Lecture presentation - The University of North Carolina at Chapel Hill

Alternatives to Difference Scores:

Polynomial Regression and

Response Surface Methodology

Jeffrey R. Edwards

University of North Carolina

1

Outline

I.

Types of Difference Scores

II.

Questions Difference Scores Are Intended To

Address

III.

Problems With Difference Scores

IV.

An Alternative Procedure

V.

Analyzing Quadratic Regression Equations

Using Response Surface Methodology

VI.

Moderated Polynomial Regression

VII. Mediated Polynomial Regression

VIII. Difference Scores As Dependent Variables

IX.

Answers to Frequently Asked Questions

2

Types of Difference Scores



Univariate:

 Algebraic difference: (X – Y)



Absolute difference: |X – Y|



Squared difference: (X – Y) 2



Multivariate:



Sum of algebraic differences: Σ(X i

– Y i

) = D 1

 Sum of absolute differences: Σ|X i

– Y i

| = |D|



Sum of squared differences: Σ(X i

– Y i

) 2 = D 2



Euclidean distance: (Σ(X i

– Y i

) 2 )

½

= D



Profile correlation: C(X i

,Y i

)/S(X)S(Y) = r

X i

,Y i

= Q

3

Questions Difference Scores are Intended to Address



How well do characteristics of the job fit the needs or desires of the employee?



To what extent do job demands exceed or fall short of the abilities of the person?



Are prior expectations of the employee met by actual job experiences?



What is the degree of similarity between perceptions or beliefs of supervisors and subordinates?



Do the values of the person match the culture of the organization?



Can novices provide performance evaluations that agree with expert ratings?

4

Problems with Difference Scores:

Reliability



When component measures are positively correlated, difference scores are often less reliable than either component.



The reliability of an algebraic difference is:



(x

 y)

=

 2 x r xx

+  2 y r yy



2

 2 x

+  2 y



2 r xy r xy

 x



 y x

 y



To illustrate, if X and Y have unit variances, have reliabilities of .75, and are correlated .50, the reliability of X – Y equals .50.

5


Conceptual Ambiguity



It might seem that component variables are reflected equally in a difference score, given that the components are implicitly assigned the same weight when the difference score is constructed.



However, the variance of a difference score depends on the variances and covariances of the component measures, which are sample dependent.



When one component is a constant, the variance of a difference score is solely due to the other component, i.e., the one that varies. For instance, when P-O fit is assessed in a single organization, the P-O difference solely represents variation in the person scores.

6


Confounded Effects



Difference scores confound the effects of the components of the difference.



For example, an equation using an algebraic difference as a predictor can be written as:

Z = b

0

+ b

1

(X – Y) + e



In this equation, b

1 can reflect a positive relationship for X, an negative relationship for

Y, or some combination thereof.

7


Untested Constraints



Difference scores constrain the coefficients relating X and Y to Z without testing these constraints.



The constraints imposed by an algebraic difference can be seen with the following equations:

Z = b

0



Expansion yields:

+ b

1

(X – Y) + e

Z = b

0

+ b

1

X – b

1

Y + e

8





Now, consider an equation that uses X and Y as separate predictors:

Z = b

0

+ b

1

X + b

2

Y + e



Using (X – Y) as a predictor constrains the coefficients on X and Y to be equal in magnitude but opposite in sign (i.e., b

1

= –b

2

).



This constraint should not be simply imposed on the data but instead should be treated as a hypothesis to be tested.

9





The constraints imposed by a squared difference can be seen with the following equations:

Z = b

0



Expansion yields:

+ b

1

(X – Y) 2 + e

Z = b

0

+ b

1

X 2 – 2b

1

XY + b

1

Y 2 + e



Thus, a squared difference implicitly treats Z as a function of X 2 , XY, and Y 2 .

10





Now, consider a quadratic equation using X and Y:

Z = b

0

+ b

1

X + b

2

Y + b

3

X 2 + b

4

XY + b

5

Y 2 + e



Comparing this equation to the previous equation shows that (X – Y) 2 imposes four constraints:

 b

1

 b

2

 b

3

 b

3

= 0

= 0

= b

5

, or b

3

+ b

4

+ b

5

– b

5

= 0

= 0



Again, these constraints should be treated as hypotheses to be tested empirically, not simply imposed on the data.

11


Dimensional Reduction



Difference scores reduce the three-dimensional relationship of X and Y with Z to two dimensions.



The linear algebraic difference function represents a symmetric plane with equal but opposite slopes with respect to the X-axis and Y-axis.



The V-shaped absolute difference function represents a symmetric V-shaped surface with its minimum (or maximum) running along the X = Y line.



The U-shaped squared difference function represents a symmetric U-shaped surface with its minimum (or maximum) running along the X = Y line.

12

Two-Dimensional Algebraic Difference

Function

7

6

5

4

3

2

1

-6 -4 -2 0

(X - Y)

2 4 6

13

Three-Dimensional Algebraic Difference

Function

14

Two-Dimensional Absolute Difference

Function

7

6

5

4

3

2

1

-6 -4 -2 0

(X - Y)

2 4 6

15

Three-Dimensional Absolute Difference

Function

16

Two-Dimensional Squared Difference

Function

7

6

5

4

3

2

1

-6 -4 -2 0

(X - Y)

2 4 6

17

Three-Dimensional Squared Difference

Function

18


Dimensional Reduction



These surfaces represent only three of the many possible surfaces depicting how X and

Y may be related to Z.



This problem is compounded by the use of profile similarity indices, which collapse a series of three-dimensional surfaces into a single two-dimensional function.

19

An Alternative Procedure



The relationship of X and Y with Z should be viewed in three dimensions, with X and Y constituting the two horizontal axes and Z constituting the vertical axis.



Analyses should focus not on two-dimensional functions relating the difference between X and Y to Z, but instead on three-dimensional surfaces depicting the joint relationship of X and Y with Z.



Constraints should not be simply imposed on the data, but instead should be viewed as hypotheses that, if confirmed, lend support to the conceptual model upon which the difference score is based.

20

Data Used for Illustration



Data were collected from 373 MBA students who were engaged in the recruiting process.



Respondents rated the actual and desired amounts of various job attributes and the anticipated satisfaction concerning a job for which they had recently interviewed.



Actual and desired measured had three items and used 7point response scales ranging from “none at all” to “a very great amount.” The satisfaction measured had three items and used a 7-point response scale ranging from

“strongly disagree” to “strongly agree.”



The job attributes used for illustration are autonomy, prestige, span of control, and travel.

21

Confirmatory Approach



When a difference scores represents a hypothesis that is predicted a priori, the alternative procedure should be applied using the confirmatory approach.



The R 2 for the unconstrained equation should be significant.



