16 The article “More on Planning Experiments to Increase Research Efficiency”

Example 11.16 The article “More on Planning Experiments to Increase Research Efficiency”

  (Industrial and Eng. Chemistry, 1970: 60–65) reports on the results of a quarter-

  replicate of a 2 5 experiment in which the five factors were

  A 5 condensation

  temperature, B 5 amount of material B, C 5 solvent volume,

  D 5 condensation

  time , and

  E 5 amount of material E . The response variable was the yield of the

  chemical process. The chosen defining contrasts were ACE and BDE, with generalized interaction (ACE)(BDE) 5 ABCD . The remaining 28 main effects and interactions can now be partitioned into seven groups of four effects each, such that the effects within a group cannot be assessed separately. For example, the generalized interactions of A with the nonestimable effects are (A)(ACE) 5 CE, (A)(BDE) 5 ABDE , and (A)(ABCD) 5 BCD , so one alias group is 5A, CE, ABDE, BCD6 . The complete set of alias groups is

  5A, CE, ABDE, BCD6

  5B, ABCE, DE, ACD6

  5C, AE, BCDE, ABD6

  5D, ACDE, BE, ABC6

  5E, AC, BD, ABCDE6

  5AB, BCE, ADE, CD6

  5AD, CDE, ABE, BC6

  ■

  Once the defining contrasts have been chosen for a quarter-replicate, they are used as in the discussion of confounding to divide the 2 p treatment conditions into four

  groups of 2 p22 conditions each. Then any one of the four groups is selected as the set of conditions for which data will be collected. Similar comments apply to a 12 r

  replicate of a 2 p factorial experiment.

  Having made observations for the selected treatment combinations, a table of signs similar to Table 11.10 is constructed. The table contains a row only for each of the treatment combinations actually observed rather than the full 2 p rows, and there

  CHAPTER 11 Multifactor Analysis of Variance

  is a single column for each alias group (since each effect in the group would have the same set of signs for the treatment conditions selected for observation). The signs in each column indicate as usual how contrasts for the various sums of squares are computed. Yates’s method can also be used, but the rule for arranging observed con- ditions in standard order must be modified.

  The difficult part of a fractional replication analysis typically involves decid- ing what to use for error sum of squares. Since there will usually be no replication (though one could observe, e.g., two replicates of a quarter-replicate), some effect sums of squares must be pooled to obtain an error sum of squares. In a half-replicate

  of a 2 8 experiment, for example, an alias structure can be chosen so that the eight main effects and 28 two-factor interactions are each confounded only with higher- order interactions and that there are an additional 27 alias groups involving only higher-order interactions. Assuming the absence of higher-order interaction effects, the resulting 27 sums of squares can then be added to yield an error sum of squares, allowing 1 df tests for all main effects and two-factor interactions. However, in many cases tests for main effects can be obtained only by pooling some or all of the sums of squares associated with alias groups involving two-factor interactions, and the corresponding two-factor interactions cannot be investigated.

  Example 11.17 The set of treatment conditions chosen and resulting yields for the quarter-replicate (Example 11.16

  of the 2 5 experiment were

  23.2 15.5 16.9 16.2 23.8 23.4 16.8 18.1 The abbreviated table of signs is displayed in Table 11.15.

  With SSA denoting the sum of squares for effects in the alias group {A, CE, ABDE, BCD},

  Table 11.15 Table of Signs for Example 11.17 A B C D E AB AD

  Similarly, (the SSB 5 53.56, SSC 5 10.35, SSD 5 .91 SSEr 5 10.35 ⬘ differenti- ates this quantity from error sum of squares SSE), SSAB 5 6.66 , and SSAD 5 3.25, giving SST 5 4.65 1 53.56 1 c 1 3.25 5 89.73 . To test for main effects, we use SSE 5 SSAB 1 SSAD 5 9.91 with 2 df. The ANOVA table is in Table 11.16.

  Since F .05,1,2 5 18.51 , none of the five main effects can be judged significant. Of course, with only 2 df for error, the test is not very powerful (i.e., it is quite likely to fail to detect the presence of effects). The article from Industrial and Engineering Chemistry from which the data came actually has an independent estimate of the

  11.4 2 p Factorial Experiments

  Table 11.16 ANOVA Table for Example 11.17 Source

  df Sum of Squares

  Mean Square f

  A 1 4.65 .94 B 1 53.56 10.80 C 1 10.35 2.09

  standard error of the treatment effects based on prior experience, so it used a some- what different analysis. Our analysis was done here only for illustrative purposes, since one would ordinarily want many more than 2 df for error.

  ■ As an alternative to F tests based on pooling sums of squares to obtain SSE,

  a normal probability plot of effect contrasts can be examined.