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Sampling strategies for clustered behavioural elements of short duration
Applied Animal Behaviour Science, 19 (1988) 315-319 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
315
S a m p l i n g S t r a t e g i e s for Clustered B e h a v i o u r a l E l e m e n t s of Short D u r a t i o n J. JANSEN' and J.A. HOEKSTRA2 TNO Institute of Applied Computer Science, P.O. Box 100, 6700 AC Wageningen (The Netherlands) (Accepted for publication 25 August 1987)
ABSTRACT Jansen, J. and Hoekstra, J.A., 1988. Sampling strategies for clustered behavioural elements of short duration: Appl. Anita. Behav. Sci., 19: 315-319. This paper is concernedwith samplingstrategies in the study of behaviouralelements of short duration. These elements often occur in clusters. A semi-Markovmodel is used to describe the time pattern. It is shown that under the semi-Markovmodel, a more precise estimate of the frequencyof the behaviouris obtained if the total observationtime is split into many short sessions,
instead of one or a few long sessions. Sessions have to be kept separate enough to guarantee independent observations. The theory is applied to data concerning sexual activity of fattening bulls.
INTRODUCTION M a n y behavioural elements of short duration occur in clusters, which appear in an unpredictable way. In these cases it is difficult to obtain accurate estimates of the m e a n frequency of the behavioural el em ent under study. This is illustrated in Fig. 1 for t he occurrence of sexual activity in fattening bulls, which consists of four elements; m o u n t i n g on the front or back, or put t i ng the head on th e shoulder or t he croup of a group-mate (H.K. Wierenga, personal communication, 1985). In this figure the variance calculated from four 3-h recordings of individual bulls is pl ot t ed against t he corresponding mean. It may be concluded t h a t four recordings results are a very inaccurate meas u r e m e n t of the average sexual activity of individual bulls. Obviously, precision is increased if animals are observed for a longer period of time. However, if the 'Present address: Institute of Horticultural Plant Breeding, P.O. Box 16, 6700 AA, Wageningen, The Netherlands. 2Present address: National Institute of Public Health and Environmental Hygiene, P.O. Box 1, 3720 BA Bilthoven, The Netherlands.
0168-1591/88/$03.50
© 1988 Elsevier Science Publishers B.V.
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Fig. 1. Illustration of the variability in measurementsof sexual behaviour of fattening bulls. Variance of four measurementsplotted against the correspondingmean. total time for measuring is fixed, it may be possible to choose between one or just a few long observation sessions, or m a n y short observation sessions. In order to be able to compare different observation strategies, we use a model for the time pattern of the behaviour under consideration. A semi-Markov model is t h o u g h t to be appropriate for behavioural elements occurring in clusters. The variance of observed frequencies is derived as a function of the number of observation sessions and their length. The efficiency of sampling strategies is compared using this variance formula. This paper is m e a n t to draw attention to possibilities of improving the accuracy of measurements in animal behaviour studies. It is not meant to give general guidance, as in m a n y investigations different types of behaviour are studied, each requiring its own sampling strategy. In t h a t case a compromise has to be found. SEMI-MARKOV MODEL A semi-Markov process can be considered as a mixture of two Poisson processes, one with a short average waiting time, ml, and one with a long average waiting time, m2. An animal is supposed to be in one of the two intervals corresponding with the two Poisson processes. After each event, i.e. the occur-
317
rence of one of the four behavioural elements defined previously, the animal may change from an interval with average waiting time ml to an interval with average waiting time m2, or vice versa. The transition probabilities are 1 - a, and 1 - a 2 , respectively. This is a 2-state Markov model (or process). This model should be considered as a first approximation to complex reality where several events occur. The model was fitted to four observation series of the bulls, each containing more t h a n 40 events. The model describes the global features of the data satisfactorily. The estimates of al, were about 0.8, of a2 0.2, of m, 0.01 h and of m2 0.2 h. For more details see J a n s e n and Hoekstra (1986). If a, is close to 1, events will appear in clusters of high frequency with longer intervals in between. The process leads to an extremely variable number of events, as was observed with the bulls. VARIANCE OF OBSERVED FREQUENCIES
The quantity of interest is N ( t ) , the number of events in an observation period of length t. The expected or mean value of N ( t ) under a semi-Markov model is E[N(t)] =t/m
in which m = p , m , + p2m2, p, = ( 1 - a2 ) / (2 - al - a2 ) = 1 - P 2 (Cox and Lewis, 1966). The variance of N ( t ) is V(t) =var[N(t) ] = (a+bt-ae-rt) /m
in which a, b and r are functions of ml, m2, al and a2. Using eqn. ( 1 ), the standard error of the mean frequency (number of events per hour), obtained from k independent observation sessions of t h, can be found as S ( t , k ) = { V ( t ) / k t 2} 1/2