The coefficients in the unconstrained equation should follow the pattern indicated by the difference score.



The constraints implied by the difference score should not be rejected.



The set of terms one order higher than those in the unconstrained equation should not be significant.

22

Confirmatory Approach Applied to the

Algebraic Difference



The unconstrained equation is:

Z = b

0

+ b

1

X + b

2

Y + e



The constrained equation used to evaluate the third condition is:

Z = b

0

+ b

1

(X – Y) + e



The equation that adds higher-order terms used to evaluate the fourth condition is:

Z = b

0

+ b

1

X + b

2

Y + b

3

X 2 + b

4

XY + b

5

Y 2 + e

23

Example: Confirmatory Test of Algebraic

Difference for Autonomy



Unconstrained equation:

Dep Var: SAT N: 360 Multiple R: 0.356 Squared multiple R: 0.127

Adjusted squared multiple R: 0.122 Standard error of estimate: 1.077

Effect Coefficient Std Error Std Coef Tolerance t P(2 Tail)

CONSTANT 5.835 0.077 0.000 . 75.874 0.000

AUTCA 0.445 0.062 0.413 0.737 7.172 0.000

AUTCD -0.301 0.071 -0.244 0.737 -4.235 0.000

Analysis of Variance

Source Sum-of-Squares df Mean-Square F-ratio P

Regression 60.133 2 30.067 25.930 0.000

Residual 413.953 357 1.160

24





Unconstrained surface:

25





The first condition is met, because the R 2 from the unconstrained equation is significant.



The second condition is met, because the coefficients on X and Y are significant and in the expected direction.



For the third condition, testing the constraints imposed by the algebraic difference is the same as testing the difference in R 2 between the constrained and unconstrained equations.

26





Constrained equation:




CONSTANT 5.937 0.061 0.0 . 97.007 0.000

AUTALD 0.393 0.058 0.339 1.000 6.825 0.000



Regression 54.589 1 54.589 46.586 0.000

Residual 419.498 358 1.172

27





Constrained surface:

28





The general formula for the difference in R 2 between two regression equations is:



( R

2 

R

2

) /( df

 df )

F U

( 1



C

R

2

) /

C df

U

U U



The test of the constraint imposed by the algebraic difference for autonomy is:

(.

127

(



1

.



115

.

)

127

/(

)

358

/



357

357 )



4 .

91 , p



.

05



The constraint is rejected, so the third condition is not satisfied.

29





For the fourth condition, the unconstrained equation for the algebraic equation is linear, so the higher-order terms are the three quadratic terms X 2 , XY, and Y 2 .



Testing the three quadratic terms as a set is the same as testing the difference in R 2 between the linear and quadratic equations.

30





Quadratic equation:




CONSTANT 5.825 0.083 0.000 . 70.161 0.000

AUTCA 0.197 0.100 0.182 0.273 1.966 0.050

AUTCD -0.293 0.106 -0.238 0.315 -2.754 0.006

AUTCA2 -0.056 0.047 -0.086 0.444 -1.177 0.240

AUTCAD 0.276 0.080 0.396 0.178 3.453 0.001

AUTCD2 -0.035 0.063 -0.054 0.242 -0.553 0.581



Regression 79.951 5 15.990 14.362 0.000

Residual 394.135 354 1.113

31





The test of the higher-order terms associated with the algebraic difference for autonomy:

(.

169

(



1

.

127



.

)

169

/(

)

357

/



354

354 )



5 .

96 , p



.

05



The higher-order terms are significant, so the fourth condition is not satisfied.

32


Absolute Difference




Z = b

0

+ b

1

X + b

2

Y + b

3

W + b

4

WX + b

5

WY + e




Z = b

0

+ b

1

|X – Y| + e




Z = b

0

+ b

1

X + b

2

Y + b

3

W + b

4

WX + b

5

WY + b

6

X 2 + b

7

XY + b

8

Y 2 + b

9

WX 2 + b

10

WXY + b

10

WY 2 + e

33

Example: Confirmatory Test of Absolute








CONSTANT 6.233 0.152 0.000 . 41.136 0.000

AUTCA -0.150 0.184 -0.139 0.082 -0.818 0.414

AUTCD 0.183 0.188 0.148 0.102 0.970 0.333

AUTW -0.349 0.201 -0.148 0.329 -1.737 0.083

AUTCAW 0.752 0.209 0.490 0.129 3.605 0.000

AUTCDW -0.554 0.219 -0.406 0.093 -2.537 0.012



Regression 75.381 5 15.076 13.386 0.000

Residual 398.705 354 1.126

34






35








The second condition is not met, because the coefficients on X and Y are not significant, and in the expected direction.



For the third condition, testing the constraints imposed by the absolute difference is the same as testing the difference in R 2 between the constrained and unconstrained equations.

36









CONSTANT 6.212 0.087 0.000 . 71.122 0.000

AUTABD -0.531 0.082 -0.323 1.000 -6.464 0.000



Regression 49.555 1 49.555 41.788 0.000

Residual 424.532 358 1.186

37






38





The test of the constraints imposed by the absolute difference for autonomy is:

(.

159



.

( 1



105 ) /(

.

159 )

358



354 )



5 .

68 , p



.

05

/ 354



The constraints are rejected, so the third condition is not satisfied.

39





For the fourth condition, the unconstrained equation for the absolute equation is piecewise linear, so the higher-order terms are the six quadratic terms X 2 , XY, Y 2 , WX 2 , WXY, and

WY 2 .



Testing the six quadratic terms as a set is the same as testing the difference in R 2 between the piecewise linear and piecewise quadratic equations.

40





Piecewise quadratic equation:




CONSTANT 6.193 0.206 0.000 . 30.124 0.000

AUTCA -0.438 0.548 -0.407 0.009 -0.799 0.425

AUTCD 0.256 0.505 0.207 0.014 0.506 0.613

AUTW -0.534 0.276 -0.225 0.172 -1.931 0.054

AUTCAW 0.672 0.608 0.438 0.015 1.105 0.270

AUTCDW -0.373 0.592 -0.273 0.013 -0.631 0.529

AUTCA2 0.146 0.312 0.225 0.010 0.468 0.640

AUTCAD -0.092 0.618 -0.133 0.003 -0.150 0.881

AUTCD2 0.107 0.350 0.169 0.008 0.307 0.759

AUTCA2W -0.088 0.325 -0.082 0.026 -0.272 0.786

AUTCADW 0.325 0.641 0.368 0.004 0.507 0.613

AUTCD2W -0.219 0.371 -0.342 0.007 -0.589 0.556



Regression 87.940 11 7.995 7.205 0.000

Residual 386.146 348 1.110

41





The test of the higher-order terms associated with the absolute difference for autonomy is:

(.

185



.

159 ) /( 354



( 1



.

