Organization of song in cardinals

n. Behav.,1971, 19, 1-17

ORGANIZATION OF SONG IN CARDINALS BY ROBERT E. LEMON & CHRISTOPHER CHATFIELD Department of Biology, McGill University, Montreal 110, Quebec, Canada, and

School of Mathematics, Bath University of Technology, Claverton Down, Bath BA2 7A.Y, England stract. North American birds known as cardinals have a repertoire of several distinctive song types, ~h consisting of repetitions of sound patterns called syllables. Through the use of statistical models, s paper considers the rules by which this behaviour is organized. Included are considerations of the ~tors determining the repetitions of the syllables in the songs, the repetitions of the songs in series or uts, the associations of different types of syllables within the songs and the associations between different song types. An attempt is made to interpret the organization of the repertoire in terms conflicting tendencies for facilitation and inhibition.

ae North American cardinal, Richmondena ~rdinalis, is a common song bird of the eastern ld southern portions of the continent, and its ud singing is a prominent feature of woodnds and gardens, especially in spring. The nging of each bird is based on a repertoire of istinctive song patterns which are usually milar among neighbouring birds, although ften different between birds of different localities Lemon 1965, 1966). This paper considers in some detail the ways i which cardinals organize their singing beaviour, especially the sequences of song types nd the repetitions of these song types when the ateractions between individuals are minimal. lince the approach is quantitative, mathematical nodels are developed fairly thoroughly. Methods l'he birds studied were wild male cardinals iving on the campus of the University of Western Ontario, London, Ontario, or in the :mmediate vicinity of this area. Since most of Lhe birds were colour-marked, it was possible to identify them throughout the season and from year to year. Cardinals are highly territorial, occupying their territories from mid-February or March to the end of August, and the birds are referred to by name of territory and often by the year of occupancy, such as Medway 1964. The singing of many birds was recorded on tape and then the songs were analysed by a Kay Electric sound spectrograph. Much information, however, especially relating to the sequences of different song types, was gathered by listening to the birds sing and then recording the data in a notebook. In the field, times were recorded at 1-min intervals, often making possible estimates to the nearest 89

Two birds were studied in detail, these being Medway in 1964 and Chambers in 1965. The two birds were studied because they sang more than most which facilitated the acquisition of data and also because they appeared to represent extremes of organization, particularly with reference to the switching between song types. It is assumed throughout this paper that we are dealing with steady-state behaviour which in statistical terms means that the probabilities of different song types do not change with time. Since it is known that the amount of song can vary with the season, the singing of Medway is considered only before nesting began, that is between 1 March and 11 April 1964. Chambers was unmated and his singing was recorded from 17 May to 14 June 1965. The data from both birds were examined to see if there were any differences in absolute and relative amounts of the different song types in the first and second halves of the sample periods, but no differences were apparent. With both birds the data were collected in 289 nearly daily samples. A second threat to the steady-state assumption is that the sequence of song types and the amount of song of a cardinal may be influenced by others (Lemon & Lemon 1968). The interactions that the two birds, Chambers and Medway, had with others appeared to be minimal. Chambers' neighbours sang very little that late in the season, and most of Medway's interactions appeared to be at low motivation. Definitions

In dealing with song, the following terminology will be useful. A singing cardinal produces sounds which are readily classed into types called syllables. Syllables are normally repeated with intervals of silence between them lasting either

2

ANIMAL

BEHAVIOUR,

19,

1

3Or-

20

IO

l ' - ~ ' l

0

,,,zo- I~ ] =

C

I0

~'o

r-~'~ %

,

,--,

,--, A

I0

9

'

.

o.' o

,.o

3.0

~.o

~.o

,.o

,,.o

,2~",~!o

I n t e r v a l s between successive s y l l a b l e s ( s )

Fig. 1. Frequency distributions of time intervals between certain successive syllables of the bird Medway 1964, based on measurements from tape recordings. The bimodal distributions permit one to designate the shorter intervals as separating syllables within utterances, whereas the longer intervals are said to separate syllables in successive utterances.

r r r r r r F r r

LLLL EM

t

N "r v 0

I

0..5

I

I

LO

1.5

f f

2.0

S

Fig. 2. Sound spectrograms of utterances of the ten song types of the bird Medway. The different letters designate the different types of syllables as illustrated for song type WBW.

,~.~,

t

LEMON & CHATFIELD: ORGANIZATION OF SONG IN CARDINALS

less than 0.5 s or longer than 1.0 s; in other words the intervals between successive syllables exhibit a bimodal distribution, examples of which are shown in Fig. 1. Successive syllables separated by shorter intervals are said to belong to the same utterance, whereas syllables separated by longer intervals are said to belong to different utterances. Within an utterance the syllables may be of one or more types, although usually not more than two; hence there are permutations of syllables which are called song types. Figure 2 gives examples of utterances of the ten song types employed by the male Medway. Seven of the song types consist of repetitions Of just one syllable, while three, WBW, UY and EM, consist of two types of syllable. For example, in Fig. 2 the song type WBW is illustrated by an utterance in which a W-syllable is followed by a B-syllable which in turn is followed by four Wsyllables. An examination of the syllables shown in Fig. 2 indicates that some such as B, S and D have intervals of silence in them. These intervals are shorter than those between successive syllables (Lemon 1965). These subdivisions of the syllables are called subsyllables. A cardinal normally sings series of utterances of the same song type, often for several minutes, and such a series is called a bout. Sometimes, especially at the beginnings of bouts, syllables normally found in utterances after certain introductory ones may be absent; for example, a bird may sing utterances of E without the usual M syllables following, although these are eventually added to utterances. In such a case we

consider the Es to be representatives of song type EM. Results Number of Syllables per Utterance The number of syllables per utterance may vary greatly but in general it is related to the duration of the syllable (Fig. 3). For shorter syllables, the distributions of the number of

! (A)

g

40~-I [

s

C 846

N =Z 6 6

~ ~o 20

~ m 0

2 468101214160

g

I 2 34.5678910

Syllables

per

utterance

(B) c 6 A p r i l 1961

8

o 9 41J 9 1 4 9 1 4 9 1 4 9 1 4 9 1 4 9 1 4 9 OO O9

6 O9 oc

9

4 o 9149

~

z

c~

0

9

9

Lc ~ (C)

I 50

20 [ ,, :29

I0 ....

I :50

I 40

I 9

:51

9

I :32

:~

9

o9

9

8--

9

9

e

0

9

3 March t963

6--

9 t

c [3

9 4--

9 S"

m

2

9

-oo

9

,

g

[

,O

=o >, m

9

9

~:28 2

9

I0 y9 p9

3

2O

Sequence W"

M,

C,

9

I

I

I

8:43

:44

:45

50 40 of utterances I I :46:47:48

I

I :49 ~,m.,es'l',

Time 3~

,E o, Io

L__ o.2o

I 0.30

Me(In d u r a t i o n of s y l l a b l e

U

__

l 0.40

A9 B9

I 0.50

(s)

Fig. 3. The relationship between the duration of the syllable and the number of syllables per utterance (bird Medway 1964).

