Polynomial Regression and Response Surface Methods - Dr. Jeff Edwards

Why is it important to scale-center variables when conducting polynomial regression in response surface analysis?

To facilitate the interpretation of the regression results and simplify the response surface computations and plots
To reduce the standard error of the regression coefficients
To normalize the distribution of the predictor variables
To eliminate the need for higher-order terms in the model

Why are commensurate measures important in polynomial regression used for response surface analysis?

Because they ensure the predictor variables are measured on the same scale, as required in congruence research
Because they reduce the number of predictors required in the model
Because they eliminate the need to center or scale the variables before analysis
Because they allow for the use of categorical variables in polynomial regression

What does the term congruence refer to in the context of response surface methodology?

It refers to the fit, match, similarity, or agreement between two constructs
It describes the degree to which a variable mediates the relationship between two others
It indicates the statistical significance of a regression coefficient
It refers to the curvature of the response surface along the line of incongruence

Which of the following is an example of complementary fit?

The match between employee needs and job rewards
The similarity between supervisor and subordinate goals, personality, and other beliefs and attributes
The congruence between the values of the person and the organization
The alignment between coworkers’ communication styles and interpersonal preferences

Which of the following is a problem with using difference scores in research?

They impose untested constraints that should be evaluated empirically
They increase the number of predictors in the model, leading to overfitting
They eliminate the need for centering or scaling variables
They improve the interpretability of interaction effects

What is a benefit of using polynomial regression as an alternative to difference scores?

Because it models the joint relationship of X and Y with Z in three dimensions, allowing for richer hypothesis testing and interpretation
Because it simplifies the model by reducing the number of predictors to a single composite score
Because it allows researchers to impose constraints directly on the data to improve model fit
Because it eliminates the need to test for interaction or curvature effects

In a typical response surface plot used in polynomial regression, how are the variables X, Y, and Z represented?

In a typical response surface plot used in polynomial regression, how are the variables X, Y, and Z represented?
X and Y are on the vertical axis, and Z is on the horizontal axis
X is on the horizontal axis, Y is on the vertical axis, and Z is held constant
All three variables are plotted on the same axis using color gradients

When is it appropriate to use a confirmatory approach with a difference score in response surface analysis?

When the difference score represents a hypothesis that is predicted a priori
When the difference score is derived from exploratory factor analysis
When the variables are not commensurate and require transformation
When the response surface shows a significant interaction effect

Which of the following best describes the conditions that must be met to justify a confirmatory analysis using a difference score in response surface methodology?

The R² must be significant, the coefficients must match the difference score pattern, the implied constraints must hold, and higher-order terms must be non-significant
The predictors must be uncorrelated, and the response surface must be flat
The predictors must be standardized, and the interaction term must be significant
The difference score must be normally distributed and centered at zero

What does it mean if the constraint b1+b2=0 is not significantly different from zero?

It suggests that the constraint implied by the difference score is supported, but does not prove that b1 and b2 are equal in size
It confirms that b1 and b2 are equal in magnitude and opposite in sign
It indicates that the response surface is flat along the line of incongruence
It proves that the interaction term b4 is not significant

When is it appropriate to use an exploratory approach?

When no clear a priori hypothesis exists, and the goal is to identify potentially meaningful patterns in the data
When testing a well-defined theoretical model with strong a priori hypotheses
When seeking to confirm the predictive power of a pre-specified set of polynomial terms
When the researcher wants to ensure the model will generalize well to new samples

What are the three primary shapes of a surface in response surface methodology?

Concave, convex, and saddle-shaped
Flat, curved, and twisted
Linear, quadratic, and cubic
Circular, elliptical, and parabolic

What best characterizes a saddle-shaped surface in response surface methodology?

It curves upward in one direction and downward in the perpendicular direction.
It curves upward in all directions from a central point.
It curves downward in all directions from a central point.
It remains flat across all values of the predictor variables.

What does a stationary point represent, and how does its interpretation depend on the shape of the surface?

A point where the slope is zero in all directions, which could represent a minimum, maximum, or saddle point depending on the surface's curvature.
A point where the response surface is always flat and constant across all values of the predictors.
A point that only occurs on convex surfaces and always represents a minimum.
A point where the slope is highest and used to determine the steepest ascent path.

What best describes the first principal axis in a convex surface?

The direction along which the upward curvature is greatest
The direction along which the downward curvature is greatest
The direction that lies parallel to the X or Y axis, depending on the model
The direction where no curvature occurs because the surface is flat

When developing congruence hypotheses, which of the following questions is NOT part of the general approach?

How does the response (Z) vary as X decreases independently of Y?
How does the response (Z) vary as X increases toward Y?
How does the response (Z) vary as X exceeds Y?
How does the response (Z) vary as X and Y jointly increase?

Which of the following statements about difference scores as dependent variables is TRUE?

When the dependent variable is an algebraic difference and both components are endogenous, specialized procedures are needed.
Problems with difference scores only occur when they are used as independent variables, not as dependent variables.
Alternative procedures for difference scores as dependent variables are fundamentally the same as those for difference scores as independent variables.
Difference scores as dependent variables are never influenced by independent variables.

In most cases, which higher-order terms are sufficient to capture theoretically meaningful effects in a response surface model?

Quadratic terms (X², XY, Y²)
Only linear terms (X and Y)
Cubic and quartic terms
Only product terms like XY

Why are the coefficients on X and Y (b₁ and b₂) considered scale-dependent in a response surface model that includes quadratic terms (X², XY, Y²)?

Because they represent the slope of the surface at the origin, which depends on how X and Y are scaled or centered.
Because quadratic terms remove the effect of scale on these coefficients.
Because b₁ and b₂ always represent the overall average effect of X and Y across all values.
Because changing the scale of X and Y does not affect b₁ and b₂.

Does asking respondents to directly compare how much X deviates from Y avoid the problems associated with difference scores?

No, because respondents may still implicitly calculate the difference, and the approach confounds X and Y.
Yes, because it removes the need for the researcher to calculate the difference.
Yes, because direct comparison items separate the effects of X and Y clearly.
No, because difference scores are always more reliable than direct comparisons.

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Polynomial Regression and Response Surface Methods – Dr. Jeff Edwards

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