21 Factorial experiments
Factorial experiments are designed to investigate the effects of two or more factors, or input parameters, on the output response of a process. Factorial experiment design, often referred to simply as factorial design, is a systematic approach that outlines the steps necessary to implement such experiments effectively. A key objective of this methodology is to estimate the effects of various factors on the output of a process while minimizing the number of observations required, thereby optimizing the process outcome efficiently.
In factorial experiments, the effects of varying the levels of different factors influencing the process output are systematically studied. Each complete trial or replication of the experiment considers all possible combinations of the levels of these factors. An effective factorial design ensures that the maximum amount of information about the influence of input variables on the output is obtained with the minimum number of experimental runs.
For example, a factorial experiment in a study on the rooting of cuttings might involve two factors, each at two levels, such as two types of hormones applied at two different doses. This setup is known as a \(2 \times 2\) (or \(2^2\)) factorial experiment. The treatments in such an experiment would include four possible combinations, corresponding to all the level permutations of the two factors.
The total number of treatments in a factorial experiment is determined by the product of the number of levels for each factor. For instance, in a \(2^2\) factorial design, the number of treatments is \(2 \times 2 = 4\). Similarly, for a \(2^3\) factorial design, the treatments total \(2 \times 2 \times 2 = 8\). As the number of factors or their levels increases, the total number of treatments grows exponentially. For example, a factorial experiment involving five clones, four spacings, and three weed-control methods would require \(5 \times 4 \times 3 = 60\) treatments.
The rapid increase in the number of treatments with added factors or levels underscores the need for cautious use of factorial experiments due to their size, complexity, and associated costs. Undertaking overly large experiments without preliminary investigation may lead to challenges in financing, securing adequate experimental areas, and controlling factors like soil heterogeneity.
A tree breeder, for instance, may collect 30 clones from a neighboring country to study their adaptability to the local environment, which is variable in soil fertility, moisture levels, and other factors. Ideally, such an experiment could involve a factorial design, combining these 30 clones with factors like fertilizer, moisture levels, or population density. However, including just one additional factor, such as nitrogen with three levels, would inflate the treatments to \(30 \times 3 = 90\), leading to significant practical difficulties in execution.
A more efficient approach would involve initial single-factor experiments to screen the clones. This preliminary study might identify a subset of five promising clones. These selected clones could then be tested in a factorial design with three nitrogen levels, reducing the number of treatments to \(5 \times 3 = 15\) rather than the original 90. This strategy not only simplifies experimentation but also optimizes resource allocation while maintaining robust insights for subsequent detailed studies.
The change produced in the process output for a change in the “level” of a given factor is referred to as the main effect of that factor. Table 21.1 provides an example of a simple factorial experiment involving two factors, each with two levels. These levels are commonly denoted as “low” and “high,” symbolized by “-” and “+” in factorial designs, respectively.
| A (-) | A (+) | |
|---|---|---|
| B (-) | 20 | 40 |
| B (+) | 30 | 52 |
The main effect of a factor is essentially the average change in the output response as that factor transitions from “-” to “+”. It is calculated as the average of two values: the change in output when the factor goes from low to high while the other factor remains low, and the change in output when the factor goes from low to high while the other factor remains high.
For the example in Table 21.1, the output is 20 (minimum) when both A and B are at their “-” levels, and 52 (maximum) when both A and B are at their “+” levels.
The main effect of A is calculated as follows. When B remains “-”, the change in output as A transitions from “-” to “+” is \(40 - 20 = 20\). When B remains “+”, the change is \(52 - 30 = 22\).
Thus, the main effect of A is the average of these changes:
\[\text{Main effect of A} = \frac{20 + 22}{2} = 21\]
Similarly, the main effect of B is calculated as follows. When A remains “-”, the change in output as B transitions from “-” to “+” is \(30 - 20 = 10\). When A remains “+”, the change is \(52 - 40 = 12\).
Thus, the main effect of B is:
\[\text{Main effect of B} = \frac{10 + 12}{2} = 11\]
In this example, factor A has a greater influence on the process output, with a main effect of 21 compared to factor B’s main effect of 11.
Aside from main effects, factors can also exhibit interaction effects. Interaction effects occur when two or more factors interact, causing changes in the process output. These effects become significant when the influence of one factor depends on the levels of another. In Table 21.1, since the effects of A (B) vary across the levels of B (A), factors A and B are interacting.
Interaction is defined as the failure of the differences in response to changes in the levels of one factor to retain the same order or magnitude across all levels of another factor. Alternatively, factors are said to interact if the effect of one factor changes as the levels of another factor are altered.
Graphical representation can illustrate interaction. If two factors are non-interacting, their effects will result in parallel lines on a graph. Conversely, interaction effects will manifest as non-parallel lines.
If interactions exist, as they often do, experiments must be designed to estimate and test these effects. Varying only one factor at a time will not suffice. Instead, multilevel, multifactor experiments are necessary.
Factorial experiments involve running factorial combinations and interpreting the mathematical output responses of these combinations. This allows researchers to identify which factors most significantly affect the process and make improvements or corrections accordingly.
Factorial experiments are defined as experiments in which the effects (main effects and interactions) of more than one factor are studied together. If there are \(n\) factors \(F_1, F_2, \ldots, F_n\), where the \(i^{th}\) factor has \(s_i\) levels, the total number of treatment combinations is given by \(\prod_{i=1}^{n} s_i\).
Factorial experiments can be classified into two types:
Symmetrical factorial experiments: All factors have the same number of levels i.e. si are equal.
Asymmetrical factorial experiments: At least two factors have different numbers of levels. It is also known as mixed factorials.
The term complete factorial is used when the treatments include all combinations of all the selected levels of the factors used. Incomplete or fractional factorials are used when only a part of all the combinations are tested.
Factorial experiments not only provide insights into individual factor effects but also their interactions. They offer the additional advantage of economizing on experimental resources. Compared to factor-by-factor experiments, factorial designs achieve the same level of precision with fewer resources.
21.1 Representation of the levels of the factors
Factors are usually represented by lower case letters like a, b, c, n, p, k etc. and the levels by adding suffixes like n0, n1, n2 etc. For example a nitrogen fertilizer is applied at three doses say 40, 60 and 80 kg/ha then these levels are denoted by n0, n1 and n2 . An irrigation treatment of two levels applied (Without and with irrigation) can be represented as i0 and i1 . The treatment ‘application of 60 kg Nitrogen per ha with irrigation’ will be represented by the combination ( taking corresponding levels from each of the factor as n1 i1). Thus for the above experiment involving application of three doses of nitrogen under the irrigated and non irrigated condition will make up 6 treatment combinations namely n0i0, n1i0, n2i0, n0i1, n1i1, n2i1.
21.2 Main effects and interactions
When the response of a factor of interest is expected to differ under different levels of the other factors, then we have to go for factorial experiments. The interaction effect can be measured only when the factors are tested simultaneously in the same experiment.
Thus for the above six combinations, the effect of irrigation at a lower level is the sum of the combinations i.e. n0i0 + n1i0 + n2i0 = denoted as (I)o and at the higher level of irrigation = n0i1 + n1i1 + n2i1 = (I)1
The difference between the sums (I)0 and (I)1 give a comparison between the response from the two levels of irrigation.