Fig. 4. (A) Frequency distributions of the number of syllables per utterance of song types S and C for bird Medway 1964. (B) The sequence of utterances in a bout of song type C of Medway, showing the initial increase in the numbor of syllables per utterance. (C) The same as (2), but an example of song type Q, showing the absence of a constant number of syllables per utterance.

ANIMAL

BEHAVIOUR,

syllables per utterance usually show a negative skew, as exemplified by song types S and C (Fig. 4A). Skewed distributions of this type result when in successive utterances the number of syllables increases to a maximum, especially if the bird has been silent for some time. An example of this phenomenon is illustrated in Fig. 4B: silent from 09.24 to 09.28 hours, Medway began singing Cs with the number of repetitions of the syllable increasing steadily until a constant value of seven syllables per utterance was reached. An asymptotic number of syllables per utterance was not always maintained, however, as Fig. 4C illustrates; here the number of Q syllables fluctuated greatly. Sometimes a bird did not increase the number of syllables in successive utterances, but instead maintained a low number. On the other hand he sometimes sang utterances of exceptional length, with the shorter syllables being repeated most, an extreme being U2Y79 from Medway on 5 March 1964. Exceptionally long utterances were recorded of most of the different song types. An exception to the other syllables of Medway and of most other birds in the area of study was syllable B which was sung not more than once in utterances of WBW. Some birds omitted B occasionally. The duration of a syllable is correlated with the duration of the interval which follows, longer syllables having longer intervals. Also, the successive intervals in an utterance are not equal, but rather there is a tendency to accelerate until a minimum interval is reached. We have not attempted to illustrate these matters here.

The Number of Utterances per Bout The different song types in a cardinal's repertoire are repeated in series of utterances to form a bout. This section is concerned with finding a model to describe the variation in the number of utterances per bout for the different song types [see Cane (1961) for a discussion of such models]. One property the model must describe is that basic to the nature of bouts, namely, that utterances of the same song type are repeated in long series. This property, called property (1), is clearly visible in transition matrices of utterances of song type i followed by song type j, shown for the birds Chambers and Medway in Tables I and II, from the large entries on the major diagonal from upper left to lower right. There are two other properties that our model must describe. One of these, called property

19,

1

(2), is that the probability of occurrence of a particular song type is positively correlated with the mean number of utterances per bout for that song type (Fig. 5A and B). The other, called property (3), is the relationship between the number of bouts for each song type and the mean number of utterances per bout: the number of bouts rises to a maximum and then decreases as the mean number of utterances per bout increases (Fig. 5C and D). The simplest hypothesis about successive utterances is that they are independent, but although this hypothesis accounts for properties (2) and (3) (Cane 1961), it will not satisfy property (1). To deal with property (1), we must introduce the concept of the conditional probability, p*, of repeating a particular song type, which is the probability that an utterance of that song type will occur given that the previous utterance was of the same type. The strong tendency to repeat song types many times means thatp* is generally much higher than the non-conditional probability, p. (Whereas for the independence model, we find p* = p.) We will now study a model in which we assume that the conditional probability p~ of repeating song type i is higher than the non-conditional probability Pi and that this conditional probability does not depend on the number of times it has been sung previously. Then we have: the expected number of utterances = N p i the expected number of bouts = N p i (1-p~) and the mean number of utterances per bout: = Npi / Upi (1-p~), ----- 1 / (1-p~). where N---- total number of bouts of all types. The quantities Pi and p* may be estimated from the observations in the transition matrix to give : /~i_~_ nii + ni. N ~, nli Pi---nii + hi.' where hi. = number of times song type i is repeated and ni. = number of times song type i is not repeated. The above model was constructed in order to

LEMON

& CHATFIELD:

ORGANIZATION

O F S O N G IN C A R D I N A L S

5

Table I. A M a t r i x Showing the Frequency with which Utterances o f Particular Song Types are Followed by Utterances of the Same or Different Song Types of the Bird Chambers in 1965 preceding event EM

Following event EM

ES

S

UY

A

T

TA

D

Q

DQ QD

QDQ

WBW

C

CS

Row totals

-

-

1141

1123

2

4

3

2

ES

-

-

2

-

-

S

3

-

1720

3

4

I

-

-

3

-

-

-

24

7

-

I765

UY

1

1

3

1586

4

3

-

2

2

-

-

-

1

4

-

1607

A

2

-

-

2

646

172

8

1

1

-

-

-

1

-

833

T

2

-

4

4

161

1009

6

2

1

-

1189

12

-

-

-

15

-

-

-

435

1

.

.

.

.

.

.

.

.

.

.

.

.

2

TA

-

-

-

-

2

1

.

-

-

1

1

-

-

205

190

19

19

-

Q

2

-

1

3

-

-

189

270

34

17

1

1

6

-

524

D Q

1

-

-

-

23

28

5

6

-

2

1

-

66

Q

-

-

-

-

-

17

21

5

9

-

QDQ

-

-

-

WBW

9

-

11

D

.

.

D

-

.

6

.

.

.

-

2

4

-

-

-

. 6

C

3

11

2

1

1

-

3

CS

-

-

3

-

-

--

--

1

Column totals 1145

3

1760

1606

834

1188

15

442

.

.

.

.

521

63

.

i

52 -

1669

2

-

1697

1

1049

5

1082

2

1

7

.

52

. -

-

.

.

1707

I

G r a n d total 6 10416

1073

Table II. A M a t r i x Showing the Frequency with which Utterances of Particular Song Types Followed Utterances of the Same or Different Song Types of the Bird M e d w a y in 1964 (The Expected Values in Parentheses are Calculated to Test the Off-Diagonal Elements for Iudependence) Preceding event W B W WBW

954

Following event UY

EM

2 (1.1)

4 (1 '4)

C

D

1 (2'4)

3 (1"9)

S

A

T

1 (1 "9)

2 (2"2)

1 (2.1)

P

Row totals

2 (2.1)

0 (1 "3)

16 13

Q

UY

2 (1.3)

969

3 (1"1)

1 (2'0)

2 (1"6)

1 (1"6)

0 (1"8)

0 (1"7)

4 (1.7)

0 (1'0)

EM

3 (1-7)

2 (1-1)

959

2 (2"6)

1 (2.1)

0 (2"1)

2 (2"3)

3 (2-2)

3 (2.2)

1 (1.4)

17

C

3 (2"2)

I (1"5)

0 (1"9)

1013

3 (2"7)

13 (2"7)

1 (3"0)

1 (2"8)

0 (2"8)

0 (1"8)

22

D

2 (1 "9)

1 (1 "3)

3 (1 "6)

