RCB 3-way factorial

Three way Factorial arrangement on a RCB

Sometimes the researcher will want to look at the effect of three different factors superimposed on each other. This is often handled with a 3-way factorial arrangement (but also look at the split-split plot design). The 3-way can also be applied to the CRD and Latin square designs.

Field marks:

Treatments are combinations of three factors, such as chemical compositions and methods of applying chemicals. This is an arrangement of treatments within a RCB design.
Treatments are assigned at random within blocks of adjacent subjects, each treatment once per block.
The number of blocks is the number of replications.
Any treatment can be adjacent to any other treatment, but not to the same treatment within the block.

Sample layout:
Different colors represent different combinations of treatments; each drawn box represents a block. There are 3 blocks (I-III) and combinations of 3, 2, & 2 treatments (A-C, changes in red intensity; 1-2, changes in green intensity; and a-b, changes in blue intensity) in this example.
RCB 3 way factorial sample layout

Block I     Block II     Block III
C1a  B2a    C2a  C1a     B2a  B1a
B1a  A2a    C1b  B1a     B2b  A2a
A2b  A1a    B2b  A2a     A2b  A1a
B1b  B2b    A1b  A1a     C1b  C2b
A1b  C2a    B1b  A2b     C1a  A1b
C2b  C1b    C2b  B2a     B1b  C2a

ANOVA table format:

Source of variation	Degrees of freedom^a	Sums of squares (SSQ)	Mean square (MS)	F
Blocks (B)	b-1	SSQ_B	SSQ_B/(b-1)	MS_B/MS_E
First factor (X)	x-1	SSQ_X	SSQ_X/(x-1)	MS_X/MS_E
Second factor (Y)	y-1	SSQ_Y	SSQ_Y/(y-1)	MS_Y/MS_E
Third factor (Z)	z-1	SSQ_Z	SSQ_Z/(z-1)	MS_Z/MS_E
First X Second (XxY)	(x-1)*(y-1)	SSQ_XxY	SSQ_XxY/((x-1)*(y-1))	MS_XxY/MS_E
First X Third (XxZ)	(x-1)*(z-1)	SSQ_XxZ	SSQ_XxZ/((x-1)*(z-1))	MS_XxZ/MS_E
Second X Third (YxZ)	(y-1)*(z-1)	SSQ_YxZ	SSQ_YxZ/((y-1)*(z-1))	MS_YxZ/MS_E
First X Second X Third (XxYxZ)	(x-1)(y-1)(z-1)	SSQ_XxYxZ	SSQ_XxYxZ/((x-1)(y-1)(z-1))	MS_XxYxZ/MS_E
Error (E)	(xyz-1)*(b-1)	SSQ_E	SSQ_E/((xyz-1)*(b-1))
Total (Tot)	xyz*b-1	SSQ_Tot
^awhere x=number of treatments in the first factor, y=number of treatments in the second factor, z=number of treatments in the third factor and b=number of blocks or replications.

Sample ANOVA table:

Source of variation	Degrees of freedom	Sums of squares (SSQ)	Mean square (MS)	F
Blocks	2	146.38	73.19	117.43^a
First	2	41.97	20.98	33.67^a
Second	1	195.70	195.70	313.99^b
Third	1	66.84	66.84	107.23^b
First X Second	2	0.04	0.02	0.03^a
First X Third	2	0.23	0.12	0.18^a
Second X Third	1	6.28	6.28	10.08^b
First X Second X Third	2	0.69	0.35	0.55^a
Error	22	13.71	0.62
Total	35	471.85
^aF test with 2,22 degrees of freedom at P=0.05 is 3.44 ^bF test with 1,22 degrees of freedom at P=0.05 is 4.30

Sample SAS GLM statements:

PROC GLM;
  CLASS BLOCKS FIRST SECOND THIRD;
  MODEL WHATEVER = BLOCKS FIRST SECOND FIRST*SECOND FIRST*THIRD
                   SECOND*THIRD FIRST*SECOND*THIRD;
RUN;

Compare with:

Randomized Complete Block (RCB): Treatments are from a single set.
Two-way Factorial arrangement on a RCB: treatments are combinations of two sets of factors.
Split-split Plot on a RCB: one set of treatments is assigned at random to experimental units; a second set is assigned to at random to sub-units; and a third set to sub-units of the second.
Split-block design: one set of treatments is randomized in one directios; a second set is randomized in a second direction.
Regression--comparing trends from different treatments: a set of treatments from a quantitative range are crossed on treatments from a qualitative set.
RCB repeated in time: one set of treatments is examined at different times.

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Thursday, August 24, 2000