The use of analysis of covariance in the

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J. Fish BioI. (1984) 24, 201-213

The use of analysis of covariance in the back-calculation
of growth in fish
J. R. BARTLETT,P. F. RANDERSON,
R. WILLIAMS
AND D. M. ELLIS*
Department of Applied Biology and *Department of Mathematics, UWIST,
King Edward VII Avenue, Cardiff
(Accepted I 7 May 1983)

I. INTRODUCTION
Techniques for the back-calculation of the growth histories of fish populations

have been developed since the beginning of this century. Notable among the early
workers were Dahl ( 1909), Lea ( I 9 lo), Lee (1 9 12) and Monastyrsky ( 1930). Reviews of the subject are contained in Lee (1920), Graham (1 929), Lagler ( I 956),
Hile (1970), Weatherley (1972) and Bagenal & Tesch (1978). The basic process
remains unchanged from that used by Dahl and Lea and comprises three stages:
( 1) An indicator calcified structure such as a body scale, otolith, cleithrum or
cross-section of a bone is chosen. This should contain marks indicating yearly
increments of growth providing a clear representation of the growth history of the
structure and, by inference, of the fish.
(2) Data are collected, from a sample of fish, on body length and of indicatorstructure size, e.g. scale radius along a particular axis, at the time of capture.
(3) A plot of body length against structure size is used to establish the form
of the relationship. This may be through or not through the origin, curved, sigmoidal or a complex of different linear or curved relationships (Wooland & Jones,
1975).

This paper relates to the problems associated with the use of linear regression
analysis when attempts are made to establish the body length : indicator-structure
relationship. Linear regression analysis may be inadequate and produce inaccurate results for two reasons; first, it is an attempt to fit a straight line through the
data regardless of the true form of the relationship and second, it does not allow
for objective comparison between the fit of regression lines with alternative transformations of the same data. Carlander (1981) has noted that a sample of fish,
taken at one time, contains a series of year classes and deviations from a linear
or curved body length :indicator-structure relationship may arise unless

regressions are calculated for each age class.
These problems are illustrated by considering a case in which a plot of body
length and indicator-structure data shows a slightly curved trend (Fig. 1).
Although linear regression analysis of these data may show a significant relationship, the fitted line would not accurately follow the curve shown by the data. In
many cases analysis of the body length : indicator-structure relationship would
20 I
0022-1 I12/84/020201+ 13 $03.00/0

0 1984 The Fisheries Society of the British Isles

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J. R. BARTLETT ET A L .

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0

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25

I

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50

75

Scale radius (arbitrary optical unlts)

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FIG.1. Relationship between body length and scale-radius from a sample of81 roach, Rufilus ru/ilus.
showing regression lines fitted to the untransformed data (dashed line) and to the age-group
means of the logarithmically transformed data. 0: 8 I individual fish; 0:means for age groups
1-9.

end at this point and back-calculation would be performed using one of the
expressions (I), (2) or ( 3 ) .
sn

L,, =-L
S

( I ) Lea(1910)

(2) Lee (1 920)

L, = a+b S,

( 3 ) Fraser (1 9 16)

where L= length of fish when sample was obtained; L,= body length at the time
of formation of the nth annulus; S,=length of indicator structure at the nth
annulus; S= length of indicator structure (e.g. total scale radius) when sample was
obtained; and a and b are parameters of the regression line.
The linear regression method will not indicate whether a transformation is
necessary and, if so, the type. Transformation may be needed to allow for a curved
relationship (Fig. l), for non-constant variation about the relationship, or for both.
A second regression analysis would fit a straight line through the transformed data
and analysis of variance (ANOVA) might show an improved level of significance.

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203

B A C K - C A L C U L A T I O N OF GROWTH

Unfortunately, it would not then be possible for an investigator to determine
which of the regressions offered the more accurate description of the true body :
indicator relationship and hence which of the two sets of back calculated results
was the more realistic. If ANOVA of the two regressions both show a significant
F-ratio, it is not then possible to say that one is the more significant and likely
to be reproducible in further samples. Therefore the choice of expression for use
in back-calculation could only be made on an arbitrary basis.

