The split plot design is specifically suited for a two-factor experiment wherein levels of one of the factors require large plot size for execution and also show large differences in their effects. In such a situation, the experiment will consist of a set of large plots called main plots in which levels for the main plot factor are assigned. Each main plot is divided into subplots to which the second factor, called the subplot factor, is assigned. Thus, each main plot becomes a block for the subplot treatments (i.e., the levels of the subplot factor). The assignment of the main plot factor can, in fact, follow any of the patterns like the completely randomized design, randomized complete block, or latin square design but here, only the randomized complete block is considered for the main plot factor because it is perhaps the most appropriate and the most commonly used design for forestry experiments.

With a split plot design, the precision for the measurement of the effects of the main plot factor is sacrificed to improve that of the subplot factor. Measurement of the main effect of the subplot factor and its interaction with the main plot factor is more precise than that obtainable with a randomized complete block design. On the other hand, the measurement of the effects of the main plot treatments (i.e., the levels of the main plot factor) is less precise than that obtainable with a randomized complete block design.

4.6.1. Layout

There are two separate randomization processes in a split plot design-one for the main plots and another for the subplots. In each replication, main plot treatments are first randomly assigned to the main plots followed by a random assignment of the subplot treatments within each main plot.

For illustration, a two-factor experiment involving four levels of nitrogen (main plot treatments) and three eucalyptus clones (subplot treatments) in three replications is used. Here, fertilzer levels were chosen for the main plots mainly for the convenience in its application, easiness in controlling the leaching effect and to detect the presence of possible interaction between fertilizers and the clones. The steps in the randomization and layout of a split plot design are shown, using a as the number of main plot treatments, b as the number of subplot treatments, and r as the number of replications.

Step 1. Divide the experimental area into r = 3 blocks, each of which is further divided into a = 4 main plots, as shown in Figure 4.9.

Step 2. Following the RCBD randomization procedure with a = 4 treatments and r = 3 replications randomly assign the 4 nitrogen treatments to the 4 main plots in each of the 3 blocks. The result may be as shown in Figure 4.10.

Step 3. Divide each of the ra = 12 main plots into b = 3 subplots and following the RCBD randomization procedure for b = 3 treatments and ra = 12 replications, randomly assign the 3 clones to the 3 subplots in each of the 12 main plots. The result may be as shown in Figure 4.11.

Main plots					Main plots					Main plots
1	2	3	4		1	2	3	4		1	2	3	4
Replication I					Replication II					Replication III

Figure 4.9. Division of the experimental area into three blocks (replications) each consisting of four main plots, as the first step in laying out of a split plot experiment involving three replications and four main plot treatments.

n3

n1

n0

n2

n1

n0

n3

n2

n0

n1

n2

n3

Replication I

Replication II

Replication III

Figure 4.10. Random assignment of four nitrogen levels (n₀, n₁, n₂ and n₃) to the four main plots in each of the three replications of Figure 4.9.

n3

n1

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n5

n2

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n1

n2

n3

v2

v1

v1

v2

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v1

v4

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v3

v1

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v3

v2

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v1

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Replication I

Replication II

Replication III

Figure 4.11. A sample layout of a split plot design involving three eucalyptus clones(v₁, v₂ and v₃) as subplot treatments and four nitrogen levels (n₀, n₁, n₂ and n₃) as main plot treatments, in three replications.

Note that the field layout of a split plot design as illustrated by Figure 4.11 has the following important features: (i) The size of the main plot is b times the size of the subplot. In our example with 3 varieties (b = 3) the size of the main plot is 3 times the subplot size (ii) Each main plot treatment is tested r times whereas each subplot treatment is tested ar times. Thus, the number of times a subplot treatment is tested will always be larger than that for the main plot and is the primary reason for more precision for the subplot treatments relative to the main plot treatments. In our example, each of the 4 levels of nitrogen is tested 3 times but each of the 3 clones is tested 12 times.

4.6.2. Analysis of variance

The analysis of variance of a split plot design is divided into the main plot analysis and the subplot analysis. The computations are shown with the data from a two-factor experiment in eucalyptus involving two silvicultural treatments (pit size) and four fertiliser treatments. The data on height of plants after one year of planting are shown in Table 4.25.

