three approaches to using lenthy psychometric scales in sem

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DOI: 10.1177/0146621609338592 2010; 34; 122 originally published online Aug 17, 2009; Applied Psychological Measurement

Chongming Yang, Sandra Nay and Rick H. Hoyle Parceling, Latent Scoring, and Shortening Scales

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Three Approaches toUsing Lengthy Ordinal Scalesin Structural Equation ModelsParceling, Latent Scoring, and Shortening ScalesChongming YangSandra NayRick H. HoyleDuke University

Lengthy scales or testlets pose certain challenges for structural equation modeling (SEM) if all

the items are included as indicators of a latent construct. Three general approaches to modeling

lengthy scales in SEM (parceling, latent scoring, and shortening) have been reviewed and eval-

uated. A hypothetical population model is simulated containing two exogenous constructs with

14 indicators each and an endogenous construct with four indicators. The simulation generates

data sets with varying numbers of response options, two types of distributions, factor loadings

ranging from low to high, and sample sizes ranging from small to moderate. The population

model is varied to incorporate one of the following: (a) single parcels, (b) various parcels as

indicators of two exogenous constructs, (c) latent scores as observed exogenous variables, and

(d) four and six individual items as indicators of two exogenous constructs. The dependent

variables evaluated are biases in the covariance and partial covariance population parameters.

Biases in these parameters are found to be minimal under the following conditions: (a) when

parcels of indicators of five response options are used as indicators of two latent exogenous

constructs, (b) when latent scores are used as observed variables at sample sizes above 100 and

with indicators that are relatively less skewed in the case of dichotomous indicators, and (c)

when four or six individual items with high or diverse factor loadings are used as indicators of

two exogenous constructs. These findings provide guidelines for resolving the inconsistency of

findings from applying various approaches to empirical data.

Keywords: testlets; latent scores; scale length; growth modeling; structural equation

modeling

Lengthy ordinal scales or testlets pose challenges for structural equation modeling

(SEM) if all the items are used as indicators of a latent construct. For instance, a

model could have too many parameters to estimate relative to the available sample size,

resulting in reduced power to detect important parameters. In addition, it might not fit the

data sufficiently well because individual items may have less than ideal measurement

Applied Psychological

Measurement

Volume 34 Number 2

March 2010 122-142

� 2010 The Author(s)

10.1177/0146621609338592


Authors’ Note: This research was supported by National Institute on Drug Abuse (NIDA) Grant P30

DA023026. Its contents are solely the responsibility of the authors and do not necessarily represent the official

views of NIDA. Please address correspondence to Chongming Yang, Social Science Research Institute, Duke

University, Durham, NC 27708; e-mail: [email protected].

122



properties, leading to the rejection of a plausible model. Three general approaches that

can be used to address these challenges are parceling, shortening, and transforming a

lengthy scale into latent score variables through preliminary analyses. Each approach has

advantages and disadvantages. The research reported here was designed to evaluate these

approaches under certain data conditions to determine which ones best reflect the true

model parameters, particularly the covariance of the exogenous constructs and partial cov-

ariances between the exogenous and endogenous constructs.

Parceling

The prevailing approach to incorporating a lengthy scale into SEM has been using the

mean or sum of the scale as an indicator of a latent construct (Cattell, 1956). Empirical

justifications for parceling include increasing reliability, achieving normality, adapting to

small sample sizes, reducing idiosyncratic influences of individual items, simplifying

interpretation, and obtaining better model fit (Bandalos & Finney, 2001). Methods for par-

celing include parceling all items into a single parcel, splitting all odd and even items into

two parcels, balancing item discrimination and difficulty across three or four parcels (e.g.,

item–construct balance; Little, Cunningham, Shahar & Widaman, 2002), randomly select-

ing a certain number of items to create three or four parcels (e.g., Kishton & Widaman,

1994; Nasser & Wisenbaker, 2003), and parceling items that have similar factor loadings

(i.e., contiguity; Cattell & Burdsal, 1975). Desirable conditions for parceling that have

been identified so far include having more than 12 items (Marsh, Hau, Balla, & Grayson,

1998) and having items that reflect a unidimensional construct (Hall, Snell, & Foust,

1999).

Psychometrically, parceling has been controversial despite the above advantages and its

prevalence in practice (Bandalos & Finney, 2001; Little et al., 2002). One concern was

that parceling results in a loss of information about the relative importance of individual

items (Marsh & O’Neill, 1984), because items are implicitly weighted equally in parcels

(Bollen & Lennox, 1991). Another concern was that parceling of ordinal scales results in

indicators with undefined values, potentially changing the original relations between the

indicators and latent variables, for instance, from nonlinear to linear relations (Coanders,

Satorra, & Saris, 1997). Parceling binary or trichotomous items could result in limited

range as opposed to the latent trait scale, thereby biasing variance and covariance para-

meters in SEM (Wright, 1999). Compared with using individual items, parceling could

underestimate the relations of the latent variables if the reliability of the scale is low

(Shevlin, Miles, & Bunting, 1997).

Empirically, the effects of parceling ordinal indicators on the covariance and partial cov-

ariance of latent constructs have been unknown under some conditions. Previous studies on

the effects of parceling on estimates of covariance parameters have typically considered

only parceling of continuous indicators with equal discrimination functions or factor load-

ings (e.g., Alhija & Wisenbaker, 2006; Bandolas, 2002; Hau & Marsh, 2004). Under these

conditions, certain parceling strategies yielded little difference in the covariance of two

latent constructs. However, most lengthy ordinal scales do not have equal item discrimina-

tion functions. In the case of continuous indicators, equal loadings imply that the latent

Yang et al. / Modeling Lengthy Measures 123



construct can explain equal variances and covariances within each parceling condition.