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11. THE ANALYSIS OF COVARIANCE

It is in situations such as this that another statistical technique, analysis of
covariance, may hold an advantage over these conventional methods. Analysis
of covariance can be used to derive a clear description of the trends in the data
by comparing the relationship between body length and indicator-structure size
in three ways. Regression analysis is performed first, on the data as a whole,
second on groups of the data divided according to a third variable (covariate), the
age of fish at capture, and finally on the mean values of body length and indicatorstructure size for each age group.

The analysis of covariance distinguishes four models dependent upon the source
of the significant F-ratio. These alternatives are illustrated in Fig. 2 and are
summarized as follows:
(1) The slopes of the lines fitted through the separate subsets of the data differ
significantly from one another [Fig. 2(a)]. Further investigation may indicate a
pattern in this variation as in Fig. 2(b). An overall regression line should not be
fitted to the data as they stand, neither should back-calculation be performed.
However, back-calculation may be possible following transformation of the data.
For instance if the data showed the curved trend of Fig. 2(b), then transformation
to logarithms might, on subsequent analysis of covariance, demonstrate an overall
linear trend.
(2) The slopes of the lines through individual age groups do not differ significantly but their mean values show a significant deviation from a straight line [Fig.
2(c)]. In this case the lines in the separate groups are parallel. Back-calculation’
cannot be performed on the data in their present form.
(3) Where neither of the sources is significant, the slope of the line fitted
through the means may differ significantly from the parallel slope of the lines
within the age groups [Fig. 2(d)]. Such a result suggests a spread ofthe data which
would not be taken into account by the use in back-calculation of an expression
based on the overall regression line. This situation could arise from a pattern
similar to that illustrated in Fig. 2(e). Here the line fitted through the mean values

of all age groups is shown as OM. Assuming that an individual fish were to follow,
throughout its life, a body length : indicator-structure relationship which deviated
from this overall relationship by a constant degree, then the growth of that fish
could be represented by another line such as OB. If all fish followed separate
body : indicator relationships and the line through the means, OM, was merely
a summary of all these lines, then this would account for the spread of the data
seen in each age group. If a further assumption is made that the individual body
length : indicator-structure relationships of all fish in a sample were to share a

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204

J . R. BARTLETT E T A L .

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Indicator-structure size

FIG.2 . An illustration of the possible models distinguished by analysis of covariance.+: Mean value
and regression line through the data of one age group. (See text for an explanation of lines
OB, OM and OC).

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common intercept on the ordinate (Fraser, 19 16) then back-calculation could be
performed using expression (2). This expression derives only the intercept term,
a,. from the line through the means. Its other terms comprise the individual’s
own body length : indicator-structure relationship.
These assumptions are similar to those developed by Hile (1941) in work on
the rock bass, Ambloplites rupestris. Having fitted a regression line to the body
length : scale radius relationship, Hile found that some fish had scales which were
significantly larger or smaller than would have been expected from their body

lengths. Making the assumptions that such a fish had deviated to a corresponding
degree from the overal I relationship throughout its life, Hile avoided inaccurate
estimates of the fish’s previous body lengths by correcting the observed annular
radii to conform to their expected values, using the expression:

BACK -CA LCU LATlON OF GROWTH

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where Ao=observed radius of the nth annulus; Ar=expected (or corrected) radius
of the nth annulus; S,=observed scale radius; &,=expected scale radius of a fish
of given body length.
Back-calculation was then carried out using the equation describing the sample

regression line. Whereas the assumptions involved in the use of expression (2)
and in Hile’s correction method (4) are identical, the former has the advantage
in being the more direct calculation.
(4)Where no other source in an analysis of covariance gives a significant
F-ratio then that due to the fit of the overall line should do so [Fig. 2(0]. Backcalculation could then be performed using the equation for the line:

L, = a,+b, S,
where a,= intercept on ordinate (body length), i.e.
a, =

(5)

L, - b, S,

(6)

and b,= slope of the overall regression line.

