How do you show that a design is a of projectivity 3 b ortho

How do you show that a design is (a) of projectivity 3, (b) orthogonal, if the design is a two-level design for seven factors in 12 runs?

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Suppose on using the above design, you fins suspect factors P, Q, R, and U are active. How could you augment the design to make it a 2^4 factorial?

Solution

there is 7(=8-1) main effect in the model so it is (2³,2) design. hence it is orthogonal and projectivity 3

Factorial Fit: OBSERVED versus P, Q, R, S, T, U, V

Estimated Effects and Coefficients for OBSERVED (coded units)

Term Effect Coef
Constant 32.50
P -8.67 -4.33
Q -7.00 -3.50
R -4.67 -2.33
S -3.33 -1.67
T 7.17 3.58
U 2.17 1.08
V 14.33 7.17
P*Q 24.00 12.00
P*S -7.00 -3.50
P*T -22.50 -11.25
P*U 18.50 9.25

S = * PRESS = *

Analysis of Variance for OBSERVED (coded units)

Source DF Seq SS Adj SS Adj MS F P
Main Effects 7 731.18 905.66 129.379 * *
P 1 102.08 75.11 75.111 * *
Q 1 4.32 1.42 1.420 * *
R 1 72.51 0.53 0.527 * *
S 1 104.17 4.76 4.762 * *
T 1 446.88 11.85 11.853 * *
U 1 0.51 2.56 2.561 * *
V 1 0.71 154.08 154.083 * *
2-Way Interactions 4 1913.73 1913.73 478.433 * *
P*Q 1 1512.90 384.00 384.000 * *
P*S 1 162.00 16.33 16.333 * *
P*T 1 10.67 168.75 168.750 * *
P*U 1 228.17 228.17 228.167 * *
Residual Error 0 * * *
Total 11 2644.92