Thus, it is not surprising that parceling has performed well in reflecting the covariance or

correlation of latent constructs. In the case of ordinal indicators, a recent study parceled 12

ordinal items with six categories, normal distribution, and loadings varying from 0.65 to

0.85, and found no difference in predictive utility in a one-exogenous and one-endogenous

construct model (Sass & Smith, 2006). Although this finding was supportive of parceling,

such results could have arisen from the fact that ordinal indicators in the study approached

the properties of normal continuous indicators (Coanders et al., 1997) and both exogenous

and endogenous indicators were parceled identically. Given the limited conditions con-

trolled for in previous studies, it remains unclear to what extent various parceling strategies

have reflected the true covariances and partial covariances of a multiple exogenous con-

structs model when ordinal indicators have a broader range of categories (e.g., two, three,

five, and seven) and varied discrimination functions. It could be assumed that the partial

covariance parameters would not be unduly biased by parceling normally distributed multi-

ple-category indicators, because such indicators have approximately linear relations with

their latent construct; moreover, parceling as a linear transformation typically does not

change the indicators’ linearity and have performed well (Coanders et al., 1997; Ferrando,

2009). However, no assumptions could be made about the magnitude and directions of

biases of parceling indicators of two or three categories and varied measurement qualities.

Latent Scoring

Lengthy ordinal scales can be transformed through item response theory (IRT) model-

ing into latent scores for further modeling (Hambleton, Swaminathan, & Rogers, 1991).

IRT describes the probabilities that individuals will respond to a set of test items given a

particular level of ability or personality trait. Samejima (1969) proposed the following

two-parameter model (ai and bk) for ordinal scales:

P= 1

1þ expð�aiðyj � bk�1ÞÞ� 1

1þ expð�aiðyj � bkÞÞ,

where P is the probability that person j responds to a particular category k of an item at a

given trait level yj, k = 1, 2, . . . , m categories, ai = a discrimination parameter of an item,

bk is a threshold at which person j has a .50 probability for the chosen response category,

and exp stands for exponentiation. After estimating unknown ai, bk, and yj, the latent score

(yj) for each individual can be saved for subsequent modeling. Latent scores obtained

through this process are theoretically intervals with a normal distribution that best reflects a

population. Such a transformation is more likely to eliminate artifactual effects (Embretson,

1996) and detect legitimate effects (Fletcher, 2005) than using raw scores.

The two-parameter IRT model can be equivalent to confirmatory factor analysis (CFA)

with categorical indicators (Takane & de Leeuw, 1987) within the advanced latent vari-

able modeling framework of Mplus (L. K. Muthen & Muthen, 1998-2006). CFA in Mplus

typically models the probability (P) of choosing a response category (m) given the indivi-

dual’s latent trait level (Z) with a probit link (�) and a residual (d):

124 Applied Psychological Measurement



Pðm= 1|ZÞ=�½ð−tþ lZÞd�1=2�,

where thresholds (t), factor loadings (l), and factor scores (Z) are conceptually equivalent

to the threshold (b), the discrimination parameter (a), and the latent trait score (y) in the

two-parameter IRT model, respectively (Reise, Widaman, & Pugh, 1993). Although cer-

tain transformations are needed to obtain exactly the same parameter estimates of a(=ld�1=2) and b (=t/l) from a typical two-parameter IRT modeling (L. K. Muthen &

Muthen, 2006), factor scores obtained from CFA with categorical indicators are equiva-

lent to the latent scores from an IRT modeling. Because CFA has been recommended as

the first step of SEM to ensure unidimensionality and measurement quality (Anderson &

Gerbing, 1988), and factor scores are by-products of this process, it would be of practical

value to ascertain whether using factor scores to reduce test length for SEM would result

in accurate estimates of covariance and partial covariance of latent constructs. (Factor

scores and latent scores are used interchangeably hereafter.)

Sample size could be an important factor to consider in using factor scores. Item

response modeling requires large samples (N > 350) to yield accurate parameter estimates

(Reise & Yu, 1990). Similarly, small samples that are typical of social science research

might yield less accurate parameter estimates from CFA with categorical indicators.

Although the minimum sample size for CFA with categorical indicators depends on model

size, distribution of variables, strength of relationships, and proportions of missing values

(B. O. Muthen & Muthen, 2002), factor scores could be assumed to perform well in repre-

senting the true relations among latent variables with relatively large samples.

Shortening a Scale

A few indicators of a latent construct with certain content and predictive validity may be

selected from a lengthy scale to yield a shortened scale (Moore, Halle, Vandivere, & Marina,

2002), which can be easily incorporated into SEM. According to behavior domain theory

(Guttman, 1955; McDonald, 1996), an underlying construct could have an infinite number

of indicators. Therefore, a shortened scale is merely a smaller sample of all possible indica-

tors. One need not be overly concerned that the shortened scale may not be commensurate

with the large scale in its content validity, because neither is a perfect measure. Empirically,

a 6-item scale selected from 27 items with empirical data could be equivalent to the full

scale (Moore et al., 2002). It has been shown that a smaller number of continuous indicators

with moderate diversity of loadings performed as well as six indicators with high diversity

of loadings in recovering the true correlation of the two constructs (Little, Lindenberger, &

Nesselroade, 1999), although it was unclear to what extent such finding could be general-

ized to ordinal scales. Short scales can be used in large-scale surveys, in which many con-

structs are assessed, if they have desirable measurement properties and similar predictive

utility to their large source scales (e.g., Stephenson, Hoyle, Palmgreen, & Slater, 2003).