Table 4.25. Data on height (cm) of Eucalyptus tereticornis plants from a field trial under split plot design.

Let A denote the main-plot factor (pit size) and B, the subplot factor (fertiliser treatments). Carry out the analysis of variance as follows:

Step 1. Construct an outline of the analysis of variance for a split plot design as follows.

Table 4.26. Schematic representation of ANOVA of a split plot experiment.

Source of	Degree of	Sum of	Mean square
variation	freedom (df)	squares (SS)		Computed F
Replication	r - 1	SSR	MSR
A	a - 1	SSA	MSA
Error (a)	(r - 1)(a - 1)	SSEa	MSEa
B	b - 1	SSB	MSB
AB	(a - 1)(b - 1)	SSAB	MSAB
Error (b)	a(r - 1)(b - 1)	SSEb	MSEb
Total	rab - 1	SSTO

Step 2. Construct two tables of totals as:

(i) The replication x factor A two-way table of totals, with the replication totals, Factor A totals and grand total : For our example, the replication x pit size table of totals ((RA)_ki), with the replication totals (R_k), pit size totals (A_i) and the grand total (G) computed, is shown in Table 4.27.

Table 4.27. The replication x pit size table of height totals computed from data in Table 4.25.

Pit size	Rep. I	Rep. II	Rep. III	(Ai)
p0	168.84	222.74	122.28	513.86
p1	78.26	193.76	167.83	439.85
Rep. total (Rk)	247.10	416.50	290.10
Grand total (G)				953.70

(ii) The factor A x factor B two-way table of totals, with factor B totals : For our example, the pit size x fertilizer treatment table of totals (AB), with the fertilizer treatment totals (B_j) computed, is shown in Table 4.28.

Table 4.28. The pit size x fertilizer treatment table of height totals computed from data in Table 4.25

	Fertilizer treatment
Pit size	f0	f1	f2		f3
p0	123.73	141.29		137.62		111.22
p1	132.66	85.62		132.60		88.97
Total (Bj)	256.39	226.91		270.22		200.19

Step 3. Compute the correction factor and sums of squares for the main plot analysis as follows. Let y_ijk refer to the response observed ith main plot, jth subplot in the rth replication.

(4.37)

=

SSTO =

(4.38)

= [(25.38)² + (46.56)² + … + (26.55)²] - 37897.92

= 6133.10

SSR = (4.39)

=

= 1938.51

SSA = (4.40)

=

= 228.25

SSE_a = (4.41)

=

= 1161.70

Step 4. Compute the sums of squares for the subplot analysis as:

SSB = (4.42)

=

= 488.03

SSAB = (4.43)

= - 37897.92 - 488.03 - 1161.70

= 388.31

SSE_b = SSTO - SSR - SSA - SSB - SSAB-SSE_a (4.44)

= 6133.10 - 1938.51 - 228.25 - 488.03 - 388.31

= 3090.00

Step 5. For each source of variation, compute the mean square by dividing the SS by its corresponding df. The F value for each effect that needs to be tested is to be computed by dividing each mean square by the corresponding error term as shown in Table 4.26.

Step 6. Enter all values obtained from Steps 3 to 5 in the ANOVA table as shown in Table 4.29; and compare each of the computed F values with its corresponding tabular F values and indicate its significance or otherwise by the appropriate asterisk notation. For each effect whose computed F value is not less than 1, obtain the corresponding tabular F value, from Appendix 3, with f₁ = df of the numerator MS and f₂ = df of the denominator MS, at the prescribed level of significance. For example, the tabular F value for testing the AB effect is 3.49 at 5% level of significance for 3 and 12 degrees of freedom.