111. THE METHOD OF ANALYSIS OF COVARIANCE

The steps in the analysis of covariance are given as an aid to those unfamiliar
with the technique. After the basic sums of squares and products have been
obtained from each age group of raw data, calculation of the covariance statistics
is not lengthy.
The first step in an analysis of covariance is to separate the sample data into
subsets according to the age of the fish at capture. The analysis will then examine
the variation of the body length : indicator-size relationship that exists between
successively older groups of fish.
Regression analyses of body length ( y ) on indicator-structure size (x) are then
performed on the data of each subset. The regression statistics are then tabulated
as:

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2x9 ZY>29, zx2,~ X Y Sj(x,x),
,
sj(YFY)v),Si(x,Y)

where S,(x,y)= individual age group’s corrected sum of products.
CXCY

Si(x,Y)= ~

X Y - ___

n

n=number of fish in that age group at capture, similarly:

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J.

R. B A R T L E T T E T A L .

These statistics are then summed across all age groups and the following derived
from them.

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where N=total number of fish in the sample. Sl(x,x)and S,(x,y) are then found
similarly and Al as:
At = S,(Y,Y)-

S,(x,x) is calculated as:

(St( X > Y ) Y

s,( X J )

(9)

S,(X,X) = ~ S i ( X , X )

Sa(x,y)and S,(y,y) are found similarly. Then Aa is calculated as:

Although not directly required in the analysis the slopes of certain lines can
be obtained from the above statistics. For instance the slope of the regression line
through a single age group’s results, bj is:

If it is decided that parallel lines may be used in the subsets of data the common
slope, b,, is calculated as:

Similarly the slope of the regression line fitted through the mean values of all age
groups, b,, is:

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BACK-CALCULATION OF GROWTH

207

TABLE
I. Compilation of the analysis of covariance result table
Degrees of
freedo m

*

Sums of squares

square
Mean

P-ratio

N- 1
I

&CV3Y 1

ss/df

ms/rms

I

At- A m - Aa

ss/df

ms/rms Fig. 2(d)

Deviation of the
means from a
straight line (b,)

K-2

Am

ss/df

ms/rms Fig. 2(c)

Between slopes (hi)

K- 1

A0 - A,

ms/rms Fig. 2(a)

Residual

N-2K

A,

ss/df
ss/df

Source
Total

Due to the
overall
line (h,)

Difference of
B, and h,

-

-

Where N=number of fish in the sample; K=number of age groups present in the sample; ss=sum
of squares; df= degrees of freedom; rns = mean square; rms = residual rns.
*Appropriate model if the F ratio is significant.

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If all results may be fitted by a single regression, the line has the slope b,:
SSXSY)

b, = ___
S/(XJ)

The results table of the analysis of covariance is then compiled in the form shown
in Table I.
As in a table of ANOVA, each mean square term is calculated as the ratio of
the sum of squares by the appropriate degrees of freedom. Analysis of covariance
is then completed by calculation of the ratio of each mean square value to the
residual mean square. The models are tested in the order 2(a), 2(c), 2(d) 2(f) using
the F-ratios indicated in Table I.

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IV. AN EXAMPLE OF THE USE OF ANALYSIS OF COVARIANCE IN
BACK-CALCULATION

Body length, scale radii and annular radii were recorded on a sample of 81
roach, Rutilus rutilus. Linear regression analysis of data for body length 01)on
scale radius (x) gave the following results:
Slope of the regression line = b ' = 0.2856
Intercept of the regression line = a ' = 2.9678
The ANOVA of the regression produced an F-ratio:
Fjq = 3225.2 (P