Shortening a widely used scale has raised other issues besides concerns about content

validity. One issue was how to determine the minimum number of indicators. From the

perspective of model identification, Kenny (1979) advocated that four indicators are the

best for a latent construct and ‘‘anything more is gravy.’’ Shortening a large scale to fewer




than four items might undermine its content validity, although its predictive validity could

be retained (Little et al., 1999). Another issue was which strategies to apply to reduce the

scale length. One could rely on the magnitude of correlations between the endogenous

and exogenous construct indicators, as in the study by Moore et al. (2002), or on the width

of the behavioral domain the indicators reflect (Little et al., 1999), or on discrimination

functions (factor loadings) because measurement errors of the exogenous constructs

attenuate associations with other constructs (Bollen & Lennox, 1991). Based on these

findings and considerations, shortening an ordinal scale to four or six indicators appeared

reasonable. Therefore, in this study, four or six items with different measurement proper-

ties were selected from a hypothetical lengthy ordinal scale to examine their efficacy in

recovering population parameters.

In sum, it is not known how well various approaches to incorporating lengthy ordinal scales

into SEM would reflect the true covariance and partial covariance of latent constructs under

conditions of varied discrimination functions or factor loadings, different numbers of response

category, distributions, sample sizes, and full SEM models.. An efficient method to gain such

knowledge was to simulate all these conditions with artificial data generated from a known

population model, so that the various approaches could be applied to these artificial data and

evaluated against the population model (Paxton, Curran, Bollen, Kirby, & Chen, 2001). Find-

ings from these simulations can also be applied to empirical data to solve practical issues.

Design and Analysis

Population Model, Parameters, Scales, and Data

The hypothetical population model was set to have two exogenous constructs (F1 and

F2) and one endogenous construct (F3), as shown in Figure 1. This model allows estima-

tion of the covariance between the exogenous constructs and the effects of the exogenous

constructs on the endogenous construct (partial covariances), as has been used in previous

simulation studies (e.g., Bandalos, 2002; B. O. Muthen, 1984). Each exogenous construct

was measured by 14 items (X1-X14 and X15-X28), and the endogenous construct was

measured by four items (Y1-Y4). The factor loadings and structural coefficients are also

shown in Figure 1. The population parameters estimated included the partial covariance

from F1 to F3 (g1 = 0.50), the partial covariance from F2 to F3 (g2 = 0.40), and the covar-

iance between F1 and F2 (f= 0.20). The variances of the exogenous constructs and dis-

turbance (d) of the endogenous construct were assigned at 0.50 and 0.60, respectively.

Multivariate continuous data were first generated from the population model and then

categorized into two-, three-, five-, and seven-category ordinal variables. The cut points

(thresholds) of the underlying dimension, which are equivalent to z values of a normally

distributed random variable, were selected to specify the proportion of each response cate-

gory and thereby vary the distributions of the categorized variables. The threshold and dis-

tribution of each response category are listed in Table 1. Sample sizes considered were

100, 350, and 600, which reflected the size of typical clinical studies and medium to large

intervention studies. Using the Mplus program (Version 4), 100 data sets were generated

for each of the 24 conditions (2 levels of distribution× 4 levels of response category× 3




levels of sample size). For the sake of comparisons, various approaches to modeling the

lengthy scales (parceling, latent scoring, and shortening) were viewed as repeated treat-

ments of the same data sets that had been generated and categorized for different condi-

tions. The acceptable range of average bias in this study was set at ±0.14, rounding to 0.1

in absolute value.

