Response variable(s) in any experiment can be found to be affected by a number of factors in the overall system some of which are controlled or maintained at desired levels in the experiment. An experiment in which the treatments consist of all possible combinations of the selected levels in two or more factors is referred as a factorial experiment. For example, an experiment on rooting of cuttings involving two factors, each at two levels, such as two hormones at two doses, is referred to as a 2 x 2 or a 2^{2} factorial experiment. Its treatments consist of the following four possible combinations of the two levels in each of the two factors.
Treatment combination 

Treatment number 
Hormone 
Dose (ppm) 
1 
NAA 
10 
2 
NAA 
20 
3 
IBA 
10 
4 
IBA 
20 
The term complete factorial experiment is sometimes used when the treatments include all combinations of the selected levels of the factors. In contrast, the term fractional factorial experiment is used when only a fraction of all the combinations is tested. Throughout this manual, however, complete factorial experiments are referred simply as factorial experiments. Note that the term factorial describes a specific way in which the treatments are formed and does not, in any way, refer to the design used for laying out the experiment. For example, if the foregoing 2^{2} factorial experiment is in a randomized complete block design, then the correct description of the experiment would be 2^{2} factorial experiment in randomized complete block design.
The total number of treatments in a factorial experiment is the product of the number of levels of each factor; in the 2^{2} factorial example, the number of treatments is 2 x 2 = 4, in the 2^{3} factorial, the number of treatments is 2 x 2 x 2 = 8. The number of treatments increases rapidly with an increase in the number of factors or an increase in the levels in each factor. For a factorial experiment involving 5 clones, 4 espacements, and 3 weedcontrol methods, the total number of treatments would be 5 x 4 x 3 = 60. Thus, indiscriminate use of factorial experiments has to be avoided because of their large size, complexity, and cost. Furthermore, it is not wise to commit oneself to a large experiment at the beginning of the investigation when several small preliminary experiments may offer promising results. For example, a tree breeder has collected 30 new clones from a neighbouring country and wants to assess their reaction to the local environment. Because the environment is expected to vary in terms of soil fertility, moisture levels, and so on, the ideal experiment would be one that tests the 30 clones in a factorial experiment involving such other variable factors as fertilizer, moisture level, and population density. Such an experiment, however, becomes extremely large as factors other than clones are added. Even if only one factor, say nitrogen or fertilizer with three levels were included, the number of treatments would increase from 30 to 90. Such a large experiment would mean difficulties in financing, in obtaining an adequate experimental area, in controlling soil heterogeneity, and so on. Thus, the more practical approach would be to test the 30 clones first in a singlefactor experiment, and then use the results to select a few clones for further studies in more detail. For example, the initial singlefactor experiment may show that only five clones are outstanding enough to warrant further testing. These five clones could then be put into a factorial experiment with three levels of nitrogen, resulting in an experiment with 15 treatments rather than the 90 treatments needed with a factorial experiment with 30 clones.
The effect of a factor is defined to be the average change in response produced by a change in the level of that factor. This is frequently called the main effect. For example, consider the data in Table 4.12.
Table 4.12. Data from a 2x2 factorial experiment

Factor B 

Level 
b_{1} 
b_{2} 

a_{1} 
20 
30 

Factor A 

a_{2} 
40 
52 
The main effect of factor A could be thought of as the difference between the average response at the first level of A and the average response at the second level of A. Numerically, this is
That is, increasing factor A from level 1 to level to 2 causes an average increase in the response by 21 units. Similarly, the main effect of B is
If the factors appear at more than two levels, the above procedure must be modified since there are many ways to express the differences between the average responses.
The major advantage of conducting a factorial experiment is the gain in information on interaction between factors. In some experiments, we may find that the difference in response between the levels of one factor is not the same at all levels of the other factors. When this occurs, there is an interaction between the factors. For example, consider the data in Table 4.13.
Table 4.13. Data from a 2x2 factorial experiment
Factor B 

