Statistical design: factorial, RSM, contour designs MCQs With Answer

Statistical design: factorial, RSM, contour designs MCQs With Answer

For M. Pharm students, mastering experimental design is essential for robust formulation development and process optimization. This quiz focuses on factorial designs for systematic screening, Response Surface Methodology (RSM) for modeling curvature, and contour-based approaches for interpreting response behavior and multiresponse optimization. You will encounter concepts such as design resolution, aliasing, central composite and Box–Behnken designs, rotatability, center points, steepest ascent, desirability functions, and overlay contour plots. The questions are designed to test both theoretical understanding and practical application in modern pharmaceutics, including how to choose designs under constraints, interpret response surfaces effectively, and make data-driven decisions to optimize critical quality attributes and process parameters with confidence.

Q1. In a full factorial 2^k design with four factors at two levels each, how many experimental runs are required (excluding replicates and center points)?

• 8
• 16
• 24
• 32

Correct Answer: 16

Q2. What is the primary advantage of a fractional factorial design during early formulation screening?

• It provides exact estimates of all interactions up to three-factor terms
• It reduces the number of runs while efficiently estimating main effects when higher-order interactions are negligible
• It maximizes replication to estimate pure error
• It ensures rotatability of the response surface

Correct Answer: It reduces the number of runs while efficiently estimating main effects when higher-order interactions are negligible

Q3. Which statement best describes a Resolution IV fractional factorial design?

• Main effects are aliased with two-factor interactions
• Main effects are clear of two-factor interactions, but two-factor interactions may be aliased with each other
• Two-factor interactions are clear of all aliasing
• Main effects are aliased only with three-factor interactions

Correct Answer: Main effects are clear of two-factor interactions, but two-factor interactions may be aliased with each other

Q4. Which of the following is not an essential component of a Central Composite Design (CCD)?

• Factorial (or fractional factorial) points
• Axial (star) points
• Center points
• Blocking factors

Correct Answer: Blocking factors

Q5. For a k-factor rotatable CCD, which expression gives the axial distance (α) ensuring rotatability?

• α = 2^(k/2)
• α = (2)^(k/4)
• α = √k
• α = 1/k

Correct Answer: α = (2)^(k/4)

Q6. Which feature is characteristic of a Box–Behnken Design (BBD)?

• It includes all 2^k factorial corner points
• It places design points only at the center of the factor space
• It avoids cube corners and uses midpoints of edges and center points
• It always requires α > 1 for axial points

Correct Answer: It avoids cube corners and uses midpoints of edges and center points

Q7. When is a Box–Behnken Design generally preferred over a Central Composite Design in pharmaceutical process development?

• When extreme combinations of factor levels are unsafe or infeasible
• When a linear model is sufficient and curvature is irrelevant
• When estimating cubic terms is mandatory
• When only categorical factors are present

Correct Answer: When extreme combinations of factor levels are unsafe or infeasible

Q8. What is the primary purpose of including center points in factorial or RSM studies?

• To estimate instrument calibration error exclusively
• To estimate pure error and test for curvature in the response
• To increase the number of treatment combinations only
• To enforce orthogonality of main effects

Correct Answer: To estimate pure error and test for curvature in the response

Q9. In a contour plot of response Y versus factors X1 and X2, which pattern most strongly suggests a significant X1×X2 interaction?

• Concentric circular contours centered at the design center
• Parallel straight contours aligned with one axis
• Tilted elliptical contours not aligned with either axis
• Uniformly spaced horizontal contour bands

Correct Answer: Tilted elliptical contours not aligned with either axis

Q10. In RSM, the method of steepest ascent (or descent) is used primarily to:

• Estimate pure error at the design center
• Move iteratively in the direction of maximum increase (or decrease) in the response based on a first-order model
• Detect outliers in residual plots
• Assess the rotatability of a quadratic model

Correct Answer: Move iteratively in the direction of maximum increase (or decrease) in the response based on a first-order model

Q11. Why are factors commonly coded to the range −1 to +1 in factorial and RSM studies?

• To increase the number of levels for each factor
• To standardize scales, improve numerical stability, and simplify interpretation of effects
• To ensure that all models are rotatable
• To avoid the need for replication

Correct Answer: To standardize scales, improve numerical stability, and simplify interpretation of effects

Q12. A valid lack-of-fit test in regression modeling requires:

• At least one replicated design point to estimate pure error
• At least one center point without replication
• No replication, but a large number of unique runs
• Blocking across all factors

Correct Answer: At least one replicated design point to estimate pure error

Q13. A 2^(3−1) fractional factorial with generator I = ABC has which property?

• Resolution II; main effects aliased with each other
• Resolution III; main effects aliased with two-factor interactions
• Resolution IV; main effects clear of two-factor interactions
• Resolution V; two-factor interactions clear of each other

Correct Answer: Resolution III; main effects aliased with two-factor interactions

Q14. In factorial experiments, blocking is typically used to:

• Increase the number of treatment combinations
• Confound low-order effects with blocks to preserve estimation of high-order interactions
• Control nuisance variability (e.g., day, batch) by confounding it with higher-order interactions
• Guarantee orthogonality in all models

Correct Answer: Control nuisance variability (e.g., day, batch) by confounding it with higher-order interactions

Q15. For three continuous factors requiring a quadratic model while avoiding extreme corner points, which design is most efficient?

• Full factorial 2^3
• Plackett–Burman design
• Box–Behnken Design
• Latin square

Correct Answer: Box–Behnken Design

Q16. A rotatable design has which defining property?

• The design includes all axial points at |α| = 1
• The prediction variance is constant at all points equidistant from the design center
• All main effects are orthogonal to two-factor interactions
• The model guarantees no lack-of-fit

Correct Answer: The prediction variance is constant at all points equidistant from the design center

Q17. Which statement about a face-centered CCD is correct?

• It uses α = (2)^(k/4)
• Star points fall outside the factor range
• It sets α = 1, placing star points at the centers of each face of the cube
• It is always more efficient than BBD for all k

Correct Answer: It sets α = 1, placing star points at the centers of each face of the cube

Q18. In multiresponse optimization using desirability, the overall desirability (D) is typically computed as:

• The arithmetic mean of the responses
• The unweighted sum of individual desirabilities
• The weighted geometric mean of the individual desirabilities (each between 0 and 1)
• The maximum of individual desirabilities

Correct Answer: The weighted geometric mean of the individual desirabilities (each between 0 and 1)

Q19. Canonical analysis of a fitted quadratic model in RSM is primarily used to:

• Compute the standard errors of regression coefficients
• Determine the nature of the stationary point using eigenvalues of the second-order coefficient matrix
• Test for normality of residuals
• Select the appropriate blocking structure

Correct Answer: Determine the nature of the stationary point using eigenvalues of the second-order coefficient matrix

Q20. Overlay contour plots are most useful in pharmaceutics when you need to:

• Identify the single factor with the largest main effect
• Visualize feasible regions satisfying multiple response constraints simultaneously (“sweet spot”)
• Estimate pure error at the design center
• Guarantee orthogonality of model terms

Correct Answer: Visualize feasible regions satisfying multiple response constraints simultaneously (“sweet spot”)

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