結構方程模型多向度構念題目組合表徵之構念意涵

Graduate Institute of Psychology College of Science

National Taiwan University Master Thesis

Nature of Multidimensional Constructs Represented by Item Parcels in Structural Equation Modeling

Yo-Lin Chen

Advisor: Li-Jen Weng, Ph.D.

106 6

June, 2017

item parcels

structural equation modeling Little Rhemtulla

Gibson Schoemann 2013 Cole Perkins Zelkowitz 2016

Cole Williams O’Boyle 2008

Sterba MacCallum 2010

Nature of Multidimensional Constructs Represented by Item Parcels in Structural Equation Modeling

Yo-Lin Chen

Abstract

Item parcels, represented as summation or average over items, can be used as indicators of latent variables in structural equation modeling (SEM). Little et al.

(2013) and Cole et al. (2016) indicated that the nature of multidimensional constructs represented by parcels can be different from that assumed by the researchers and thus affects the results of SEM analysis. Researchers should therefore examine the nature of multidimensional constructs represented by parcels prior to the analysis. Cole et al.

and Williams and O’Boyle (2008) indicated that many of the constructs investigated in psychological research are multidimensional, consisting of several facets, and item parcels have been frequently adopted to be indicators for these constructs in SEM.

The present study therefore extended the algebraic derivation of the covariance matrix of item parcels in Sterba & MacCallum (2010) from unidimensional to

multidimensional constructs to provide a theoretical framework for examining the nature of multidimensional constructs implied by parcels. The effects of parceling strategy, correlations among facets, and factorial complexity of items on the nature of multidimensional constructs represented by parcels were discussed using this

framework and illustrated by a numerical simulation example. Researchers may apply the framework proposed in this study to clarify the nature of the multidimensional constructs inferred from item parcels through examining the covariance matrix among

parcels so as to avoid misleading results from SEM analysis.

Keywords: multidimensional constructs, structural equation modeling, item parcels

... 1

... 15

... 18

... 34

... 38

... 46

... 49

Item parceling Cattell 1956, 1974; Cattell & Burdsal, 1975

structural equation modeling

Bandalos, 2002, 2008; Bandalos & Finney, 2001; Kim & Hagtvet, 2003; Williams &

O’Boyle, 2008 Hall Snell

Foust 1999 nature of

construct

Cole, Perkins, & Zelkowitz, 2016; Williams & O’Boyle, 2008

multidimensional construct

facet

facet-representative strategy domain-

representative strategy Coffman &

MacCallum, 2005; Cole et al., 2016; Kishton & Widaman, 1994; Little, Rhemtulla, Gibson, & Schoemann, 2013

Sterba MacCallum 2010

Little Cole

Little Cole

Sterba MacCallum

Little Cole

Cole 2016

Marsh, Ludtke, Nagengast, Morin, & Von Davier, 2013

Bollen, 1989

Bandalos & Finney, 2001; Cattell, 1956; Cole et al., 2016; Hall et al., 1999; Little, Cunningham, Shahar, & Widaman, 2002; Little et al., 2013; Sterba

& Rights, 2016; Williams & O’Boyle, 2008 Bandalos Finney 2001

Bandalos Finney 2001

maximum likelihood estimation

method Jackson, Gillaspy, &

Pure-Stephenson, 2009; Kline, 2011; Schreiber, 2008

asymptotic theory

Bandalos Finney idiosyncratic features of the items

Bagozzi & Edwards, 1998; Bagozzi & Heatherton, 1994;

Bandalos & Finney, 2001; Cole et al., 2016; Little et al., 2002; Marsh et al., 2013;

Marsh & O’Neill, 1984; Rhemtulla, 2016; Rogers & Schmitt, 2004; Sterba & Rights, 2016; Williams & O’Boyle, 2008 Bagozzi

Marsh latent

means measurement invariance differential item

functioning

Bandalos, 2002, 2008; Cole et al., 2016; Little et al., 2002;

