General-Full Factorial Design With Replication

[Jain, Ch. 23]

Model

Consider k factors with any number of levels. Model is sum of:

• (mean of all observations)

• main effects

• two-factor interactions

• three-factor interactions

• ...

• error (if replicas are used).

Example for three factors A,B,C at levels a, b, and c:

where , , and:

:

Observed response in th replication with factors A, B, C at levels i,j,k, respectively

:

effects of factors A, B, C, respectively

:

interaction of factors A (at level i) and B (at level j)

:

difference between and mean response of all replicas at factor levels A=i, B=j, C=k

Analysis

Simply generalize two factor design:

• effect and interaction computation

• sum of squares,

• degrees of freedom, and

• F-test.

For example, effect requires mean taken along one dimension of multi-dimensional data table:

Example: Four Factor, Three Level Design Without Replication

Response variable is page swaps.

See Table 23.1 for data. Ratio , so use of data.

See Table 22.3 for log data.

Compute:

1. means of values at each factor level

2. means of numbers from item 1 above

3. effects: difference between each number in item 1 and item 2:

   

   See Table 23.4 for all effects.

4. interactions

5. SSY, SS0, SSA, SSD, SSP, SSM, SSAD, ..., SSADP, ..., SSADPM

6. 

7. percent variation: SSA/SST, SSD/SST, ..., SSADPM/SST. Example:

   

   See Table 23.5 for results.

ANOVA Table Interpretation

Conclusions:

• All first order effects exceed 1% variation, so include them.

• In first-order interaction, DM is 4.0% of the 4.91% total; so ignore all first-order interactions except DM.

• Second- and third-order effects are about 1% total; ignore them.

So our resulting model is a lot simpler than adding terms:

where is D M interaction. See Table 23.6 for values.

Or:

Note: Jain's D,M interaction matrix is wrong - above is correct.

Informal Methods

Observation Method

If you only ask, ``which factor-level combination yields best response,'' and either a high or low response is best:

do no math - just find experiment with highest response(s)

See Table 23.8 for example.

Ranking Method

Variation of observation method:

1. Sort factor-level-response table by response variable

2. Look for pattern in each factor's levels at top and bottom of table. Make conclusions.

See Table 23.9 for example.

Observation Method

Normally we compute SSA/SST, SSB/SST, ... to compare percent variation for each factor.

Shortcut:

1. Create table:
   • rows are factors

   • columns are levels, and

   • cells are mean response (never log response) for a factor-level combination.

2. Add column containing max-min value for each row.

3. Sort rows by decreasing values for item 2 above. Result is ranking of factor importance.

See Table 23.10 for an example. ``2056'' in first row of Table 23.10 is computed from data in Table 23.2:

``1725'' in first row of Table 23.10 is