7 (2"9)

849

1 (2"3)

2 (2"6)

0 (2"5)

3 (2"5)

0 (1 "5)

19

S

0 (1 "8)

1 (1 "2)

2 (1"5)

5 (2"7)

1 (2"2)

719

3 (2-4)

4 (2"3)

1 (2"3)

i (1 "4)

18

A

3 (1 "8)

0 (1 "2)

0 (1 "5)

3 (2"7)

1 (2"2)

0 (2"2)

1064

9 (2"3)

2 (2"3)

0 (1 "4)

18

T

1 (2"0)

1 (1 "3)

0 (1"7)

2 (3"2)

6 (2-4)

0 (2"4)

9 (2"7)

693

1 (2"6)

0 (1"6)

20

Q

1 (2'0)

2 (1"3)

0 (1.7)

1 (3"0)

0 (2"4)

2 (2"4)

2 (2"7)

1 (2"6)

494

11 (1 "6)

20

P

1 (1"3)

1 (0"9)

2 (1.1)

1 (2'0)

2 (1"6)

1 (1"6)

0 (1"8)

1 (1"7)

4 (1"7)

222

13

Column totals 16

4

0.100

11 0.067

14 0.086

23 0.150

19 0.121

19 0.121

21 0.135

20 0-128

20 0-128

13

176 G r a n d total 0.080 8141

6

ANIMAL

BEHAVIOUR,

19,

1

80 (A)

GC

60

40

40

0

UY

4t

TM

_{P

v', o.to4 L-o.~o~

'

~ Q.12

i __J 0.16

PrOb

III 0

t

Chambers 1965 N= 10416 r = + 0.785

{o

0

"~' 2C

,~

(B)

Medwoy 1964 N : 8141 9 = +0.926

b il

icy

0 of

~,k,~" D O I i~ 1 7 6I 0.04 .occ

A+ I 0.08

u r r enc

Te l 0.12

I

t 0.16

I----J 0.20

e

-

C

a 6o

[C)

Medway

\

1964.

UY ~k-e =,, WBW " * ~ " '

4Z}

Do

4O

6s eA

el Y

(O) C h a m b e r s

1965

EM+ ~oWBW C+

4E

I0

20

"~," ',''

~'2 ' & Number

o of

9 DQ QD ~>_~L TA __...a~e--OL~ I 4-0 80 bouts

I

"-"'----------'~A-.:----.---.D Q I I i i i er-i ~2o ~6o 200 2aO

Fig. 5. The relationship between the probabilities of occurrence of different song types and the number of bouts of the same with the mean number of utterances per bout, based on data from the birds Chambers and Medway. Vertical bars indicate standard errors.

account for property (1), but unless there is a relationship between p and p*, the model is not sufficient to explain properties (2) and (3). Fortunately there is a strong positive correlation betweenp and p* botk for Medway and especially for Chambers. This follows from the fact that a 'large' value of nil yields a 'large' estimate for both Pi and p~. Because of this correlation, we can show that our conditional model will also explain properties (2) and (3). The estimated values ofp and p* are shown in Table III. For Chambers, the relationship is approximately linear with p* --- 10ft. As p* increases from 0 to I, I/(l-p*) increases from 1 to oo. Thus 1/0-p*) is positively correlated with p* and hence with p. This gives property (2). The expected number ofbouts--Np (l-p*) Np* (l-p*) / 10. As p* increases from 0 to {, Np* ( l - p * ) / 1 0 increases from 0 to N/40, but as p* increases form { to 1, Np* ( l - p * ) / 1 0 decreases from N/40 to 0. Thus Np* (l-p*) / 10 has a maximum with respect to p* and hence

also with respect to I / ( l - p * ) since the latter function is positively correlated with p*. This gives property (3). For Medway, the relationship between p and p* is not linear. Nevertheless, the correlation between the two variables is high enough to result in both properties (2) and (3). However, the maximum value of the number of bouts is not as well defined for Medway as for Chambers (Fig. 5C and D). The model described above emphasizes the similarity in organization between Chambers and Medway. It is important to realize, however, that there are also major differences. For example, in Table III, the values offi* for Medway always exceed 0.94, whereas for Chambers they go as low as 0.06. As the mean number of utterances per bout is positively correlated with p*, the mean number of utterances per bout is always quite high for Medway but varies much more for Chambers (Fig. 5C and D). Up to now we have been concerned with con-

L E M O N & CHATFIELD: O R G A N I Z A T I O N O F SONG I N C A R D I N A L S

Table ]l-[. A Comparison of Estimates of the Overall Probability of Singing an Utterance of a Particular Song Type (p) and the Conditional Probability of Repeating the Song Type (p*) Chambers

and then an utterance of a different type is p~(r-1) (1-p~). Thus the expected number of bouts of length r is given by" er = N i P ~ ( r - l )

Medway

7

(1-p~)

(r = 1, 2 , . . . )

where Ni = total number of bouts of song type i Song type

p

~*

Song type

p

/)*

=zf r r

EM

0.11

0.98

WBW

0.12

0.98

S

0.17

0.98

UY

0.12

0.99

UY

0"15

0.98

EM

0-12

0-98

A

0-08

0.77

C

0-13

0.98

T

0.11

0.84

D

0.11

0.98

TA

0.0015

0.06

S

0.09

0.97

D

0.04

0.47

A

0.13

0.98

Q

0.05

0.55

T

0.09

0.97

DQ

0.006

0.08

Q

0.06

0.96

QD

0.005

0.17

P

0.03

0.94

WBW

0.16

0-98

C

0.10

0.97

structing a model to describe the number of utterances per bout which will explain properties (1), (2) and (3). We have seen that the conditional probability p* of repeating a particular song type is generally much higher than the nonconditional probability of the same song type. In addition we have seen that the values of p and p* are positively correlated. One important question remains, namely whether the conditional probability is constant or changes as the bout progresses. Our~ earlier results were obtained on the assumption that the conditional probability does not depend on the number of times the song type has previously been sung. We will now test this assumption by examining the frequency distributions of the number of utterances per bout. For each song type we can find the observed frequency of bouts which contain 1, 2, 3. . . . . or in general r utterances of the same song type. The distribution will be denoted by f l , f2, f3 . . . . . where J~=number of bouts in which the song is repeated r times. If our hypothesis is true, the probability that an initial utterance of song type i will be followed by (r-l) utterances of the same type

This discrete distribution is called a geometric distribution. The parameter p , may be estimated by equating the mean of the observed distribution, x, to the mean of the theoretical geometric distribution which is 1 / (1-pi); this gives PA. i = 1--1/x. The observed mean x is calculated from: x = Zrfr/zfr.

