Appendix
EXPERIMENTAL DESIGN FOR RESEARCH ON FISH NUTRITION

Comparative feeding trials are basic to nutrition research with animals, including fish. In these trials, comparable groups of animals are subjected to specified treatments over a predetermined period of time, and the results or consequences of these imposed treatments, as measured by one or more criteria, are used as the basis for estimating nutritional values of feeds, or the nutritional needs of the animals, or the elucidation of the function of some part of the normal metabolism of the animals. The most widely used experimental plan for answering questions in applied nutrition is “factorial design”.

1. Factorial Design - Basic Concepts

Factorial design in nutrition research is essentially a scheme for obtaining as much information from the experiment as possible by taking advantage of the fact that if those causes of variation between animals or groups are balanced, they “cancel out” and do not disturb differences between such groups with respect to experimental treatment imposed on one of them.

As an example, consider the hypothetical case of a feeding test involving tilapia in which 3 experimental dietary treatments are to be compared. For simplicity, suppose 48 of the fish used are grouped into 3 lots of 16 fish each. Assume further that these 3 lots of fish, that averaged out as nearly alike as possible with respect to sex, age and genetic origin, recorded live weight gains over the experimental period as shown in Table 1a.

Table 1a

GAIN OF INDIVIDUAL TILAPIA FROM A FEEDING TEST INVOLVING A COMPARISON OF THREE DIETS

By grouping the test animals in this fashion, it was thought that any sex effects had been cancelled out. However, by so doing, the possibility of measuring how much sex may be a factor in the diet effect has inadvertently been lost. On the other hand, males and females could have been grouped separately, as in Table 1b and information obtained as to the effects of sex.

Table 1b

INCREASED COMPARISONS POSSIBLE BY SUB-GROUPING THE DIET LOTS INTO MALE AND FEMALE TILAPIA

Case I

Treatment

In Table 1b, the effect of sex in this test has been factored out and measured. In addition, whether the difference in the diets are affected by, or depend on, sex can now be observed. This information might be important if, in another case, all male or all female tilapia were to be fed.

Thus, it is noted that male tilapia have, on the average, gained more than the females -a difference of 1.27 vs. 1.21 g/day, or about 5 percent. This is consistently so in each lot; hence, the sex of tilapia will not substantially alter the general differences between the diets. However, there is about as much difference between sexes as there is between rations, and unless sexes are balanced, false conclusions regarding diets might be made. For example, if the diet of Lot I was fed to male tilapia, and that of Lot II to female tilapia, the sex effect would make some of the real ration differences. Or, if females were fed the Lot I diet and compared with males fed Lot II diet, the differences between ration and sex would be added to give a high false figure.

“Factoring” has increased the information to be had from the 48 tilapia. The only penalty is in the reduced numbers of animals per sub-group, and hence a possible reduction in the reliability of the averages in the smaller groups.

Now, it can be further assumed that the 48 tilapia were the progeny of two spawners, and that the lots had been “balanced” in this respect also. By segregating the fish in accordance with this possible cause of variation, it is possible to extract a little more information from them. Also, if it were assumed that there had been a difference in spawning dates so that, again, half the fish were, say, two weeks older than the others - this factor of initial age had been “balanced” in the 3 original lots, it may also be factored out. This would mean the raising and feeding of the 24 pairs of tilapia in separate aquaria according to similarity in sex, age and breeding, as illustrated in Table 1c.

Table 1c

GAINS OF TILAPIA GROUPED ACCORDING TO DIET, SEX, AGE AND BREEDING

Case II

The main comparisons of the first hypothetical factorially designed experiment are shown in Table 1d.

Table 1d

INDEPENDENT COMPARISON OF GROUPS AND THE EFFECT OF FACTORING ON THE DEGREES OF FREEDOM FOR ERROR (RESIDUAL)

Case III

A generalization can now be made from diet comparisons to include quantitative statements of the effects which sex, or initial age, or the breeding of tilapia may have in modifying the general effects of the rations themselves. These interactions are, in some trials, of greater use in avoiding the pitfalls of erroneous conclusions than in the standard deviation for “experimental error”.

The extent of the extra information the factorial designs have yielded is specifically indicated in the statement of the variance analysis which can be made in each case. Each fraction of the variance which has been partitioned can be tested for significance against that for “within” group (Table 1e).

The limitations of the simple design of Case I becomes clearly evident when the variance analysis is examined.

Case II is somewhat more efficient, showing that significant differences exist not only between the three diets, but also that sex affects the extent of the response to tilapia, and that there is no interaction between sex and kind of diet.

Case III represents a more complete analysis and brings out the fact that there are real differences in weight gains due to initial ages of fish, and to the spawner from which they come. In addition, the more detailed accounting for the causes of variation and, hence, a smaller unaccounted-for variance between fish “within groups”, it becomes evident that there is a tendency for sex difference to be different on one diet than on another. Thus, there is almost as much difference between male and female tilapia on diet III as on diet I. There is also a slight interaction between sex and spawner.

Table 1e

PARTITION OF THE VARIANCE OF DAILY GAINS AND THE RATIOS OF ERROR TO TREATMENT VARIANCES ACCORDING TO THE EXTENT OF THE “FACTORING OUT” OF THE VARIABLES POSSIBLE IN THE THREE DESIGNS

It can be generally concluded that the extent of difference in the weight gains of the fish on the three diets would depend to some degree on the sex of the fish, and this, in turn, could be modified a little by the genetic origin. On the other hand, it is certain that the differences in age are not complicated by sex or genetic differences.

