DoE Implementation - Design, Analysis & Interpretation
Step 1: We have 3 factors (A, B, C), each at 2 levels (-1 = Low, +1 = High)
Step 2: Total runs needed = 2³ = 8 experiments
Step 3: We use standard order based on binary counting (000, 001, 010, 011, 100, 101, 110, 111)
Step 4: Convert binary to factorial notation: 0 → -1, 1 → +1
Factor A: Compression Force (15 kN = -1, 25 kN = +1)
Factor B: Binder Concentration (2% = -1, 6% = +1)
Factor C: Lubricant Level (0.5% = -1, 1.5% = +1)
Response: Tablet Hardness (kp)
| Run | Standard Order | A (Compression) | B (Binder) | C (Lubricant) | AB | AC | BC | ABC | Hardness (kp) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | -1 | -1 | -1 | +1 | +1 | +1 | -1 | |
| 2 | 8 | +1 | -1 | -1 | -1 | -1 | +1 | +1 | |
| 3 | 5 | -1 | +1 | -1 | -1 | +1 | -1 | +1 | |
| 4 | 3 | +1 | +1 | -1 | +1 | -1 | -1 | -1 | |
| 5 | 7 | -1 | -1 | +1 | +1 | -1 | -1 | +1 | |
| 6 | 4 | +1 | -1 | +1 | -1 | +1 | -1 | -1 | |
| 7 | 6 | -1 | +1 | +1 | -1 | -1 | +1 | -1 | |
| 8 | 2 | +1 | +1 | +1 | +1 | +1 | +1 | +1 |
Randomization Note: The run order should be randomized to protect against bias. The table shows one possible random sequence.
Step 1: A main effect measures the average change in response when moving from low (-1) to high (+1) level
Step 2: Sum all responses where factor is at high level (+1)
Step 3: Sum all responses where factor is at low level (-1)
Step 4: Calculate the difference and divide by half the total number of runs
Step 1: For AB interaction, we look at how the effect of A changes at different levels of B
Step 2: Calculate the effect of A when B is at high level
Step 3: Calculate the effect of A when B is at low level
Step 4: The AB interaction is the difference between these two A effects, divided by 2
Positive AB Interaction: The effect of compression force (A) is stronger when binder concentration (B) is high
Negative AB Interaction: The effect of compression force (A) is weaker when binder concentration (B) is high
No Interaction: The effect of compression force (A) is the same regardless of binder level (B)
Step 1: We partition the total variance into different sources
Step 2: Sources include: Main effects (A, B, C), Two-factor interactions (AB, AC, BC), Three-factor interaction (ABC), and Error
Step 3: Calculate Sum of Squares (SS) for each source
Step 4: Calculate degrees of freedom (df) for each source
Step 5: Calculate Mean Squares (MS = SS/df) and F-ratios (F = MS_effect/MS_error)
Step 1: Start with the overall mean (intercept)
Step 2: Add significant main effects (coefficient = Effect/2)
Step 3: Add significant interactions (coefficient = Effect/2)
Step 4: Final model: Y = b₀ + b₁X₁ + b₂X₂ + b₃X₃ + b₁₂X₁X₂ + ...
Step 1: 2^k designs assume linear relationships (no curvature)
Step 2: But many pharmaceutical responses are curved (think dissolution vs time)
Step 3: Center points (0,0,0) help detect if curvature exists
Step 4: If center point response differs significantly from factorial point average, curvature is present
Imagine we're studying polymer concentration effect on dissolution rate:
The center point shows much higher dissolution, indicating a curved relationship that 2^k design would miss!
Scenario: You're optimizing a capsule formulation with two factors:
Tasks:
Given Data: Using the 2² design from Exercise 1, dissolution times (minutes) were:
Calculate:
Challenge: Using your calculated effects from Exercise 2: