Response surface plot and contour plot representation MCQs With Answer

Response surface plot and contour plot representation are essential tools in Response Surface Methodology (RSM) used in B. Pharm formulation development and process optimization. Response surface plots provide 3D visualization of a measured response across two factors, while contour plots show isoresponse lines for easier interpretation of interactions, curvature and optimum regions. These visualizations help interpret quadratic models, identify stationary points (minima, maxima or saddle), guide steepest ascent experiments, and evaluate design choices such as CCD or Box–Behnken. Understanding contour spacing, ridge analysis, canonical analysis and model diagnostics improves formulation decisions and ensures robust product quality. Now let’s test your knowledge with 30 MCQs on this topic.

Q1. What does a response surface plot typically display?

• A 3D surface showing the relationship between two factors and a response
• A 2D bar chart of factor main effects
• A time-series plot of a single factor
• A histogram of residuals

Correct Answer: A 3D surface showing the relationship between two factors and a response

Q2. What is the primary purpose of a contour plot in RSM?

• To display residuals against predicted values
• To show lines of equal response (isoresponse) for two factors
• To list factor levels used in the experiment
• To perform ANOVA tests

Correct Answer: To show lines of equal response (isoresponse) for two factors

Q3. Which model form is most commonly used in RSM for response surface and contour plotting?

• First-order linear model
• Second-order (quadratic) polynomial model
• Exponential decay model
• Nonparametric spline model

Correct Answer: Second-order (quadratic) polynomial model

Q4. How does a Box–Behnken design differ from a central composite design (CCD)?

• Box–Behnken includes axial/star points at extreme corners
• Box–Behnken omits factorial corner points and uses midpoints of edges
• CCD cannot estimate quadratic terms
• CCD is only for one factor

Correct Answer: Box–Behnken omits factorial corner points and uses midpoints of edges

Q5. On a contour plot, what do closely spaced contour lines indicate?

• A flat region with little change in response
• A steep gradient or rapid change in the response
• That the model is invalid
• That factors are uncorrelated

Correct Answer: A steep gradient or rapid change in the response

Q6. In canonical analysis of a quadratic response surface, what does a positive definite Hessian indicate?

• A saddle point
• A local maximum
• A local minimum
• No stationary point exists

Correct Answer: A local minimum

Q7. What is the purpose of the path of steepest ascent (or descent) in RSM?

• To validate the final model with new factors
• To move experimental conditions toward an improved response quickly
• To estimate pure error from replicates
• To convert coded units to actual units

Correct Answer: To move experimental conditions toward an improved response quickly

Q8. What does an overlay plot show in formulation optimization?

• The histogram of residuals for the model
• Regions in factor space that satisfy multiple response criteria simultaneously
• The interaction coefficients only
• The ANOVA table

Correct Answer: Regions in factor space that satisfy multiple response criteria simultaneously

Q9. Canonical analysis is used to:

• Transform the quadratic model to identify and classify the stationary point
• Fit only linear effects to the data
• Perform bootstrapping of coefficients
• Scale factors to their original units

Correct Answer: Transform the quadratic model to identify and classify the stationary point

Q10. Circular contour lines typically indicate which condition between two factors?

• Strong interaction between factors
• No interaction and equal curvature in both directions
• Model misspecification
• Highly skewed residuals

Correct Answer: No interaction and equal curvature in both directions

Q11. Which software package is commonly used in R for response surface analysis?

• ggplot2
• rsm
• survival
• caret

Correct Answer: rsm

Q12. In coded units for RSM, the center point value is usually:

• -1
• 0
• 1
• 10

Correct Answer: 0

Q13. Which plot feature best indicates a strong interaction between two factors?

• Parallel straight contour lines
• Non-parallel or tilted contour lines forming ellipses
• Single isolated point
• Uniform color across the plot

Correct Answer: Non-parallel or tilted contour lines forming ellipses

Q14. A saddle point on a response surface means:

• There is a clear global maximum in the studied region
• Response increases in all directions from that point
• Response increases in one direction and decreases in another
• The experiment has no variance

Correct Answer: Response increases in one direction and decreases in another

Q15. Which terms must be included in a second-order RSM model for two factors?