185 ) / 348

348 )



1 .

85 , p



.

05



The higher-order terms are not significant, so the fourth condition is satisfied.

42


Squared Difference




Z = b

0

+ b

1

X + b

2

Y + b

3

X 2 + b

4

XY + b

5

Y 2 + e




Z = b

0

+ b

1

(X – Y) 2 + e




Z = b

0

+ b

1

X + b

2

Y + b

3

X 2 + b

4

XY + b

5

Y 2 + b

6

X 3 + b

7

X 2 Y + b

8

XY 2 + b

9

Y 3 + e

43

Example: Confirmatory Test of Squared








CONSTANT 5.825 0.083 0.000 . 70.161 0.000

AUTCA 0.197 0.100 0.182 0.273 1.966 0.050

AUTCD -0.293 0.106 -0.238 0.315 -2.754 0.006

AUTCA2 -0.056 0.047 -0.086 0.444 -1.177 0.240

AUTCAD 0.276 0.080 0.396 0.178 3.453 0.001

AUTCD2 -0.035 0.063 -0.054 0.242 -0.553 0.581



Regression 79.951 5 15.990 14.362 0.000

Residual 394.135 354 1.113

44






45








The second condition is not met, because the coefficients on X and Y are significant, and the coefficients on X 2 and Y 2 are not significant.



For the third condition, testing the constraints imposed by the squared difference is the same as testing the difference in R 2 between the constrained and unconstrained equations.

46









CONSTANT 5.993 0.067 0.000 . 89.830 0.000

AUTSQD -0.183 0.030 -0.310 1.000 -6.162 0.000



Regression 45.463 1 45.463 37.972 0.000

Residual 428.623 358 1.197

47






48





The test of the constraint imposed by the squared difference for autonomy is:

(.

169



.

( 1



096 ) /(

.

169 )

358



354 )



7 .

77 , p



.

05

/ 354



The constraint is rejected, so the third condition is not satisfied.

49





For the fourth condition, the unconstrained equation for the squared equation is quadratic, so the higher-order terms are the four cubic terms X 3 , X 2 Y, XY 2 , and Y 3 .



Testing the four cubic terms as a set is the same as testing the difference in R 2 between the quadratic and cubic equations.

50





Cubic equation:




CONSTANT 5.757 0.109 0.000 . 52.736 0.000

AUTCA 0.364 0.119 0.337 0.190 3.055 0.002

AUTCD -0.312 0.120 -0.253 0.245 -2.609 0.009

AUTCA2 0.043 0.095 0.066 0.109 0.456 0.649

AUTCAD 0.356 0.175 0.511 0.037 2.033 0.043

AUTCD2 -0.075 0.126 -0.117 0.060 -0.594 0.553

AUTCA3 -0.104 0.037 -0.442 0.094 -2.817 0.005

AUTCA2D 0.052 0.066 0.167 0.052 0.794 0.428

AUTCAD2 -0.030 0.089 -0.098 0.028 -0.338 0.736

AUTCD3 0.003 0.053 0.011 0.046 0.047 0.962



Regression 90.233 9 10.026 9.142 0.000

Residual 383.853 350 1.097

51





The test of the higher-order terms associated with the squared difference for autonomy is:

(.

190



.

169 ) /(

( 1



.

190 )

354

/



350

350 )



2 .

27 , p



.

05



The higher-order terms are not significant, so the fourth condition is satisfied.

52

Analyzing Quadratic Regression Equations

Using Response Surface Methodology



Response surface methodology can be used to analyze features of surfaces corresponding to quadratic regression equations. These analyses are useful for three reasons:



Constraints imposed by difference scores are usually rejected, which makes it necessary to interpret unconstrained equations.



Many conceptually meaningful hypotheses cannot be expressed using difference scores.



Response surfaces can themselves serve as the basis for developing and testing hypotheses.

53

Key Features of Response Surfaces:

Stationary Point



The stationary point is the point at which the slope of the surface relating X and Y to Z is zero in all directions.



For convex (i.e., bowl-shaped) surfaces, the stationary point is the overall minimum of the surface with respect to the Z axis.



For concave (i.e., dome-shaped) surfaces, the stationary point is the overall maximum of the surface with respect to the Z axis.



For saddle-shaped surfaces, the stationary point is where the surface is flat with respect to the Z axis.

54


Stationary Point



The coordinates of the stationary point can be computed using the following formulas:

X

0

= b

2

4 b b

4



3 b

5

2

 b

1 b b

2

4

5

Y

0

= b

1

4 b

4 b



3 b

5

2

 b

2 b

2

4 b

3



X

0 and Y

0 are the coordinates of the stationary point in the X,Y plane.

55

Example: Stationary Point for Autonomy



Applying these formulas to the equation for autonomy yields:

X

0

=

(



0 .

293

4 (

)(



0 .

0 .

276

056 )(

)





0 .

2 ( 0 .

035 )

197



)(



0

0 .

276

2

.

035 )

= 0 .

982

Y

0

=

( 0 .

197 )(

4 (

0 .

276 )



0 .

056 )(





0

2 (

.



0 .

293 )(

035 )



0 .



0 .

276

2

056 )

=



0 .

315

56

Example: Stationary Point for Autonomy

Stationary Point

57


Principal Axes



The principal axes describe the orientation of the surface with respect to the X,Y plane. The axes are perpendicular and intersect at the stationary point.



For convex surfaces, the upward curvature is greatest along the first principal axis and least along the second principal axis.



For concave surfaces, the downward curvature is greatest along the second principal axis and least along the first principal axis.



For saddle-shaped surfaces, upward curvature is greatest along the first principal axis, and the downward curvature is greatest along the second principal axis.

58


First Principal Axis



An equation for the first principal axis is:

Y

 p

10

 p

11

X



The formula for the slope of the first principal axis

(i.e., p

11

) is: p

11

= b

5

 b

3



( b

3

 b

5

) 2  b 2

4 .

b

4



Using X

0

, Y

0

, and p

11

, the intercept of the first principal axis (i.e., p

10

) can be calculated as follows: p

10



Y

0

 p

11

X

0

59

Example: First Principal Axis for

Autonomy



Applying these formulas to the equation for autonomy yields: p

11

=



0 .

035



(



0 .

056 )



[



0 .

056



(



0 .

035 )]

2 

0 .

276

2

= 1 .

079

0 .

276 p

10

=



0 .

315



( 1 .

079 )( 0 .

982 ) =



1 .

