4.4 Full Factorial Experiments

Full Factorial Experiments Concept and Purpose Full factorial experiments are designed experiments where all possible combinations of factor levels are tested. They reveal: - Main effects of each factor on the response - Interaction effects between factors - How the response behaves across the entire experimental region Full factorials are central when you must understand a process thoroughly, not just screen factors. A factor is an input you can control (temperature, pressure, speed). A level is a specific setting of that factor (e.g., 50°C and 70°C). A response is the output you measure (yield, defect rate, cycle time). In a full factorial, every combination of factor levels is run at least once, often with replication. --- Structure of Full Factorial Designs Factors, Levels, and Design Size For a full factorial with: - k factors - L levels per factor (if equal) - Total runs (no replication): - If all factors have 2 levels: runs = 2^k - For general case: runs = L₁ × L₂ × … × Lk Common full factorial types: - 2^k designs: each factor has 2 levels (low, high) - 3^k designs: each factor has 3 levels (low, medium, high) - Mixed-level designs: e.g., 2 × 3 × 4 Example: A 2^3 experiment (3 factors, 2 levels each) has 2 × 2 × 2 = 8 unique treatment combinations. Coding Factor Levels Coding simplifies analysis and interpretation: - For 2-level factors: - Low level → −1 - High level → +1 - For 3-level factors (common coding): - Low → −1 - Medium → 0 - High → +1 Why coding matters: - Simplifies computation of effects - Makes coefficients comparable in magnitude - Helps with interpreting interactions and regression models --- Main Effects and Interactions Main Effects A main effect is the change in the mean response when a factor moves from its low to high level, averaging over all other factors. For 2-level factors: - Main effect of factor A ≈ mean at A high − mean at A low Interpretation: - Large main effect: factor has strong influence on the response - Small main effect: factor has weak influence (possibly negligible) Interaction Effects An interaction occurs when the effect of one factor depends on the level of another factor. - No interaction: the change caused by factor A is the same at all levels of factor B - With interaction: the change caused by A is different at different levels of B In 2^k designs: - AB interaction: combined effect of A and B beyond their individual main effects - ABC interaction: combined effect of A, B, and C beyond all lower-order effects Practically: - Interactions show whether adjusting one factor’s setting is beneficial only under certain conditions of other factors. - Often, low-order interactions (2-factor) are important, while very high-order interactions are usually small. --- Model Structure and Notation General Linear Model for Full Factorial For 2^2 design (factors A and B): - y = β₀ + βA·xA + βB·xB + βAB·xA·xB + ε For 2^3 design (factors A, B, C): - y = β₀ + βA·xA + βB·xB + βC·xC + βAB·xA·xB + βAC·xA·xC + βBC·xB·xC + βABC·xA·xB·xC + ε Where: - y = response - xA, xB, xC = coded factor levels (−1, +1) - β terms = coefficients (effects) to be estimated - ε = random error term (assumed normal with mean 0) Hierarchy Principle Model hierarchy means: - If an interaction term is included, its associated main effects must also be included. - If a three-way interaction is important, keep all relevant two-way interactions and main effects. This preserves logical and interpretable models. --- Planning a Full Factorial Experiment Defining Objectives Before selecting design: - Clearly define the response(s) to improve or understand - Identify candidate controllable factors - Decide the feasible levels based on process, cost, and safety - Specify precision needs (how small an effect you must detect) Clarify the main goal: - Understanding the system - Optimizing the response - Testing specific hypotheses Selecting Factors and Levels Factors should be: - Controllable during the experiment - Believed to influence the response - Chosen within safe and realistic ranges Guidelines for levels: - Two levels: good for identifying main effects and interactions, but not curvature - Three or more levels: needed to detect curvature and model non-linear effects Avoid picking levels that are: - So close that changes in response are masked by noise - So extreme that results are not practically useful or unsafe Choosing Sample Size and Replicates Total runs = number of treatment combinations × number of replicates. Replicates help: - Estimate pure error - Increase power to detect effects - Check process stability Consider: - Resource limits (time, material, cost) - Required detection threshold for effects - Expected process variability Balance between: - Enough replication for reliable estimates - Not so many runs that the experiment becomes impractical --- Randomization, Replication, and Blocking Randomization Randomization means running the treatment combinations in random order. Purpose: - Protects against hidden time-related variables - Makes statistical assumptions about independence more plausible Implementation: - Use random tables or software to shuffle run order - If there are constraints (e.g., warm-up periods), use restricted randomization while maintaining as much randomness as possible Replication Replication is repeating the entire treatment combination: - Provides an unbiased estimate of experimental error - Improves precision of effect estimates - Allows formal significance testing Do not confuse replication with repeated measurements on the same run; repeated measurements reduce measurement error but not process variation. Blocking Blocking is used when there is a known nuisance factor that cannot be controlled, but can be grouped. Examples: - Different days - Different machines - Different batches of material Idea: - Within each block, run a complete or nearly complete set of factor combinations - Separate variation due to blocks from variation due to factors of interest In full factorials, blocking can be: - Complete blocks: each block contains all treatment combinations - Incomplete blocks: each block contains only a subset, chosen carefully Blocking improves sensitivity to detect factor effects by reducing unexplained variation. --- Analyzing Full Factorial Experiments Computing Effects in 