Similarly the comparison among the totals (N)0 = (n0i0+ n0i1), (N)1 = (n1 i0 + n1 i1), and (N)2 = (n2io+ n2i1) represent the main effect of nitrogen. (For these three totals representing three levels of N will give rise to 2 independent comparisons i.e. two degrees of freedom for nitrogen; and the two levels of irrigation will give rise to one degree of freedom for irrigation.
Thus the main effect of a factor represents the average change in response when the level of that factor is increased from a lower to a higher level.
To study interaction we have to find the effect of one factor in the presence of another: say effect of irrigation under different levels of nitrogen: at n0 level: (n0i1 - n0i0) at n1 level: (n1i1 - n1i0) at n2 level : (n2i1 - n2i0)
Comparison among these quantities indicate equality (or inequality) of the effects of irrigation at different levels of n. This sort of comparisons indicates whether the factors act independently or they interact to influence the yield of the experimental unit on which they appear simultaneously. Effects representing such comparisons (n0i1 - n0i0) - (n1i1- n1i0) are called interaction effects. Here we can write two such independent comparisons among the combinations of two factors N and I taken at 3 and 2 levels respectively, to get (3-1)×(2-1) = 2 degree of freedom for the interaction N × I or NI.
Interaction of two factors the failure of the levels of one factor to retain the order and magnitude of performance (within random sampling errors) throughout all levels of the second factor.
(Differences like: (n0i1 - n0i0) are called as simple effects of a factor (say irrigation in this case)
21.3 Factorial experiments 2n series
When we take all possible combinations of n factors each at two levels each, this will make up a 2n factorial experiment. The two levels of a factor are usually denoted by suffixes 0 and 1 attached to the factors represented by small letters. The simplest 2n factorial is obtained when n = 2. i.e. (22 factorial experiment). If the two factors are A and B each at two levels we get a 22 factorial experiment with four treatment combinations represented as
\[a_0b_0, a_0b_1, a_1b_0, a_1b_1\]
If these four treatment combinations are allocated ( at random) to a single block of a Randomised block design, and if there are r such blocks we get a 22 factorial experiment in a Randomised block design.
21.3.1 Analysis of 22 factorial experiments
Let the two factors are A and B each at two levels .Then the four treatment combinations represented as \[a_0b_0, a_0b_1, a_1b_0, a_1b_1\] Then the two main effects of A and B and interaction A×B can be worked out by making use of the following relations Work out the total responses of the treatment combinations \((A)_{i=0} = a_0b_0 + a_0b_1\)
\((A)_{i=1} = a_1b_0 + a_1b_1\)
Then main effect of \(A = \frac{1}{2r}[(A)_{i=1}-(A)_{i=0}]\)
Sum of squares due to \(A = \left(\frac{(A)_{i=0}^2}{2r}+\frac{(A)_{i=1}^2}{2r}\right)-C.F=\frac{1}{4r}[(A)_{i=1}-(A)_{i=0}]^2\)
Similarly, \((B)_{j=0} = a_0b_0 + a_1b_0\)
\((B)_{j=1} = a_0b_1 + a_1b_1\)
Then main effect of \(B = \frac{1}{2r}[(B)_{j=1}-(B)_{j=0}]\)
Sum of squares due to \(B = \left(\frac{(B)_{j=0}^2}{2r}+\frac{(B)_{j=1}^2}{2r}\right)-C.F=\frac{1}{4r}[(B)_{j=1}-(B)_{j=0}]^2\)
For interaction A×B find,
\((AB)_{i+j=0} = a_0b_0 + a_1b_1\)
\((AB)_{i+j=1} = a_0b_1 + a_1b_0\)
(Here the sum of the suffixes are to be reduced modulo 2; i.e. 0 ≡ 2 ≡ 4 etc. and 1≡ 3≡ 5 and so on ) Then interaction effect of \(A \times B = \frac{1}{2r}[(AB)_{i+j=0}-(AB)_{i+j=1}]\)
Sum of squares due to \(A\times B = \frac{1}{4r}[(AB)_{i+j=0}-(AB)_{i+j=1}]^2\)
(Obviously interaction A×B is same as interaction B ×A).
Analysis is same as that of a Randomised block design except that the sum of squares due to treatments will be split into (2n-1) = (22-1) = 3 components each having one degree of freedom; Each of these one degree of freedom will correspond to one main effect or interaction. Each of the corresponding mean square is compared against Mean error sum of squares.
| Source | df |
|---|---|
| Blocks | r-1 |
| Treatments (3) | |
| A | 1 |
| B | 1 |
| AxB | 1 |
| Error | 3(r-1) |
| Total | 4r-1 |
If the interaction turns out to be significant a two way table (factor A versus factor B)of treatment means are prepared and comparison can be made.
In that case Critical difference can be obtained as = \(t_\alpha \times \sqrt{\frac{2MSE}{r}}\)
21.3.2 Analysis of 23 factorial experiments
There will be three factors are A ,B and C each at two levels. Then the eight treatment combinations are represented as
\[a_0b_0c_0, a_0b_0c_1 , a_0b_1c_o , a_0b_1c_1 , a_1b_0c_o, a_1b_0c_1 , a_1b_1c_0 , a_1b_1c_1\] If these eight combinations are allotted at random to a single block of a RBD and if there are r such blocks( replications); we get a 23 factorial design in r blocks. Then there are three main effects A, B and C ; \((^3_2) = 3\), two factor interactions (A×B, A×C and B×C) and \((^3_3) = 1\), three factor interaction (A×B×C).
These can be worked out by making use of the following relations:
The total responses of the treatment combinations representing levels of the factors in the following way. (Sum of suffixes i, j etc. representing levels of the factors A, B etc. to be reduced modulo 2)
\((A)_{i=0} = a_0b_0c_0 + a_0b_0c_1+ a_0b_1c_0 + a_0b_1c_1\)
\((A)_{i=1} = a_1b_0c_0+ a_1b_0c_1 + a_1b_1c_0 + a_1b_1c_1\)
The main effect of \(A = \frac{1}{4r}[(A)_{i=1}-(A)_{i=0}]\)
Sum of squares due to \(A = \frac{1}{8r}[(A)_{i=1}-(A)_{i=0}]^2\)
\((B)_{j=0} = a_0b_0c_0 + a_0b_0c_1+ a_1b_0c_0 + a_1b_0c_1\)
\((B)_{j=1} = a_0b_1c_0+ a_0b_1c_1 + a_1b_1c_0 + a_1b_1c_1\)
Then main effect of \(B = \frac{1}{4r}[(B)_{j=1}-(B)_{j=0}]\)
Sum of squares due to \(B = \frac{1}{8r}[(B)_{j=1}-(B)_{j=0}]^2\)
The main effect and Sum of squares due to factor C also can be found in a similar way.
