0-15). (3-231, ('7-32). and (12-481, respectively. Similar intervals for n = 30 are (0-71, (3- 121, (7- 19), (12-281, and (19-4L). respec- tively. The upper limits of these intervals, which are generally of most interest in estimating disease incidence. compare favorably with those in Table 1. It appears then that If the number of groups is at least 30. the simpler method given by equations (5) and (6) could safely be utilized.

Except when individuals are examined, the estimate of 89 given by equation (2) is biased. The expected value of the estimate of W is:

where

T4er.r:. 1. Estimated percentage disease incidcnze fc9r 12 and 30 samples of five fish each.

909 confidence limits Number of Nu111ber sasnples Point

sanmples disease positive estimate I,c~wer Upper

For various valucs of disease incidence, R. Table 2 illus- trates the magnitude of the bias, E(R) - 62, for various num- bers of groups. The estin~ate of R is expcctcd to bc an aver- estimate (this does not mean, caf course, that any particular estinsate exceeds the true value). 'The bias is snsall relative to the lnaagnitiade of incidence for Iow levels of distzase inci- dence; it's importance will dcpcnd on the particular case.

Keeping the total sarasple size at 60, Table 2 also indicates the decrease in bias obtainktblc by increasing the nun~ber of

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y T

exas

A&

M U

nive

rsity

on

11/1

4/14

For

pers

onal

use

onl

y.

2196 CAN. J . FISH. AQUAT. SCH., VOL. 40, 1983

A

TABLE 2. Bias, 100(E(R) - R ) . in estimating percentage disease incidence for groups of five fish each.

Nutnber of groups

"Bias based on 20 groups of three fish each.

groups and decreasing the number of individuals per group to 20 and 3, respectively. Again, in any particular case, im- provement in the estimate would have to be considered in relation to the cost and time of analyzing additional samples.

Considering the bias and width of confidence intervals, the estimator is not very good for those cases in Table I where most of the groups are observed to be disease positive. Im- proved estimates require additional sampling.

Further, if disease incidence is large, pooling material from several fish may lead to less precise estimates than does making disease determinations on individuals (Table 3). For each hypothetical disease incidence colasidered, it is assumed that 100 groups of fish are analyzed; and that in one case each group is a single fish and in the other, a pool s f five fish. For comparative purposes it is also assumed that the expected number of disease positive groups is observed in each case. Using the width of the 90% confidence interval to judge precision, this example indicates that for disease incidence up to about 20% there is a significant gain realized by pooling individuals into one sample. However, for 30% and above there is little to be gained or an :actual loss in precision (Table 3).

Optimal group size, k t . in terms of max~rnum precision can be examined in general by minimizing V(R) (equation 6) over k for k 3 1 and fixed n and U . This is most easily done by minimizing (U - 1)/k2 over k numerically since optimal k is independent of the value of n. For various values of R , pairs of (100W,k1) are: ( I , I S ) , (B.5,105), (2,791, (352). (5,38), (10,15), (20,7), (30,4), (40,3), (50,2), a n d ( > 70,1).

A limiting factor to group size is the ability to detect a diseased individual among a possible k - l disease negative individuals in a group of size k. Another factor is the increased bias d R related to increasing group size for %xed n. For example if R = 0.05 and n = 12, the bias (R - 0.05) is 0.802, 0.003, 0.063, and 0.359 for k values of 5, 10 ,3 1 , and 58, respectively. While the optimum group size is 31 for an R of 0.05, repeated sampling of 12 groups of 31 individuals each produce estimates of R that vary around 0.113 (0.05 + 0.063) rather than 0.05.

TABLE 3. Width of 90% confidence intervals for various percentage disease incidences ( I @OR), number of individuals per group ( k ) , and expected disease p a - itive groups (Exj observed from a total of 100 groups.

90% confidence limits

Lower

7.6 5 .5

14.1 13.7 24.6 22.5 32.5 31.8 40.3 41.4 45.$ 51.3

- lok).

Width

5.2 10.9 8.0

14.0 10.9 15.9 14.9 16.9 21.4 17.2 32.2 16.9

Increasing the number of groups sampled decreases the bias. In the above example if k is held at its optimum value (i.e. k = 3I), the bias (dg - 0.05) is 0.063, 0.034, 0.013, 0.006, and 0.003 for n values of 12. 15, 20. 25. and 30, respectively. This example as well as others where the inci- dence level is as low as 0.01 indicate that a sample size (n) of 30 is sufficient to limit the bias to 0.003 when the optimum group size is used.

