Better to compute systematically:

: Main effects A, B, C positive; interactions AB, BC positive; AC negligible. Block effect significant but aliased with ABC. Example 3: (2^4) Design in 4 Blocks (Confounding ABC and ABD) Problem : Construct a (2^4) design (A, B, C, D) in 4 blocks of 4 runs each, confounding ABC and ABD. Find all confounded effects.

Compute: 25 1 +22 (-1)+20*(-1)+30 1 +24 (-1)+28 1 +32 1 +35*(-1) = 25 -22 -20 +30 -24 +28 +32 -35 = (25-22=3; 3-20=-17; -17+30=13; 13-24=-11; -11+28=17; 17+32=49; 49-35=14).

B: -25-22+20+30-24-28+32+35 = (-47+20=-27; -27+30=3; 3-24=-21; -21-28=-49; -49+32=-17; -17+35=18) ✅

C: -25-22-20-30+24+28+32+35 = (-47-20=-67; -67-30=-97; -97+24=-73; -73+28=-45; -45+32=-13; -13+35=22) ✅

(A = (-1, +1, -1, +1, -1, +1, -1, +1) ) (B = (-1, -1, +1, +1, -1, -1, +1, +1)) (C = (-1, -1, -1, -1, +1, +1, +1, +1))

Order: (1), a, b, ab, c, ac, bc, abc.

So ABC contrast = 14. This is the difference between Block 1 and Block 2? Let’s check block totals:

Thus, in this design, we cannot estimate ABC, ABD, or CD separately from block differences. When a design is replicated in blocks but different effects are confounded in different replicates, we have partial confounding . This allows estimation of all effects, but with reduced precision for the confounded ones.

If you have specific problem numbers from your textbook, I can provide the exact step‑by‑step solutions.

AC: (+1,-1,+1,-1,-1,+1,-1,+1) = 25-22+20-30-24+28-32+35 = (25-22=3; 3+20=23; 23-30=-7; -7-24=-31; -31+28=-3; -3-32=-35; -35+35=0) ✅

Effect B: Contrast = (-y_(1) - y_a + y_b + y_ab - y_c - y_ac + y_bc + y_abc) = (-25 -22 +20 +30 -24 -28 +32 +35) = (-47 +50=3 -24=-21 -28=-49 +32=-17 +35=18) → Wait, recalc carefully:

Block 1: (1)=25, ab=30, ac=28, bc=32 Block 2: a=22, b=20, c=24, abc=35