The interactions also can be worked out by taking the treatment combinations satisfying the respective equations as shown below:
For Example for interaction B×C:
\((BC)_{j+k=0} = a_0b_0c_0 + a_1b_0c_0+ a_0b_1c_1 + a_1b_1c_1\)
\((BC)_{j+k=1} = a_0b_0c_1+ a_1b_0c_1 + a_0b_1c_0 + a_1b_1c_0\)
The interaction effect \(B\times C = \frac{1}{4r}[(BC)_{j+k=0}-(BC)_{j+k=1}]\)
Sum of squares due to \(B\times C = \frac{1}{8r}[(BC)_{j+k=0}-(BC)_{j+k=1}]^2\)
For interaction A×B×C:
\((ABC)_{i+j+k=0} = a_0b_0c_0 + a_0b_1c_1+ a_1b_0c_1 + a_1b_1c_0\)
\((ABC)_{i+j+k=1} = a_0b_0c_1 + a_0b_1c_0+ a_1b_0c_0 + a_1b_1c_1\)
The interaction effect \(A\times B\times C = \frac{1}{4r}[(ABC)_{i+j+k=1}-(ABC)_{i+j+k=0}]\)
Sum of squares due to \(A\times B\times C = \frac{1}{8r}[(ABC)_{i+j+k=1}-(ABC)_{i+j+k=0}]^2\)
The other sum of squares can be obtained in a similar way.
| Source | df |
|---|---|
| Blocks | r-1 |
| Treatments (7) | |
| A | 1 |
| B | 1 |
| C | 1 |
| AxB | 1 |
| AxC | 1 |
| BxC | 1 |
| AxBxC | 1 |
| Error | 7(r-1) |
| Total | 8r-1 |
If the two factor interactions(A×B, A×C or B×C) turns out to be significant a two way table of treatment means of that two factors is prepared and comparison can be made.
In that case Critical difference can be obtained as = \(t_\alpha \times \sqrt{\frac{2MSE}{r\times 2}}=t_\alpha \times \sqrt{\frac{MSE}{r}}\)
When the three factor interaction A×B×C is significant a three way table of treatment means is prepared and comparison among the mean values can be made. The Critical difference in that case is \(t_\alpha \times \sqrt{\frac{2MSE}{r}}\)
21.3.3 Analysis of 24 and higher factorial experiments
For 24 factorial there are 16 treatment combinations. If these 16 combinations are allotted to a single block of a RBD, and if there are r such blocks, we get designs for 24 factorial experiment. The 15 degrees of freedom due to treatments is bifurcated to 15 components, namely 4 main effects, 6 two factor interactions, 4 three factor interactions and one four factor interaction. The sum of squares can be obtained by the above method or by one of methods discussed below.
When the number of factors are more than 4 the RBD as such cannot be used for conducting the experiment since the number of treatment combinations become very large. There will be difficulty in getting homogeneous blocks accommodating large number of treatment combinations. Special techniques like fractional factorials, Confounding etc. are adopted in those circumstances. This will be discussed in later portion of this chapter.
21.3.4 Other methods for writing treatment combinations
We have seen that for 23 factorial experiment, the eight treatment combinations can be represented as
\(a_0b_0c_0, a_0b_0c_1 , a_0b_1c_0 , a_0b_1c_1 , a_1b_0c_0, a_1b_0c_1 , a_1b_1c_0 , a_1b_1c_1\)
There is another standard method of writing the treatment combinations in 2n series experiments. The lower level of all the factors is represented by the symbol (1) and the higher level by the letter itself without any suffix. The combinations are now expressed as product of (1) and the corresponding letters. Thus the above 8 combinations can written as
\((1), c, b, bc, a, ac, ab, abc\)
| RUN | Comb. | M | A | B | AB | C | AC | BC | ABC |
|---|---|---|---|---|---|---|---|---|---|
| 1 | (1) | + | - | - | + | - | + | + | - |
| 2 | a | + | + | - | - | - | - | + | + |
| 3 | b | + | - | + | - | - | + | - | + |
| 4 | ab | + | + | + | + | - | - | - | - |
| 5 | c | + | - | - | + | + | - | - | + |
| 6 | ac | + | + | - | - | + | + | - | - |
| 7 | bc | + | - | + | - | + | - | + | - |
| 8 = 23 | abc | + | + | + | + | + | + | + | + |
Standard way of writing the treatment combinations by this method
Large number of treatment combinations can be expressed very easily by this method. Write (1) first. Take the higher level of the first factor say a and multiply it by (1) to get the second combination a. Now take the second factor b and multiply it with the first two combinations to get two more combinations i.e. (1), a, b and ab. (Thus in the case of a 22 factorial ab ≡ a1b1 while for a 23 factorial ab ≡ a1b1c0).
If there is a third factor say, c then multiply the first four combinations by c to get four more combinations to get a total of 8 combinations.
(1), a, b, ab, c, ac, bc, abc
If we have a 24 factorial then multiply the above 8 combinations by factor ‘d’ to get 8 more combinations.
(1), a, b, ab, c, ac, bc, abc, d, ad, bd, abd, cd, acd, bcd, abcd
This method can be continued to write any number of treatment combinations.
Fisher’s algebraic method of computation of main effects and interactions
By the above first notation we can express various effects and interactions in the following way:
For the 23 factorial experiment we have seen that
\((A)_{i=0} = a_0b_0c_0 + a_0b_0c_1+ a_0b_1c_0 + a_0b_1c_1\)
\((A)_{i=1} = a_1b_0c_0+ a_1b_0c_1 + a_1b_1c_0 + a_1b_1c_1\)
The main effect of \(A = \frac{1}{4r}[(A)_{i=1}-(A)_{i=0}]\)
By the above notation, main effect of \(A= \dfrac{a + ac + ab + abc - (1) - c - b - bc}{4r} = \dfrac{(a-1)(b+1)(c+1)}{4r}\)
Similarly we can obtain the interaction \(A \times B = \dfrac{(a-1)(b-1)(c+1)}{4r}\)
And interaction \(A \times B \times C = \dfrac{(a-1)(b-1)(c-1)}{4r}\)
The method can be extended to any factorial of the 2n series.
(The above expression for \(A \times B \times C\) can be expressed as \(\dfrac{(a_1-a_0)(b_1-b_0)(c_1-c_0)}{4r}\) so that the expansion will give treatment combinations in the original form like a0b0c0 etc.)
Yates’s method of estimating sum of squares
This is a tabular method of computing effects / interactions and sum of squares in 2n series factorial experiments. We have to prepare a table of (n+2) columns by the following method.
In the first column of the table enter the treatment combinations in the above standard order i.e. like (1) a, b, ab, c, ac, bc, abc etc. selecting the factors in a certain order. (The standard order must not be altered). In the second column of the table enter the sum of responses corresponding to these treatment combinations from r replications.
Now we have to develop n more additional columns other than the first two by the following method. Group the entries of the second column into pairs starting from the top. The first half of the entries of the third column are the sum of these pairs; and the second half of the entries of the third column are the difference of the pairs of the previous column, difference being taken as: lower entry minus upper entry.