An alternative to the geometric distribution is the logarithmic series distribution for which the expected number of bouts of length r is given by e r = - - N i q r / r In(l-q) ( r = 1, 2 . . . . ). The parameter q ( 0 < q < 1) can also be estimated from the observed mean ~, and Williamson & Bretherton (1964) provide a table for t) for different values of x. For this distribution, the conditional probability rises as the sequence length increases. Theoretical geometric and logarithmic series distributions were fitted to the observed frequency distributions of song types D, Q, A and T of Chambers (Fig. 6). A geometric distribution fits well the observed frequency distribution of song type D whereas logarithmic series distributions fit better for song types Q and A. (Remember, the higher the Z 2 value, the lesser the agreement between observed and expected values.) In the latter examples the agreement with logarithmic series distributions indicates that the longer a bout has lasted, the higher the conditional probability that the next utterance will also be of the same song type. In the case of D, however, the logarithmic series distribution gives a slightly poorer fit than the geometric distribution so that the above phenomenon does not apply to all song types. Furthermore the distribution of Chambers T differs in that both the geometric and logarithmic series distributions fit poorly. The reason for this is that the conditional probability of repeating a song type increases substantially after two utterances have been sung but eventually starts to decrease after about ten utterances have been sung. This may be demonstrated by estimating the conditional probabilities as

8

ANIMAL

12C

~6.6

x-~ = 0.46

BEHAVIOUR,

L/

P>O.05

1

: 0.77

P >O.05(6DJE) lo~~ =0.67 J X2= 9.9 I•

19,

X~ = 40.7 #< 0.001(11D.E) o~ ~ =0.92

I

P>O.05 X~=17"7

4C U

2

3 4

5

6

7 8+

2

3

4

5

6 ?

8 9 1011/13/17+ 12 14 i.O

(.9

o.s

J~ o J~ o

0.6

q

Q 12C L3<

I !

x - ~ =0,52 XZ=12,9(6D,E] P
8G-•

o - ~ = 0.75 X2= 1.7 P> 0.05

,85

~

X2= 73.4 P <0.001( 16D,F.) ~ ~

r

o-~/ : 0 . 9 5 X z = 60.5

c

P
0

I

23

45

67/9+

8 Utter

I.I 2__5 an c e s

o

•o • X

- 0.4

o

0

4

5 6

per

7

B

9 I0 11

.:,o 12 13 14 15 1 6 t 7 / 2 1 +

G E O t.)

20

bouf

Fig. 6. Observed frequency distributions of the number of utterances per bout of the song types D, Q, A and T of the bird Chambers. Also shown are values of theoretical distributions: • geometric series; O, logarithmic series. For song type T the conditional probablity of repeating a song type is also shown. follows. Let Cr denote the number of bouts in which the song type is repeated at least r times.

Thus Cr = Z

:3.

j>>r Then an estimate of the conditional probability of repeating the song type given that there have been r previous utterances of the same song type is given by Cr+l / Cr. Values of this ratio for Chambers T are given in Fig. 5. The first value C 2 / C ] is only 0.67. The average of the next seven values is about 0.9, but the average of the next ten values is about 0.8. A pattern similar to that of Chambers T was noted also for Chambers C, WBW and S and for Medway S, A and C, although the data are not presented. We have not attempted to fit a theoretical distribution to any of these since the

sample sizes are quite small and the distributions are quite attenuated. In any case the changes in the conditional probability (first an increase, then a decrease) make it extremely difficult to find a simple model which is appropriate. To sum up, we have constructed a model for the number of utterances per bout which depends on the fact that the conditional probability of repeating a song type is generally much higher than the non-conditional probability. We have seen also that this conditional probability is generally not constant, but depends on the number of utterances of the same song type which have previously been sung. It appears that there is usually an increasing tendency for the particular song type to be repeated as the bout progresses, but that eventually the tendency to repeat wanes. For at least two song types,

LEMON & CHATFIELD: ORGANIZATION OF SONG IN CARDINALS however, Q and A of Chambers, the tendency to repeat does not wane, while for Chambers D the conditional probability is approximately constant throughout. Further remarks on bouts are appended to the next section. The Sequence of Bouts of Different Song Types

Having examined the distributions of the number of utterances per bout for different song types, we now turn our attention to the sequence of bouts, that is, the sequence of switches from one song type to another. We would like to know if successive bouts of song are independent or if when a bird chooses its next song type its choice is affected by one or more of its immediately preceding song types. In statistical terms we say that a bout is an event and that a sequence o f events in which each depends just on the r previous events is an rth order Markov chain (e.g. Wilks 1962). The sequence of events considered here is somewhat unusual in that two successive events (bouts) are always different. The frequency with which a bout of one song type is followed by a bout of a different song can be found from transition matrices such as Tables I and II. N o w we are only concerned with the entries in cells lying off the major diagonal. An inspection of Table I for Chambers shows immediately that switches between certain song types occur very frequently: song types D, Q, DQ, QD and Q D Q form one group and A, T and T A form another. It is clear from this visual inspection that the off-diagonal entries in Table I are not arranged independently.

9

In the case of Medway (Table Ii), the offdiagonal entries are much smaller and a visual inspection is not sufficient to decide if they are arranged independently; instead we shall apply a %2 goodness-of-fit test. Since the diagonal entries in the transition matrix are omitted from our analysis, the usual test has to be modified somewhat (see Appendix I and Chatfield 1969). I n carrying out this test, we must remember that the goodness-of-fit statistic will approximate a chi-square distribution only if less than 20 per cent of the cells have an expected value of less than five and also if none of the cells has an expected value of less than one. Unfortunately all the expected values in Table I I turn out to be less than three. In order to increase these values, it was necessary to reduce the number of categories by combining the song types in pairs. This was done as follows: U E G = U Y + EM CDG= C+D SAG = S + A TQG= T+Q P W G = P + WBW The result of this regrouping, shown in Table IV, is that only two of the cells have expected values less than five, so that the %2 approximation is now valid. The test statistic turns out to be 36.4, which is significant at the 1 per cent level. Thus we have strong evidence that the off-diagonal entries in Table IV, and hence also in Table II, are not arranged independently. Having ruled out the independence hypothesis for both Chambers and Medway, the next step

Table IV. A Rearrangement of the Data in Table H, as Outlined in the Text Showing Observed and Expected Values, the Latter in Parentheses

Preced-

Following event

ing event

UEG

CDG

SAG

TQG

PWG

Row totals

3 (7.6)

10 (8.1)

6 (5.5)

25

UEG

--

6 (6.5)

CDG

5 (4.8)

--

17 (9.4)

4 (10.1)

5 (6.8)

31

SAG

3 (5.1)

10 (8.5)

--

16 (12.4)

4 (7.3)

33

TQG

3 (5.9)

9 (9.8)

13 (11.5)

--

13 (8.3)

38

PWG Column totals

9 (4-3)

7 (7.2)

4 (8.5)

--

28

/~j

20 0.154

%2 = 36.4, I1 D.F., P<0.01.