• Only linear terms for both factors
• Linear, squared (quadratic) and cross-product interaction terms
• Only squared terms
• Only cross-product term

Correct Answer: Linear, squared (quadratic) and cross-product interaction terms

Q16. What is the purpose of a lack-of-fit test in RSM?

• To estimate factor levels
• To assess whether the proposed model adequately fits the experimental data
• To calculate coded unit values
• To determine sample size

Correct Answer: To assess whether the proposed model adequately fits the experimental data

Q17. On a contour plot, each contour line is labeled with:

• The factor combination index
• The response value corresponding to that line
• The p-value for interaction
• The standard error

Correct Answer: The response value corresponding to that line

Q18. Ridge analysis in RSM is performed when:

• The stationary point lies within the experimental region
• The stationary point lies outside the experimental region
• There is no interaction terms in the model
• Only linear models are used

Correct Answer: The stationary point lies outside the experimental region

Q19. A central composite design (CCD) typically includes which special points?

• Only center points
• Axial (star) points, factorial points, and center points
• Only Box–Behnken midpoints
• Only 3-level factorial points

Correct Answer: Axial (star) points, factorial points, and center points

Q20. Which commercial software is widely used for RSM and response surface plots in pharmaceutical development?

• Design-Expert
• Excel Solver
• SPSS Amos
• Photoshop

Correct Answer: Design-Expert

Q21. In a 3D response surface plot, color shading usually represents:

• The number of replicates
• The magnitude of the response value
• The residual degrees of freedom
• The coded units of factors

Correct Answer: The magnitude of the response value

Q22. What does rotatability of a design ensure in RSM?

• Equal prediction variance at all points in the design
• Equal prediction variance at points equidistant from the center
• No interactions between factors
• That axial points are unnecessary

Correct Answer: Equal prediction variance at points equidistant from the center

Q23. How is a coded factor value typically calculated?

• (Actual value) × (mean)
• (Actual − center)/half-range
• (Actual + center) / 2
• Square root of the actual value

Correct Answer: (Actual − center)/half-range

Q24. A strong quadratic effect with negligible interaction often produces which contour shape?

• Highly irregular polygons
• Concentric circles or ellipses centered near the optimum
• Parallel straight lines
• Random scatter points

Correct Answer: Concentric circles or ellipses centered near the optimum

Q25. Contour plots are also known as:

• Isoresponse plots
• Pareto charts
• Box plots
• Time–response curves

Correct Answer: Isoresponse plots

Q26. Adding unnecessary higher-order terms to an RSM model may cause:

• Reduced model variance and perfect prediction
• Overfitting, increasing variance and reducing predictive ability
• Better generalization to new data always
• No change to model performance

Correct Answer: Overfitting, increasing variance and reducing predictive ability

Q27. Why are multiple center points included in design for RSM?

• To estimate pure error and detect curvature
• To increase the number of factors
• To avoid quadratic terms entirely
• To reduce the number of experiments

Correct Answer: To estimate pure error and detect curvature

Q28. For two factors x1 and x2, which general form represents a full quadratic response surface?

• z = b0 + b1x1 + b2x2
• z = b0 + b1x1 + b2x2 + b11x1^2 + b22x2^2 + b12x1x2
• z = b0 × e^(b1x1+b2x2)
• z = b0 + b1/x1 + b2/x2

Correct Answer: z = b0 + b1x1 + b2x2 + b11x1^2 + b22x2^2 + b12x1x2

Q29. The Hessian matrix used to classify stationary points contains:

• First derivatives (linear coefficients) only
• Second derivatives (curvature) coefficients of the quadratic form
• Observed responses only
• P-values of the model terms

Correct Answer: Second derivatives (curvature) coefficients of the quadratic form

Q30. Which approach is used for simultaneous optimization of multiple responses in RSM?

• ANOVA only
• Desirability function
• Principal component regression only
• Simple averaging of responses

Correct Answer: Desirability function

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