2^k Designs For 2^k designs with coded factors −1 and +1, main and interaction effects can be computed using contrast formulas. For main effect of factor A: - Effect(A) = (average response at A = +1) − (average response at A = −1) For AB interaction effect: - Group runs based on the product xA·xB: - When xA·xB = +1, compute mean response M+ - When xA·xB = −1, compute mean response M− - Effect(AB) = M+ − M− These effects can also be related to coefficients: - For coded designs, βA, βB, etc. are typically half of the corresponding effect estimates (depending on convention). ANOVA for Full Factorials Analysis of variance (ANOVA) partitions the total variability into components: - Total SS: total sum of squares for all responses - Factor SS: sum of squares for each main effect - Interaction SS: sum of squares for each interaction - Error SS: residual variation not explained by the model Key outputs: - F-statistics and p-values for each effect - Percentage of total variation explained by each factor and interaction - Overall R² and adjusted R² Interpretation: - Effects with small p-values are statistically significant - Consider both significance and practical magnitude - Check whether important interactions are present before focusing on main effects alone Residual Analysis Assumptions: - Errors are independent - Errors have constant variance - Errors are normally distributed (for inference accuracy) Check via: - Residuals vs fitted values plot (look for constant spread, no pattern) - Residuals vs run order (look for stability over time) - Normal probability plot of residuals (check approximate straight line) If assumptions are violated: - Consider transforming the response (e.g., log, square root, Box–Cox) - Re-examine the process for special causes or unmodeled factors - Use alternative methods if necessary --- Interaction Plots and Effect Plots Main Effect Plots Main effect plot: - X-axis: factor level (coded or actual) - Y-axis: mean response at each level Interpretation: - Flat line: factor has little or no main effect - Steep line: factor has strong main effect - Direction of slope indicates how the factor influences the response Interaction Plots Interaction plot: - X-axis: levels of one factor - Separate lines: levels of a second factor - Y-axis: mean response Interpretation: - Parallel lines: little or no interaction - Non-parallel lines: interaction present - Crossing lines: strong interaction, especially important Interactions guide: - Whether to optimize jointly over factors rather than one at a time - How robust a chosen setting is to changes in other factors Pareto and Normal Plots of Effects For 2^k designs, effect screening is often visualized using: - Pareto plot of standardized effects: - Bars ordered by magnitude of effect - Reference line to indicate approximate significance threshold - Normal probability plot of effects: - Effects that deviate from the straight-line trend are likely important These tools help: - Quickly identify which main effects and interactions are most influential - Focus attention when many potential effects exist --- Curvature and Higher-Level Full Factorials Curvature Detection 2-level full factorials assume local linearity. To detect curvature: - Add center points (all factors at mid-level) to a 2-level full factorial - Compare: - Average response at factorial points - Average response at center points If there is a statistically significant difference, the relationship is curved. 3-Level and Mixed-Level Designs When curvature is suspected or important: - Use 3-level full factorials (3^k) to model quadratic behavior: - Include squared terms (e.g., A²) in regression - Allow estimation of response surfaces within the design region - Use mixed-level designs when: - Some factors only have 2 feasible settings - Others can vary at 3 or more levels These designs require more runs, so planning and resource constraints are critical. --- Practical Considerations and Common Pitfalls When Full Factorials Are Appropriate Full factorials are well-suited when: - The number of factors is modest and runs are manageable - Interactions are likely and must be understood - A complete picture of factor-response relationships is needed As factor count grows, full factorials can become large: - 2^5 = 32 combinations - 2^7 = 128 combinations Trade-offs must be evaluated against time, cost, and stability of the process. Common Pitfalls Key issues to avoid: - Too many factors: - Design becomes large and costly - Process or environment might change during long experiments - Poor choice of factor levels: - Levels too narrow: effects hidden in noise - Levels unrealistic: conclusions not useful for operations - Ignoring interactions: - Making decisions only on main effects when interactions are present can lead to incorrect settings - Lack of randomization: - Confounds effects with time trends or drifting conditions - Insufficient replication: - Cannot estimate error reliably - Significance tests become unreliable - No residual diagnostics: - Violated assumptions go unnoticed - Confidence intervals and p-values may be misleading Using Results to Optimize and Control From the fitted model, you can: - Predict the response at any combination of factors within the design region - Identify combinations that: - Maximize or minimize the mean response - Achieve a target value - Keep the process robust to variation in some factors Use insights from: - Main effects to set overall factor directions - Interactions to fine-tune combinations - Curvature terms (if present) to find interior optima rather than boundary solutions --- Summary Full factorial experiments systematically test all combinations of factor levels, enabling clear estimation of: - Main effects of each factor - Interactions among factors - (With additional levels) curvature in factor-response relationships Effective use involves: - Careful selection of factors, levels, and replication - Randomization and, when needed, blocking - Proper analysis using effect estimates, ANOVA, and residual diagnostics - Interpretation of main effect and interaction plots to guide decisions When designed and analyzed correctly, full factorial experiments provide a complete, statistically sound understanding of how controllable inputs drive process outputs, supporting reliable optimization and control.