Little et al., 2013; Marsh et al., 2013; Rhemtulla, 2016 Hall 1999 Bandalos secondary factor nuisance factor

Little

Bandalos, 2002, 2008; Hall et al., 1999;

Marsh, Hau, Balla, & Grayson, 1998; Nasser & Wisenbaker, 2003; Nasser &

Wisenbaker, 2006; Rogers & Schimitt, 2004; Sass & Smith, 2006

Cole et al., 2016; Williams & O’Boyle, 2008 Williams O’Boyle 2001 2007

human resource management 75

44%

Cole

depression self-esteem

Coffman MacCallum 2005

latent variable model path analysis model

Little 2013

Coffman MacCallum 2005

Little 2013 Cole 2016 hierarchical

model high-order factor model

first-order factor second-order factor

residual uniqueness

Cole 2016

subfactor symptom

Coffman &

MacCallum, 2005; Cole et al., 2016; Kishton & Widaman, 1994; Little et al., 2013;

Williams & O’Boyle, 2008

homogeneous parceling, Coffman & MacCallum, 2005; Cole et al., 2016 internal consistent parceling, Kishton

& Widaman, 1994 isolated uniqueness strategy, Hall et al., 1999

heterogeneous parceling, Cole et al., 2016 distributed uniqueness strategy, Hall et al., 1999

Coffman & MacCallum, 2005; Cole et al., 2016; Kishton & Widaman,

1994 Little 2013 Cole 2016

Rhemtulla 2016

Hall 1999

Little 2013 Cole

2016 Little

Cole Cole

Kishton Widaman 1994

Cole Marsh 2013

Hall 1999

Hall

Hall

Hall

Hall 1999

Hall

Hall

Bandalos 2002, 2008 Rogers Schmitt 2004

Hall

Cole et al.,

2016; Little et al., 2013 Cole

T A B C

T

T A B C

Little

Little

Little

Cole 2016

Cole

Cole

Kishton

Widaman 1994 Kishton Widaman

.23 .15 .19 Kishton Widaman

Cole 2016 Kishton Widaman 1994

Cole

Little 2013 Cole 2016

Cole et al., 2016; Marsh et al., 2013

Toronto Alexithymia Scale-20, TAS-20; Bagby, Parker, & Taylor, 1994

Kooiman, Spinhoven, & Trijsburg, 2002;

Moriguchi et al., 2007

Gignac 2006

emotional intelligence life satisfaction

Sterba MacCallum 2010

Sterba MacCallum

Sterba MacCallum 2010 MacCallum Tucker 1991

MacCallum Widaman Zhang Hong 1999

y_i m × 1

deviation scores i F q × 1

y_i = Λ_iF + ϵ_i (1)

Λi m × q E(F) = 0 ϵi m × 1

E(ϵi) = 0 E(ϵiF^') = 0

(1) y_i Σi

Σ_i = E(y_iy_i^') = Λ_iΦ_iΛ_i^' + Θ_ϵi (2) Φi = E(FF^') q × q Θϵi = E(ϵiϵi')

m × m

A n × m selection matrix y_i n × 1

y_p p y_p = Ay_i y_p Σp

Σ_p = E(y_py_p^')

= E(Ay_iy_i^'A^' )

= AΣ_iA^'

= AΛ_iΦ_iΛ_i^'A^' + AΘ_ϵiA ^'

= Λ_pΦ_iΛ_p^'+ Θ_ϵp (3)

Λp = AΛi n × q Θϵp = AΘ_ϵiA^' n × n

Θϵi

Θϵp

A Φ_i

Φ_p Φ_i

Sterba MacCallum 2010

Sterba MacCallum

Coffman MacCallum 2005 Little

2013 Cole 2016

Little Cole

Coffman MacCallum

Sterba MacCallum 2010

Coffman & MacCallum, 2005; Cole et al., 2016; Little et al., 2013

Cole Little

Coffman MacCallum 2005

1 D1 D4

1 Coffman MacCallum D1

D4

Coffman MacCallum

2

1 2 0.6 0.1 0.9

1 18 2

´ 9 10 100000

Coffman & MacCallum, 2005 10

3

A_f A_d

A_f = 1 3

1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1

(4)