Table 4.29. ANOVA of data in Table 4.20 from a split plot design

Source of	Degree of	Sum of	Mean	Computed	Tabular F
variation	freedom	squares	square	F	5%
Replication	2	1938.51	969.26
A	1	228.25	228.25	0.3930ns	4.75
Error (a)	2	1161.70	580.85
B	3	488.03	162.68	0.6318ns	3.49
AB	3	388.31	129.44	0.5027ns	3.49
Error (b)	12	3090.00	257.50
Total	23	37897.92

^ns Nonsignificant at 5% level

Step 7. Compute the two coefficients of variation, one corresponding to the main plot analysis and another corresponding to the subplot analysis.

(4.45)

(4.46)

=

The value of cv(a) indicates the degree of precision attached to the main plot factor. The value of cv(b) indicates the precision of the subplot factor and its interaction with the main-plot factor. The value of cv(b) is expected to be smaller than that of cv(a) because, as indicated earlier, the factor assigned to the main plot is expected to be measured with less precision than that assigned to the subplot. In our example, the value of cv(b) is smaller than that of cv(a) but both of them were high enough to mask any possible treatment differences turning the all the factor effects in the ANOVA nosignificant.

4.6.3. Comparison of treatments

In a split plot design, there are four different types of pair comparisons. Each requires its own set of LSD values. These comparisons are:

Type-(1). Comparisons between two main plot treatment means averaged over all subplot treatments.

Type-(2). Comparison between two subplot treatment means averaged over all main plot treatments.

Type-(3). Comparison between two subplot treatment means at the same main plot treatment.

Type-(4). Comparison between two main plot treatment means at the same or different subplot treatments (i.e., means of any two treatment combinations).

Table 4.30 gives the formula for computing the appropriate standard error of the mean difference ( ) for each of these types of pair comparison.

Table 4.30. Standard error of the mean difference for each of the four types of pair comparison in a split plot design.

Note : E_a = MSE_a, E_b = MSE_b, r = number of replications, a = number of main plot treatments, and b = number of subplot treatments.

When the computation of s__d involves more than one error term, such as in comparisons of Type-(4), the tabular t values from Appendix 2 cannot be used directly and weighted tabular t values are to be computed. The formula for weighted tabular t values in such a case is given below.

Weighted tabular t value = (4.47)

where t_a is the t value for Error (a) df and t_b is the t value for Error (b) df.

As an example, consider the 2 x 4 factorial experiment whose data are shown in Table 4.25. Although the analysis of variance (Table 4.29) shows all the three effects (the two main effects and the interaction effect) as nonsignificant, for the purpose of illustration, consider the case where there is a significant interaction between pit size and fertiliser indicating the variation in fertilizer effect with the changing pit size. In such a case, comparison between the means of pit size levels pooled over all fertilizer levels or that between fertilizer levels averaged over all levels of pit size will not be valid. The more appropriate comparisons will be those between fertilizer means under the same pit size levels or between pit size means at same fertilizer level. Thus the steps involved in the computation of the LSD for comparing two subplot means at the same main plot treatment are:

Step 1.Compute the standard error of the difference between means following the formula for Type-(3) comparison of Table 4.30.

=

Step 2. Following the formula compute the LSD value at 5% level of significance using the tabular t value with 12 degrees of freedom of Error(b).

Step 3.Construct the pit size x fertilizer two-way table of mean differences in height as shown in Table 4.31. Compare the observed differences in the mean height among the fertilzer levels at each pit size with the LSD value computed at Step 2 and identify significant differences if any.

Table 4.31. Difference between mean height of eucalyptus plants with four fertilizer levels at the pit size of 30 cm x 30 cm x 30 cm based on the data in Table 4.25.

	Difference in mean height (cm) at p0
	f0	f1	f2	f3
f0	0.00	-5.86	-4.63	4.17
f1		0.00	1.23	10.03
f2			0.00	8.80
f3				0.00
	Difference in mean height (cm) at p1
	f0	f1	f2	f3
f0	0.00	15.68	0.02	14.56
f1		0.00	-15.66	-1.12
f2			0.00	14.54
f3				0.00