Figure 1

Population Model

F2

x28

x27

x26

x25

x24

x23

x22

x21

x20

x19

x18

x17

x16

x15

.40

.43

.46

.50

.53

.56

.60

.63

.66

.70

.73

.76

.80

.83

F1

x14

x13

x12

x11

x10

x9

x8

x7

x6

x5

x4

x3

x2

x1e1

e2

e3

e4

e5

e6

e7

e8

e9

e10

e11

e12

e13

e14

e15

e16

e17

e18

e19

e20

e21

e22

e23

e24

e25

e26

e27

e28

.43

.46

.53

.56

.60

.63

.66

.70

.73

.76

.80

.83

.86

.90

F3

y1 e29

y2 e30

y3 e31

y4 e32

.70

.70

.70

.70

.50

.40

d

.20




Tab

le1

Pop

ula

tion

Ind

icato

rL

oad

ings,

Item

sC

hose

nfo

rE

ach

Parc

el,

an

dIt

emL

oad

ings

of

Sh

ort

ened

Sca

les

Po

pu

lati

on

Par

cels

Sh

ort

ened

Sca

leo

fF

ou

rIn

dic

ato

rsS

ho

rten

edS

cale

of

Six

Ind

icat

ors

Ind

icat

or

Lo

adin

g

Od

d-E

ven

Sp

lit

Bal

ance

Ran

do

m

Fo

ur

Ran

do

m

Six

Co

nti

gu

ou

s

Fo

ur

Co

nti

gu

ou

s

Six

Div

ersi

tyH

igh

Med

ium

Lo

wD

iver

sity

Hig

hM

ediu

mL

ow

X1

.90

AA

AA

AA

.90

.90

.90

.90

X2

.86

BB

AB

AA

.86

.86

X3

.83

AC

CC

AA

.83

.83

X4

.80

BC

DD

AB

.80

.80

.80

X5

.76

AB

BE

BB

.76

.76

.76

.76

X6

.73

BA

BE

BB

.73

.73

.73

X7

.70

AA

DF

BC

.70

.70

.70

X8

.66

BB

BF

BC

.66

.66

X9

.63

AC

AB

CD

.63

.63

.63

.63

X1

0.6

0B

CB

CC

D.6

0.6

0.6

0.6

0

X1

1.5

6A

BA

AC

E.5

6.5

6.5

6

X1

2.5

3B

AC

DD

E.5

3.5

3

X1

3.4

6A

CA

FD

F.4

6

X1

4.4

3B

BD

ED

F.4

3.4

3.4

3

X1

5.8

3C

DD

GE

G.8

3.8

3.8

3.8

3

X1

6.8

0D

EB

HE

G.8

0.8

0

X1

7.7

6C

FC

HE

G.7

6.7

6

X1

8.7

3D

FA

GE

H.7

3.7

3.7

3

X1

9.7

0C

ED

IF

H.7

0.7

0.7

0

X2

0.6

6D

DA

JF

H.6

6.6

6.6

6.6

6

X2

1.6

3C

DB

KF

I.6

3.6

3.6

3

X2

2.6

0D

EC

KF

I.6

0.6

0

X2

3.5

6C

FA

JG

J.5

6.5

6.5

6.5

6

X2

4.5

3D

FA

LG

J.5

3.5

3.5

3

X2

5.5

0C

ED

IG

K.5

0.5

0.5

0.5

0

X2

6.4

6D

DB

KH

K.4

6.4

6

X2

7.4

3C

FC

HH

L.4

3

X2

8.4

0D

EB

LH

L.4

04

0.4

0

No

te:

Th

esa

me

lett

ers

ina

colu

mn

des

ign

ate

ind

icat

ors

for

the

sam

ep

arce

l.

128



Parcel and Latent Score Creation and ItemSelection for Shortened Scales

Parcels were created from indicators of only the exogenous constructs in this study. A

single parcel was created for each of F1 and F2 by taking the means of all 14 indicators of

F1 and F2, respectively. Two odd–even parcels were created for each of F1 and F2 by tak-

ing the means of the odd- and even-numbered indicators. Three item-to-construct balance

parcels were created by selecting an item for each parcel in the order of discrimination func-

tions and repeating the selection process in a reversed order, as indicated by letters in the

Balance column in Table 1. Four random parcels were created by randomly selecting indi-

cators without replacement. Four and six adjacent-loading parcels were created by taking

the means of indicators that had contiguous loadings. The final indicators for each parcel are

listed in Table 1 and denoted by capital letters. Latent scores for the two exogenous con-

structs (F1 and F2) were obtained by fitting a measurement model of the three constructs of

the population model to the generated categorical data. The indicators were specified as

categorical and the latent (factor) scores produced from this process were saved in the data

files. Four shortened scales were created by selecting four or six individual indicators of

each of F1 and F2. The factor loadings of these exogenous construct indicators were chosen

to be diverse, high, moderate, and low, as listed in Table 1.

Model Estimation

In the case of single parcels or latent scores, the population model was altered to two

single parcels or two latent scores as the observed exogenous variables (in place of F1 and

F2) predicting the endogenous construct (F3). With two or more parcels, the population

model was altered to use parcels as indicators of F1 and F2. Means and standard devia-

tions of the parameter estimates and the number of converged solutions were recorded in

this process. Mean biases (parameter estimate – population parameter) were reported or

calculated for subsequent analyses (Paxton et al., 2001).

The performances of various estimation methods in SEM have been compared for mod-

els with categorical indicators. The number of categories typically selected has been two,

three, five, or seven. Under various distributions, the weighted least squares estimator with

degrees of freedom adjusted for means and variances (WLSMV) in Mplus has proven to

be fairly robust to various categorical indicators with large sample sizes (B. O. Muthen &

Kaplan, 1985, 1992). To eliminate any effects from the estimation method, indicators of

the endogenous latent variables were specified as categorical and all the models were esti-

mated with the WLSMV estimator.

Empirical Data

The empirical data were adopted from the Child Development Project, an ongoing long-

itudinal study of children’s social and emotional development (Lansford et al., 2006). A

total of 585 families were recruited from two cohorts in consecutive years, 1987 and

1988, from Nashville and Knoxville, Tennessee, and Bloomington, Indiana. Data collec-

tion began the year before the children entered kindergarten and data have been collected




annually ever since. A linear growth model of aggression was chosen for the empirical

example, because it could provide an opportunity to examine the effects of various

approaches on the mean levels of latent constructs. Aggression was measured from kin-

dergarten through adolescence using the Child Behavior Checklist (Achenbach, 1978),

with less than 30% attrition at the final measurement of aggression. Our main interest

involved examining whether findings from the simulation study could be applied to resol-

ving any practical issues.

Results

Based on converged solutions, the average biases of each approach in the three para-

meters (g1, g2, and f) for each data condition were recorded and are depicted in Table 2.

The standard deviations and numbers of converged solutions for each condition are listed

in Table 3.

Parceling

Data in Table 2 reveal a clear pattern and three key findings: First, any parceling of

ordinal indicators of two or three response categories overestimated the partial covar-

iances but underestimated the covariance of the exogenous constructs. Second, parceling

of indicators with seven response categories underestimated the partial covariances (g1

and g2) by 0.10 to 0.24 but more accurately reflected the covariance of the two exogenous

constructs (f). Third, odd–even split, random parceling (4 or 6 parcels), and item–

construct-balance parceling of indicators of five response categories biased the estimates

of the effects (g1 and g2) maximally by 0.1 in absolute value, and biased the correlation of

the two constructs maximally by 0.13 in absolute value. Parceling of indicators with con-

tiguous factor loadings attenuated the partial covariances (g1 and g2) by more than 0.1,

which appeared to be slightly less advantageous than other parceling approaches. Thus,

parceling (odd–even split, random, and item-construct balance) was most effective when

categorical indicators had five response categories and parcels were further used as indica-

tors of the latent constructs.

