Allocation of Variation

Old Procedure (for 2^k Factorial Design)

  1. Calculate three sum of squares terms -- SSA, SSB, and SSAB -- and their sum: 
        SST = SSA + SSB + SSAB.

  2. Calculate fraction of variation explained by A, B, and AB: 
        SSA/SST,  SSB/SST,  and SSAB/SST


Definition: SSE

The sum of errors is always 0.

The sum of squared errors (SSE) is not:



Example of SSE

Recall last sign table: 

Sum of errors for experiment 1 = e_{11} + e_{12} +e_{13} 
			       = 0 + 3 - 3 
			       = 0.

SSE for all experiments = 0^2 + 3^2 + (-3)^2 + ... + 4^2 
			= 102


Definitions: SSY and SS0

SSY = sum_{i,j} y_{ij}^2

SS0 = (2^2)r(q_0^2)



Computing SSE

Computing SSA, SSB, and SSB is easy: (2^2)r(q_A^2), etc.

But computing SSE from definition is painful -- lots of terms to sum!

Instead, compute SSE from other SS* terms (SST, SSA, SSB, SSAB) and from SS0:

SST = SSA + SSB + SSAB + SSE

SST = SSY - SS0

Thus SSE = SSY - (SS0 + SSA + SSB + SSAB)



New Procedure (for (2^k)r)

  1. Compute SSY

  2. Compute SS0, SSA, SSB, SSAB

  3. Compute SSE from preceeding SS* terms

  4. Compute SST = SSY - SS0

  5. Compute SSA/SST, SSB/SST, SSAB/SST, and SSE/SST


Example of Allocating Variation for (2^k)r Factorial Design

For the memory-cache study. 

SSY   = 15^2 + 18^2 + 12^2 + 45^2+... = 27,204
SSO   = (2^2)r(q_0^2)  = 4*3*41^2     = 20,172
SSA   = (2^2)r(q_A^2)  = 4*3*21.5^2   =  5,547
SSB   = (2^2)r(q_B^2)  = 4*3*9.5^2    =  1,083
SSAB  = (2^2)r(q_AB^2) = 4*3*5^2      =    300
SSE   = SSY - (SS0+SSA+SSB+SSAB)      =    102
SST   = SSY - SS0                     =  7,032

Thus: