CRD factorial

Two way Factorial arrangement on a CRD

Field marks:

Treatments are combinations of two factors, such as chemical compositions and methods of applying chemicals. This is an arrangement of treatments within a CRD design.
Replications of treatments are assigned completely at random to independent experimental subjects.
Adjacent subjects could potentially have the same combination of treatments.

Sample layout:
Different colors represent different combinations of treatments; each combination is randomly assigned to a subject. There are 4 replications (1-4) and 4 treatment combinations (Aa, Ba, Ab and Bb) in this example.
CRD factorial sample layout

Aa1  Ba1  Ab1  Aa2

Bb1  Aa3  Bb2  Ab2 

Ba2  Bb3  Ab3  Ba3

Ab4  Aa4  Ba4  Bb4

ANOVA table format:

Source of variation	Degrees of freedom^a	Sums of squares (SSQ)	Mean square (MS)	F
First factor (F1)	f-1	SSQ_F1	SSQ_F1/(f-1)	MS_F1/MS_E
Second factor (F2)	s-1	SSQ_F2	SSQ_F2/(s-1)	MS_F2/MS_E
First X Second (FxS)	(f-1)*(s-1)	SSQ_FxS	SSQ_FxS/((f-1)*(s-1))	MS_FxS/MS_E
Error (E)	fs(r-1)	SSQ_E	SSQ_E/(fs(r-1))
Total (Tot)	fsr-1	SSQ_Tot
^awhere f=number of treatments in the first factor, s=number of treatments in the second factor and r=number of replications.

Sample ANOVA table:

Source of variation	Degrees of freedom	Sums of squares (SSQ)	Mean square (MS)	F
First	1	80.30	80.30	259.43^a
Second	1	34.50	34.50	111.47^a
First X Second	1	0.22	0.22	0.70^a
Error	12	63.71	0.31
Total	15	118.73
^aF test with 1,12 degrees of freedom at P=0.05 is 4.75

Sample SAS GLM statements:

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

Compare with:

Complete Randomized Design (CRD): Treatments are from a single set.
Randomized Complete Block (RCB): Treatments are from a single set assigned at random once to each spatial block.
Two-way Factorial arrangement on a RCB: combinations of treatments are assigned at random once to each block.
Split Plot on a CRD: one set of treatments is assigned at random to experimental units; a second set is assigned to at random to sub-units.
Regression--comparing trends from different treatments: A set of treatments from a quantitative range are crossed on treatments from a qualitative set.
Analysis of covariance--controlling variation with a second measureable factor: additional variation is controlled by measuring a second variable on each subject.

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Wednesday, August 16, 2000