375

60

Example: First Principal Axis for

Autonomy

First Principal Axis

61


Second Principal Axis



An equation for the second principal axis is:

Y

 p

20

 p

21

X



The formula for the slope of the second principal axis

(i.e., p



X

0

, Y

0

21

) is: p

21

= b

, and p

21

5

 b

3



( b

3

 b

5

)

2  b

2

4 .

b

4 can be used to obtain the intercept of the second principal axis (i.e., p

20

) as follows: p

20



Y

0

 p

21

X

0

62

Example: Second Principal Axis for

Autonomy



Applying these formulas to the equation for autonomy yields: p

21

=



0 .

035



(



0 .

056 )



[



0 .

056



(



0 .

035 )]

2



0 .

276

2

=



0 .

927

0 .

276 p

20

=



0 .

315



(



0 .

927 )( 0 .

982 ) = 0 .

594

63

Example: Second Principal Axis for

Autonomy

Second Principal Axis

64


Shape Along the Y = X Line



The shape of the surface along a line in the X,Y plane can be estimated by substituting the expression for the line into the quadratic regression equation.



To estimate the slope along the Y = X line, X is substituted for Y in the quadratic regression equation, which yields:

Z = b

0

+ b

1

X + b

2

X + b

3

X 2 + b

4

X 2 + b

5

X 2 + e

= b

0

+ (b

1

+ b

2

)X + (b

3

+ b

4

+ b

5

)X 2 + e



The term (b

3

+ b

4

+ b

5

) represents the curvature of the surface along the Y = X line, and (b

1

+ b

2

) is the slope of the surface at the point X = 0.

65

Example: Shape Along Y = X Line for

Autonomy



For autonomy, the shape of the surface along the Y = X line is:

Z = 5.825 + [0.197 + (–0.293)]X

+ [–0.056 + 0.276 + (–0.035)]X 2 + e



Simplifying this expression yields:

Z = 5.825 – 0.096X + 0.185X

2 + e



The surface is curved upward along the Y = X line and is negatively sloped at the point X = 0

(the curvature is significant at p < .05).

66

Example: Shape Along Y = X Line for

Autonomy

Contours Show

Shape Along

Y = X Line

67


Shape Along Y = –X Line



To estimate the slope along the Y = –X line, –

X is substituted for Y in the quadratic regression equation, which yields:

Z = b

0

+ b

1

X – b

2

X + b

3

X 2 – b

4

X 2 + b

5

X 2 + e

= b

0

+ (b

1

– b

2

)X + (b

3

– b

4

+ b

5

)X 2 + e



The term (b

3

– b

4

+ b

5

) represents the curvature of the surface along the Y = –X line, and (b

1

– b

2

) is the slope of the surface at the point X =

0.

68

Example: Shape Along Y = –X Line for

Autonomy



For autonomy, the shape of the surface along the Y = –X line is:

Z = 5.825 + [0.197 – (–0.293)]X

+ [–0.056 – 0.276 + (–0.035)]X 2 + e



Simplifying this expression yields:

Z = 5.825 + 0.490X – 0.367X

2 + e



The surface is curved downward along the Y =

–X line and is positively sloped at the point X

= 0 (both are significant at p < .05).

69

Example: Shape Along Y = –X Line for

Autonomy

Contours Show

Shape Along

Y = –X Line

70


Shape Along First Principal Axis



To estimate the slope along the first principal axis, p

10

+ p

11

X is substituted for Y:

Z

 b



0

 b

5 b

1

X

( p

10



 b

2

( p p

11

X )

10

2



 e p

11

X )

 b

3

X

2  b

4

 b

0





( b

3 b

2

 p

10

 b

4 p

11 b

5

2 p

10

 b

5



2 p

11

( b

1

) X 2



 b

2 p

11 e

 b

4 p

10

X ( p

10



 p

11

2 b

5 p

10 p

11

X )

) X



The composite terms preceding X 2 and X are the curvature of the surface along the first principal axis and the slope of the surface at the point X = 0.

71

Example: Shape Along First Principal

Axis for Autonomy



For autonomy, the shape of the surface along the first principal axis is:

Z



5 .

825





[ 0 .

197

(



0 .

293 )(



1 .

375 )



(



0 .

293 )( 1 .

079 )





(



2 (



0 .

035 )(



0 .

035 )(



1 .

375

2

( 0



1 .

375 )( 1 .

079 )] X

.

276 )(



1 .

375 )

)





6 .

[



0

162

.

056



0 .



( 0 .

276 )( 1 .

079 )

395 X



0 .

201 X

2 

 e

(



0 .

035 )( 1 .

079 2 )] X 2



The surface is curved upward along the first principal axis and is negatively sloped at the point X = 0 (both are significant at p < .05).

 e

72

Example: Shape Along First Principal

Axis for Autonomy

Contours Show

Shape Along

First Principal

Axis

73


Shape Along Second Principal Axis



To estimate the slope along the second principal axis, p

20

+ p

21

X is substituted for Y:

Z

 b

0



 b

5 b

1

X

( p

20



 b

2

( p

20 p

21

X ) 2

 p

21

X )

 e

 b

3

X

2  b

4

 b

0





( b

3 b

2

 p

20 b

4

 p

21 b

5 p 2

20

 b

5

 p

2

21

( b

1

) X

2

 b



2 e p

21

 b

4 p

20

X ( p

20



 p

21

2 b

5 p

20 p

21

X )

) X



The composite terms preceding X 2 and X are the curvature of the surface along the second principal axis and the slope of the surface at the point X = 0.

74

Example: Shape Along Second Principal

Axis for Autonomy



For autonomy, the shape of the surface along the second principal axis is:

Z



5 .

825





[ 0 .

197

(



0 .

293 )( 0 .

594 )



(



0 .

293 )(



0 .



(

927 )



0 .

035 )( 0 .

594



( 0 .

276 )(

2 )

0 .

594 )



2 (



0 .

035 )( 0 .

594 )(



0 .

927 )] X





5 .

[



0

639

.

056



0 .



( 0 .

276 )(



0 .

927 )

671 X



0 .

342 X

2  e



(



0 .

035 )(



0 .

927

2

)] X

2  e



The surface is curved downward along the second principal axis and is positively sloped at the point X =

0 (both are significant at p < .05).

75

Example: Shape Along Second Principal

Axis for Autonomy

Contours Show

Shape Along

Second

Principal Axis

76


Tests of Significance



The formulas for shapes along predetermined lines such as Y = X and Y = –X can be tested using procedures for testing weighted linear combinations of regression coefficients.



For example, a t-test for b

1 dividing b

1

+ b

2

+ b

2 is obtained by by its standard error, or the square root of the variance of b

1

+ b

2

:

S ( b

1

 b

2

)





The variances of b

1

V ( b

1

) and b

2



V ( b

2

)



2 C ( b

1

, b

2

) are the squares of their standard errors, and the covariance of b

1 and b

2 their correlation times their standard errors.

is

77





Weighted linear combinations of regression coefficients can also be tested using routines available in many statistical packages.