RESULTS
In Table I quantities e ( t ) = S ( t , 3 / t ) / S ( 3 , 1 ) are given for a semi-Markov model with different parameter values, for various values of t. The quantity e (t) gives the standard error based on 3 / t independent small periods of length t relative to the standard error of one 3-h period. The small periods have to be kept separate enough to guarantee independence. Differences in efficiency due to splitting the entire interval become more pronounced the larger the expected waiting time rn2, given the value of ml. In our example with m2 = 0.2, splitting is not very useful, but it should be realized t h a t most bulls showed very long periods of inactivity. High values of al and a2 also favour short observation
318
0
>
319
intervals. The reason for this is that if al and a2 are close to 1, once an animal is one of the two types of interval, it will very probably stay there for a long period of time. In this situation in particular, it is worthwhile spreading the total observation time in order to obtain a sample from both situations. As shown in Table I for given values of m2 and t, the quantities e (t) become smaller if the value of al increases. However, this is not so if m2--0.2 and t = 0.08 and 0.17. No explanation could be found for this inconsistency. DISCUSSION
In this paper a 2-state Markov model is used to simulate the variance of the frequency of certain events under different sampling schemes. The model describes the features of the behaviour of the bulls sufficiently well to give confidence of the comparisons in a qualitative sense. The relationship between the variance and the mean of the number of events found in this paper shows similarities with the power relationship found by Taylor (1961) for many biological phenomena where aggregation occurs. It can be concluded that in cases of clustered events, one should spread the total observation time. This is also sensible from an intuitive point of view. If an animal is encountered in an interval with a short average time, this will lead to a high number of events, whereas if the animal has just started a period in an interval with a long average waiting time, no events will occur for some time. So the first minutes of an observation session contain most of the information. By using many observation sessions, separated far enough in time to guarantee independence, one gets an average over both types of interval. This paper is meant to draw attention to the possibility that accuracy may be improved by choosing a proper sampling scheme. Much work has still to be done in order to be able to give general guidelines for choosing a sampling scheme in animal behaviour studies. ACKNOWLEDGEMENT
We gratefully acknowledge the help of Dr. H.K. Wieringa in supplying data.
REFERENCES Cox, D.R. and Lewis, P.A.W., 1966. The Statistical Analysis of Series of Events. Chapman and Hall, London. Jansen, J. and Hoekstra, J.A., 1986. Comparison of sampling strategies under a semi-Markov model. ITI-TNO Report A21. Taylor, L.R., 1961. Aggregation, variance and mean. Nature (London), 189: 732-735.
315
S a m p l i n g S t r a t e g i e s for Clustered B e h a v i o u r a l E l e m e n t s of Short D u r a t i o n J. JANSEN' and J.A. HOEKSTRA2 TNO Institute of Applied Computer Science, P.O. Box 100, 6700 AC Wageningen (The Netherlands) (Accepted for publication 25 August 1987)
ABSTRACT Jansen, J. and Hoekstra, J.A., 1988. Sampling strategies for clustered behavioural elements of short duration: Appl. Anita. Behav. Sci., 19: 315-319. This paper is concernedwith samplingstrategies in the study of behaviouralelements of short duration. These elements often occur in clusters. A semi-Markovmodel is used to describe the time pattern. It is shown that under the semi-Markovmodel, a more precise estimate of the frequencyof the behaviouris obtained if the total observationtime is split into many short sessions,
instead of one or a few long sessions. Sessions have to be kept separate enough to guarantee independent observations. The theory is applied to data concerning sexual activity of fattening bulls.
INTRODUCTION M a n y behavioural elements of short duration occur in clusters, which appear in an unpredictable way. In these cases it is difficult to obtain accurate estimates of the m e a n frequency of the behavioural el em ent under study. This is illustrated in Fig. 1 for t he occurrence of sexual activity in fattening bulls, which consists of four elements; m o u n t i n g on the front or back, or put t i ng the head on th e shoulder or t he croup of a group-mate (H.K. Wierenga, personal communication, 1985). In this figure the variance calculated from four 3-h recordings of individual bulls is pl ot t ed against t he corresponding mean. It may be concluded t h a t four recordings results are a very inaccurate meas u r e m e n t of the average sexual activity of individual bulls. Obviously, precision is increased if animals are observed for a longer period of time. However, if the 'Present address: Institute of Horticultural Plant Breeding, P.O. Box 16, 6700 AA, Wageningen, The Netherlands. 2Present address: National Institute of Public Health and Environmental Hygiene, P.O. Box 1, 3720 BA Bilthoven, The Netherlands.
0168-1591/88/$03.50
© 1988 Elsevier Science Publishers B.V.