A_d = 1 3

1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1

(5)

R 3.4.0 R Core Team, 2017

MASS Venables & Ripley, 2002 mvrnorm

moments Komsta &Novomestky, 2015

lavaan Rossel, 2012 1

150 10

χ² CFI

comparative fit index; Bentler, 1990 TLI Tucker-Lewis index; Tucker & Lewis, 1973 RMSEA root mean square error of approximation; Steiger & Lind, 1980 SRMR standardized root mean square residual; Bentler, 1995

Sterba MacCallum 2010

(1) y_i = Λ_iF + ϵ_i F

Coffman MacCallum 2005 Little 2013 Cole

2016

D r × 1

Γi q × r ζi q × 1

F = Γ_iD + ζ_i (6)

E(D) = 0 E(ζi) = 0 (1)

y_i = Λ_iF + ϵ_i = Λ_i(Γ_iD + ζ_i) + ϵ_i (7)

y_i Σ_i

Σ_i = E(y_iy_i^')

= Λ_iΦ_iΛ_i^' + Θ_ϵi

= Λ_i(Γ_iΦΓ_i^' + Ψ_i)Λ_i^' + Θ_ϵi (8)

Φ = E(DD^') r × r Ψ_i = E(ζ_iζ_i^') q × q

Sterba MacCallum 2010 A n × m y_i n × 1

y_p y_p = Ay_i y_p

Σ_p

Σ_p = E(y_py_p^')

= E(Ay_iy_i^'A^' )

= AΣ_iA^'

= A(Λ_iΦ_iΛ_i^' + Θ_ϵi)A^'

= AΛ_iΦ_iΛ_i^'A^' + AΘ_ϵiA ^'

= AΛ_i(Γ_iΦΓ_i^' + Ψ_i)Λ_i^'A^' + AΘ_ϵiA ^'

= AΛ_iΓ_iΦΓ_i^'Λ_i^'A^' + AΛ_iΨ_iΛ_i^'A^' + AΘ_ϵiA ^' (9) (9)

Φ Ψ_i

Θ_ϵi X Y Z

X = AΛ_iΓ_iΦΓ_i^'Λ_i^'A^' (10) Y = AΛ_iΨ_iΛ_i^'A^' (11) Z = AΘ_ϵiA ^' (12)

Y

X Z Y

X Φ

X

Z

Θ_ϵi A

Θ_ϵi

Z

Y

Y Ψ_i

Y

Y

Y Y

Y Y

0 0

0

Y

0

Y 0

Sterba MacCallum 2010

Little 2013 Cole 2016

Little Cole

Little

Little Y

0

0

Little Y

Y

Cole 2016

Y Y

Cole Y Little

2013 Cole

0

Little 2013 Cole 2016

Y

Y

Little Cole

Coffman MacCallum 2005 1

0.6 0.7 0.8 0.3 0.4 0.5

1 Coffman MacCallum

Coffman MacCallum

1

X Y Z 1 × 1

1

1

1 2 y1 y9

D1

X Y Z

1 y1 y9 9 × 1

y_i

y_i^' = y1 y₂ y₃ y₄ y₅ y₆ y₇ y₈ y₉ (13) F1 F2 F3

Λ_i

Λ_i^' =

λ₁₁ λ₂₁ λ₃₁ 0 0 0 0 0 0

0 0 0 λ₄₂ λ₅₂ λ₆₂ 0 0 0

0 0 0 0 0 0 λ₇₃ λ₈₃ λ₉₃

(14)

D₁ ζ₁

ζ₂ ζ₃ Γ_i Φ

Ψ_i

Γ_i^' = γ11 γ₂₁ γ₃₁ (15) Φ = var(D₁) (16)

Ψ_i =

var(ζ₁) 0 0

0 var(ζ₂) 0

0 0 var(ζ₃)

(17)

