3.4 Design of Experiments Foundations

3.4 Design of Experiments Foundations

Design of Experiments (DOE) is a structured, mathematically optimal methodology used to plan, conduct, analyze, and interpret controlled tests. Unlike legacy One-Factor-At-a-Time (OFAT) experimentation—which changes a single variable while holding all others constant—DOE allows quality engineers to systematically evaluate multiple input factors simultaneously, determine their individual impact, and discover complex factor interactions with a minimal number of experimental runs.

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1. Three Core Principles of DOE

Sir Ronald Fisher established the three fundamental pillars of experimental design:

+-----------------------------------------------------------------------------------+
|                                 CORE DOE PRINICIPLES                              |
+-------------------+-------------------------------+-------------------------------+
|   RANDOMIZATION   |          REPLICATION          |            BLOCKING           |
+-------------------+-------------------------------+-------------------------------+
| Protects against  | Provides estimate of          | Isolates known, controllable  |
| lurking variables | experimental error variance   | nuisance variables (e.g.,     |
| and time-order    | and increases statistical     | shift, raw material batch,    |
| process drift.    | power / precision.            | test operator).               |
+-------------------+-------------------------------+-------------------------------+

1. Randomization: Running experimental trials in a completely randomized sequence. Randomization balances out the effects of unmeasured or uncontrolled lurking variables (such as ambient temperature change, operator fatigue, or machine warm-up drift) across all treatment combinations.
2. Replication: Repeating independent experimental runs for each factor combination. Replication allows the experimenter to estimate pure experimental error variance ($\sigma^2_{\text{error}}$) and increases the statistical power to detect true effect signals. Note: Repeated measurements of the same physical unit are NOT true replications; they are repeated measures. True replication requires setting up the experimental run again from scratch.
3. Blocking: A design technique used to isolate and eliminate the variance caused by known, controllable nuisance factors that are not of primary interest (e.g., different raw material batches, different test shifts, or distinct lab instruments). By grouping homogeneous runs into "blocks," nuisance variation is filtered out of the error term.

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2. Key DOE Terminology

• Factor ($k$): An independent variable manipulated by the experimenter (e.g., Temperature, Pressure, Cure Time).
• Level: The specific values or settings assigned to a factor. In standard 2-level designs, levels are coded as Low ($-1$ or $-$) and High ($+1$ or $+$).
• Response Variable ($Y$): The measured output or dependent quality characteristic (e.g., Tensile Strength, Yield %, Surface Roughness).
• Treatment Combination: A specific combination of factor levels applied during an experimental run.
• Confounding / Aliasing: When the effects of two or more factors or interactions cannot be separated mathematically.

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3. Full $2^k$ Factorial Designs

A Full Factorial Design tests all possible combinations of factor levels. When evaluating $k$ factors, each at $2$ levels, the total number of experimental runs required for a single replicate $N$ is:

$$N = 2^k$$

• For $k = 2$ factors: $N = 2^2 = 4$ runs.
• For $k = 3$ factors: $N = 2^3 = 8$ runs.
• For $k = 4$ factors: $N = 2^4 = 16$ runs.
• If $n$ replicates are performed, total runs $= n \times 2^k$.

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4. Calculating Main Effects and Interactions (Step-by-Step)

Mathematical Definitions

• Main Effect: The average change in response $Y$ observed when moving a factor from its Low level ($-$) to its High level ($+$): $$\text{Effect}(A) = \bar{Y}_{A+} - \bar{Y}_{A-}$$
• Interaction Effect ($AB$): Occurs when the effect of Factor $A$ depends on the level setting of Factor $B$. Mathematically, it is half the difference between the effect of $A$ at the High level of $B$ and the effect of $A$ at the Low level of $B$: $$\text{Effect}(AB) = \frac{1}{2} \left( [\bar{Y}_{A+B+} - \bar{Y}_{A-B+}] - [\bar{Y}_{A+B-} - \bar{Y}_{A-B-}] \right)$$

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Worked Numerical Problem ($2^2$ Design)

A process quality engineer investigates a polymer molding process to maximize Tensile Strength ($Y$ in MPa).

• Factor A: Mold Temperature (Low $- = 150^\circ\text{C}$, High $+ = 180^\circ\text{C}$)
• Factor B: Injection Pressure (Low $- = 30\text{ psi}$, High $+ = 50\text{ psi}$)

The resulting experimental response table for 4 single-replicate runs is:

Step 1: Calculate Main Effect of Factor A

$$\bar{Y}_{A+} = \frac{Y_{\text{Run 2}} + Y_{\text{Run 4}}}{2} = \frac{52 + 70}{2} = \frac{122}{2} = 61.0$$ $$\bar{Y}_{A-} = \frac{Y_{\text{Run 1}} + Y_{\text{Run 3}}}{2} = \frac{40 + 46}{2} = \frac{86}{2} = 43.0$$ $$\text{Effect}(A) = 61.0 - 43.0 = +18.0\text{ MPa}$$

Interpretation: Increasing Mold Temperature from 150°C to 180°C increases average tensile strength by 18.0 MPa.

Step 2: Calculate Main Effect of Factor B

$$\bar{Y}_{B+} = \frac{Y_{\text{Run 3}} + Y_{\text{Run 4}}}{2} = \frac{46 + 70}{2} = \frac{116}{2} = 58.0$$ $$\bar{Y}_{B-} = \frac{Y_{\text{Run 1}} + Y_{\text{Run 2}}}{2} = \frac{40 + 52}{2} = \frac{92}{2} = 46.0$$ $$\text{Effect}(B) = 58.0 - 46.0 = +12.0\text{ MPa}$$

Interpretation: Increasing Injection Pressure from 30 psi to 50 psi increases average tensile strength by 12.0 MPa.

Step 3: Calculate Interaction Effect AB

$$\bar{Y}_{AB+} = \frac{Y_{\text{Run 1}} + Y_{\text{Run 4}}}{2} = \frac{40 + 70}{2} = \frac{110}{2} = 55.0$$ $$\bar{Y}_{AB-} = \frac{Y_{\text{Run 2}} + Y_{\text{Run 3}}}{2} = \frac{52 + 46}{2} = \frac{98}{2} = 49.0$$ $$\text{Effect}(AB) = 55.0 - 49.0 = +6.0\text{ MPa}$$

Graphical Representation of Interaction

On an interaction plot ($Y$ vs. Factor A level), non-parallel lines indicate the presence of a statistical interaction between Factor A and Factor B:

 Tensile Strength (MPa)
   70 |                                    *(A+, B+ = 70)
   60 |                                   /
   50 |                *(A+, B- = 52)    /
   40 |               /                 *(A-, B+ = 46)
   30 |   *(A-, B- = 40)
      +-------------------------------------------
          Low Temp (150°C)       High Temp (180°C)
          Line B- (30 psi)  ---- 
          Line B+ (50 psi)  -----

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5. CQE Exam Tips & Key Pitfalls