Latent Scoring

An analysis of variance confirmed that the number of response categories produced

some difference in g1, F(3, 14)= 40.96, p< .01, and g2, F(3, 14)= 18.99, p< .01; type of

distribution produced some difference in g1, F(1, 14)= 16.34, p< .01; and the two effects

were dependent on each other in g1, evidenced by a significant interaction effect between

the number of response categories and type of distribution, F(3, 14)= 10.27, p< .01. As

shown in Table 2, latent scoring biased the two direct effect estimates maximally by 0.08

in absolute value in all the data conditions, except in situations with binary indicators and

small samples or extremely skewed binary indicators. The average bias in the covariance

of F1 and F2 (f) was consistently between 0.06 and 0.11 across all the conditions. Thus,




Tab

le2

Ab

solu

teB

ias

inP

op

ula

tion

Mod

elP

ara

met

ers

of

Each

Ap

pro

ach

toL

ength

yS

cale

sU

nd

erD

iffe

ren

tD

ata

Con

dit

ion

s

Dat

aS

etC

on

dit

ion

s

Ap

pro

ach

es

Lat

ent

Sco

res

Sin

gle

Par

cel

Ev

en-S

pli

tP

arce

lsIt

em−C

on

stru

ctB

alan

ce

Cat

.D

istr

ibu

tio

n(T

hre

sho

lds)

Ng 1

g 2f

g 1g 2

fg 1

g 2f

g 1g 2

f

Tw

o7

0,3

0(0

.52

44

)1

00

.06

.15

−.0

9.4

7.4

2−.

19

0.9

70

.76

−.1

90

.91

0.9

2−.

19

35

0.0

4.0

8−.

10

.42

.39

−.1

90

.82

0.7

3−.

19

0.7

50

.73

−.1

9

60

0.0

5.0

6−.

11

.41

.37

−.1

90

.81

0.6

8−.

19

0.7

60

.68

−.1

9

85

,1

5(1

.03

6)

10

0.1

7.1

9−.

10

.63

.55

−.2

01

.26

1.2

5−.

19

1.2

61

.21

−.1

9

35

0.1

3.1

4−.

11

.66

.56

−.2

01

.25

1.1

1−.

20

1.2

11

.09

−.1

9

60

0.1

6.1

1−.

11

.68

.58

−.2

01

.28

1.1

3−.

20

1.2

81

.09

−.1

9

Th

ree

60

,3

0,1

0(0

.25

53

,1

.28

2)

10

0.0

1.0

2−.

08

.10

.10

−.1

70

.29

0.2

6−.

17

0.2

90

.28

−.1

7

35

0.0

3.0

7−.

11

.09

.11

−.1

80

.29

0.3

0−.

18

0.2

60

.28

−.1

7

60

0.0

3.0

5−.

10

.09

.11

−.1

80

.28

0.2

8−.

17

0.2

70

.28

−.1

7

70

,2

5,5

(0.5

24

4,1

.64

5)

10

0.0

2.0

3−.

06

.10

.16

−.1

70

.36

0.3

8−.

17

0.3

10

.41

−.1

7

35

0.0

5.0

8−.

11

.19

.19

−.1

80

.45

0.4

2−.

18

0.4

10

.43

−.1

8

60

0.0

3.0

6−.

10

.18

.19

−.1

80

.42

0.4

2−.

18

0.4

00

.40

−.1

8

Fiv

e1

02

54

02

51

0

(−1

.28

2,−0

.38

53

,

0.6

74

5,

1.2

82

)

10

0.0

5.0

3−.

10

−.1

4−.

12

−.1

2−0

.03

−0.0

4−.

12

−0.0

3−0

.04

−.1

1

35

0−.

02

.02

−.0

9−.

15

−.1

1−.

11

−0.0

7−0

.03

−.1

1−0

.08

−0.0

3−.

11

60

0−.

01

.02

−.1

0−.

15

−.1

1−.

11

−0.0

7−0

.03

−.1

1−0

.08

−0.0

4−.

11

25

,4

0,2

0,1

0,

5

(−0

.67

45

,0

.38

53

,

1.0

36

,1

.64

5)

10

0−.

02

.04

−.0

9−.

16

−.1

1−.

12

−0.0

6−0

.02

−.1

1−0

.08

−0.0

1−.

11

35

0−.

01

.02

−.1

0−.

16

−.1

1−.

12

−0.0

7−0

.01

−.1

2−0

.08

−0.0

3−.

11

60

0.0

0.0

1−.

10

−.1

5−.

11

−.1

2−0

.06

−0.0

2−.

11

−0.0

7−0

.04

−.1

1

Sev

en5

,1

0,2

0,3

0,

20

,1

0,5

(−1

.64

5,−1

.03

6,−0

.38

53

,

0.3

85

3,

1.0

36

,1

.64

5)

10

0−.

02

.04

−.0

9−.

26

−.1

8−.

04

−0.1

9−0

.12

−.0

3−0

.14

−0.1

2−.

03

35

0−.

02

.02

−.0

9−.

24

−.1

8−.