32 0.258

37 0.303

8 (9.1) 38 0.325

28 0.220

155

ANIMAL

10

BEHAVIOUR,

is to test the hypothesis that successive bouts form a first order Markov chain, this being the simplest alternative model. One way of doing this is to calculate the expected frequencies of different triplets, assuming that successive bouts form a first order Markov chain, and to compare these expected values with the observed frequencies. A triplet here is defined to be a sequence of three bouts of song types, with, by definition, a switch between the first and second song types and also between the second and third. The method of estimating the expected frequencies is given in Appendix II (p. 16). Because the number of possible triplets for Chambers and Medway is very large, (for example, for Chambers the number of possible triplets is 15 • 142), the •2 approximation for the goodness-of-fit test is not valid (Chatfield 1969). Thus we have limited ourselves to a visual inspection of those triplets for which the first

19,

1

and third song types are the same: for example, an A T A or a QDQ. The observed and expected values of these triplets are given in Table V for both Chambers and Medway. Also approximate 95 per cent probability limits are provided (see Appendix II for details), although these are unreliable when the expected value is less than about five. It can be seen that all the observed frequencies for Chambers lie within the expected ranges, except D Q D which lies just outside. For Medway, there is also good agreement. As the frequencies for Medway are quite small, we also compare the total values of observed and expected frequencies, which turn out to be practically identical. The sample size being smaller, it was also feasible for Medway to look at all other triplets. Again there is good agreement. The triplet D - C - S which occurs most frequently has the largest expected frequency, because D - C and C-S frequently occurred as pairs.

Table Vo Observed and Expected Triplets of Song Types Where the First and Third Items are the Same from Medway in 1964 and Chambers in 1965, with 95 per cent Probability Intervals in Parentheses

Medway Triplet

Observed

Chambers Expected

Q P Q P QP

4 3

2.8(0to4) 1.8(0to4)

A T A

2

3.1 (1 to 7)

T A T

3

3.3(lto7)

S C S C SC

2 2

2.5(0to15) 3.1(0to5)

D C D

1

C D C

I

0.8(0to5) 0.9(0to3)

Total

18

18.3

Largest: D C S

5

3.4(1 to 7)

Triplet Q D D Q

D Q Q D QD D QD Q

QD Q QD

D DQ D DQ D DQ Q DQ Q QD DQ QD DQ QD DQ A T A T A T A TA A TA A T A T TAT TA T TA

Observed 141 151 4 16 2

Expected 150 136 7.0 15.3

(140 to 160) (124 to 148) (2 to 11) (10 to 21)

3-7 ( 0 t o 8 )

5 4 4 l 1

7.4 1.4 8.1 0.7 0.5

(3 to 12) (0to5) (4 to 13) (0to3) (0to3)

152 144 4 1 1 1

151 145 6.5 0.9 0.8 0.1

(143 to 159) (138 to 152) (4 to 8) (0to3) (0to3) (0to1)

S WBW S WBW S WBW

8 8

9"2 (4 to 15) 5.8 (2 to 9)

SC S C SC

3 3

2.3 (0to5) 1.70to5)

WBW EM WBW EM WBW EM

l 1

2.9 (0 to 6) 1.9 (0 to 5)

C CS C CS C CS

2 1

1.6 (0to2) 0.3 (0to 1)

S EM S UY S UY

1 1

0.6 (0to2) 0.2 (0 to 2)

LEMON & CHATFIELD" ORGANIZATION OF SONG IN CARDINALS The above evidence leads us to the conclusion that the sequence of bouts is a first order Markov chain for both Chambers and Medway, meaning that the occurrence of a song type depends on the immediately previous song type. An immediate consequence of our Markov chain model is that we can expect some pairs of song types to occur much more frequently than predicted by the independence model. An examination of Tables I and II will reveal those pairs of song types which are strongly associated for Chambers and Medway and these have been listed in Table VI. Data was also available from other birds in the same locality but, as the samples were much shorter, it was only possible in these cases to find the most obvious associations. They are also listed in Table VI. The

11

samples from these birds were too small to test statistically. An examination of Table VI reveals that most of the associations are two-way, so that not only does T tend to follow A but also A tends to follow T. It can also be seen that combinations of two strongly associated song types often occur in the same utterance, for example, AT and QDQ. Another feature of interest is that different birds seem to form the same associations, providing that the birds have the same repertoires, which they generally do. The most common associations are those between A and T, and C and S. As a consequence of a first order Markov dependency, if a song type such as C is associated with song type S, and S in turn is associated

Table VI. Observed Associations Between Song Types in the Repertoires of Certain Cardinals at London, Ontario (The ArrowIndicates the Directionof the Observed Sequence. Parentheses enclosePermutations of the Associates;Brackets enclose Associated Groups) Bird

Association >D---~C~-'---~S

Other song types

Medway 1964

A-~c---'-~T

Q<-'--~P

UY, EM, WBW

Chambers 1965

A~---(AT)--~T Q~r--(QD,QDQ, DQ)--~D C<--(CS)--~S S'~-(WSW, SWS)--~W EM~V-(WEW, EWE)--~W S,~-(ES)--~E C<-'---~ [D, Q] UY-"~ T]

[A,

UY, BW, EM, C, D, A

Bridge, 1 9 6 1

T~-(TQ)-~Q

Boiler House 1962

A<--T D~----~Q

Huron Swale 1962

A-~--~T C~--(CS)---~S EM~-(ES)--~S

C'~--(CS)----~ S

(BW,6---~W)

Q.qf--(PQ, QP)--~P

UY, EM UY

~r"~x4, (pD ~) D~e"

Bridge 1963 River 1963

Q~---~p,

C~-(CS, SC)---~S D,~---(DW)--~-P" A'~-----~T

Q~--~---~p

C-~--(CS)---~S

UY

f M'~-(MT)-~T

(DO, QD)

\/ (D)

M~----~W

Typical sequences of Chambers: EM~------~WBW'~----~S~---'~C'~-----~[D, Q] NN~ [A,~T] "~- UY

EM, W. A

(DP, PD)

12

ANIMAL

BEHAVIOUR,

with say W B W , then triplets o f the f o r m C - S W B W will often occur. Other three-way associtions are Q - P - D for H u r o n Swale, 1962, W - C - S for Bridge, 1963, and D - P - Q for River, 1963. Such three-way associations are quite consistent with the first order M a r k o v chain model. One can proceed further to build up such typical sequences o f song types as that o f Chambers shown at the b o t t o m o f Table VI. I n contrast to those song types which had obvious associations, there were some which were not usually clearly associated with any other song type, in particular UY, E M and WBW.