Θ_ϵi

Θ_ϵi =

var(ϵ₁) 0 0 0 0 0 0 0 0

0 var(ϵ₂) 0 0 0 0 0 0 0

0 0 var(ϵ₃) 0 0 0 0 0 0

0 0 0 var(ϵ₄) 0 0 0 0 0

0 0 0 0 var(ϵ₅) 0 0 0 0

0 0 0 0 0 var(ϵ₆) 0 0 0

0 0 0 0 0 0 var(ϵ₇) 0 0

0 0 0 0 0 0 0 var(ϵ₈) 0

0 0 0 0 0 0 0 0 var(ϵ₉)

(18)

y_i (4) (5)

1 2

X Y Z

X var(D₁)

Z

Y 1 Y

0

var(D₁) 2 Y

var(ζ₁) var(ζ₂) var(ζ₃)

var(D₁) var(ζ₁) var(ζ₂) var(ζ₃)

Γ_i Y Ψ_i

Ψ_i Ψ_i

Ψ_i var(ζ₁) var(ζ₂) var(ζ₃)

1

0 Y

Sterba MacCallum 2010 Hall

1999

var(ζ₁) var(ζ₂) var(ζ₃)

Cole 2016 Kishton Widaman

1994 Cole

Kishton Widaman

Marsh et al., 2013

Kooiman et al., 2002; Moriguchi et al., 2007

Gignac, 2006 Cole et al., 2016; Little et al., 2013

2 y1 y9

Λ_i

Λ_i^' =

λ₁₁ λ₂₁ λ₃₁ λ₄₁ 0 0 0 0 0

0 0 0 λ₄₂ λ₅₂ λ₆₂ λ₇₂ 0 0

λ₁₃ 0 0 0 0 0 λ₇₃ λ₈₃ λ₉₃

(19)

y1 y4 y7 Γ_i Φ Ψ_i Θ_ϵi (15)

(18) (4) (5)

3 4 X Y Z

(19) X

var(D1) Z

Y 3

Y

Y 0

var(D₁) var(ζ₁) var(ζ₂) var(ζ₃) 4

Y 2

var(ζ₁) var(ζ₂) var(ζ₃) 0

(19)

Λ_i Γ_i

Γ_i

Γ_i

Y

Y

Y

18 2 ´ 9

10

0.02 0.98~1.02 -0.05~0.03

-0.12~0.12 0.1 0.3

0.1

0.3

31 13 12 10

1 3

(4) (5) 1

5

Cole 2016

0.1

Cole

Cole

Cole

6

CFI TLI RMSEA SRMR

CFI TLI RMSEA SRMR 0.1

1

1

2

3 (4) (5)

2

2

7

8

0.1

2

Coffman MacCallum 2005 Little 2013 Cole 2016

Sterba MacCallum 2010

X Y Z

X

Z

Y Y

Little 2013 Cole 2016

Little Cole

Cole et al., 2016; Marsh et al., 2013

Coffman MacCallum 2005

Kishton Widaman 1994

0.19 1

Cole 2016

Little 2013 Cole 2016

Coffman MacCallum 2005

subscale

Coffman MacCallum 2005 Little 2013 Cole

2016

parcel-allocation variability combination scheme

Sterba, 2011; Sterba & MacCallum, 2010; Sterba & Rights, 2016

Sterba, 2011; Sterba & Rights, 2016 Marsh et al., 1998;

Rogers & Schmitt, 2004 Bandalos, 2002, 2008; Hall et al., 1999;

Rogers & Schmitt, 2004

R

1

X 1

9

λ₁₁ + λ₂₁ + λ₃₁ ²γ₁₁²

λ₁₁ + λ₂₁ + λ₃₁ λ₄₂ + λ₅₂ + λ₆₂ γ₁₁γ₂₁ λ₁₁ + λ₂₁ + λ₃₁ λ₇₃ + λ₈₃ + λ₉₃ γ₁₁γ₃₁

λ₄₂ + λ₅₂ + λ₆₂ ²γ₂₁²

λ₄₂ + λ₅₂ + λ₆₂ λ₇₃ + λ₈₃ + λ₉₃ γ₂₁γ₃₁

Sym.