04

−0.1

8−0

.13

−.0

3−0

.20

−0.1

3−.

02

60

0−.

01

.01

−.1

0−.

24

−.1

8−.

04

−0.1

8−0

.13

−.0

3−0

.19

−0.1

3−.

02

14

,3

4,2

2,1

4,

9,5

,2

(−1

.08

0,−0

.05

15

,0

.52

44

,

0.9

94

5,

1.4

76

,2

.05

4)

10

0.0

3−.

02

−.0

9−.

25

−.2

1−.

05

−0.1

6−0

.16

−.0

4−0

.17

−0.1

5−.

04

35

0.0

0.0

0−.

10

−.2

4−.

19

−.0

5−0

.18

−0.1

3−.

04

−0.1

9−0

.14

−.0

3

60

0.0

0.0

2−.

10

−.2

4−.

19

−.0

5−0

.18

−0.1

3−.

04

−0.1

8−0

.13

−.0

3

No

te:

Cat

.=n

um

ber

of

cate

go

ries

.

131



Tab

le2

(con

tin

ued

)

Ap

pro

ach

es

Ran

do

miz

edto

Fo

ur

Par

cels

Ran

do

miz

edto

Six

Par

cels

Fo

ur

Par

cels

of

Co

nti

gu

ou

sL

oad

ing

s

Six

Par

cels

of

Co

nti

gu

ou

sL

oad

ing

s

Fo

ur

Item

so

f

Div

erse

Lo

adin

gs

Fo

ur

Item

so

f

Hig

hL

oad

ing

s

g 1g 2

fg 1

g 2f

g 1g 2

fg 1

g 2f

g 1g 2

fg 1

g 21

f

1.2

60

.97

−.1

91

.04

0.8

7−.

19

0.7

1.6

7−.

19

.76

0.6

5−.

19

.20

.27

−.0

8.0

2−.

01

−.0

9

0.9

70

.76

−.1

90

.62

1.1

3−.

19

0.5

9.5

6−.

19

.58

0.5

6−.

19

−.1

2.0

3−.

08

−.0

6−.

08

−.0

9

0.9

10

.74

−.1

90

.95

0.6

6−.

19

0.6

0.5

2−.

19

.57

−0.3

0−.

19

−.0

9−.

06

−.0

8−.

11

−.0

7−.

09

1.4

41

.61

−.2

01

.69

1.2

5−.

20

1.0

8.9

5−.

19

.95

1.0

7−.

19

.08

.19

−.0

9.0

2−.

01

−.0

8

1.4

61

.16

−.2

01

.49

1.0

8−.

20

0.9

9.8

2−.

19

.95

0.7

9−.

19

−.0

7.0

3−.

09

−.0

6−.

01

−.0

9

1.5

11

.17

−.2

01

.57

1.0

5−.

20

0.9

7.8

1−.

19

.95

0.7

9−.

19

−.0

5−.

04

−.1

0−.

08

−.0

5−.

09

0.4

10

.35

−.1

80

.45

0.2

7−.

17

0.1

9.1

6−.

16

.15

0.1

6−.

16

.05

−.1

0−.

07

−.0

7.0

0−.

06

0.3

80

.30

−.1

80

.40

0.2

6−.

18

0.1

6.1

8−.

16

.15

0.1

7−.

16

−.1

1.0

0−.

10

−.1

0−.

03

−.1

0

0.3

90

.30

−.1

80

.39

0.2

5−.

18

0.1

5.1

7−.

16

.14

0.1

5−.

16

−.1

0−.

04

−.0

9−.

09

−.0

7−.

09

0.4

90

.36

−.1

80

.49

0.4

1−.

17

0.2

1.2

4−.

15

.21

0.2

4−.

15

.31

−.5

6−.

04

−.2

2.2

0−.

04

0.5

90

.45

−.1

90

.59

0.4

1−.

18

0.2

8.2

8−.

17

.26

0.2

7−.

17

−.0

7.0

0−.

10

−.0

9−.

02

−.0

9

0.5

40

.45

−.1

80

.55

0.4

1−.

18

0.2

5.2

8−.

17

.24

0.2

6−.

17

−.1

1−.

05

−.0

9−.

11

−.0

8−.

09

0.0

2−0

.01

−.1

20

.06

−0.0

4−.

12

−0.1

0−.

08

−.0

8−.

10

−0.0

9−.

07

.01

.21

−.1

1−.

04

−.0

3−.

09

−0.0

2−0

.02

−.1

20

.00

−0.0

5−.

12

−0.1

4−.

09

−.0

7−.

15

−0.1

0−.

06

−.1

0−.

04

−.0

9−.

10

−.0

5−.

09

−0.0

2−0

.02

−.1

2−0

.01

−0.0

5−.

12

−0.1

4−.

10

−.0

7−.

15

−0.1

1−.

06

−.0

9−.

07

−.0

9−.

11

−.0

5−.

09

−0.0

20

.01

−.1

20

.02

0.0

0−.

12

−0.1

3−.

06

−.0

7−.

13

−0.0

7−.

07

−.0

5−.

09

−.0

8−.

09

−.0

3−.

08

−0.0

2−0

.02

−.1

30

.00

−0.0

4−.

12

−0.1

5−.

09

−.0

8−.

15

−0.1

0−.

07

−.0

8−.

07

−.0

9−.

10

−.0

6−.

09

−0.0

1−0

.02

−.1

20

.00

−0.0

4−.

12

−0.1

3−.

09

−.0

7−.

14

−0.1

0−.

07

−.1

0−.

05

−.0

9−.

08

−.0

7−.