The tendency for certain song types to be closely associated has an important side effect, namely that the frequency distributions o f the n u m b e r o f utterances per b o u t tend to be similar between associates, although often different between non-associates. This tendency can be seen clearly in Fig. 6 where the distributions o f associates D, Q and A, T were presented. A further outcome is the similarity in the n u m b e r o f bouts as seen especially for the associated song types, D - Q , A - T , C - S and W B W - E M for Chambers in Fig. 5. Also, as a corollary, the closer the association between song types the shorter are the bouts as a result o f the more frequent switching. Effects o f T i m e on the S e q u e n c e o f S o n g T y p e s

Time is often an important influence in behaviour and one might expect it to be so in the

19,

1

sequence o f song types in cardinals. One aspect of the importance o f time is that intervals between closely associated song types tend to be shorter than the average interval between all song types as judged f r o m the data o f Medway (Table VII). The only exceptions are C - D , which Table VH. Mean Duration in Minutes of Intervals Between Frequently Associated Song Types and for all Switches of Medway in 1964

Source of interval

N

Mean

Standard error

Range

P- Q Q- P

4 11

1.25 2.82

0.595 2.32

0.5- 3-0 0.5 - 26.0

A- T T- A

8 8

4.00 7.19

2.29 2.14

0.5 - 14.0 0.5 - 16.0

C- S S-C

13 5

1.3t 3.30

0.806 1-23

0-5 - 11-0 0 5 - 6.5

D - C C- D

7 3

6.07 9.50

1.83

0.5 - 11.0

T- D

6

10"33

5"07

(D-T)

o

Total sample

175

0.5 - 27.5

9.43

0.5 - 29"0

0"618

only had a sample size 3, T - D which m a y have a large frequency of association by chance for its reverse D - T did not occur even once, unlike the situation for the other c o m m o n associations.

Table VHL The Observed Frequencies with which Song Types A and Q of Medway were Repeated, Followed by an Associated Song Type or Followed by Some Other Song Type, Classified by whether or not the Interval Between Utterances was Less than or Greater than 89rain (the Distribution of Intervals Exceeding 89rain is also Given for Repetitious of the Two Song Types)

Following A A

Following Q T

Other

Q

P

Other

<89rain

1039

7

1

<89min

489

9

2

> 89min

16

1

7

> 89min

8

2

7

Distribution of intervals >89min for repetitions of A Time (rain)

Frequency

Distributions of intervals >89 rain for repetitions of Q Time (min)

Frequency

1 -3

7

1 -3

4

3-6

4

3-6

2

6-9

2

6-9

2

9+

3

Total

16

Total

8

LEMON & CHATFIELD: ORGANIZATION OF SONG IN CARDINALS Table VIII, under sections 'Following A' and 'Following Q', enables us to determine other aspects of the effect of the length of the interval between successive utterances. If the interval is less than 89 rain, the song type is almost certain to be repeated, but if a switch occurs it is most likely to be to an associate. On the other hand, if the interval exceeds 89 min, there is a much higher probability of switching, especially to a non-associate. Long intervals of silence can occur within bouts as shown at the bottom of Table VIII. Nevertheless, long intervals are exceptional. Discussion From the results presented it is obvious that singing in cardinals is highly organized behaviour. The repertoires of sounds used in the songs are grouped into definite permutations called song types, and the sequence of the song types tends to follow a first order Markovian relationship in that the occurrence of a particular song type is determined to a great extent by the previous song type. This relationship is most often revealed in frequent associations between pairs of particular song types. The relationship is not fixed but rather it is most probable if the time interval at the switch in song types is relatively short. There can be considerable variations between different birds in the probabilities of associations between song types, even where the associations are the same. The two levels of association, that is between syllables within song types and between song types, appear to reflect the same underlying process. This is suggested especially from the data of Chambers where the closely associated song types Q, D and A, T are often combined also within the same utterances as song types QD, DQ and TA. Yet there are some quantitative differences in the two levels of association. Within a song type most associations are oneway in that one syllable type normally precedes the other, as in EM. Occasionally there are exceptions to this: another bird, Western Road 1961 to 1963 alternated U and Y in each utterance before singing a final group of Ys as in UYUYUYYYYY. Perhaps the basis for the one-way tendency is the habit of accelerating the singing of syllables as the utterance progresses. Such a tendency appears to favour the use of longer syllables at the beginning of an utterance, followed by shorter syllables. In Texas most song types consist of two kinds of syllable and the longer is usually the first (Lemon &

13

Herzo.g 1969), while in Ontario the most widely occurnng two-syllable song types have the longer first, although here there are many local exceptions (Lemon 1966). Where the song types of Chambers were closely associated such that the two associates were combined in the same song type, with Q and D there was no obvious tendency for one to precede the other, whereas with T and A, TA was the only permutation noted. Therefore, in these instances there is no consistent relationship between the sequence and duration of syllables. In contrast to the associations within song types, associations between song types tend to be two-way, although the directions of the dependencies are not necessarily equiprobable, as may be seen for the bird Chambers in song types A, T and D, Q (Table I). In addition to the associations between song types, the habit o f repeating utterances of the same song type in long sequences is further evidence of organization. These bouts are often too long to be explained by a model in which successive utterances are considered independent. Instead, the model must also incorporate the idea of conditional probability that once a song type is uttered it will tend to be repeated. This conditional probability is found to be generally much higher than the non-conditional probability of that song type. Further complications arise from the fact that the conditional probability does not appear to remain constant as the bout progresses. Hinde (1958) in reference to the chaffinch, Fringilla coelebs, suggested that the alternation between song types might be discussed in terms of two opposing tendencies, one a facilitative tendency to repeat utterances of the same song type, the other to inhibit such repetitions. We may apply these ideas here, even more broadly. Evidence for facilitation may be seen in the tendency for utterances to increase in the number of syllables, especially at the beginnings of bouts following periods of silence. Eventually this tendency is inhibited and the number of syllables reaches an asymptote, although occasionally the inhibition is removed and exceptionally long utterances occur. This inhibition normally lasts only a few seconds and the song type is again repeated. Actually within each utterance there is further evidence of facilitation in the gradual shortening of the intervals between successive syllables. Facilitation is seen further in the tendency to repeat utterances in series. Here the facilitative