Example for 22 factorial experiment
| Treatment combination | Total of response from r replications | Process 1st step | Process 2nd step | Estimation of effects |
|---|---|---|---|---|
| (1) | Y1 | Y1+Y2 | Y1+Y2 + Y3+Y4 | \((\rightarrow)^2/4r\) = CF |
| a | Y2 | Y3+Y4 | Y2-Y1 + Y4-Y3 | \((\rightarrow)^2/4r\) = SS for A |
| b | Y3 | Y2-Y1 | Y3+Y4 - Y1-Y2 | \((\rightarrow)^2/4r\) = SS for B |
| ab | Y4 | Y4-Y3 | Y4-Y3 - Y2+Y1 | \((\rightarrow)^2/4r\) = SS for AB |
\(\rightarrow\) denote value in the just previous cell
Now continue grouping of the entries into pairs for the third column and repeat the process to develop the fourth column. The procedure must be continued to n columns other than the first two in the case of a 2n factorial. When the process is completed, the first entry of the last column (the row starting with (1)) will be the grand total. The second entry will be the numerator of the estimate of main effect represented by first treatment combination, third will be numerator part of the estimate of the main effect of second factor, Fourth will be numerator part of the estimate of the interaction effect of first and second factors, and so on.
Thus in general we can say that (i) The first entry of the last column after the Yates’s process will give the grand total of the observations (ii) The other values of the last column divided by 2(n-1)r will give the respective effects and interactions (represented by the first column) (iii) Square of the entries of the last column divided by 2nr will give the respective sum of squares of the effects and interactions (represented by the first column).
21.4 Factorial experiments of 3n series
When we take all possible combinations of n factors each at three levels this will make up a 3n factorial experiment. The three levels of a factor are usually denoted by suffixes 0 1, and 2 attached to the factors represented by small letters. The simplest 3n factorial is obtained when n = 2. i.e. (32 Factorial experiment). The two factors say A and B each at three levels give nine treatment combinations represented as
\(a_0b_0, a_0b_1, a_0b_2, a_1b_0, a_1b_1, a_1b_2, a_2b_0, a_2b_1, a_2b_2\)
If these nine treatment combinations are allocated (at random) to a single block of a Randomised block design and if there are r such blocks we get a 32 factorial experiment in a Randomised block design.
21.4.1 Analysis of 32 factorial experiments
The two main effects A and B and interaction A×B can be worked out by making use of the following relations
The total responses of the treatment combinations
\((A)_{i=0} = a_0b_0 + a_0b_1 + a_0b_2\)
\((A)_{i=1} = a_1b_0 + a_1b_1 + a_1b_2\)
\((A)_{i=2} = a_2b_0 + a_2b_1 + a_2b_2\)
Then main effect of A is comparison among these three totals (or mean values) with two degree of freedom.
Comparison of the type (A)i=1 − (A)i=0 involving two levels is called as a linear (AL)comparison and (A)i=2 +(A)i=0 − 2(A)i=1 will make a quadratic comparison(AQ).
Sum of squares due to \(A=\frac{(A)_{i=0}^2+(A)_{i=1}^2+(A)_{i=2}^2}{3r}-CF\)
Similarly, \((B)_{j=0} = a_0b_0 + a_1b_0 + a_2b_0\)
\((B)_{j=1} = a_0b_1 + a_1b_1 + a_2b_1\)
\((B)_{j=2} = a_0b_2 + a_1b_2 + a_2b_2\)
Sum of squares due to \(B=\frac{(B)_{j=0}^2+(B)_{j=1}^2+(B)_{j=2}^2}{3r}-CF\)
| Treatments | [1] | [a] | [a2] | [b] | [ab] | [a2 b] | [b2] | [ab2] | [a2b2] | Divisor |
|---|---|---|---|---|---|---|---|---|---|---|
| M | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | 9r |
| AL | -1 | 0 | +1 | -1 | 0 | +1 | -1 | 0 | +1 | 6r |
| AQ | +1 | -2 | +1 | +1 | -2 | +1 | +1 | -2 | +1 | 18r |
| BL | -1 | -1 | -1 | 0 | 0 | 0 | +1 | +1 | +1 | 6r |
| ALBL | +1 | 0 | -1 | 0 | 0 | 0 | -1 | 0 | +1 | 4r |
| AQBL | -1 | +2 | -1 | 0 | 0 | 0 | +1 | -2 | +1 | 12r |
| BQ | +1 | +1 | +1 | -2 | -2 | -2 | +1 | +1 | +1 | 18r |
| ALBQ | -1 | 0 | +1 | +2 | 0 | -2 | -1 | 0 | +1 | 12r |
| AQBQ | +1 | -2 | +1 | -2 | +4 | -2 | +1 | -2 | +1 | 36r |
The interaction A×B with four degrees of freedom can be estimated using four components viz. AL×BL; AL×BQ; AQ×BL and AQ×BQ (L for linear and Q for quadratic parts. But in general the component wise estimations of interactions were not made).
The sum of square due to A×B interaction with four degrees of freedom can be worked out as follows:
Prepare a two way table of treatment totals of factors A and B over all replications. The Interaction sum of square can be obtained as = Total sum of squares due to this table - (Sum of squares due to Factor A + Sum of squares due to factor B).
| Source | df |
|---|---|
| Blocks | r-1 |
| Treatments (8) | |
| A | 2 |
| B | 2 |
| AxB | 4 |
| Error | 8(r-1) |
| Total | 9r-1 |
If the main effects A or B turns out to be significant then the mean values over the levels of the factors are worked out and compared by using the critical difference
\(CD=t_{\alpha} \cdot \sqrt{\frac{2MSE}{r\times 3}}\)
If the interaction(A×B) turns out to be significant a two way table of treatment means of that two factors is prepared and comparison can be made then the \(CD=t_{\alpha} \cdot \sqrt{\frac{2MSE}{r}}\)
21.4.2 Analysis of 33 factorial experiments
The 3×3×3 factorial contains 27 treatment combinations. For field studies, as such the 27 treatment combinations cannot be accommodated in a single block of a Randomised block design. (Special type of incomplete blocks called confounded factorial designs are used in such situations, that is, experiments involving a large number of treatment combinations e.g. 33, 34, 35, 25, 26, 27 etc.). Here the 27 treatment combinations give rise to 26 degree of freedom. For three factors A, B and C the break up of the degree of freedom are as follows:
| Treatment Effects | df |
|---|---|
| Main effects | |
| A | 2 |
| B | 2 |
| C | 2 |
| Two factor interactions | |
| AxB | 4 |
| AxC | 4 |
| BxC | 4 |
| Three factor interactions | |
| AxBxC | 8 |
21.5 Analysis of asymmetrical factorial experiments
Asymmetrical or mixed factorials are those involving factors at different (unequal) levels. The simplest among them is the 2×3 factorial experiment. Here the first factor say A has got two levels denoted as a0 and a1 and the second factor say, B has got three levels denoted as b0, b1 and b2. Hence we have six treatment combinations involving both these factors denoted as a0b0, a0b1, a0b2, a1b0, a1b1 and a1b2. If these six treatment combinations are allotted at random to each block of a randomised block design, and if there are r such blocks we will get a 2×3 factorial in r blocks (or replications).
| Levels of A/B | a0 | a1 | Total |
| b0 | a0b0 | a1b0 | \(B_{j=0}\) |
| b1 | a0b1 | a1b1 | \(B_{j=1}\) |
| b2 | a0b2 | a1b2 | \(B_{j=2}\) |
| Total | \(A_{i=0}\) | \(A_{i=1}\) | Grand total |
Sum of squares due to \(A=\frac{(A_{i=0}^2 + A_{i=1}^2)}{3r}-C.F\)
Sum of squares due to \(B=\frac{(B_{j=0}^2 + B_{j=1}^2 + B_{j=2}^2)}{2r}-C.F\)
Sum of squares due to interaction A×B= Total sum of squares based on the above table minus (Sum of squares due to A + Sum of squares due to B).
