STAM101 :: Lecture 20 :: 2cube factorial experiments in RBD – lay out – analysis
2cube Factorial Experiment in RBD
2cube factorial experiment mean three factors each at two levels. Suppose the three factors are A, B and C are tried with two levels the total number of combinations will be eight i.e. a0b0c0, a0b0c1, a0b1c0, a0b1c1, a1b0c0, a1b0c1, a1b1c0 and a1b1c1.
The allotment of these eight treatment combinations will be as allotted in RBD. That is each block is divided into eight experimental units. By using the random numbers these eight combinations are allotted at random for each block separately.
The analysis of variance table for three factors A with a levels, B with b levels and C with c levels with r replications tried in RBD will be as follows:
Sources of Variation 
d.f. 
SS 
MS 
F 
Replications 
r1 
RSS 
RMS 

Factor A 
a1 
ASS 
AMS 
AMS / EMS 
Factor B 
b1 
BSS 
BMS 
BMS / EMS 
Factor C 
c1 
CSS 
CMS 
CMS / EMS 
AB 
(a1)(b1) 
ABSS 
ABMS 
ABMS / EMS 
AC 
(a1)(c1) 
ACSS 
ACMS 
ACMS / EMS 
BC 
(b1)(c1) 
BCSS 
BCMS 
BCMS / EMS 
ABC 
(a1)(b1)(c1) 
ABCSS 
ABCMS 
ABCMS / EMS 
Error 
(r1)(abc1) 
ESS 
EMS 

Total 
rabc1 
TSS 


Analysis
 Arrange the results as per treatment combinations and replications.
Treatment combination 
Replication 
Treatment Total 

a0b0c0 




T1 
a0b0c1 




T2 
a0b1c0 




T3 
a0b1c1 




T4 
a1b0c0 




T5 
a1b0c1 




T6 
a1b1c0 




T7 
a1b1c1 




T8 
As in the previous designs calculate the replication totals to calculate the CF, RSS, TSS, overall TrSS in the usual way. To calculate ASS, BSS, CSS, ABSS, ACSS, BCSS and ABCSS, form three two way tables A X B, AXC and BXC.
AXB two way table can be formed by taking the levels of A in rows and levels of B in the columns. To get the values in this table the missing factor is replication. That is by adding over replication we can form this table.
A X B Two way table
B A 
b0 
b1 
Total 
a0 
a0 b0 
a0 b1 
A0 
a1 
a1 b0 
a1 b1 
A1 
Total 
B0 
B1 
Grand Total 
ASS=
A X C two way table can be formed by taking the levels of A in rows and levels of C in the columns
A X C Two way table
C A 
c0 
c1 
Total 
a0 
a0 c0 
a0 c1 
A0 
a1 
a1 c0 
a1 c1 
A1 
Total 
C0 
C1 
Grand Total 
B X C two way table can be formed by taking the levels of B in rows and levels of C in the columns
B X C Two way table
C B 
c0 
c1 
Total 
b0 
b0 c0 
b0 c1 
B0 
b1 
b1 c0 
b1 c1 
B1 
Total 
C0 
C1 
Grand Total 
CFASSBSSCSSABSSACSSBCSS
ESS = TSSRSS ASSBSSCSSABSSACSSBCSSABCSS
By substituting the above values in the ANOVA table corresponding to the columns sum of squares, the mean squares and F value can be calculated.
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