Another approach is to test the reduction in R 2 produced by the constraint represented by the weighted linear combination of coefficients.



For instance, to jointly test (b

1

+ b

2

) and (b

3 b

4

+ b

5

), we set both quantities equal to zero and impose the resulting constraints.

+

78





The expression b

1

+ b

2

= 0 implies b

2

Likewise, the expression b

3 b

5

= –b

3

– b

4

+ b

4

+ b

5

= 0 implies

. Imposing these constraints on the quadratic regression equation yields:

= –b

1

.

Z = b

0

+ b

1

X – b

1

Y + b

3

X 2 + b

4

XY + (–b

3



The expression simplifies to:

– b

4

)Y 2 + e

Z = b

0

+ b

1

(X – Y) + b

3

(X 2 – Y 2 ) + b

4

(XY – Y 2 ) + e



The reduction in R 2 from this equation relative to the R 2 from the quadratic equation is a joint test of b

1

+ b

2

= 0 and b

3

+ b

4

+ b

5

= 0.

79





X

0

, Y

0

, p

10

, p

11

, p

20

, p

21

, and slopes along the principal axes are nonlinear combinations of regression coefficients. For these quantities, significance tests can be conducted using the bootstrap, as follows:



A large number (e.g., 10,000) of samples of size N are randomly drawn with replacement .



Each sample is used to estimate the quadratic regression equation.



The coefficients from each sample are used to compute X

0

,

Y

0

, p

10

, p

11

, p

20

, and p

21

.



The distributions of X

0

, Y

0

, p

10

, p

11

, p

20

, and p

21 construct confidence intervals.

are used to

80

Example: Testing Response Surface

Features for Autonomy



A joint test of (b

1

+ b

2

) and (b

3

+ b

4

+ b

5

), which represent the slope at the point X = 0 and the curvature along the Y = X line, is yielded by the following commands:

MGLH

MOD SAT=CONSTANT+AUTCA+AUTCD+AUTCA2+AUTCAD+AUTCD2

EST

HYP

AMA [0 1 1 0 0 0;,

0 0 0 1 1 1]

TEST

81





For autonomy, this test yields the following result:

Hypothesis.

A Matrix

1 2 3 4 5 6

1 0.0 1.000 1.000 0.0 0.0 0.0

2 0.0 0.0 0.0 1.000 1.000 1.000

Test of Hypothesis

Source SS df MS F P

Hypothesis 16.878 2 8.439 7.580 0.001

Error 394.135 354 1.113

82





Separate tests of (b

1

+ b

2

) and (b

3

+ b yielded by the following commands:

4

+ b

5

) are

MGLH


EST

HYP

AMA [0 1 1 0 0 0]

TEST

HYP

AMA [0 0 0 1 1 1]

TEST

83





For autonomy, the results are:

A Matrix

1 2 3 4 5 6

1 0.0 1.000 1.000 0.0 0.0 0.0

Test of Hypothesis

Source SS df MS F P

Hypothesis 1.068 1 1.068 0.959 0.328

Error 394.135 354 1.113

A Matrix

1 2 3 4 5 6

1 0.0 0.0 0.0 1.000 1.000 1.000

Test of Hypothesis

Source SS df MS F P

Hypothesis 11.740 1 11.740 10.545 0.001

Error 394.135 354 1.113

84





Likewise, a joint test of (b

1

– b

2

) and (b

3

– b

4

+ b

5

), which represent the slope at the point X = 0 and the curvature along the Y = – X line, is yielded by the following commands:

MGLH


EST

HYP

AMA [0 1 -1 0 0 0;,

0 0 0 1 -1 1]

TEST

85





For autonomy, this test yields the following result:

Hypothesis.

A Matrix

1 2 3 4 5 6

1 0.0 1.000 -1.000 0.0 0.0 0.0

2 0.0 0.0 0.0 1.000 -1.000 1.000

Test of Hypothesis

Source SS df MS F P

Hypothesis 39.512 2 19.756 17.744 0.000

Error 394.135 354 1.113

86





Separate tests of (b

1

– b

2

) and (b

3

– b yielded by the following commands:

4

+ b

5

) are

MGLH


EST

HYP

AMA [0 1 -1 0 0 0]

TEST

HYP

AMA [0 0 0 1 -1 1]

TEST

87





For autonomy, the results are:

A Matrix

1 2 3 4 5 6

1 0.0 1.000 -1.000 0.0 0.0 0.0

Test of Hypothesis

Source SS df MS F P

Hypothesis 8.105 1 8.105 7.279 0.007

Error 394.135 354 1.113

A Matrix

1 2 3 4 5 6

1 0.0 0.0 0.0 1.000 -1.000 1.000

Test of Hypothesis

Source SS df MS F P

Hypothesis 6.588 1 6.588 5.917 0.015

Error 394.135 354 1.113

88





In SYSTAT, the bootstrap is implemented with the following commands:

MGLH


SAVE AUTBOOT.SYD/COEF

EST/SAMPLE=BOOT(10000)



These commands will produce a large output file with the results of all 10,000 regressions and a system file containing 10,000 sets of coefficients.



The coefficients are used to construct confidence intervals (Mooney & Duval, 1993; Stine, 1989).

89

X

0

Y

0 p

10 p

11 p

20 p

21





For autonomy, the 95% confidence intervals for X

0

, Y

0

, p

10

, p

11

, p

20

, p

21 are:

Value CI

L

0.982

–0.315

0.199

–3.480

–1.375

–11.423

1.079

0.594

–0.927

0.688

–1.167

–1.449

CI

U

5.142

0.239

–0.359

2.123

1.120

–0.466

90

Interpretation of Results for Autonomy



The surface was saddle-shaped.



The slope of the first principal axis did not differ from 1, and the intercept of first principal axis was negative, meaning that the axis ran parallel to the Y = X line but was shifted to the right.



The slope and intercept of the second principal axis did not differ from –1 and 0, respectively. Thus, the axis did not differ from the

Y = –X line.



The location of the first principal axis combined with the slope along the second principal axis indicate that satisfaction increased as actual autonomy increased toward desired autonomy, continued to increase as actual autonomy exceeded desired autonomy, and began to decrease when actual autonomy exceeded desired autonomy by about one unit.



Within the range of the data, satisfaction increased at an increasing rate as actual and desired autonomy both increased along the first principal axis.

91




In some cases, the effect represented by a quadratic regression equation is believed to be moderated by another variable.



Incorporating the moderator variable V into a quadratic regression equation yields:

Z = b

0

+ b

1

X + b

2

Y + b

3

X 2 + b

4

XY + b

5

Y 2 + b

6

V + b

7

XV + b

8

YV + b

9

X 2 V + b

10

XYV + b

11

Y 2 V + e



Moderation is tested by assessing the increment in

R 2 yielded by the terms XV, YV, X

Y 2 V.