The total Sum of squares, Block sum of squares (or replication sum of squares) and error sum of squares are calculated as usual. The ANOVA will be as follows:
| Source | df |
|---|---|
| Blocks | r-1 |
| Treatments (5) | |
| A | 1 |
| B | 2 |
| AxB | 2 |
| Error | 5(r-1) |
| Total | 6r-1 |
If the main effects A or B turns out to be significant then the mean values over the levels of the factors are worked out and compared by using the critical difference
CD for computing mean values of A levels = \(t_{\alpha} \cdot \sqrt{\frac{2MSE}{r\times 3}}\)
CD for computing mean values of B levels = \(t_{\alpha} \cdot \sqrt{\frac{2MSE}{r\times 2}}\)
If the interaction(A×B) turns out to be significant the mean values are obtained from the above two way table and comparison can be made then \(CD=t_{\alpha} \cdot \sqrt{\frac{2MSE}{r}}\)
21.6 Confounding
When the number of factors or their levels increases, the treatment combinations grow rapidly. A \(2^5\) factorial has 32 treatment combinations. To test these in a Randomised Complete Block Design (RCBD), every block would need to hold all 32 treatment combinations as one complete replication. Finding 32 plots uniform in soil fertility, moisture, and slope is very difficult, because the larger a block, the more variation it is likely to contain, and this variation inflates the experimental error and reduces precision.
The way out is to use smaller blocks. Instead of one big block of 32 plots making up a complete replication, we split each replication into two smaller blocks of 16 plots each. Provided real heterogeneity exists within the original 32-plot block for this finer division to remove, smaller blocks tend to be more homogeneous, and the experimental error goes down accordingly; the gain is not automatic, since a split that does not align with the actual pattern of field variation may bring little benefit. But splitting comes at a price, and understanding that price is the whole idea of confounding.
The core idea
To split the treatments into two blocks, we need a rule that decides which treatment combination goes into which block. We use one of the interaction contrasts to make this split. When we do that, the difference between the two block totals becomes exactly the same, up to sign, as that interaction’s contrast, so that the two are algebraically identical and cannot be separated. We can no longer tell whether a difference between the blocks is due to the blocks (soil, field position) or due to that interaction.
That interaction is said to be confounded with blocks, i.e. its contrast is mixed up with the block contrast and the interaction can no longer be estimated separately from the blocks. In exchange, every other main effect and interaction is now estimated within smaller, more homogeneous blocks, so their precision improves relative to a single large block.
This is the trade-off: we deliberately sacrifice information on one interaction (usually an unimportant one or higher order) to gain precision on everything else.
A worked example: \(2^3\) factorial
Take three factors: A (Nitrogen), B (Phosphorus), C (Potassium), each at two levels (low = absent, high = present). This gives \(2^3 = 8\) treatment combinations, written in standard notation:
\[(1),\ a,\ b,\ ab,\ c,\ ac,\ bc,\ abc\]
where a letter present means “high level of that factor.” Suppose our field only has uniform patches of 4 plots, not 8. We must split the 8 treatments into two blocks of 4. We decide to confound the highest-order interaction, \(ABC\), because a three-factor interaction is usually of least practical interest.
The rule: a treatment goes into one block or the other depending on the sign it carries in the ABC contrast. Treatments where the letters \(a\), \(b\), \(c\) appear an even number of times together go in one block; an odd number in the other.
| Block 1 (even) | Block 2 (odd) |
|---|---|
| (1) | a |
| ab | b |
| ac | c |
| bc | abc |
Now look at what happened. From Section 21.3.2, the interaction effect of \(A\times B\times C\) is \(\dfrac{1}{4r}\) times the ABC contrast, where the contrast itself is
\[ABC = (a+\ b+\ c+\ abc)-((1)+\ ab+\ ac+\ bc) \tag{21.1}\]
The interaction contrast is the raw, unnormalized difference between the sums shown above; dividing this contrast by \(4r\) gives the interaction effect proper, which estimates the average change in response due to the interaction. It is the contrast, not the divided effect, that is directly comparable to a difference between block totals, since block totals are themselves raw sums of the data.
The difference between the two block totals here is
\[\text{Block 1} - \text{Block 2} = ((1)+\ ab+\ ac+\ bc)-(a+\ b+\ c+\ abc) \tag{21.2}\]
Comparing Equation 21.1 and Equation 21.2, the block difference is exactly the negative of the ABC contrast, that is, \(\text{Block 1} - \text{Block 2} = -ABC\). Since the two quantities are the same linear combination of the data up to sign, they are algebraically identical, and we cannot tell whether an observed difference between the two blocks is due to the blocks themselves or due to the ABC interaction.
So if Block 1 sits on richer soil than Block 2, that soil difference is inseparable from the ABC contrast, and we have lost the ability to estimate ABC. But the main effects A, B, C and the two-factor interactions AB, AC, BC are all balanced across the two blocks (each appears equally in both), so their contrasts are orthogonal to the block effect and remain independently estimable, and now within blocks of 4 uniform plots instead of 8 uneven ones.
Complete vs partial confounding
With more than one replication, we get a choice.
Complete confounding: confound the same interaction (say ABC) in every replication. ABC is then lost entirely; no replication can recover it.
Partial confounding: confound a different interaction in each replication. For example, confound ABC in replication I, AB in replication II, BC in replication III. Each confounded interaction is then estimated only from the replications in which it was left unconfounded, for instance, ABC from replications II and III alone, rather than from all three. Since fewer replications contribute to each confounded interaction’s estimate, its effective replication, and hence its precision, is reduced compared to an effect that is unconfounded throughout, but the interaction is at least recoverable, unlike in complete confounding.
Advantages of confounding
- Reduces experimental error by breaking the material into small, homogeneous blocks.
- Increases the precision of the main effects and important interactions we care about, compared with an ordinary RBD.
Disadvantages of confounding
- Precision is gained by losing information on the confounded interaction(s), partly (partial) or entirely (complete).
- Confounded contrasts are effectively replicated fewer times, so they are estimated with lower precision.
- Careless confounding can throw away an effect that actually matters; only genuinely unimportant interactions should be confounded.
- The analysis is more involved and becomes tricky if observations are missing.
The one line to remember: confounding trades away information on an unimportant (usually higher-order) interaction in order to shrink block size and sharpen the estimates of everything else.
21.7 Chapter Summary
Fill in the blanks
Answers are given at the end of the chapter.
A factorial experiment is used to study the effects of __________ or more factors simultaneously.
In a factorial experiment, all possible combinations of the levels of the factors are called __________ combinations.