λ₇₃ + λ₈₃ + λ₉₃ ²γ₃₁²

var D₁

Y 1

9 λ₁₁ + λ₂₁ + λ₃₁ ²var ζ₁ 0

0

0

λ₄₂ + λ₅₂ + λ₆₂ ²var ζ₂ 0

0 0

λ₇₃ + λ₈₃ + λ₉₃ ²var ζ₃

Z 1

9 var ε₁ + var ε₂ + var(ε₃) 0

0

0

var ε₄ + var ε₅ + var(ε₆) 0

0 0

var ε₇ + var ε₈ + var(ε₉)

2

X 1 9

λ₁₁γ₁₁ + λ₄₂γ₂₁ + λ₇₃γ₃₁ ² λ₁₁γ₁₁ + λ₄₂γ₂₁ + λ₇₃γ₃₁ λ₂₁γ₁₁ + λ₅₂γ₂₁ + λ₈₃γ₃₁ λ₁₁γ₁₁ + λ₄₂γ₂₁ + λ₇₃γ₃₁ λ₃₁γ₁₁ + λ₆₂γ₂₁ + λ₉₃γ₃₁

λ₂₁γ₁₁ + λ₅₂γ₂₁ + λ₈₃γ₃₁ ² λ₂₁γ₁₁ + λ₅₂γ₂₁ + λ₈₃γ₃₁ λ₃₁γ₁₁ + λ₆₂γ₂₁ + λ₉₃γ₃₁

Sym.

λ₃₁γ₁₁ + λ₆₂γ₂₁ + λ₉₃γ₃₁ ²

var D₁

Y 1 9

λ₁₁²var ζ₁ + λ₄₂²var ζ₂ + λ₇₃²var ζ₃ λ₁₁λ₂₁var ζ₁ + λ₄₂λ₅₂var ζ₂ +

λ₇₃λ₈₃var ζ₃

λ₁₁λ₃₁var ζ₁ + λ₄₂λ₆₂var ζ₂ + λ₇₃λ₉₃var ζ₃

λ₂₁²var ζ₁ + λ₅₂²var ζ₂ + λ₈₃²var ζ₃ λ₂₁λ₃₁var ζ₁ + λ₅₂λ₆₂var ζ₂ +

λ₈₃λ₉₃var ζ₃

Sym.

λ₃₁²var ζ₁ + λ₆₂²var ζ₂ + λ₉₃²var ζ₃

Z 1 9

var ε₁ + var ε₄ + var(ε₇) 0

0

0

var ε₂ + var ε₅ + var(ε₈) 0

0 0

var ε₃ + var ε₆ + var(ε₉)

3

X 1 9

λ₁₁γ₁₁ + λ₂₁γ₁₁ + λ₃₁γ₁₁ + λ₁₃γ₃₁ ² λ₁₁γ₁₁ + λ₂₁γ₁₁ + λ₃₁γ₁₁ + λ₁₃γ₃₁ λ₄₂γ₂₁ + λ₅₂γ₂₁ + λ₆₂γ₂₁ + λ₄₁γ₁₁ λ₁₁γ₁₁ + λ₂₁γ₁₁ + λ₃₁γ₁₁ + λ₁₃γ₃₁ λ₇₃γ₃₁ + λ₈₃γ₃₁ + λ₉₃γ₃₁ + λ₇₂γ₂₁

λ₄₂γ₂₁ + λ₅₂γ₂₁ + λ₆₂γ₂₁ + λ₄₁γ₁₁ ² λ₄₂γ₂₁ + λ₅₂γ₂₁ + λ₆₂γ₂₁ + λ₄₁γ₁₁ λ₇₃γ₃₁ + λ₈₃γ₃₁ + λ₉₃γ₃₁ + λ₇₂γ₂₁

Sym.