2 V, XYV, and

92




The moderated quadratic regression equation can be rewritten to show simple surfaces at selected levels of the moderator variable, as follows:

Z = (b

0

+ b

6

V) + (b

1

+ b

7

V)X + (b

2

+ b

8

V)Y +

(b

3

+ b

9

V)X 2 + (b

4

+ b

10

V)XY + (b

5

+ b

11

V)Y 2 + e



The compound coefficients on the terms X, Y, X 2 ,

XY, and Y 2 can be tested using procedures for testing weighted linear combinations of regression coefficients.

93

Example: Moderated Polynomial

Regression for Autonomy

 Quadratic equation with importance as a moderator:




CONSTANT 5.514 0.481 0.000 . 11.455 0.000

AUTCA 0.409 0.487 0.379 0.012 0.841 0.401

AUTCD -0.740 0.518 -0.595 0.014 -1.429 0.154

AUTCA2 0.181 0.292 0.278 0.012 0.620 0.536

AUTCAD 0.595 0.489 0.855 0.005 1.218 0.224

AUTCD2 -0.225 0.306 -0.353 0.010 -0.736 0.462

AUTI 0.062 0.101 0.051 0.343 0.614 0.540

AUTCAI -0.050 0.103 -0.242 0.009 -0.479 0.632

AUTCDI 0.103 0.115 0.454 0.009 0.890 0.374

AUTCA2I -0.046 0.054 -0.408 0.011 -0.862 0.389

AUTCADI -0.047 0.088 -0.392 0.004 -0.533 0.594

AUTCD2I 0.021 0.059 0.200 0.008 0.360 0.719



Regression 88.023 11 8.002 7.158 0.000

Residual 385.670 345 1.118

94





The test of the increment in R 2 yielded by the five moderator terms is:

(.

186



.

169 ) /( 350



345 )

( 1



.

186 ) / 345



1 .

44 , p



.

05



The increment in R 2 is not significant, so moderation is not supported.

95





Simple quadratic equations at low, medium, and high levels of importance:

Low

Medium

High

X Y X 2 XY Y 2

0.21

-0.33

** -0.00

0.41

* -0.14

0.16

-0.23

-0.05

0.36

** -0.12

0.11

-0.13

-0.09

0.32

** -0.10

96





Simple surface for low importance:

97





Simple surface for medium importance:

98





Simple surface for high importance:

99

Mediated Polynomial Regression



On occasion, the effect represented by a quadratic regression equation is believed to be mediated by

(i.e., transmitted through) another variable.



Mediation can be analyzed using two regression equations, one that regresses the mediator on the five quadratic terms, and another that regresses the outcome on the five quadratic terms and the mediator:

M = a

0

+ a

1

X + a

2

Y + a

3

X 2 + a

4

XY + a

5

Y 2 + e

M

Z = b

0

+ b

1

M + b

2

X + b

3

Y + b

4

X 2 + b

5

XY + b

6

Y 2 + e

Z

100




The mediated effect represented by these two equation can be derived by substituting the equation for M into the equation for Z to obtain a reduced form equation:

Z = b

0

+ b

1

(a

0

+ a

1

X + a

2

Y + a

3

X 2 + a

4

XY + a

5

Y 2 + e

M

)

+ b

2

X + b

3

Y + b

4

X 2 + b

5

XY + b

6

Y 2 + e

Z



Distribution yields:

Z = b

0

+ a

0 b

1

+ a

1 b

1

X + a

2 b

1

Y + a

3 b

1

X 2 + a

4 b

1

XY + a

5 b

1

Y 2 + b

1 e

M

+ b

2

X + b

3

Y + b

4

X 2 + b

5

XY + b

6

Y 2

+ e

Z

101




Collecting like terms yields:

Z = (b

0

+ a

0 b

1

) + (b

2

+ a

1 b

1

)X + (b

3

+ a

2 b

1

)Y +

(b

4

+ a

3 b

1

)X 2 + (b

5

+ a

4 b

1

)XY + (b

6

+ a

5 b

1

)Y 2 +

(e

Z

+ b

1 e

M

)



The compound coefficients on X, Y, X 2 , XY, and Y 2 capture the portion of the quadratic effect mediated by

M as the products a

1 b

1

, a

2 b

1

, a

3 b

1

, a

4 b

1

, and a

5 b

1

.



The portion of the quadratic effect that bypasses M is captured by b

2

, b

3

, b

4

, b

5

, and b

6

.



These coefficients can be analyzed separately and jointly to examine the mediated quadratic effect.

102

Example: Mediated Polynomial


 Quadratic equation with intent to take the focal job as the outcome variable:

Dep Var: INT N: 360 Multiple R: 0.276 Squared multiple R: 0.076



CONSTANT 5.851 0.092 0.000 . 63.319 0.000

AUTCA 0.161 0.111 0.142 0.273 1.449 0.148

AUTCD -0.244 0.119 -0.187 0.315 -2.056 0.041

AUTCA2 -0.076 0.052 -0.110 0.444 -1.438 0.151

AUTCAD 0.197 0.089 0.267 0.178 2.211 0.028

AUTCD2 0.008 0.070 0.013 0.242 0.121 0.904



Regression 40.397 5 8.079 5.858 0.000

Residual 488.231 354 1.379

103



 Quadratic equation with satisfaction as the mediator variable:




CONSTANT 5.825 0.083 0.000 . 70.161 0.000

AUTCA 0.197 0.100 0.182 0.273 1.966 0.050

AUTCD -0.293 0.106 -0.238 0.315 -2.754 0.006

AUTCA2 -0.056 0.047 -0.086 0.444 -1.177 0.240

AUTCAD 0.276 0.080 0.396 0.178 3.453 0.001

AUTCD2 -0.035 0.063 -0.054 0.242 -0.553 0.581



Regression 79.951 5 15.990 14.362 0.000

Residual 394.135 354 1.113

104



 Quadratic equation with intent to take the focal job as the outcome variable and satisfaction as the mediating variable:

Dep Var: INT N: 360 Multiple R: 0.760 Squared multiple R: 0.578



CONSTANT 1.074 0.242 0.000 . 4.445 0.000

SAT 0.820 0.040 0.777 0.831 20.480 0.000

AUTCA 0.000 0.076 0.000 0.270 0.001 0.999

AUTCD -0.003 0.081 -0.002 0.308 -0.038 0.969

AUTCA2 -0.030 0.036 -0.044 0.443 -0.842 0.401

AUTCAD -0.030 0.061 -0.040 0.173 -0.484 0.629

AUTCD2 0.037 0.047 0.055 0.242 0.780 0.436



Regression 305.506 6 50.918 80.556 0.000

Residual 223.122 353 0.632

105





The compound coefficients are:

 b

0

 b

2

 b

3

 b

4

 b

5

 b

6

+ a

0 b

1

+ a

1 b

1

+ a

2 b

1

+ a

3 b

1

+ a

4 b

1

+ a

5 b

1

= 1.07 + 5.83 × 0.82 = 1.07 + 4.78 = 5.85

= 0.00 + 0.20 × 0.82 = 0.00 + 0.16 = 0.16

= –0.00 – 0.29 × 0.82 = –0.00 – 0.24 = –0.24

= –0.03 – 0.06 × 0.82 = –0.03 – 0.05 = –0.08

= –0.03 + 0.28 × 0.82 = –0.03 + 0.23 = 0.20

= 0.04 – 0.04 × 0.82 = 0.04 – 0.03 = 0.01



The individual coefficients can be tested using the reported standard errors, and the products of coefficients can be tested using the bootstrap.