For practical purposes it appears that we can define a large sample size (n) as 36) so that equation 5 can be safely used for computing cesnfidence intervals and for determining sansple sizes (230) needed to limit the width of those intervals to a desired value.

The problem addressed by Ossiander and Wederneyer (1973) is the sample size in terms of individuals needed to detect a given disease incidence. When a disease positive group of one or more individuals is found, disease deter- minations can stop with reasonable assurance that the inci- dence Bevel is at least the specified value. When all groups are disease negative it is nnlikely that the incidence level is as great as the specified value. If a lower or upper hound is a11 that is needed for the problem at hand, their sampling scheme is the most economical. One needs only to determine a max- imum group size such that a diseased individual can be detected.

Here we are concerned with examining the expected pre- cision (width of confidence interval) in estimating disease incidence for various combinations of group size, k , and num- ber of groups, a z , sampled. One half the width of 90% con- fidence intervalss calculated from ( 5 ) , are;hown in Table 4 for values of k and ra in which the bias IW(R - R) is less than 0.5 and for incidence levels of B , 2 - 5 , and 10%. In most cases the bias is 3 3 . 2 for n s 30 and <0.1 for az > 30.

If d , is a value in Table 4 corresponding to a disease incidence of say 100Rl. and if the associated M I groups of k , fish each are sampled, we can assert with probability 0.90 that the estimate 100k will be within (II units of 10dlW,. In this way. Table 4 provides some insight to the precision that can be expected for difkrent sampling levels and group sizes

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y T

exas

A&

M U

nive

rsity

on

11/1

4/14

For

pers

onal

use

onl

y.

NOTES

TABLE 4 . One half width of 9 Q 8 confidence intervals for disease incidence.

Percent disease incidence (IOOR)

1 .0 2.0 5.0 10.0 Number

of Group size ( k ) Group size 4k) Group size ( k ) Group size ( k ) groups

( n ) 1 5 10 159 1 5 I0 79 1 5 10 31 1 5 10 15

when planning a sampling program. Equation 5 also provides a way to determine the necessary

sampling levels, n, when a certain level of precision, d, is needed. This is given by:

where Z, is the value of the standard norinal distribution for desired confidence 1004 1 - 2aW. The most economical (smallest) value of n is obtained by using the optimal group size, k ' , in equation 7. The value of the unknown disease incidence, R , is also needed to determine 41. However, if some notion of the range of R is known. a conservative sample size (i.e. maximum) is obtained by using the upper limit of that range. This is illustrated in Table 4 by observing the decrease

in interval width with decreasing R for fixed k . While equations 5 and 7 are useful in planning a sampling

program and do provide reasonable confidence limits under certain restrictions, those limits are symmetric a b u t R and will always be less than the respective limits calculated from equations 3 and 4. For these reasons the latter method is superior in reporting results.

Acknnwledgmunhs - 'The statistical review and suggestions of W . Reed are appreciated.

OSSIANDER, H. J . , ANT) G . WEDEMEYER. 1973. Computer program for sa~nple sizes required to determine disease incidence in fish populations. J. Fish. Wes. Board Can. 30: 1383- 1384.

WETHERILL, 6. B. 198 1 . Intermediate statistical methods. Chapman and Hall Ltd., London.

Seasonal Variation in Weight-Specific Growth Rates, Feeding Rates, and Growth Efficiencies in Microgadus torncod

I. h/I. SAH~INAS AND I. A, MGLAREN

SALBNAS, I. M . , AND I. A. MCLAREN. 1983. Seasonal variation in weight-specific growth rates, feeding rates, and growth efficiencies in ~lgicrogadus fomcod. Can. J . Fish. Aquaa. Sci. 48: 2195-2200.

Among Microgadus ~ C P I ~ C ' O ~ from Cumberland Basin, Bay of Fundy , exponents of the weight - length relationship (liver. gonads removed) ranged from 2.74 in April, after spawning, to 3.47 in September, after summer feeding. This implies that growth rate per unit of weight is seasonally higher in larger fish, although this diminishes on an annual basis after 2 yr in these short-lived (<4 yr) fish. In feeding experiments larger fish after spawning increased feeding (weight-specific ration declined with weight in September, not in April -May) and grew faster, using proportionately more food for growth (growth efficiency not correlated with weight in September, positively so in April-May).

Printed in Canada (56988) Imprim6 au Canada (36988)

Can

. J. F

ish.

Aqu

at. S

ci. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y T

exas

A&

M U

nive

rsity

on

11/1

4/14

For

pers

onal

use

onl

y.