λ₇₃γ₃₁ + λ₈₃γ₃₁ + λ₉₃γ₃₁ + λ₇₂γ₂₁ ²

var D₁

Y 1 9

λ₁₁ + λ₂₁ + λ₃₁ ²var ζ₁ + λ₁₃ ²var ζ₃ λ₁₁ + λ₂₁ + λ₃₁ λ₄₁var ζ₁

λ₇₃ + λ₈₃ + λ₉₃ λ₁₃var ζ₃

λ⁴² + λ₅₂ + λ₆₂ ²var ζ₂ + λ₄₁ ²var ζ₁ λ₄₂ + λ₅₂ + λ₆₂ λ₇₂var ζ₂

Sym.

λ₇₃ + λ₈₃ + λ₉₃ ²var ζ₃ + λ₇₂ ²var ζ₂

Z 1

9 var ε₁ + var ε₂ + var(ε₃) 0

0

0

var ε₄ + var ε₅ + var(ε₆) 0

0 0

var ε₇ + var ε₈ + var(ε₉)

4

X 1 9

λ₁₁γ₁₁ + λ₄₂γ₂₁ + λ₇₃γ₃₁ + λ₁₃γ₃₁ + λ₄₁γ₁₁ + λ₇₂γ₂₁

2

λ₁₁γ₁₁ + λ₄₂γ₂₁ + λ₇₃γ₃₁ +

λ₁₃γ₃₁ + λ₄₁γ₁₁ + λ₇₂γ₂₁ λ₂₁γ₁₁ + λ₅₂γ₂₁ + λ₈₃γ₃₁ λ₁₁γ₁₁ + λ₄₂γ₂₁ + λ₇₃γ₃₁ +

λ₁₃γ₃₁ + λ₄₁γ₁₁ + λ₇₂γ₂₁ λ₃₁γ₁₁ + λ₆₂γ₂₁ + λ₉₃γ₃₁

λ₂₁γ₁₁ + λ₅₂γ₂₁ + λ₈₃γ₃₁ ² λ₂₁γ₁₁ + λ₅₂γ₂₁ + λ₈₃γ₃₁ λ₃₁γ₁₁ + λ₆₂γ₂₁ + λ₉₃γ₃₁

Sym.

λ₃₁γ₁₁ + λ₆₂γ₂₁ + λ₉₃γ₃₁ ²

var D₁

Y 1 9

λ₁₁ + λ₄₁ var ζ₁ + λ₄₂ + λ₇₂ var ζ₂ + λ₇₃ + λ₁₃ var ζ₃

λ₁₁ + λ₄₁ λ₂₁var ζ₁ + λ₄₂ + λ₇₂ λ₅₂var ζ₂ + λ₇₃ + λ₁₃ λ₈₃var ζ₃

λ₁₁ + λ₄₁ λ₃₁var ζ₁ + λ₄₂ + λ₇₂ λ₆₂var ζ₂ + λ₇₃ + λ₁₃ λ₉₃var ζ₃

λ²¹²var ζ₁ + λ₅₂²var ζ₂ + λ₈₃²var ζ₃

λ₂₁λ₃₁var ζ₁ + λ₅₂λ₆₂var ζ₂ + λ₈₃λ₉₃var ζ₃

Sym.

λ₃₁²var ζ₁ + λ₆₂²var ζ₂ + λ₉₃²var ζ₃

Z 1 9

var ε₁ + var ε₄ + var(ε₇) 0

0

0

var ε₂ + var ε₅ + var(ε₈) 0

0 0

var ε₃ + var ε₆ + var(ε₉)

5

φ21 γ1 γ2 β1

.3 .6 .6 .6

M SD M(SE)^a M SD M(SE) M SD M(SE) M SD M(SE)

