Let's think step-by-step about how to construct a factorial design. A 2^k factorial design systematically explores all possible combinations of k factors, each at 2 levels (typically coded as -1 for low and +1 for high).
Step 1: Understand what we're building
- We have 3 factors (k=3), so we need 2Β³ = 8 experimental runs
- Each factor will be at low (-1) or high (+1) levels
- We want to systematically explore ALL combinations
Step 2: Use Yates' standard order algorithm
- Factor A: Alternates every run (-, +, -, +, -, +, -, +)
- Factor B: Alternates every 2 runs (-, -, +, +, -, -, +, +)
- Factor C: Alternates every 4 runs (-, -, -, -, +, +, +, +)
Step 3: This creates a balanced, orthogonal design
| Run | A: Binder % | B: Compression Force | C: Lubricant % | Response: Hardness (kp) |
|---|---|---|---|---|
| 1 | -1 (2%) | -1 (500 kg) | -1 (0.5%) | Yβ |
| 2 | +1 (4%) | -1 (500 kg) | -1 (0.5%) | Yβ |
| 3 | -1 (2%) | +1 (1000 kg) | -1 (0.5%) | Yβ |
| 4 | +1 (4%) | +1 (1000 kg) | -1 (0.5%) | Yβ |
| 5 | -1 (2%) | -1 (500 kg) | +1 (1.5%) | Yβ |
| 6 | +1 (4%) | -1 (500 kg) | +1 (1.5%) | Yβ |
| 7 | -1 (2%) | +1 (1000 kg) | +1 (1.5%) | Yβ |
| 8 | +1 (4%) | +1 (1000 kg) | +1 (1.5%) | Yβ |
Now let's understand how to calculate effects from our factorial design:
Understanding the logic: We want to know the average effect of changing binder from low to high level.
Step 1: Identify runs where A = +1
Runs 2, 4, 6, 8: Yβ + Yβ + Yβ + Yβ
Step 2: Identify runs where A = -1
Runs 1, 3, 5, 7: Yβ + Yβ + Yβ
+ Yβ
Step 3: Calculate the contrast
Contrast_A = (Yβ + Yβ + Yβ + Yβ) - (Yβ + Yβ + Yβ
+ Yβ)
Step 4: Convert to average effect
Effect_A = Contrast_A Γ· 4 (since we have 4 observations at each level)
Scenario: We conducted a 2Β³ design to study tablet hardness with results:
| Run | A | B | C | Hardness (kp) |
|---|---|---|---|---|
| 1 | -1 | -1 | -1 | 6.2 |
| 2 | +1 | -1 | -1 | 8.5 |
| 3 | -1 | +1 | -1 | 7.8 |
| 4 | +1 | +1 | -1 | 10.1 |
| 5 | -1 | -1 | +1 | 5.1 |
| 6 | +1 | -1 | +1 | 7.2 |
| 7 | -1 | +1 | +1 | 6.5 |
| 8 | +1 | +1 | +1 | 8.9 |
Calculation:
Effect_A = [(8.5 + 10.1 + 7.2 + 8.9) - (6.2 + 7.8 + 5.1 + 6.5)] Γ· 4
Effect_A = [34.7 - 25.6] Γ· 4 = 9.1 Γ· 4 = 2.275 kp
Interpretation: Increasing binder from 2% to 4% increases tablet hardness by an average of 2.3 kp.
Understanding interactions: An interaction means the effect of factor A depends on the level of factor B.
Step 1: Identify runs where A and B have the same sign
Both negative: Runs 1, 5 (A=-1, B=-1)
Both positive: Runs 4, 8 (A=+1, B=+1)
Step 2: Identify runs where A and B have opposite signs
A negative, B positive: Runs 3, 7 (A=-1, B=+1)
A positive, B negative: Runs 2, 6 (A=+1, B=-1)
Step 3: Calculate the interaction contrast
AB_contrast = (Yβ + Yβ + Yβ
+ Yβ) - (Yβ + Yβ + Yβ + Yβ)
Step 4: Convert to interaction effect
Effect_AB = AB_contrast Γ· 4
Problem: We need to run our 8 experiments, but can only do 4 per day.
Poor approach: Run experiments 1-4 on Monday, 5-8 on Tuesday
Risk: Any difference between days gets confounded with factor C
Good approach: Randomize within blocks
Day 1 (random order): Runs 3, 7, 1, 5
Day 2 (random order): Runs 6, 2, 8, 4
Excel implementation: =RAND() function to assign random numbers, then sort
Let's think about why we sometimes need three levels instead of two:
Problem with 2-level designs:
- They can only detect linear relationships
- Real pharmaceutical responses often have curves (optimums)
Solution with 3-level designs:
- Low level: -1
- Middle level: 0
- High level: +1
- Now we can fit quadratic (curved) models!