Theoretically, the complete block designs, such as RCBD are applicable to experiments with any number of treatments. However, these complete block designs become inefficient as the number of treatments increases, because the blocks loose their homogeneity due to the large size. An alternative class of designs for single-factor experiments having a large number of treatments is the incomplete block designs. As the name implies, each block in an incomplete block design does not contain all treatments and so a reasonably small block size can be maintained even if the number of treatments is large. One consequence of having incomplete blocks in the design is that the comparison of treatments appearing together in a block will be made with greater precision than those not doing so. This difficulty can be circumvented by seeing that in the overall design, every pair of treatments appear together in some block or other equal number of times. Such designs are called balanced designs. Since it calls for large number of replications to achieve complete balance, one may go for partially balanced designs wherein different levels of precision for comparison are admitted for different groups of treatments. One commonly used class of incomplete block designs in forestry experiments is the class of lattice designs wherein the number of treatments is a perfect square and the blocks can be grouped into complete sets of replications. A special case of lattice designs is the simple lattice designs which is discussed in the following.

4.7.1. Simple lattice design

Simple lattice design is also called double lattice or square lattice design. As the number of treatments is to be a perfect square, the design may be constructed for number of treatments such as 9, 16, 25, 36, 49, 64, 81, 121 etc. This design needs two replications and is only a partially balanced design in the sense that the treatments form two groups and the degree of precision of treatment comparison differs between these two groups. The construction and layout of the design are exemplified for the case of 25 treatments.

Step 1. Assign numbers 1 to 25 to the treatments at random. This is necessary to avoid any kind of bias of unknown origin affecting treatment effects.

Step 2. Arrange the treatment numbers from 1 to 25 in the form of a square as given in Figure 4.12.

Figure 4.12. Initial arrangement of treatments in simple lattice design

Step 3. Group the treatments by rows. This results in the groupings (1, 2, 3, 4, 5), (6, 7, 8, 9, 10), (11, 12, 13, 14, 15), (16, 17, 18, 19, 20) and (21, 22, 23, 24, 25). Each block will now constitute a group of treatments for one block and five such blocks constitute one complete replication. This special way of row-wise grouping is generally known as X-grouping or A-grouping.

Step 4. Again group the treatments by columns. This results in the groupings (1, 6, 11, 16, 21), (2, 7, 12, 17, 22), (3, 8, 13, 18, 23), (4, 9, 14, 19, 24) and (5, 10, 15, 20, 25). Each group will now constitute a group of treatments for one block and five such blocks will constitute one complete replication. This special way of column-wise grouping is generally known as Y-grouping or B-grouping.

The X-grouping and Y-grouping ensure that treatments that have occurred together in the same block once do not appear together in the same block again. The above two sets of groupings will appear as in Figure 4.13 before randomization is done.

Replication I ( X- grouping )

Block No. 1	1	2	3	4	5
Block No. 2	6	7	8	9	10
Block No. 3	11	12	13	14	15
Block No. 4	16	17	18	19	20
Block No. 5	21	22	23	24	25

Replication II ( Y-grouping)

Block No.6	1	6	11	16	21
Block No.7	2	7	12	17	22
Block No.8	3	8	13	18	23
Block No.9	4	9	14	19	24
Block No.10	5	10	15	20	25

Figure 4.13. Two replications of a simple lattice design before randomizaion

Step 5.Within each replication, the treatment groups are allocated at random to the different blocks. This randomization is to be done separately for each replication. The allocation of treatments to plots within each block should also be randomized. The randomization should be done separately for each group independently for each replication. Finally while laying down the replications in the field, the X and Y replications should themselves be randomized over the field locations. This procedure of treatment and replication allocations ensures elimination of any kind of unknown systematic variations affecting the treatment effects. As a result of complete randomization the actual layout plan might appear as shown in Figure 4.14.

Figure 4.14. Randomized layout plan of simple lattice design.

If the blocks within each replication are contiguous, under certain conditions, this will allow the analysis of the whole experiment as a RCBD. It was mentioned already that simple lattice design requires a minimu m of two replications, one with X-grouping and other with Y-grouping of the treatments. If more than two replications are desired for this design, then it should be in multiples of two only, as both groups (X and Y) will have to be repeated equal number of times. The procedures of treatment allocations remain the same as above.