Levels 
b_{1} 
b_{2} 

a_{1} 
20 
40 

Factor A 

a_{2} 
50 
12 
At the first level of factor B, the factor A effect is
A = 5020 = 30
and at the second level of factor B, the factor A effect is
A = 1240 = 28
Since the effect of A depends on the level chosen for factor B, we see that there is interaction between A and B.
These ideas may be illustrated graphically. Figure 4.5 plots the response data in Table 4.12. against factor A for both levels of factor B.
Figure 4.5. Graphical representation of lack of interaction between factors.
Note that the b_{1} and b_{2} lines are approximately parallel, indicating a lack of interaction between factors A and B.
Similarly, Figure 4.6 plots the response data in Table 4.13. Here we see that the b_{1} and b_{2} lines are not parallel. This indicates an interaction between factors A and B. Graphs such as these are frequently very useful in interpreting significant interactions and in reporting the results to nonstatistically trained management. However, they should not be utilized as the sole technique of data analysis because their interpretation is subjective and their appearance is often misleading.
Figure 4.6. Graphical representation of interaction between factors.
Note that when an interaction is large, the corresponding main effects have little practical meaning. For the data of Table 4.13, we would estimate the main effect of A to be
= 1
which is very small, and we are tempted to conclude that there is no effect due to A. However, when we examine the effects of A at different levels of factor B, we see that this is not the case. Factor A has an effect, but it depends on the level of factor B i.e., a significant interaction will often mask the significance of main effects. In the presence of significant interaction, the experimenter must usually examine the levels of one factor, say A, with level of the other factors fixed to draw conclusions about the main effect of A.
For most factorial experiments, the number of treatments is usually too large for an efficient use of a complete block design. There are, however, special types of designs developed specifically for large factorial experiments such as confounded designs. Descriptions on the use of such designs can be found in Das and Giri (1980).
4.4.1. Analysis of variance
Any of the complete block designs discussed in sections 4.2 and 4.3 for singlefactor experiments is applicable to a factorial experiment. The procedures for randomization and layout of the individual designs are directly applicable by simply ignoring the factor composition of the factorial treatments and considering all the treatments as if they were unrelated. For the analysis of variance, the computations discussed for individual designs are also directly applicable. However, additional computational steps are required to partition the treatment sum of squares into factorial components corresponding to the main effects of individual factors and to their interactions. The procedure for such partitioning is the same for all complete block designs and is, therefore, illustrated for only one case, namely, that of RCBD.
The stepbystep procedure for the analysis of variance of a twofactor experiment on bamboo involving two levels of spacing (Factor A) and three levels of age at planting (Factor A) laid out in RCBD with three replications is illustrated here. The list of the six factorial treatment combinations is shown in Table 4.14, the experimental layout in Figure 4.7, and the data in Table 4.15.
Table 4.14. The 2 x 3 factorial treatment combinations of two levels of spacing and three levels of age.
Age at planting 
Spacing (m) 

(month) 
10 m x 10 m 
12 m x 12m 
(a_{1}) 
(a_{2}) 

6 (b_{1}) 
a_{1}b_{1} 
a_{2}b_{1} 
12 (b_{2}) 
a_{1}b_{2} 
a_{2}b_{2} 
24 (b_{3}) 
a_{1}b_{3} 
a_{2}b_{3} 
Replication I Replication II Replication III
a_{2}b_{3} 
a_{2}b_{3} 
a_{1}b_{2} 

a_{1}b_{3} 
a_{1}b_{2} 
a_{1}b_{1} 

a_{1}b_{2} 
a_{1}b_{3} 
a_{2}b_{2} 

a_{2}b_{1} 
a_{2}b_{1} 
a_{1}b_{3} 

a_{1}b_{1} 
a_{2}b_{2} 
a_{2}b_{1} 

a_{2}b_{2} 
a_{1}b_{1} 
a_{2}b_{3} 
Figure 4.7. A sample layout of 23 factorial experiment involving two levels of spacing and three levels of age in a RCBD with 3 replications.
Table 4.15. Mean maximum culm height of Bambusa arundinacea tested with three age levels and two levels of spacing in a RCBD.
Treatment 
Maximum culm height of a clump (cm) 
Treatment 