A factorial experiment with two factors, each at two levels, is called a __________ factorial experiment.
The number of treatment combinations in a factorial experiment is obtained by taking the __________ of the number of levels of all factors.
The change in response produced by changing the level of one factor is called the __________ effect.
The effect of one factor averaged over the levels of another factor is called the __________ effect.
When the effect of one factor depends on the level of another factor, the factors are said to __________.
Parallel lines in an interaction graph indicate __________ interaction.
Non-parallel lines in an interaction graph indicate the presence of __________.
A factorial experiment in which all factors have the same number of levels is called a __________ factorial experiment.
A factorial experiment in which factors have unequal numbers of levels is called an __________ factorial experiment.
A factorial experiment containing all possible treatment combinations is called a __________ factorial.
A factorial experiment in which only a part of all possible treatment combinations is used is called a __________ factorial.
In a \(2^n\) factorial experiment, each factor has __________ levels.
A \(2^3\) factorial experiment has __________ treatment combinations.
A \(2^4\) factorial experiment has __________ treatment combinations.
A \(3^2\) factorial experiment has __________ treatment combinations.
A \(3^3\) factorial experiment has __________ treatment combinations.
In a \(2^2\) factorial experiment, the treatment effects consist of __________ main effects and __________ interaction.
In a \(2^3\) factorial experiment, there are __________ main effects.
In a \(2^3\) factorial experiment, there are __________ two-factor interactions.
In a \(2^3\) factorial experiment, there is __________ three-factor interaction.
In a \(2^2\) factorial experiment, the interaction \(A \times B\) has __________ degree of freedom.
In a \(2^3\) factorial experiment, each main effect and interaction has __________ degree of freedom.
In a \(3^2\) factorial experiment, the main effect of each factor has __________ degrees of freedom.
In a \(3^2\) factorial experiment, the interaction \(A \times B\) has __________ degrees of freedom.
For a factor at three levels, the two independent comparisons are the __________ and __________ comparisons.
The tabular method for computing effects and sums of squares in \(2^n\) factorial experiments is called __________ method.
The algebraic method of computing factorial effects was developed by __________.
The tabular method of computing factorial effects was developed by __________.
In Yates’s method, the treatment combinations are first written in __________ order.
In Yates’s method, the first entry of the final column gives the __________ total.
In a \(2^4\) factorial experiment, the treatment degrees of freedom are divided into __________ main effects, __________ two-factor interactions, __________ three-factor interactions, and __________ four-factor interaction.
When the number of treatment combinations becomes too large to fit into a homogeneous block, __________ may be used.
The process of deliberately confounding one or more interaction effects with blocks is called __________.
In confounding, higher-order interactions are generally selected because they are considered __________ important.
When the same interactions are confounded in every replication, the confounding is called __________ confounding.
When different interactions are confounded in different replications, the confounding is called __________ confounding.
The main advantage of confounding is that it helps to reduce __________ error.
A major disadvantage of confounding is the loss of information about some __________.
Short-answer questions
Define a factorial experiment.
Explain the main objectives of factorial experiments.
What is meant by a treatment combination?
Explain the advantages of factorial experiments over one-factor-at-a-time experiments.
Explain the meaning of main effect.
Explain the meaning of interaction effect.
What is a simple effect?
Explain how interaction between two factors can be identified graphically.
Distinguish between main effects and interaction effects.
Explain symmetrical and asymmetrical factorial experiments.
Distinguish between complete and incomplete factorial experiments.
Explain the representation of factor levels using suffixes.
Explain how treatment combinations are represented in a \(2^n\) factorial experiment.
Explain a \(2^2\) factorial experiment with suitable treatment combinations.
Explain a \(2^3\) factorial experiment with suitable treatment combinations.
Explain the main effects and interaction in a \(2^2\) factorial experiment.
Explain the decomposition of treatment degrees of freedom in a \(2^2\) factorial experiment.
Explain the decomposition of treatment degrees of freedom in a \(2^3\) factorial experiment.
Explain the analysis of a \(2^2\) factorial experiment in an RBD.
Explain the analysis of a \(2^3\) factorial experiment in an RBD.
Explain how the interaction \(A \times B\) is calculated in a \(2^2\) factorial experiment.
Explain how the interaction \(A \times B \times C\) is calculated in a \(2^3\) factorial experiment.
Explain the analysis of a \(2^4\) factorial experiment.
Why are factorial experiments with a large number of factors difficult to conduct in an RBD?
Explain the standard method of writing treatment combinations in a \(2^n\) factorial experiment.
Explain Fisher’s algebraic method of computation of main effects and interactions.
Explain Yates’s method of estimating sums of squares.
Describe the steps involved in Yates’s method.
Explain the interpretation of the final column in Yates’s method.
Explain a \(3^2\) factorial experiment.
Explain linear and quadratic comparisons for a factor at three levels.
Explain the analysis of a \(3^2\) factorial experiment.
Explain the degrees of freedom associated with the main effects and interaction in a \(3^2\) factorial experiment.
Explain a \(3^3\) factorial experiment and its treatment combinations.
Explain the decomposition of treatment degrees of freedom in a \(3^3\) factorial experiment.
Explain an asymmetrical \(2 \times 3\) factorial experiment.
Explain the analysis of a \(2 \times 3\) factorial experiment.
What is confounding in factorial experiments?
Why is confounding necessary in large factorial experiments?
Explain complete and partial confounding.
State the advantages of confounding.
State the disadvantages of confounding.
Why are higher-order interactions generally selected for confounding?
Explain why one-factor-at-a-time experiments cannot detect interaction effects.
Explain the contribution of Fisher to factorial experimentation.
Explain the contribution of Yates to factorial experimentation.
Numerical and conceptual questions
Answers are given at the end of the chapter.
A factorial experiment has three factors with 2, 3, and 4 levels. Find the number of treatment combinations.
A factorial experiment consists of five clones, four spacings, and three weed-control methods. Find the total number of treatment combinations.
Explain why screening a large number of clones before conducting a factorial experiment can reduce the experimental size.
A \(2^2\) factorial experiment has the following responses:
| A (-) | A (+) | |
|---|---|---|
| B (-) | 20 | 40 |
| B (+) | 30 | 52 |
Calculate the main effect of A.
Using the same data, calculate the main effect of B.
Using the same data, explain whether A and B interact.
In a \(2^2\) factorial experiment with \(r\) replications, state the degrees of freedom for blocks, A, B, \(A \times B\), error, and total.
In a \(2^3\) factorial experiment with \(r\) replications, state the degrees of freedom for all sources of variation.
In a \(3^2\) factorial experiment with \(r\) replications, state the degrees of freedom for blocks, A, B, \(A \times B\), error, and total.
In a \(2 \times 3\) factorial experiment with \(r\) replications, state the degrees of freedom for blocks, A, B, \(A \times B\), error, and total.
A nitrogen factor has three levels and irrigation has two levels. Write all the treatment combinations.
For a nitrogen factor at three levels and irrigation at two levels, explain how the main effect of nitrogen is obtained.
For the same experiment, explain how the main effect of irrigation is obtained.
Explain how the interaction between nitrogen and irrigation is assessed.