0.1 .22 .17 .18 .51 .17 .25 .55 .21 .17 .56 .17 .24

0.2 .29 .07 .06 .57 .05 .07 .58 .04 .05 .61 .05 .07

0.3 .30 .03 .03 .59 .02 .03 .59 .02 .02 .60 .03 .03

0.4 .30 .02 .02 .60 .01 .02 .60 .02 .01 .60 .01 .02

0.5 .30 .01 .01 .60 .01 .01 .60 .01 .01 .60 .01 .01

0.6 .30 .01 .01 .60 .01 .01 .60 .01 .01 .60 .01 .01

0.7 .30 .01 .01 .60 .01 .01 .60 .01 .00 .60 .01 .01

0.8 .30 .01 .01 .60 .00 .00 .60 .00 .00 .60 .00 .00

0.9 .30 .01 .00 .60 .00 .00 .60 .00 .00 .60 .00 .00

0.1 .01 .01 .01 .02 .01 .01 .02 .00 .01 .02 .00 .01

0.2 .03 .01 .01 .08 .00 .01 .08 .01 .01 .07 .01 .01

0.3 .07 .01 .01 .17 .00 .01 .17 .01 .01 .15 .01 .01

0.4 .11 .01 .01 .27 .00 .01 .27 .01 .01 .24 .01 .01

0.5 .15 .01 .01 .35 .00 .00 .35 .01 .00 .34 .01 .00

0.6 .19 .01 .00 .43 .01 .00 .43 .00 .00 .42 .00 .00

0.7 .22 .01 .00 .49 .00 .00 .49 .00 .00 .48 .00 .00

0.8 .25 .01 .00 .53 .00 .00 .54 .00 .00 .53 .00 .00

0.9 .28 .01 .00 .57 .00 .00 .57 .00 .00 .57 .00 .00

6

χ²(df=50) CFI TLI RMSEA SRMR

M SD M SD M SD M SD M SD

0.1 41.56 (.75)^a 8.42 .99 .03 1.19 .17 .00 .00 .00 .00

0.2 43.15 (.68) 9.41 1.00 .00 1.01 .01 .00 .00 .00 .00

0.3 44.86 (.62) 10.32 1.00 .00 1.00 .00 .00 .00 .00 .00

0.4 50.47 (.48) 13.29 1.00 .00 1.00 .00 .00 .00 .00 .00

0.5 50.80 (.47) 11.87 1.00 .00 1.00 .00 .00 .00 .00 .00

0.6 50.39 (.47) 12.77 1.00 .00 1.00 .00 .00 .00 .00 .00

0.7 50.55 (.44) 13.34 1.00 .00 1.00 .00 .00 .00 .00 .00

0.8 51.14 (.45) 11.41 1.00 .00 1.00 .00 .00 .00 .00 .00

0.9 46.96 (.57) 11.94 1.00 .00 1.00 .00 .00 .00 .00 .00

0.1 56.90 (.30) 9.56 1.00 .00 1.00 .00 .00 .00 .00 .00

0.2 154.72 (.00) 26.47 1.00 .00 1.00 .00 .00 .00 .01 .00

0.3 423.59 (.00) 52.28 1.00 .00 .99 .00 .01 .00 .01 .00

0.4 766.06 (.00) 80.39 .99 .00 .99 .00 .01 .00 .01 .00

0.5 1015.15 (.00) 88.19 .99 .00 .99 .00 .01 .00 .02 .00

0.6 1027.42 (.00) 71.02 .99 .00 .99 .00 .01 .00 .02 .00

0.7 835.36 (.00) 66.96 1.00 .00 1.00 .00 .01 .00 .01 .00

0.8 513.86 (.00) 53.69 1.00 .00 1.00 .00 .01 .00 .01 .00

0.9 203.69 (.00) 31.81 1.00 .00 1.00 .00 .01 .00 .00 .00

df = degrees of freedom; CFI = comparative fit index; TLI = Tucker-Lewis index; RMSEA = root mean square error of approximation; SRMR = standardized root mean square residual ^a p

7

φ21 γ1 γ2 β1

.3 .6 .6 .6

M SD M(SE)^a M SD M(SE) M SD M(SE) M SD M(SE)