combination 
Rep. I 
Rep. II 
Rep. III 
total (T_{ij}) 
a_{1}b_{1} 
46.50 
55.90 
78.70 
181.10 
a_{1}b_{2} 
49.50 
59.50 
78.70 
187.70 
a_{1}b_{3} 
127.70 
134.10 
137.10 
398.90 
a_{2}b_{1} 
49.30 
53.20 
65.30 
167.80 
a_{2}b_{2} 
65.50 
65.00 
74.00 
204.50 
a_{2}b_{3} 
67.90 
112.70 
129.00 
309.60 
Replication total (R_{k}) 
406.40 
480.40 
562.80 
G=1449.60 
Step 1. Denote the number of replication by r, the number of levels of factor A (i.e., spacing) by a, and that of factor B (i.e., age) by b. Construct the outline of the analysis of variance as follows:
Table 4.16. Schematic representation of ANOVA of a factorial experiment with two levels of factor A, three levels of factor B and with three replications in RCBD.
Source of variation 
Degrees of freedom (df) 
Sum of squares (SS) 
Mean square 
Computed f 
Replication 
r1 
SSR 
MSR 

Treatment 
ab 1 
SST 
MST 

A 
a 1 
SSA 
MSA 

B 
b 1 
SSB 
MSB 

AB 
(a1)(b1) 
SSAB 
MSAB 

Error 
(r1)(ab1) 
SSE 
MSE 

Total 
rab 1 
SSTO 
Step 2.Compute treatment totals (T_{ij}), replication totals (R_{k}), and the grand total (G), as shown in Table 4.15 and compute the SSTO, SSR, SST and SSE following the procedure described in Section 4.3.3. Let y_{ijk} refer to the observation corresponding to the ith level of factor A and jth level factor B in the kth replication.
(4.22)
SSTO (4.23)
= 17479.10
SSR (4.24)
= 2040.37
SST (4.25)
= 14251.87
SSE = SSTO  SSR  SST (4.26)
= 17479.10  2040.37  14251.87
= 1186.86
The preliminary analysis of variance is shown in Table 4.17.
Table 4.17. Preliminary analysis of variance for data in Table 4.15.
Source of variation 
Degree of freedom 
Sum of squares 
Mean square 
Computed F 
Tabular F 5% 
Replication 
2 
2040.37 
1020.187 
8.59567* 
4.10 
Treatment 
5 
14251.87 
2850.373 
24.01609* 
3.33 
Error 
10 
1186.86 
118.686 

Total 
17 
17479.10 
*Significant at 5% level.
Step 3. Construct the factor A x factor B twoway table of totals, with factor A totals and factor B totals computed. For our example, the Spacing x Age table of totals (AB) with Spacing totals (A) and Age totals (B) computed are shown in Table 4.18.
Table 4.18. The Spacing x Age table of totals for the data in Table 4.15.
Age 
Spacing 
Total 

a_{1} 
a_{2} 
(B_{j}) 

b_{1} 
181.10 
167.80 
348.90 

b_{2} 
187.70 
204.50 
392.20 

b_{3} 
398.90 
309.60 
708.50 

Total (A_{i}) 
767.70 
681.90 
G = 1449.60 
Step 4. Compute the three factorial components of the treatment sum of squares as:
SSA = (4.27)
= 408.98
SSB = (4.28)
= 12846.26
SSAB = SST  SSA  SSB (4.29)
= 14251.87  408.98  12846.26
= 996.62
Step 5. Compute the mean square for each source of variation by dividing each sum of squares by its corresponding degrees of freedom and obtain the F ratios for each of the three factorial components as per the scheme given in the Table 4.16
Step 6. Enter all values obtained in Steps 3 to 5 in the preliminary analysis of variance of Step 2, as shown in Table 4.19.
Table 4.19. ANOVA of data in Table 4.15 from a 2 x 3 factorial experiment in RCBD.
Source of variation 
Degree of freedom 
Sum of squares 
Mean square 
Computed F 
Tabular F 5% 
Replication 
2 
2040.37 
1020.187 
8.60* 
4.10 
Treatment 
5 
14251.87 
2850.373 
24.07* 
3.33 
A 
1 
12846.26 
6423.132 
3.45 
4.96 
B 
2 
408.98 
408.980 
54.12* 
4.10 
AB 
2 
996.62 
498.312 
4.20* 
4.10 
Error 
10 
1186.86 
118.686 