A \(2^4\) factorial experiment is conducted in an RBD. State the number of treatment combinations and the decomposition of treatment degrees of freedom.
A \(2^5\) factorial experiment requires 32 treatment combinations. Explain why a standard RBD may not be suitable.
Explain how confounding can be used to divide a large factorial experiment into smaller blocks.
Explain why confounding an interaction with blocks makes that interaction unmeasurable.
Explain the difference between complete and partial confounding using a suitable example.
Explain why higher-order interactions are preferred for confounding rather than main effects.
Important formulae
Number of treatment combinations in a factorial experiment:
\[ N=\prod_{i=1}^{n}s_i \tag{21.3}\]
Number of treatment combinations in a \(2^n\) factorial experiment:
\[ N=2^n \tag{21.4}\]
Number of treatment combinations in a \(3^n\) factorial experiment:
\[ N=3^n \tag{21.5}\]
Main effect of A in a \(2^2\) factorial experiment:
\[ \text{Main effect of A}=\frac{(A)_{i=1}-(A)_{i=0}}{2r} \tag{21.6}\]
Sum of squares due to A in a \(2^2\) factorial experiment:
\[ SS_A=\frac{[(A)_{i=1}-(A)_{i=0}]^2}{4r} \tag{21.7}\]
Main effect of B in a \(2^2\) factorial experiment:
\[ \text{Main effect of B}=\frac{(B)_{j=1}-(B)_{j=0}}{2r} \tag{21.8}\]
Sum of squares due to B in a \(2^2\) factorial experiment:
\[ SS_B=\frac{[(B)_{j=1}-(B)_{j=0}]^2}{4r} \tag{21.9}\]
Interaction effect of \(A \times B\) in a \(2^2\) factorial experiment:
\[ \text{Interaction effect of }A\times B=\frac{(AB)_{i+j=0}-(AB)_{i+j=1}}{2r} \tag{21.10}\]
Sum of squares due to \(A \times B\):
\[ SS_{AB}=\frac{[(AB)_{i+j=0}-(AB)_{i+j=1}]^2}{4r} \tag{21.11}\]
Critical difference for a significant interaction in a \(2^2\) factorial experiment:
\[ CD=t_\alpha\sqrt{\frac{2MSE}{r}} \tag{21.12}\]
Main effect of A in a \(2^3\) factorial experiment:
\[ \text{Main effect of A}=\frac{(A)_{i=1}-(A)_{i=0}}{4r} \tag{21.13}\]
Sum of squares due to A in a \(2^3\) factorial experiment:
\[ SS_A=\frac{[(A)_{i=1}-(A)_{i=0}]^2}{8r} \tag{21.14}\]
Interaction effect of \(B\times C\) in a \(2^3\) factorial experiment:
\[ \text{Interaction effect of }B\times C=\frac{(BC)_{j+k=0}-(BC)_{j+k=1}}{4r} \tag{21.15}\]
Sum of squares due to \(B\times C\):
\[ SS_{BC}=\frac{[(BC)_{j+k=0}-(BC)_{j+k=1}]^2}{8r} \tag{21.16}\]
Interaction effect of \(A\times B\times C\):
\[ \text{Interaction effect of }A\times B\times C=\frac{(ABC)_{i+j+k=1}-(ABC)_{i+j+k=0}}{4r} \tag{21.17}\]
Sum of squares due to \(A\times B\times C\):
\[ SS_{ABC}=\frac{[(ABC)_{i+j+k=1}-(ABC)_{i+j+k=0}]^2}{8r} \tag{21.18}\]
Critical difference for comparing means of a two-factor interaction in a \(2^3\) factorial:
\[ CD=t_\alpha\sqrt{\frac{MSE}{r}} \tag{21.19}\]
Critical difference for comparing means when the three-factor interaction is significant:
\[ CD=t_\alpha\sqrt{\frac{2MSE}{r}} \tag{21.20}\]
Linear comparison for a factor at three levels:
\[ A_L=(A)_{i=1}-(A)_{i=0} \tag{21.21}\]
Quadratic comparison for a factor at three levels:
\[ A_Q=(A)_{i=2}+(A)_{i=0}-2(A)_{i=1} \tag{21.22}\]
Sum of squares due to A in a \(3^2\) factorial:
\[ SS_A=\frac{(A)_{i=0}^2+(A)_{i=1}^2+(A)_{i=2}^2}{3r}-CF \tag{21.23}\]
Sum of squares due to B in a \(3^2\) factorial:
\[ SS_B=\frac{(B)_{j=0}^2+(B)_{j=1}^2+(B)_{j=2}^2}{3r}-CF \tag{21.24}\]
Sum of squares due to \(A\times B\) in a \(3^2\) factorial:
\[ SS_{AB}=SS_{\text{two-way table}}-(SS_A+SS_B) \tag{21.25}\]
Critical difference for comparing levels of A in a \(3^2\) factorial:
\[ CD=t_\alpha\sqrt{\frac{2MSE}{3r}} \tag{21.26}\]
Critical difference for comparing levels of A in a \(2\times3\) factorial:
\[ CD_A=t_\alpha\sqrt{\frac{2MSE}{3r}} \tag{21.27}\]
Critical difference for comparing levels of B in a \(2\times3\) factorial:
\[ CD_B=t_\alpha\sqrt{\frac{2MSE}{2r}} \tag{21.28}\]
Critical difference for comparing means when \(A\times B\) is significant:
\[ CD=t_\alpha\sqrt{\frac{2MSE}{r}} \tag{21.29}\]
Quick revision
Factorial experiment → studies two or more factors simultaneously.
Treatment combinations → all possible combinations of factor levels.
Number of treatments → product of the number of levels of all factors.
Main effect → average change in response due to a change in the level of one factor.
Simple effect → effect of one factor at a particular level of another factor.
Interaction → effect of one factor changes according to the level of another factor.
Parallel lines → no interaction.
Non-parallel lines → interaction.
Symmetrical factorial → all factors have the same number of levels.
Asymmetrical factorial → factors have unequal numbers of levels.
Complete factorial → all possible treatment combinations are included.
Incomplete or fractional factorial → only part of the possible combinations are included.
\(2^n\) factorial → \(n\) factors, each at two levels.
\(3^n\) factorial → \(n\) factors, each at three levels.
\(2^2\) → 4 treatment combinations.
\(2^3\) → 8 treatment combinations.
\(2^4\) → 16 treatment combinations.
\(3^2\) → 9 treatment combinations.
\(3^3\) → 27 treatment combinations.
In a \(2^n\) factorial, the treatment degrees of freedom are \(2^n-1\).
In a \(3^n\) factorial, the treatment degrees of freedom are \(3^n-1\).
In a \(2^2\) factorial → A, B, and \(A\times B\) each have 1 degree of freedom.
In a \(2^3\) factorial → A, B, C, \(A\times B\), \(A\times C\), \(B\times C\), and \(A\times B\times C\) each have 1 degree of freedom.
In a \(3^2\) factorial → A has 2 df, B has 2 df, and \(A\times B\) has 4 df.
For a three-level factor → 2 df can be divided into linear and quadratic components.
