The smallest run number possible to examine the main effects of 22 factors at 2 levels is:

The correct answer is B. 24. To estimate 22 main effects plus an overall mean (intercept), you need a minimum of k + 1 = 23 model parameters - but the run count must also be a valid design size. Plackett-Burman (PB) designs, the standard tool for screening main effects at 2 levels, only exist in multiples of

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(26 responses)

A
4% (1)
B
85% (22)
C
4% (1)
D
8% (2)

Explanation

To estimate 22 main effects plus an overall mean (intercept), you need a minimum of k + 1 = 23 model parameters - but the run count must also be a valid design size. Plackett-Burman (PB) designs, the standard tool for screening main effects at 2 levels, only exist in multiples of 4 (4, 8, 12, 16, 20, 24, 28…). Since 23 is not a multiple of 4, you must round up to 24 runs, which supports up to 23 factors - just enough for 22.

Why the distractors are wrong:

A (23): Although 23 ≥ k+1, it is not a valid PB or 2-level fractional factorial design size (not a multiple of 4 or a power of 2), so no such design exists.
C (44): Far exceeds the minimum; no standard design rationale produces this number for 22 factors.
D (56): Same problem - unnecessarily large and not tied to any design-size rule for this scenario.

Memory tip: For main-effects-only screening, use the rule "k + 1, rounded up to the next multiple of 4." For 22 factors: 22 + 1 = 23 → next multiple of 4 = 24.

Topics

#fractional factorial#main effects#factor screening#DOE

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