316 28
m
26 Q
2o
0 0
• k
16
m
m
14 ~ d
•
12-
we
10-
a ,o • 0
d a
m
8-
64-
B m
2-
m
ira 0
•
I m ,pmwm
0
nilO O 0
I~w
I
10 Meam v a l u e b u e d
I
1
20
I
30
I
40
on 4 reooFdtnp
Fig. 1. Illustration of the variability in measurementsof sexual behaviour of fattening bulls. Variance of four measurementsplotted against the correspondingmean. total time for measuring is fixed, it may be possible to choose between one or just a few long observation sessions, or m a n y short observation sessions. In order to be able to compare different observation strategies, we use a model for the time pattern of the behaviour under consideration. A semi-Markov model is t h o u g h t to be appropriate for behavioural elements occurring in clusters. The variance of observed frequencies is derived as a function of the number of observation sessions and their length. The efficiency of sampling strategies is compared using this variance formula. This paper is m e a n t to draw attention to possibilities of improving the accuracy of measurements in animal behaviour studies. It is not meant to give general guidance, as in m a n y investigations different types of behaviour are studied, each requiring its own sampling strategy. In t h a t case a compromise has to be found. SEMI-MARKOV MODEL A semi-Markov process can be considered as a mixture of two Poisson processes, one with a short average waiting time, ml, and one with a long average waiting time, m2. An animal is supposed to be in one of the two intervals corresponding with the two Poisson processes. After each event, i.e. the occur-
317
rence of one of the four behavioural elements defined previously, the animal may change from an interval with average waiting time ml to an interval with average waiting time m2, or vice versa. The transition probabilities are 1 - a, and 1 - a 2 , respectively. This is a 2-state Markov model (or process). This model should be considered as a first approximation to complex reality where several events occur. The model was fitted to four observation series of the bulls, each containing more t h a n 40 events. The model describes the global features of the data satisfactorily. The estimates of al, were about 0.8, of a2 0.2, of m, 0.01 h and of m2 0.2 h. For more details see J a n s e n and Hoekstra (1986). If a, is close to 1, events will appear in clusters of high frequency with longer intervals in between. The process leads to an extremely variable number of events, as was observed with the bulls. VARIANCE OF OBSERVED FREQUENCIES
The quantity of interest is N ( t ) , the number of events in an observation period of length t. The expected or mean value of N ( t ) under a semi-Markov model is E[N(t)] =t/m
in which m = p , m , + p2m2, p, = ( 1 - a2 ) / (2 - al - a2 ) = 1 - P 2 (Cox and Lewis, 1966). The variance of N ( t ) is V(t) =var[N(t) ] = (a+bt-ae-rt) /m
in which a, b and r are functions of ml, m2, al and a2. Using eqn. ( 1 ), the standard error of the mean frequency (number of events per hour), obtained from k independent observation sessions of t h, can be found as S ( t , k ) = { V ( t ) / k t 2} 1/2
RESULTS
In Table I quantities e ( t ) = S ( t , 3 / t ) / S ( 3 , 1 ) are given for a semi-Markov model with different parameter values, for various values of t. The quantity e (t) gives the standard error based on 3 / t independent small periods of length t relative to the standard error of one 3-h period. The small periods have to be kept separate enough to guarantee independence. Differences in efficiency due to splitting the entire interval become more pronounced the larger the expected waiting time rn2, given the value of ml. In our example with m2 = 0.2, splitting is not very useful, but it should be realized t h a t most bulls showed very long periods of inactivity. High values of al and a2 also favour short observation
318
0
>
319
intervals. The reason for this is that if al and a2 are close to 1, once an animal is one of the two types of interval, it will very probably stay there for a long period of time. In this situation in particular, it is worthwhile spreading the total observation time in order to obtain a sample from both situations. As shown in Table I for given values of m2 and t, the quantities e (t) become smaller if the value of al increases. However, this is not so if m2--0.2 and t = 0.08 and 0.17. No explanation could be found for this inconsistency. DISCUSSION
In this paper a 2-state Markov model is used to simulate the variance of the frequency of certain events under different sampling schemes. The model describes the features of the behaviour of the bulls sufficiently well to give confidence of the comparisons in a qualitative sense. The relationship between the variance and the mean of the number of events found in this paper shows similarities with the power relationship found by Taylor (1961) for many biological phenomena where aggregation occurs. It can be concluded that in cases of clustered events, one should spread the total observation time. This is also sensible from an intuitive point of view. If an animal is encountered in an interval with a short average time, this will lead to a high number of events, whereas if the animal has just started a period in an interval with a long average waiting time, no events will occur for some time. So the first minutes of an observation session contain most of the information. By using many observation sessions, separated far enough in time to guarantee independence, one gets an average over both types of interval. This paper is meant to draw attention to the possibility that accuracy may be improved by choosing a proper sampling scheme. Much work has still to be done in order to be able to give general guidelines for choosing a sampling scheme in animal behaviour studies. ACKNOWLEDGEMENT
We gratefully acknowledge the help of Dr. H.K. Wieringa in supplying data.
REFERENCES Cox, D.R. and Lewis, P.A.W., 1966. The Statistical Analysis of Series of Events. Chapman and Hall, London. Jansen, J. and Hoekstra, J.A., 1986. Comparison of sampling strategies under a semi-Markov model. ITI-TNO Report A21. Taylor, L.R., 1961. Aggregation, variance and mean. Nature (London), 189: 732-735.