106





Tests of individual and compound coefficients:

Direct 1 First Second Indirect Total

Intercept

Effect Stage Stage Effect Effect

1.07

** 5.83

** 0.82

** 4.78

** 5.85

**

X

Y

X 2

XY

Y 2

0.00

0.20

* 0.82

** 0.16

* 0.16

–0.00

–0.29

** 0.82

** –0.24

** –0.24

*

–0.03

–0.06

0.82

** –0.05

–0.08

–0.03

0.28

** 0.82

** 0.23

** 0.20

0.04

–0.04

0.82

** –0.03

0.01

1 The direct effect of the five quadratic terms was not significant.

107





Surface for unmediated effect:

108





Surface for direct effect:

109





Surface for first stage of indirect effect:

110





Surface for indirect effect:

111





Surface for total effect:

112

Difference Scores as Dependent Variables



Many of the problems that occur when difference scores are used as independent variables also occur when they are used as dependent variables.



Alternative procedures for difference scores as dependent variables are fundamentally different from those for difference scores as independent variables.



We will briefly consider procedures when the dependent variable is an algebraic difference and both components are endogenous, meaning they are caused by the independent variables.

113




An equation that uses an algebraic difference as a dependent variable is:

(Y

1

– Y

2

) = b

0

+ b

1

X + e



Y

1 and Y

2 may be recast as separate dependent variables in a multivariate regression analysis:

Y

1

= b

10

+ b

11

X + e

1

Y

2

= b

20

+ b

21

X + e

2

114




The correspondence between these equations can be seen by subtracting the Y equation, which yields:

2 equation from the Y

1

(Y

1

– Y

2

) = (b

10

– b

20

) + (b

11

– b

21



This subtraction shows the following:

)X + (e

1

– e

2

) b

0

= b

10

– b

20 b

1

= b

11

– b

21



These expressions reveal a fundamental ambiguity, in that b

0 and b

1 indicate the differences between the intercepts and slopes, respectively, from the Y

1 and Y

2 equations, but they provide no information regarding the absolute magnitudes of these intercepts and slopes.

115




This ambiguity is illustrated by the following examples, all of which yield the same value for b

1

.



This pattern indicates that the effects of X on Y

1 are equal in magnitude but opposite in sign: and Y

2 b

11

= b

1

/2, b

21

= –b



Here, X is positively related to Y

1

1

/2 and unrelated to Y

2

: b

11

= b

1

, b

21

= 0



Here, X is negatively related to Y

2 and unrelated to Y

1

: b

11

= 0, b

21

= –b

1



These examples show that b

1 is essentially useless for determining the effect of X on Y

1 and Y

2

.

116




The alternative procedure uses Y

1 and Y

2 jointly as dependent variables in multivariate regression equations.



The multivariate equations reveal the separate effects of

X on Y

1 and Y

2 and can be used to test whether these effects correspond to hypotheses implied when (Y

1

– Y

2

) is used as a dependent variable.



The procedure provides multivariate tests of the effects of X on Y

1 and Y

2 and differences between these effects.



Multivariate piecewise regression equations can be used as an alternative to |Y

1

– Y

2

| is used as a dependent variable.

117


Q: Which higher-order terms should I use? Are squared and product terms sufficient, or should I also use cubed terms, the products of squared and first-order terms, etc.?

A: The higher-order terms to be included in the equation depend entirely on one’s hypotheses regarding the joint relationships of

X and Y with Z. In most cases, I have found that the three quadratic terms (i.e., X 2 , XY, and Y 2 ) are sufficient to capture most theoretically meaningful effects. In exploratory analyses, I have found significant effects for cubic and quartic terms, but these rarely survive cross-validation and are often symptoms of a few outliers or influential cases in the data.

118


Q: How do I interpret the coefficients on X 2 , XY, and Y 2 ? I understand what they each mean separately, but thinking about them all together is confusing.

A: The coefficients on X 2 , XY, and Y 2 should be interpreted along with the coefficients on X and Y as a set, because these coefficients collectively describe the shape of the surface relating

X and Y to Z. Trying to interpret any one of these coefficients in the absence of the others will often yield erroneous conclusions.

Instead, surfaces indicated by quadratic regression equations should be treated as whole entities, and features of the surfaces can be tested using response surface methodology. A major motivation for applying response surface methodology was my frustration when trying to make sense of coefficients from quadratic equations. Response surface methodology makes the task much easier.

119


Q: Given that the coefficients on X and Y are scale dependent when

X 2 , XY, and Y 2 in the equation, how can I meaningfully interpret these coefficients?

A: The coefficients on X and Y (i.e., b

1 and b

2

) are indeed scale dependent. However, this simply reflects the fact that b

1 and b

2 indicate the slope of the surface where X and Y are zero (i.e., the origin of the X,Y plane). One could add or subtract arbitrary constants to X and Y and change the values of b

1 and b

2

, but doing so may shift the origins of X and Y beyond the bounds of the data, where it doesn’t make sense to estimate b in the

1 and b

2 first place. A more reasonable strategy is to scale X and Y such that their origins represent a meaningful point in the distribution of the data in the X,Y plane, such as a point midway between their means or the midpoint of their common scale.

120


Q: How large should my sample be?

A: The sample should be large enough to provide the statistical power needed to test constraints and combinations of regression coefficients required to test hypotheses. Power is important because showing support for constraints requires support for the null hypothesis (i.e., the R 2 values for the constrained and unconstrained equations do not differ). A related concern is that the sample should provide adequate dispersion of cases in the

X,Y plane. For example, if cases are skewed in the direction of

X > Y or X < Y, it will be very difficult to detect changes in the slope of the surface along the Y = –X line, which are usually of interest in congruence research. Keep in mind that skewness on either side of the Y = –X line cannot be detected by examining the distributions of X and Y separately.