09

−0.1

6−0

.10

−.0

5−0

.14

−0.1

1−.

05

−0.2

4−.

16

.04

−.2

4−0

.16

.04

−.0

5.0

5−.

09

−.1

3.0

3−.

08

−0.1

5−0

.11

−.0

5−0

.14

−0.1

3−.

05

−0.2

3−.

17

.04

−.2

4−0

.18

.06

−.0

8−.

07

−.0

8−.

10

−.0

6−.

09

−0.1

5−0

.13

−.0

5−0

.14

−0.1

4−.

04

−0.2

4−.

18

.05

−.2

4−0

.18

.06

−.1

0−.

06

−.0

9−.

10

−.0

7−.

09

−0.1

3−0

.14

−.0

6−0

.10

−0.1

5−.

05

−0.2

2−.

19

.04

−.2

2−0

.19

.04

−.0

2−.

05

−.0

8−.

02

−.1

1−.

08

−0.1

4−0

.13

−.0

6−0

.14

−0.1

4−.

05

−0.2

3−.

18

.03

−.2

4−0

.18

.05

−.0

9−.

03

−.0

9−.

08

−.0

8−.

09

−0.1

4−0

.13

−.0

6−0

.13

−0.1

4−.

05

−0.2

3−.

18

.04

−.2

4−0

.18

.05

−.0

8−.

06

−.0

9−.

09

−.0

6−.

09

132



Tab

le2

(con

tin

ued

)

Ap

pro

ach

es

Fo

ur

Item

so

f

Med

ium

Lo

adin

gs

Fo

ur

Item

so

f

Lo

wL

oad

ing

s

Six

Item

so

f

Div

erse

Lo

adin

gs

Six

Item

so

f

Hig

hL

oad

ing

s

Six

Item

so

f

Med

ium

Lo

adin

gs

Six

Item

so

f

Lo

wL

oad

ing

s

g 1g 2

fg 1

g 2f

g 1g 2

fg 1

g 2f

g 1g 2

fg 1

g 2f

.11

.13

−.1

1−.

05

.62

−.1

2−.

11

.08

−.0

8−.

02

.02

−.0

7.2

6.2

1−.

10

0.3

80

.49

−.1

2

−.0

8.0

8−.

11

.27

.26

−.1

5−.

09

−.0

5−.

08

−.1

0−.

07

−.0

9−.

06

.05

−.1

20

.08

0.2

0−.

14

−.0

2.0

0−.

12

.12

.12

−.1

5−.

10

−.0

7−.

09

−.1

0−.

08

−.0

9−.

04

−.0

2−.

12

0.0

30

.07

−.1

3

.32

−.1

8−.

08

.05

.45

−.1

1.1

2.2

2−.

08

−.0

4.1

3−.

07

.16

.17

−.0

8−0

.03

0.1

3−.

12

.04

.06

−.1

2.3

2.1

9−.

14

−.0

7−.

01

−.0

9−.

08

−.0

4−.

09

.01

.06

−.1

20

.11

0.4

5−.

14

−.0

6.0

8−.

11

.33

.15

−.1

5−.

06

−.0

5−.

09

−.0

9−.

06

−.0

9−.

03

.03

−.1

10

.18

0.0

9−.

14

.05

.26

−.0

9.4

8−.

04

−.1

3−.

03

−.0

6−.

06

−.0

8−.

07

−.0

6.0

2.1

2−.

09

0.1

50

.16

−.1

2

−.0

2.0

7−.

12

.25

.05

−.1

4−.

10

−.0

5−.

10

−.1

0−.

05

−.1

0−.

02

.02

−.1

20

.10

0.0

6−.

14

−.0

4−.

02

−.1

2.1

5.1

5−.

15

−.0

9−.

06

−.0

9−.

10

−.0

7−.

09

−.0

4−.

02

−.1

20

.09

0.0

9−.

14

.02

.09

−.0

7.1

3.2

3−.

11

−.1

2.0

3−.

03

−.1

5.1

3−.

03

.05

.01

−.0

70

.77

2.0

8−.

11

−.0

2.0

4−.

12

.25

.13

−.1

5−.

09

−.0

4−.

09

−.0

9−.

04

−.0

9−.

03

.01

−.1

20

.05

0.1

8−.

14

−.0

4.0

2−.

12

.10

.16

−.1

5−.

12

−.0

5−.

09

−.1

0−.

08

−.0

9−.

05

.01

−.1

20

.02

0.1

1−.

14

.13

.00

−.1

2.4

2.2

5−.

14

.09

−.0

8−.

10

−.0

4−.

05

−.0

9.0

8.0

1−.

12

0.2

60

.15

−.1

3

−.0

3−.

01

−.1

1.0

8.1

2−.

14

−.0

9−.

07

−.0

9−.

10

−.0

5−.

09

−.0

4−.

01

−.1

10

.03

0.1

0−.

14

−.0

3−.

04

−.1

1.1

5.0

6−.

15

−.1

0−.

06

−.0

9−.

11

−.0

6−.

09

−.0

3−.

04

−.1

10

.04

0.0

5−.

14

.06

.08

−.1

1.1

5.4

5−.

13

−.0

9−.

06

−.0

7−.

09

−.0

4−.

08

.01

.08

−.1

00

.15

−0.0

1−.

12

−.0

7.0

3−.

12

.12

.14

−.1

5−.

10

−.0

6−.

09

−.1

0−.

07

−.0

9−.

08

.01

−.1

10

.07

0.1

0−.

14

−.0

5−.

01

−.1

2.1

2.0

8−.