Yates’s method → tabular method for calculating effects and sums of squares in \(2^n\) factorial experiments.
Fisher’s algebraic method → expresses factorial effects and interactions algebraically.
Confounding → deliberately making an interaction inseparable from block effects to reduce block size.
Complete confounding → the same interaction is confounded in all replications.
Partial confounding → different interactions are confounded in different replications.
Higher-order interactions are generally preferred for confounding because main effects and lower-order interactions are usually more important.
Major advantage of confounding → smaller and more homogeneous blocks, resulting in improved precision.
Major disadvantage of confounding → loss of information about the confounded interactions.
Answers to fill in the blanks
1. Two 2. Treatment 3. \(2^2\) 4. Product 5. Main 6. Main 7. Interact 8. No 9. Interaction 10. Symmetrical 11. Asymmetrical 12. Complete 13. Fractional 14. Two 15. 8 16. 16 17. 9 18. 27 19. Two; one 20. Three 21. Three 22. One 23. One 24. One 25. Two 26. Four 27. Linear; quadratic 28. Yates’s 29. R. A. Fisher 30. Frank Yates 31. Standard 32. Grand 33. Four; six; four; one 34. Confounding 35. Confounding 36. Less 37. Complete 38. Partial 39. Experimental 40. Interactions
Solutions to numerical and conceptual questions
With three factors at 2, 3, and 4 levels, the number of treatment combinations is \(2\times3\times4=24\).
With five clones, four spacings, and three weed-control methods, the number of treatment combinations is \(5\times4\times3=60\).
Screening first identifies a smaller subset of promising clones using simple single-factor experiments. Combining this smaller subset with the remaining factors in a factorial design keeps the number of treatment combinations manageable, rather than combining every original clone with every level of every factor from the outset.
When B is “-”, the change in output as A goes from “-” to “+” is \(40-20=20\). When B is “+”, the change is \(52-30=22\). The main effect of A is \(\frac{20+22}{2}=21\).
When A is “-”, the change in output as B goes from “-” to “+” is \(30-20=10\). When A is “+”, the change is \(52-40=12\). The main effect of B is \(\frac{10+12}{2}=11\).
Yes, A and B interact. The change in output due to A is 20 when B is “-” but 22 when B is “+”; since the effect of A is not the same at both levels of B, the two factors interact (though the interaction here is small).
For a \(2^2\) factorial in \(r\) replications: Blocks \(=r-1\), A \(=1\), B \(=1\), \(A\times B=1\), Error \(=3(r-1)\), Total \(=4r-1\).
For a \(2^3\) factorial in \(r\) replications: Blocks \(=r-1\), A \(=1\), B \(=1\), C \(=1\), \(A\times B=1\), \(A\times C=1\), \(B\times C=1\), \(A\times B\times C=1\), Error \(=7(r-1)\), Total \(=8r-1\).
For a \(3^2\) factorial in \(r\) replications: Blocks \(=r-1\), A \(=2\), B \(=2\), \(A\times B=4\), Error \(=8(r-1)\), Total \(=9r-1\).
For a \(2\times3\) factorial in \(r\) replications: Blocks \(=r-1\), A \(=1\), B \(=2\), \(A\times B=2\), Error \(=5(r-1)\), Total \(=6r-1\).
With nitrogen at three levels (\(n_0,n_1,n_2\)) and irrigation at two levels (\(i_0,i_1\)), the six treatment combinations are \(n_0i_0\), \(n_1i_0\), \(n_2i_0\), \(n_0i_1\), \(n_1i_1\), \(n_2i_1\).
The main effect of nitrogen is obtained by comparing the totals \((N)_0\), \((N)_1\), and \((N)_2\), each summed over both levels of irrigation, giving two independent comparisons and two degrees of freedom for nitrogen.
The main effect of irrigation is obtained by comparing the totals \((I)_0\) and \((I)_1\), each summed over all three levels of nitrogen, giving one degree of freedom for irrigation.
The interaction is assessed by finding the effect of irrigation separately at each level of nitrogen, that is \((n_0i_1-n_0i_0)\), \((n_1i_1-n_1i_0)\), and \((n_2i_1-n_2i_0)\), and comparing these simple effects. If they are unequal, nitrogen and irrigation interact.
A \(2^4\) factorial has \(2^4=16\) treatment combinations. The 15 treatment degrees of freedom split into 4 main effects, 6 two-factor interactions, 4 three-factor interactions, and 1 four-factor interaction.
A \(2^5\) factorial has 32 treatment combinations, which is too many to fit into a single homogeneous block; soil and other conditions are unlikely to remain uniform across so many plots, so a standard RBD would inflate the experimental error. Confounding or fractional factorials are used instead to keep blocks smaller and more homogeneous.
The treatment combinations are divided into smaller groups, matching the number of blocks required per replication, by aligning the split with the contrast for a chosen interaction. Each group is then assigned to a separate block, so that block size is reduced while still including a complete replication across the blocks together.
When the interaction contrast is used to divide the treatments into blocks, the contrast between the block totals becomes identical to the interaction contrast. Since the two cannot be separated, any variation between blocks is indistinguishable from the interaction effect, making the interaction unmeasurable.
In complete confounding, the same interaction (for example, \(A\times B\times C\)) is confounded in every replication, so information on that interaction is completely lost. In partial confounding, different interactions are confounded in different replications (for example, \(A\times B\times C\) in replication 1 and \(A\times B\) in replication 2), so each confounded interaction can still be recovered from the replications in which it is not confounded.
Higher-order interactions are usually of least practical importance and are the hardest to interpret meaningfully, so losing information about them costs little. Main effects and lower-order interactions are generally of greatest interest to the experimenter and are therefore preserved rather than confounded.
Fisher, Yates, and the “one factor at a time” revolution
For a long time, the accepted wisdom in experimental science was to vary just one factor at a time, holding everything else fixed, in the belief that this was the only way to obtain clear, uncontaminated conclusions. It was Ronald A. Fisher, working at the Rothamsted Experimental Station, who overturned this idea. Fisher argued that nature is questioned most efficiently not one factor at a time, but by varying several factors together in a single organised experiment. Such a factorial experiment not only uses the data far more economically, giving information on every factor from the same set of plots, but is the only way to reveal how factors interact, something a one-factor-at-a-time study can never detect. Fisher set out these ideas in his landmark book The Design of Experiments (1935), which contains chapters devoted to the factorial design and to confounding. (Fisher 1935)
The systematic machinery of factorial analysis was then developed in detail by Frank Yates, whom Fisher had recruited to Rothamsted in 1931. In his classic 1937 monograph The Design and Analysis of Factorial Experiments, Yates laid out the methods still taught today, including the elegant tabular shortcut for computing all the factorial effects and their sums of squares that now bears his name, Yates’s algorithm. (Yates 1937) Together, Fisher and Yates transformed agricultural experimentation, and their factorial methods spread from the wheat fields of Rothamsted into medicine, industry, and the whole of modern science.
“No aphorism is more frequently repeated in connection with field trials, than that we must ask Nature few questions, or, ideally, one question, at a time. The writer is convinced that this view is wholly mistaken.”
- R. A. Fisher