121


Q: I have seen measures that ask the respondent to directly compare the degree to which X deviates from Y. Doesn’t this approach avoid the problems with difference scores?

A: Not really. Although it removes the need for the researcher to calculate the difference, it does not guarantee that the respondent will not implicitly or explicitly calculate the difference between

X and Y when providing a response (many response scales for such items prompt the respondent to do just that). If this occurs, then items that solicit direct comparisons are subject to the problems as difference scores, because these problems do not depend on who calculates the difference. Moreover, direct comparison items hopelessly confound X and Y (analogous to any “double-barreled” item) and force the researcher to take a two-dimensional view of the relationship of X and Y with Z, even when a three-dimensional view may be more informative.

122


Q: The unconstrained equations for profile similarity indices contain so many items. How do I interpret all those coefficients, and what do I do about degrees of freedom?

A: Testing the full set of constraints imposed D 1 , |D|, and D 2 does indeed require using items for all of the dimensions as predictors.

However, the items constituting profiles can often be grouped into conceptually homogeneous subsets. Scales corresponding to these subsets can then be constructed, which can drastically reduce the effective number of dimensions to be analyzed. This not only makes interpretation easier, but also reduces sample size requirements. Moreover, higher-order terms for each dimension can be tested as sets, and those that are not significant may be dropped (for an illustration of this, see Edwards, 1993). Of course, models derived in this manner should be considered exploratory, pending cross-validation.

123


Q: By not using difference scores, aren’t we ignoring “fit?”

A: Models using difference scores are simply special cases of general models containing the components of the difference.

Hence, these general models subsume those that use difference scores. The general models also permit tests of the constraints imposed by difference scores, which remain unverified when difference scores are used. Moreover, fit hypotheses can usually be restated in terms of relationships involving the variables that constitute the fit construct. By stating hypotheses in these terms, one can verify that relationships for these variables conform to patterns depicted by fit hypotheses. Thus, the use of component variables, supplemented by higher-order terms and response surface analyses, permit tests of most fit hypotheses as well as hypotheses difference scores cannot depict. This approach lets the researcher gain much and lose little, if anything at all.

124


Q: How can I apply the quadratic approach to structural equations modeling?

A: Drawing from the literature on moderated structural equation modeling, I have developed procedures for specifying and estimating quadratic structural equation models and applying response surface methodology.

These procedures require squares and products of the indicators of first-order latent variables, involve complex nonlinear constraints on parameters, and use estimation methods for nonnormal data. I hope to finish a manuscript describing this procedure in the near future.

125


Q: How do you generate those fancy graphs?

A: I have traditionally used SYSTAT, which is great for plotting three-dimensional surfaces and adding contour lines, principal axes, and so forth. Surfaces can also be plotted using Microsoft Excel, and I have developed a file that allows the user to enter coefficient estimates from a quadratic equation and the minimum and maximum values of X and Y to produce a surface. This file can be downloaded from my website at: http://public.kenan-flagler.unc.edu/faculty/edwardsj/downloads.htm

126


Q: Can you recommend empirical examples of polynomial regression in the organizational behavior literature?

A: The use of polynomial regression has grown since its introduction. Examples published through 2000 are cited in the Edwards (2001) article on difference score myths, and more recent examples are cited in the meta-analysis conducted by Kristof-Brown et al. (2005).

127


Q: Your approach looks like a real pain. Can I just pretend it doesn’t exist? Or, can I just cite your work to make it look like

I'm doing what you recommend?

A: Some researchers tenaciously cling to difference scores. Old habits die hard. As a case in point, in a 1992 Psychological

Bulletin article, Lee Cronbach lamented that researchers continue to use profile similarity indices he once advocated (Cronbach,

1955; Cronbach & Gleser, 1953) but subsequently disavowed

(Cronbach, 1958). Researchers have also developed clever ways of citing articles that criticize difference scores without following the advice in the articles. Here are some of my favorites, quoted from studies that cite Edwards (1994):

128


 “Computing a correlation across dimensions for each individual to predict outcomes of fit or congruence represents a flawed measure of fit (Edwards, 1994). However, for our purposes here, correlations across individuals within a dimension provide an appropriate measure of the relationship between person and environment.”

 “The reliabilities of the difference scores created to assess similarity were relatively high, so it seemed simpler and more understandable to keep the analysis as it was rather than to apply more complicated alternatives (e.g., Edwards, 1994).”

 “Unmet expectations were assessed by subtracting scores on each item for the early expectations from scores on each item from the current situation . . . Problems in measuring and analyzing discrepancy scores, and unmet expectations in particular, have been reported recently (Edwards, 1994) . . . these problems have not been entirely overcome here.”

129

Key References

Bohrnstedt, G. W., & Goldberger, A. S. (1969). On the exact covariance of products of random variables.

Journal of the American Statistical Association, 64, 1439-1442.

Bohrnstedt, G. W., & Marwell, G. (1978). The reliability of products of two random variables. In K. F.

Schuessler, (Ed.), Sociological Methodology 1978 (pp. 254-273). San Francisco: Jossey-Bass.

Edwards, J. R. (1994). The study of congruence in organizational behavior research: Critique and a proposed alternative. Organizational Behavior and Human Decision Processes, 58, 51-100 (erratum, 58, 323-325).

Edwards, J. R., & Parry, M. E. (1993). On the use of polynomial regression equations as an alternative to difference scores in organizational research. Academy of Management Journal, 36, 1577-1613.

Edwards, J. R. (1995). Alternatives to difference scores as dependent variables in the study of congruence in organizational research. Organizational Behavior and Human Decision Processes, 64, 307-324.

Edwards, J. R. (2001). Ten difference score myths. Organizational Research Methods, 4, 264-286.

Edwards, J. R. (2002). Alternatives to difference scores: Polynomial regression analysis and response surface methodology. In F. Drasgow & N. W. Schmitt (Eds.), Advances in measurement and data analysis (pp. 350-

400). San Francisco: Jossey-Bass.

Kristof-Brown, A. L., Zimmerman, R. D., & Johnson, E. C. (2005). Consequences of individual's fit at work: A meta-analysis of person-job, person-organization, person-group, and person-supervisor fit. Personnel

Psychology, 58, 281-342.

Mooney, C. Z., & Duval, R. D. (1993). Bootstrapping: A nonparametric approach to statistical inference .

Newbury Park, CA: Sage.

Stine, R. (1989). An introduction to bootstrap methods. Sociological Methods & Research, 18, 243-291.

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Lecture presentation - The University of North Carolina at Chapel Hill

Alternatives to Difference Scores:

Polynomial Regression and

Response Surface Methodology

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Lecture presentation - The University of North Carolina at Chapel Hill

Alternatives to Difference Scores:

Polynomial Regression and

Response Surface Methodology

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