Cost consideration:
2Β² design = 4 runs
3Β² design = 9 runs
2Β³ design = 8 runs
3Β³ design = 27 runs (!)
Scenario: We're studying how polymer concentration affects drug release rate.
| Polymer Concentration | Coded Level | Dissolution Rate (%/min) |
|---|---|---|
| 1% | -1 | 2.1 |
| 3% | 0 | 4.8 |
| 5% | +1 | 2.3 |
Observation: The middle level (3%) gives the highest dissolution rate! A 2-level design would have missed this optimum completely.
Linear model (2-level): Would predict dissolution increases with polymer
Quadratic model (3-level): Correctly identifies the optimum at 3% polymer
A clever way to detect curvature without the full expense of a 3^k design:
The idea: Add center point runs (0,0,...,0) to a 2^k design
Step 1: Run the standard 2^k factorial design
Calculate average response of factorial points: Θ³_factorial
Step 2: Run multiple center point experiments
Calculate average response of center points: Θ³_center
Step 3: Test for curvature
If |Θ³_factorial - Θ³_center| is statistically significant β curvature exists
Statistical test: t-test comparing the two means
Formula for t-statistic:
Where sΒ² is the pooled variance estimate
Data from 2Β² + 4 center points design:
| Run Type | A: Binder | B: Force | Hardness (kp) |
|---|---|---|---|
| Factorial | -1 | -1 | 6.2 |
| Factorial | +1 | -1 | 8.5 |
| Factorial | -1 | +1 | 7.8 |
| Factorial | +1 | +1 | 10.1 |
| Center | 0 | 0 | 9.8 |
| Center | 0 | 0 | 9.5 |
| Center | 0 | 0 | 9.7 |
| Center | 0 | 0 | 9.6 |
Calculation:
Θ³_factorial = (6.2 + 8.5 + 7.8 + 10.1) Γ· 4 = 8.15 kp
Θ³_center = (9.8 + 9.5 + 9.7 + 9.6) Γ· 4 = 9.65 kp
Difference = 9.65 - 8.15 = 1.5 kp
Conclusion: Center points are significantly higher β curvature exists! The optimum is near the center of our design space.
Analysis of Variance (ANOVA) helps us determine which effects are statistically significant:
Step 1: Partition the total variation
Total SS = SS(A) + SS(B) + SS(C) + SS(AB) + SS(AC) + SS(BC) + SS(ABC) + SS(Error)
Step 2: Calculate Mean Squares
MS = SS Γ· degrees of freedom
For main effects: df = 1
For interactions: df = 1 (in 2^k designs)
Step 3: Calculate F-statistics
F = MS(effect) Γ· MS(Error)
Step 4: Compare with critical F-value
If F > F_critical β effect is significant
| Source | SS | df | MS | F | p-value | Significant? |
|---|---|---|---|---|---|---|
| A (Binder) | 20.7 | 1 | 20.7 | 103.5 | <0.001 | Yes |
| B (Force) | 12.3 | 1 | 12.3 | 61.5 | <0.001 | Yes |
| C (Lubricant) | 8.1 | 1 | 8.1 | 40.5 | <0.001 | Yes |
| AB | 0.8 | 1 | 0.8 | 4.0 | 0.046 | Yes |
| AC | 0.2 | 1 | 0.2 | 1.0 | 0.317 | No |
| BC | 0.1 | 1 | 0.1 | 0.5 | 0.481 | No |
| ABC | 0.0 | 1 | 0.0 | 0.0 | 0.995 | No |
| Error | 1.6 | 8 | 0.2 | - | - | - |
Once we know which effects are significant, we build a prediction equation:
Step 1: Start with the general form
Y = bβ + bβXβ + bβXβ + bβXβ + bββXβXβ + ... + Ξ΅
Step 2: Calculate coefficients from effects
bβ = overall average of all responses
bβ = Effect_A Γ· 2
bββ = Effect_AB Γ· 2
Step 3: Include only significant terms
From our ANOVA: A, B, C, and AB are significant
Step 4: Write the final model
From our factorial design analysis:
Significant effects:
β’ Effect_A (Binder) = +2.28 kp
β’ Effect_B (Force) = +1.76 kp
β’ Effect_C (Lubricant) = -1.43 kp
β’ Effect_AB = +0.45 kp
β’ Overall mean = 7.65 kp
Model coefficients:
bβ = 7.65
bβ = 2.28 Γ· 2 = 1.14
bβ = 1.76 Γ· 2 = 0.88
bβ = -1.43 Γ· 2 = -0.72
bββ = 0.45 Γ· 2 = 0.23
Practical use: This model can predict tablet hardness for any combination of factor levels within our experimental range!
If factor levels are in cells A1 (Binder), B1 (Force), C1 (Lubricant):
=7.65 + 1.14*A1 + 0.88*B1 - 0.72*C1 + 0.23*A1*B1