A more complete accounting for the portion labelled “residue” is provided by the analysis of variance.

In Case I, all the variance due to sex, age, genetic origin and their interactions is included in the “residue” item. Further partitioning is a matter of breaking up the “residue”. The result, even with the consequent reduction in the degrees of freedom (df), is a smaller “error” variance and standard deviation for the test. This is desirable insofar as tests for the significance of differences between means of groups of the same size depend directly on the magnitude of the standard deviation. By accounting for as much of the variation as possible by complete partition of the variance this is achieved.

The particular grouping of the tilapia in the first example is a “3 × 2 × 2 × 2 factorial design”. Each of the numbers indicates the number of primary groupings or comparisons (the simplest factorial design is a 2 × 2). This example has been chosen because it illustrates two ways of dealing with three possible causes of variation in the response of fish on feeding trials. In the old method of allotment (Case I), an attempt is made to eliminate or equalize the effects of these factors by “balancing” them between lots. The second method (Cases II and III) separates them by sub-grouping, so that advantage may be taken of the extra information their quantitative consideration may yield.

2. Application of Factorial Design to Comparative Feeding Trials

As an example of application of factorial design in feeding trials with fish, let it be supposed that an experiment with common carp is intended to study the effects of three levels of protein, and that water temperatures (cool vs. hot season), and sex differences are suspected of influencing the results. Using comparable fish (i.e., of the same age) the test might be designed as in Table 2a.

Table 2a

ALLOTMENT PLAN FOR DIETARY PROTEIN TEST
(100 FISH PER LOT - 2 400 FISH TOTAL)

^a Denotes lot number

Here there is a 3 × 2 × 2 × 2 factorial design as before, with imposed treatments so grouped to cover all possible comparisons and, thus, to yield maximum information from the numbers of animals used. In this trial, the data to be collected might include weight gains and feed consumption, digestibility of the diets, and whatever carcass measurements were desired. For each criterion, the data would be entered on a form corresponding to the allotment plan, in order to facilitate its orderly examination by whatever methods were to be employed.

2.1 Stocking Density

There have been suggestions that a relationship exists between stocking density and short interval fish growth at different fish weights (that larger, or older, fish more readily suffer from the adverse effects of increases in stocking density). In this hypothetic study on the effects of dietary protein level on growth of the carp, the 100 fish in each feeding lot could be partitioned according to three stocking densities. It must be recognized also that the test must be conducted in two parts - one during the cooler months, and one during the hotter months of the year. The effects of stocking density is best seen in the plan of the analysis of variance shown in Table 2b.

2.2 Analysis of Variance for Multifactor Design

The computations involved in analysis of variance may be demonstrated by using the numerical data of the hypothetical feeding test with tilapia illustrated earlier. Before doing so, however, it might be appropriate at this point to familiarize the reader with the symbols and formulae used in such computations (Table 3a) and the setting up of tables of analysis of variance (Table 3b).

Table 2b

PLAN OF ANALYSIS OF VARIANCE ACCORDING TO FISH GROUPING USED

Table 3a

SYMBOLS AND FORMULAE USED IN ANALYSIS OF VARIANCE

Common Symbol	df	Σ(y2)	Σ(y-y)2	S2	S
Common name	Degree of freedom	Sum of squares	Mean square	Variance	Standard deviation
Actual meaning		Sum of squared deviations from zero	Sum of squared deviations from mean
Formula by definition	n-1	Σ (y2)	Σ(y-y)
Formula actually used	n-1	y2+y22+…y2n

Table 3b

ANALYSIS OF VARIANCE FOR A 3 × 2 × 2 × 2 FACTORIAL

A multi-factor design is said to be “orthogonal” if the estimations of parameters associated with any one factor are uncorrelated with those of another. This definition is partially satisfied when each level of one factor appears with the same frequency as the levels of a second factor. Factorial experiments, by definition, are orthogonal, and the Latin square design, of which the present example is one, is likewise orthogonal. Orthogonal designs possess patterns which permit the development of formulae which eliminate the necessity for repeated fittings of linear models in analysing data and, thus, reduce the computational work involved in analysing factorially designed experiments. A most important property of orthogonal designs is that the total sum of squares of deviations of the observations about their mean

Σ(y-y)²

can be partitioned into the important sums of squares, plus SSE, i.e.,

Σ(y-y)² = SS (blocks) + SS (treatments) + SSE

and SSE = Σ(y-y)² - SS (blocks) - SS (treatments)

An important relationship between the sums of square expressions in analysis of variance is:

Σ(y-y)² = Σy² - CM

where, CM, called the correction for the mean,

where n is the total number of observations.

Having identified and defined the common terms used in analysis of variance for multi-factorial experiments, the data obtained from the hypothetic feeding experiment with tilapia can be analysed.

Second and third order interactions do not lead to meaningful interpretation and their sums of squares will not be computed here, although their combined degrees of freedom have to be computed for determining residual degrees of freedom. This works out to be 9 degrees of freedom for a residue of 24 degrees of freedom.

To determine the error mean square, the total sum of squares first has to be determined which has the following relationship.

Mean squares values are obtained by dividing the sums of square of the factors by their respective degrees of freedom.

F values are then determined by dividing the mean squares of the factors by the error mean square. Table 3c shows the F values of the factorial analysis of the hypothetic example.

Table 3c

TABLE OF ANALYSIS OF VARIANCE OF HYPOTHETIC DIETARY EXPERIMENT WITH TILAPIA