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 |
r-1 |
RSS |
RMS |
|
Factor A |
a-1 |
ASS |
AMS |
AMS / EMS |
Factor B |
b-1 |
BSS |
BMS |
BMS / EMS |
Factor C |
c-1 |
CSS |
CMS |
CMS / EMS |
AB |
(a-1)(b-1) |
ABSS |
ABMS |
ABMS / EMS |
AC |
(a-1)(c-1) |
ACSS |
ACMS |
ACMS / EMS |
BC |
(b-1)(c-1) |
BCSS |
BCMS |
BCMS / EMS |
ABC |
(a-1)(b-1)(c-1) |
ABCSS |
ABCMS |
ABCMS / EMS |
Error |
(r-1)(abc-1) |
ESS |
EMS |
|
Total |
rabc-1 |
TSS |
|
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Analysis
- Arrange the results as per treatment combinations and replications.
Treatment combination |
Replication |
Treatment Total |
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a0b0c0 |
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T1 |
a0b0c1 |
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T2 |
a0b1c0 |
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T3 |
a0b1c1 |
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T4 |
a1b0c0 |
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T5 |
a1b0c1 |
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T6 |
a1b1c0 |
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T7 |
a1b1c1 |
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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
|
c0 |
c1 |
Total |
a0 |
a0 c0 |
a0 c1 |
A0 |
a1 |
a1 c0 |
a1 c1 |
A1 |
Total |
C0 |
C1 |
Grand Total |
C A
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 |
-CF-ASS-BSS-CSS-ABSS-ACSS-BCSS
ESS = TSS-RSS- ASS-BSS-CSS-ABSS-ACSS-BCSS-ABCSS
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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