0.1 .02 .01 .01 .04 .01 .01 .04 .01 .01 .03 .01 .01

0.2 .05 .01 .01 .14 .01 .01 .14 .01 .01 .12 .01 .01

0.3 .10 .01 .01 .26 .00 .01 .25 .01 .01 .24 .01 .01

0.4 .15 .01 .01 .36 .00 .01 .36 .00 .01 .35 .01 .01

0.5 .19 .01 .01 .44 .00 .00 .44 .00 .00 .43 .01 .00

0.6 .22 .01 .00 .49 .00 .00 .49 .00 .00 .49 .00 .00

0.7 .25 .01 .00 .53 .00 .00 .53 .00 .00 .53 .00 .00

0.8 .27 .00 .00 .56 .00 .00 .56 .00 .00 .56 .00 .00

0.9 .29 .00 .00 .58 .00 .00 .58 .00 .00 .58 .00 .00

0.1 .01 .00 .00 .02 .00 .00 .02 .01 .00 .02 .00 .00

0.2 .03 .01 .00 .08 .01 .00 .08 .00 .00 .07 .01 .00

0.3 .07 .00 .00 .17 .00 .00 .17 .00 .00 .14 .01 .00

0.4 .11 .00 .00 .26 .00 .00 .26 .00 .00 .23 .00 .00

0.5 .15 .00 .00 .35 .00 .00 .35 .00 .00 .32 .01 .00

0.6 .19 .00 .00 .42 .00 .00 .42 .00 .00 .40 .00 .00

0.7 .22 .00 .00 .48 .00 .00 .48 .00 .00 .47 .00 .00

0.8 .25 .00 .00 .53 .00 .00 .53 .00 .00 .52 .00 .00

0.9 .28 .00 .00 .57 .00 .00 .57 .00 .00 .57 .00 .00

8

χ²(df=50) CFI TLI RMSEA SRMR

M SD M SD M SD M SD M SD

0.1 62.73 (.17)^a 8.96 1.00 .00 1.00 .00 .00 .00 .00 .00

0.2 211.83 (.00) 31.70 1.00 .00 .99 .00 .01 .00 .00 .00

0.3 507.60 (.00) 43.01 .99 .00 .99 .00 .01 .00 .00 .00

0.4 760.84 (.00) 75.39 .99 .00 .99 .00 .01 .00 .01 .00

0.5 864.85 (.00) 61.19 .99 .00 .99 .00 .01 .00 .01 .00

0.6 793.11 (.00) 59.69 1.00 .00 1.00 .00 .01 .00 .01 .00

0.7 632.68 (.00) 52.49 1.00 .00 1.00 .00 .01 .00 .01 .00

0.8 410.40 (.00) 47.53 1.00 .00 1.00 .00 .01 .00 .01 .00

0.9 180.21 (.00) 24.41 1.00 .00 1.00 .00 .01 .00 .00 .00

0.1 70.96 (.10) 14.56 1.00 .00 1.00 .00 .00 .00 .00 .00

0.2 293.69 (.00) 44.78 1.00 .00 1.00 .00 .01 .00 .00 .00

0.3 840.92 (.00) 63.89 1.00 .00 .99 .00 .01 .00 .00 .00

0.4 1589.81 (.00) 104.30 .99 .00 .99 .00 .02 .00 .02 .00

0.5 2250.63 (.00) 117.77 .99 .00 .99 .00 .02 .00 .03 .00

0.6 2472.46 (.00) 117.61 .99 .00 .99 .00 .02 .00 .02 .00

0.7 2273.96 (.00) 115.78 .99 .00 .99 .00 .02 .00 .02 .00

0.8 1687.51 (.00) 108.00 1.00 .00 1.00 .00 .02 .00 .01 .00

0.9 803.67 (.00) 68.67 1.00 .00 1.00 .00 .01 .00 .01 .00

df = degrees of freedom; CFI = comparative fit index; TLI = Tucker-Lewis index; RMSEA = root mean square error of approximation; SRMR = standardized root mean square residual ^a p

1 Coffman MacCallum 2005

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