14

ANIMAL

BEHAVIOUR,

tendency is reflected in the increase in conditional probability as the bout progresses. This tendency is eventually inhibited also, as refleeted in the eventual decline of the conditional probability. What determines the end of a bout and the switch to another song type ? There is no evidence of different motivational states being related to any of the particular song types. Instead the changes must be related to the particular song type being sung or to other song types, especially to a closely related associate, or both. If the song type has been sung in long bouts with no frequent switches to associates, and there is a long interval before the next song type is sung, the inhibitions related to that particular song type itself seem more probable. When the associations are close, mutual inhibitions which could rise and fall rapidly would result in the alternations between song types, sometimes resulting in the combination of the two associates within the same utterance, as was apparent in the bird Chambers. This, of course, is essentially the same process as the switch within a song type where two types of syllable are involved. Conceivably self-inhibition of a particular song type could have longer lasting effects than cross-inhibitions, the former taking longer to establish but once done so taking longer to dissipate. The result would be that at any instant of time the thresholds of the song types would differ so that should the motivation for singing occur only one song pattern would be selected, that with the lowest threshold. On the other hand, once a song type has been sung, it will be some time before it is sung again, with the overall result that the total repertoire is used. The model can be elaborated further, but we shall not attempt to do so here. Many features of song in the cardinal are exhibited by other members of the family Fringillidae and even members of other families, although the organization is by no means universal. Among comembers of the subfamily Richmondeninae, the closest relative studied, the pyrrhuloxia, Pyrrhuloxia sinuata, a species which might well be considered congeneric, sings in much the same way, differing mainly by using more song types consisting of one syllable type only (Lemon & Herzog 1969). Some members of the genus Passerina, also of the Richmondeninae, have more than one songtype

19,

1

per bird, consisting of different types of syllables, often with some repetition of each type in any utterance (Thompson 1968). Other less closely related fringillids such as the Oregon junco, Junco oreganus (Konishi 1964), and chaffinches (Thorpe 1961) develop definite song types with repetitions of syllables, the latter species also sharing song types among neighbours. The sequences of song elements have not been studied in many cases since the labour involved is often prohibitive, especially where there may be many elements in the sequence. The most relevant study is that of the mistle thrush, Turdus viscivorus (Isaac & Marler 1963), a member of the Turdidae. In this species the utterances consist of distinctive syllables which are usually not repeated. The sequence of syllables is to a great extent predictable but there is some uncertainty which correlates well with the intervals of time between syllables, the shorter intervals resulting in more predictable events. Between utterances, no sequences of particular syllables were particularly frequent, although the authors suggest that the sequence appears to be ordered in some way. Appendix I: Testing Off-Diagonal Observations in a Transition Matrix for Independence

Since two successive bouts are, by definition, different, the transition matrices showing the number of times a bout of song type i is followed by a bout of song type j are exactly the same as Tables I and II except that the diagonal terms are zero. The purpose of this appendix is to describe a method of analysing the off-diagonal observations to see if they are in some sense independent. We will need the following notation. Let P(i) denote the probability that a bout is of song type i. Then 27 PO) : 1. Here we assume that i

these probabilities are independent of time, so that the sequence of bouts is a stationary process. Let PO, J) denote the probability that in two successive bouts, the first is song type i and the second song type j. A similar definition applies to P(i, j, k); and so on. These joint probabilities must be carefully distinguished from the transition or conditional probabilities which will be denoted as follows. Let P(j/i) denote the conditional probability that a bout is song type j given that the previous bout was song type i. (This is sometimes denoted by Pi(J) or Pij.) Let P(k/i, j) denote the conditional probability that a bout is song type k given

LEMON & CHATFIELD: ORGANIZATION OF SONG IN CARDINALS that the two previous bouts were song type i followed by song type j. And so on. The main problem is to decide what is meant by a random or 'independent' sequence of events when two successive events are always different. One solution would appear to be that the assumption that the transition probabilities P(j/i) are proportional to PO). Then: P010 = kP(j) (j # i). (1) But the transition probabilities are subject to the restriction: 27 P ( j i i ) = 1

J j~i which implies that k = 1 / ( 1 - P O D . This would mean that the joint probability P(i, j) is given by:

P(i, j) = P(i) P(jli) = p(i) Pfj) / (1-P(i))

(i ~ j).

(2)

But the fact that the process is stationary implies that: Z' e(i, j) = P(j) i i#j and this equation is not satisfied by formula (2). Thus the assumption contained in equation (1) is inadmissible. An alternative, more fruitful, approach is simply to assume that the observed sequence of unrepeated events is obtained from a sequence of truly independent events by removing any repeats. Let P*(i) denote the probability of song type i in the original sequence. After removing repeats, the joint probability of outcomes i and j, assuming that the original sequence is independent, is given by: P(i, j) = P*(i) e*(j) / (1-27 p*(k)2). (3)

k Then we find 27 P(i, j) = 1 as required. i,j i#j. The important point to notice about equation .(3) is that the joint probability can be expressed in the form: P(i, j) = ai bj (i r j), (4) where the quantity a i depends only on the first bout and bj depends only on the second bout. However the transition probabilities, P(jli) are proportional to P*0) and not P(j). This is similar to the idea of quasi-independence proposed by Goodman (1968).

15

In order to test the above independence hypothesis, it is necessary to estimate the values of P*(i) or equivalently of { ai} and { bi }. In the latter case, there appear to be more unknown parameters to estimate. But we note that the quantities defined by equation (4) are not unique since if { ai } and { bj } satisfy equation (4), then so will { ka i } and { bj/k }. In order to get a unique solution and to give the quantities a physical meaning we take a i = P*(i) in which case bj = P*(j) / (1-27 p*(k)2) = aj / (1-27ak2). k Then the number of independent parameters is the same as before. The values of { ai }, and hence of { bi }, will be estimated from the observed frequencies. Let nij = observed number of pairs in which a bout of song type i is followed by a bout of song type j (i # j). n = total number of switches. ni. = Z nij = number of pairs in which the first bout is song type i J j#i n.j = 27 nij = number of pairs in which the i second bout is song type j. i#j Thus n =~ 27 nij = Zni. = 27nj.. i,j i j i~j As in an ordinary contingency table, the unknown parameters will be estimated from the row and column sums. Then the expected frequency of pairs, in which song type i is followed by song type j, is given by: eij -----n d i bj (i ~ j), and these values may be compared with the observed frequencies { nij }. From equation (4) we find e(i) = 27 a i bj ~ .27aj bi, (for j # i)

J

J

and we also notice that the joint probabilities are symmetric; that is aibj ~ ajb i. This symmetry suggests using a symmetric estimate for P(i), and the simplest is: /5(i) = hi. + n.j. 2n Then estimates of a i, and hence of bj, may be found by solving the following c equation in c unknowns, where c is the number of different song types (c/>3):

