23 Factorial Experiment in RBD

23 factorial experiment

    • 23 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

    Analysis:
    Arrange the results as per treatment combinations and replications.
    Treatment combination Replication
    R1 R2 R3 …
    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
    • BSS
    • ABSS
    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

    • CSS
    • ACSS
    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
    • BCSS
    • Abcss
    • CF
    • ESS
    • 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.

Last modified: Sunday, 8 April 2012, 6:47 PM