Total 
17 
17479.10 
In a factorial experiment, as the number of factors to be tested increases, the complete set of factorial treatments may become too large to be tested simultaneously in a single experiment. A logical alternative is an experimental design that allows testing of only a fraction of the total number of treatments. A design uniquely suited for experiments involving large number of factors is the fractional factorial design (FFD). It provides a systematic way of selecting and testing only a fraction of the complete set of factorial treatment combinations. In exchange, however, there is loss of information on some preselected effects. Although this information loss may be serious in experiments with one or two factors, such a loss becomes more tolerable with large number of factors. The number of interaction effects increases rapidly with the number of factors involved, which allows flexibility in the choice of the particular effects to be sacrificed. In fact, in cases where some specific effects are known beforehand to be small or unimportant, use of the FFD results in minimal loss of information.
In practice, the effects that are most commonly sacrificed by use of the FFD are high order interactions  the fourfactor or fivefactor interactions and at times, even the threefactor interaction. In almost all cases, unless the researcher has prior information to indicate otherwise he should select a set of treatments to be tested so that all main effects and twofactor interactions can be estimated. In forestry research, the FFD is to be used in exploratory trials where the main objective is to examine the interactions between factors. For such trials, the most appropriate FFD’s are those that sacrifice only those interactions that involve more than two factors.
With the FFD, the number of effects that can be measured decreases rapidly with the reduction in the number of treatments to be tested. Thus, when the number of effects to be measured is large, the number of treatments to be tested, even with the use of FFD, may still be too large. In such cases, further reduction in the size of the experiment can be achieved by reducing the number of replications. Although the use of FFD without replication is uncommon in forestry experiments, when FFD is applied to exploratory trials, the number of replications required can be reduced to the minimum.
Another desirable feature of FFD is that it allows reduced block size by not requiring a block to contain all treatments to be tested. In this way, the homogeneity of experimental units within the same block can be improved. A reduction in block size is, however, accompanied by loss of information in addition to that already lost through the reduction in number of treatments. Although the FFD can thus be tailormade to fit most factorial experiments, the procedure for doing so is complex and so only a particular class of FFD that is suited for exploratory trials in forestry research is described here. The major features of these selected designs are that they (i) apply only to 2^{n} factorial experiments where n, the number of factors is at least 5, (ii) involve only one half of the complete set of factorial treatment combinations, denoted by 2^{n}1(iii) allow all main effects and twofactor interactions to be estimated. For more complex plans, reference may made to Das and Giri (1980).
The procedure for layout, and analysis of variance of a 2^{51 }FFD with a field experiment involving five factors A, B, C, D and E is illustrated in the following. In the designation of the various treatment combinations, the letters a, b, c,…, are used to denote the presence (or high level) of factors A, B, C,… Thus the treatment combination ab in a 2^{5} factorial experiment refers to the treatment combination that contains the high level (or presence) of factors A and B and low level (or absence ) of factors C, D and E, but this same notation (ab) in a 2^{6} factorial experiment would refer to the treatment combination that contains the high level of factors A and B and low level of factors C, D, E, and F. In all cases, the treatment combination that consists of the low level of all factors is denoted by the symbol (1).
4.5.1. Construction of the design and layout
One simple way to arrive at the desired fraction of factorial combinations in a 2^{51 }FFD is to utilize the finding that in a 2^{5} factorial trial, the effect ABCDE can be estimated from the expression arising from the expansion of the term (a1)(b1)(c1)(d1)(e1) which is
(a1)(b1)(c1)(d1)(e1) = abcde  acde  bcde + cde  abde + ade + bde  de
 abce + ace + bce  ce + abe  ae  be + e
 abcd + acd + bcd  cd + abd  ad  bd + d
+ abc  ac  bc + c  ab + a + b  1
Based on the signs (positive or negative) attached to the treatments in this expression, two groups of treatments can be formed out of the complete factorial set. Retaining only one set with either negative or positive signs, we get a half fraction of the 2^{5} factorial experiment. The two sets of treatments are shown below.
Treatments with negative signs 
Treatments with positive signs 
acde, bcde, abde, de, abce, ce, ae, be, 
abcde, bcde, abde, de, abce, ce, ae, be, 
abcd, cd, ad, bd, ac, bc, ab, 1 
abcd, cd, ad, bd, ac, bc, ab, 1 
As a consequence of the reduction in number of treatments included in the experiment, we shall not be able to estimate the effect ABCDE using the fractional set. All main effects and two factor interactions can be estimated under the assumption that all three factor and higher order interactions are negligible. The procedure is generalizable in the sense that in a 2^{6} experiment, a half fraction can be taken by retaining the treatments with either negative or positive signs in the expansion for (a1)(b1)(c1)(d1)(e1)(f1).
The FFD refers to only a way of selecting treatments with a factorial structure and the resulting factorial combinations can be taken as a set of treatments for the physical experiment to be laid out in any standard design like CRD or RCBD. A sample randomized layout for a 2^{51 }FFD under RCBD with two replications is shown in Figure 4.8.
1 de 
9 ab 
1 abce 
9 acde 