16

ANIMAL 27d it~j = hi" q- n.i

j

2n

(i = 1 to c),

BEHAVIOUR,

(5)

jr subject to the conditions Zdi= 1 i and Sj = dj [ (1-27 dk2 ). k The solution of these equations is not an easy task except in certain simple cases. For example, in the special case when all the row and column totals are the same, we find: di = 1 [ c (for all i) and ~j = 1 / (c-l) (for all j) so that eij is given by eij = n ai bj = n / c (c-l), (6) which is of course the intuitive result in that all the values of eij are equal. It has not proved possible so far to develop any general theorems concerning the existence of solutions of equations (5). However solutions can usually be obtained by numerical iteration; see Goodman (1968). We have not attempted a numerical solution of equation (5) but have taken the following to be an approximation to the true value: eij (1) ~ ni. n.j ] (n-nj.). This quantity was suggested by equation (2) and has the required property that: .Z. eij(1) = n. 1, j i~j In the case of equation (6), eij (1) is exactly the same as eij. Certain other simple cases were investigated and provided that the row and column totals are of the same order of magnitude, as they are in Table II, the values of eij (1) will be nearly equal to the values of eij. However, if there is a large variation in the row and column totals, then there may be a substantial difference between eij (1) and eij and it is advisable to find the values of eij by iteration. The goodness-of-fit test statistic was here taken to be: 27 (nij -- eij (1))2 i, j eij (1)

i#j

19,

1

and this will be (very) approximately %2 with (c2 -- 3c q- 1) D.F. In fact as the true values of eli have not been obtained, the expected value of the test statistic will be larger than the number of degrees of freedom and so the %2 approximation will be poor. This means that the level of significance of the observed test statistic must be treated with caution. It is only an approximation to the true value and it is suggested that the null hypothesis should only be rejected if the test statistic is highly significant (P<0.01). The justification for using a %2 test on ordered data depends on the work of Bartlett (1951) (see also Chatfield 1969).

Appendix H: Testing First Order Markovian Dependency A sequence of events is a first order Markov chain if the probabilities of the different events depend only on the immediately preceding event and not on earlier events. If the independence hypothesis described in Appendix I is rejected, the first order Markov chain is a suitable alternative model to describe sequential dependencies. This appendix discusses how to test whether or not successive bouts of song form a first order Markov chain. A triplet is here defined to be a sequence of three bouts, where, by definition, the second song type is different from both the first and third song types. A straight forward method of testing first order Markovian dependency is to calculate the expected frequencies of different triplets, assuming the first order Markov property, and to compare these expected values with the observed frequencies. The notation used here is the same as in Appendix I. The sequence of bouts is a first order Markov chain i f P (k[i, j) does not depend on the value of i, so that P (kLi,j) = P (klj). Then the expected frequency of triplets in which song type i is followed by song types j and k (i ~j, j ~ k) is given by: eijk = N3 e(i, j, k), = N3 P(i, j) P(kli, j), = N3 P(i, j) P(k[j), when N 3 -~ total number of triplets f n -- 1 if all the observations are in one = ~ n sample -- p if there are p lists or sub-samples. The normal method of comparing the estimated values of eijk with the observed frequencies nijk is by means of a Z 2 goodness-of-fit test

LEMON & CHATFIELD: ORGANIZATION OF SONG IN CARDINALS (Chatfield 1969; Chatfield & Lemon 1970). But with ten behaviour patterns the degrees of freedom will be so high that the Z 2 approximation will not be valid. Thus we limit ourselves here to a visual comparison of individual values. Estimates of P (i, j) and e (k/j) may be obtained from the transition matrix of pairs by: P (i, j) = nij [ n P (k/j) = njk / nj. to give eijk = N3 nij njk / nj, n. The observed and expected values will rarely agree exactly, and to assist the comparison it is helpful to construct a probability interval within which the value ofnij k should lie if our hypothesis is correct. In estimating this probability interval, we must take into account that nijk is correlated with nij and njk and in particular must be less than both. I f the sequence really is first order Markov, it can be shown that the observed number of triplets of types i, j and k, will be approximately binomial with parameters nij. and P(klj) where nij = f nijk ~-~ nij N3 / n -"- nij" Estimates of the probability interval may be found by taking P (klj) ---- njk / nj. I f the mean value of the binomial distribution is bigger than about 10, we can in turn approximate the binomial distribution with a normal distribution with the same mean and variance. Then the approximate 95 per cent probability interval for nijk is given by eijk :1:1"96 C eijk (1 - - njk. ) nj. / Alternatively if nij > 20 and P(kJj) < 0.15, we can approximate the binomial distribution with a Poisson distribution with the same mean. But if neither the normal or Poisson approximations are applicable, then the binomial distribution itself must be used. F o r both the binomial and Poisson distributions the approximate probability interval lies between r l and r2 where: Prob (nijk ~> r 1) --= 0.975, and Prob (nijk ~< rE) = 0"975. Integers r l and r2 can be found by evaluation or from the appropriate table. Tables of binomial and Poisson distributions are given for example

17

by Freund (1962). Note that if eijk is less than about 5, the above approximate probability interval will not be reliable. The true limits will be narrower than those obtained from the binomial distribution but have not been calculated owing to the amount of arithmetic involved.

Acknowledgments This study was supported by the National Research Council of Canada. Dr John Stanley did the computer programming. REFERENCES Bartlett, M. S. (1951). The frequency goodness-of-fit test for probability chains. Proc. Cambridge Philos. Soe., 47, 86-95. Cane, V. R. (1961). Some ways of describing behaviour. In: Current Problems in Animal Behaviour (Ed. by W. H. Thorpe and O. L. Zangwill). Cambridge: Cambridge University Press. Chatfield, C. (1969). Statistical techniques for examining sequential dependencies in a series of events. Research Report. Math.] S] l. Bath, Bath University of Technology Press. Chatfield, C. & Lemon, R. E. (1970). Analysing sequences of behavioural events. J. theoret. Biol., 29, 427445. Freund, J. E. (1962). Mathematical Statistics. New York: Prentice-Hall. Goodman, L. A. (1968). The analysis of cross-classified data: independence, quasi-independence, and interactions in contingency tables with or without missing entries. J. Am. Statist. Ass., 63, 10911131. Hinde, R. A. (1958). Alternative motor patterns in chaffinch song. Anim. Behav., 6, 211-218. Isaac, D. & Marler, P. (1963). Ordering of sequences of singing behaviour of mistle thrushes in relationship to timing. Anita, Behav., 11, 1792188. Konishi, M. (1964). Song variation in a population of Oregon juncos. Condor, 66, 423-436. Lemon, R. E. (1965). The song repertoires of cardinals (Richmondena cardinalis) at London, Ontario. Can. J. ZooL, 43, 569-569. Lemon, R. E. (1966). Geographic variation in the song of cardinals. Can. J. Zool., 44, 413-428. Lemon, R. E. (1968). The relation between organization and function of song in cardinals. Behaviour, 32, 158-178. Lemon, R. E. & Herzog, A. (1969). The vocal behaviour of cardinals and pyrrhuloxias in Texas. Condor, 71, 1-15. Thorpe, W. H. (1961). Bird-song: The Biology of Vocal Communication and Expression in Birds. Cambridge: Cambridge University Press. Thompson, W. (1968). The songs of five species of Passerina. Behaviour, 31, 261-287. Williamson, E. & Bretherton, M. H. (1964). Tables of the logarithmic series distribution. Ann. Math. Statist., 35, 284-297. Wilks, S. S. (1962). Mathematical Statistics. New York: Wiley. (Received 17 March 1969; revised 10 November 1969; MS. number: 879)