2 1 
10 adde 
2 cd 
10 bd 

3 acde 
11 ad 
3 be 
11 de 

4 ae 
12 abce 
4 ad 
12 bcde 

5 ce 
13 be 
5 ae 
13 ce 

6 ac 
14 bc 
6 abcd 
14 1 

7 bcde 
15 bcd 
7 abce 
15 ac 

8 bd 
16 cd 
8 bc 
16 be 
Replication I Replication II
Figure 4.8. A sample layout of a 2^{51 } FFD with two replications under RCBD.
4.5.2. Analysis of variance.
The analysis of variance procedure of a 2^{51 }FFD with 2 replications is illustrated using Yate’s method for the computation of sums of squares. This is a method suitable for manual computation of large factorial experiments. Alternatively, the standard rules for the computation of sums of squares in the analysis of variance, by constructing oneway tables of totals for computing main effects, twoway tables of totals for twofactor interactions and so on as illustrated in Section 4.4.1 can also be adopted in this case.
The analysis of 2^{51 }FFD is illustrated using hypothetical data from a trial whose layout is shown in Figure 4.8 which conforms to that of a RCBD. The response obtained in terms of fodder yield (t/ha) under the different treatment combinations is given in Table 4.21. The five factors were related to different components of a soil management scheme involving application of organic matter, fertilizers, herbicides, water, and lime.
Table 4.21.Fodder yield data from a 2^{51} factorial experiment
Treatment combination 
Fodder yield (t/ha) 
Treatment total (T_{i}) 

Replication I 
Replication II 

acde 
1.01 
1.04 
2.06 
bcde 
1.01 
0.96 
1.98 
abde 
0.97 
0.94 
1.92 
de 
0.82 
0.75 
1.58 
abce 
0.92 
0.95 
1.88 
ce 
0.77 
0.75 
1.53 
ae 
0.77 
0.77 
1.55 
be 
0.76 
0.80 
1.57 
abcd 
0.97 
0.99 
1.97 
cd 
0.92 
0.88 
1.80 
ad 
0.80 
0.87 
1.68 
bd 
0.82 
0.80 
1.63 
ac 
0.91 
0.87 
1.79 
bc 
0.79 
0.76 
1.55 
ab 
0.86 
0.87 
1.74 
1 
0.73 
0.69 
1.42 
Replication total (R_{j}) 
13.83 
13.69 

Grand total (G) 
27.52 