14

−.1

0−.

05

−.0

9−.

09

−.0

8−.

09

−.0

4.0

0−.

12

0.0

50

.10

−.1

4

−.0

3.0

4−.

11

.10

.30

−.1

4−.

06

−.0

4−.

08

−.1

2.0

4−.

09

−.0

5.2

1−.

11

−1.0

70

.64

−.1

3

−.0

4.0

2−.

11

.14

.10

−.1

4−.

10

−.0

6−.

08

−.1

0−.

06

−.0

9−.

03

.00

−.1

10

.04

0.0

8−.

13

−.0

5−.

01

−.1

1.1

0.0

8−.

14

−.1

0−.

06

−.0

9−.

10

−.0

7−.

09

−.0

6.0

1−.

12

0.0

40

.07

−.1

4

.04

.04

−.1

2.3

1.2

4−.

13

−.0

4−.

13

−.0

8−.

04

−.0

9−.

08

.07

−.0

4−.

12

0.1

30

.07

−.1

3

−.0

4.0

1−.

11

.13

.10

−.1

5−.

10

−.0

7−.

09

−.1

0−.

07

−.0

9−.

03

−.0

1−.

11

0.0

90

.09

−.1

4

−.0

3−.

02

−.1

1.1

2.0

6−.

14

−.0

9−.

06

−.0

9−.

10

−.0

6−.

09

−.0

4−.

02

−.1

10

.03

0.0

6−.

14

133



Tab

le3

Sta

nd

ard

Dev

iati

on

of

Para

met

erE

stim

ate

sU

nd

erV

ari

ou

sD

ata

Con

dit

ion

san

dA

pp

roach

esan

dC

orr

esp

on

din

gS

am

ple

Siz

es(n

)

Ap

pro

ach

es

Dat

aS

etC

on

dit

ion

sL

aten

tS

core

sS

ing

leP

arce

lsE

ven

-Sp

lit

Par

cels

Item

-Co

nst

ruct

Bal

ance

Cat

.D

istr

ibu

tio

nN

g 1g 2

fn

g 1g 2

fn

g 1g 2

fn

g 1g 2

fn

Tw

o7

0,3

01

00

0.6

50

.69

.06

10

0.4

0.3

8.0

01

00

0.8

50

.82

.00

99

0.8

90

.92

.00

10

0

35

00

.20

0.2

6.0

31

00

.19

.20

.00

10

00

.35

0.3

9.0

01

00

0.3

30

.39

.00

10

0

60

00

.16

0.1

6.0

21

00

.15

.14

.00

10

00

.28

0.2

8.0

01

00

0.2

80

.27

.00

10

0

85

,1

51

00

1.0

20

.10

.07

10

0.4

3.5

4.0

01

00

1.1

91

.62

.00

10

01

.19

1.6

1.0

01

00

35

00

.31

0.3

2.0

31

00

.28

.26

.00

10

00

.65

0.6

1.0

01

00

0.5

60

.62

.00

10

0

60

00

.25

0.2

4.0

21

00

.21

.22

.00

10

00

.43

0.4

9.0

01

00

0.5

10

.50

.00

10

0

Th

ree

20

,6

0,2

01

00

0.4

90

.62

.05

10

0.2

3.2

6.0

11

00

0.4

60

.48

.01

10

00

.46

0.5

7.0

11

00

35

00

.18

0.1

9.0

21

00

.13

.12

.00

10

00

.22

0.2

1.0

11

00

0.2

20

.21

.01

10

0

60

00

.14

0.1

6.0

21

00

.09

.10

.00

10

00

.15

0.1

9.0

01

00

0.1

60

.18

.01

10

0

70

,2

0,1

01

00

0.7

81

.37

.06

10

0.2

6.3

0.0

11

00

0.6

40

.80

.01

98

0.5

80

.77

.01

10

0

35

00

.20

0.2

2.0

21

00

.15

.16

.00

10

00

.29

0.3

0.0

01

00

0.2

80

.31

.00

10

0

60

00

.16

0.1

5.0

21

00

.12

.11

.00

10

00

.20

0.2

0.0

01

00

0.2

20

.20

.00

10

0

Fiv

e5

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5,6

0,1

5,

51

00

0.3

30

.32

.04

10

0.1

4.1

2.0

31

00

0.2

10

.20

.03

10

00

.23

0.2

0.0

41

00

35

00

.15

0.1

5.0

21

00

.07

.07

.02

10

00

.11

0.1

0.0

21

00

0.1

00

.10

.02

10

0

60

00

.11

0.1

3.0

11

00

.06

.06

.01

10

00

.08

0.0

9.0

11

00

0.0

80

.09

.01

10

0

5,1

0,2

0,4

0,

25

10

00

.29

0.4

1.0

51

00

.12

.15

.03

10

00

.22

0.2

2.0

49

90

.19

0.2

5.0

41

00

35

00

.15

0.1

6.0

31

00

.07

.07

.02

10

00

.11

0.1

2.0

21

00

0.1

10

.12

.02

10

0

60

00

.11

0.1

3.0

21

00

.05

.06

.01

10

00

.08

0.1

0.0

21

00

0.0

80

.09

.02

10

0

Sev

en5

,1

0,2

0,3

0,

20

,1

0,5

10

00

.37

0.4

1.0

41

00

.11

.12

.05

10

00

.17

0.2

0.0

61

00

0.1

80

.18

.07

10

0

35

00

.15

0.1

7.0

31

00

.06

.06

.03

10

00

.09

0.1

0.0

31

00

0.0

80

.09

.04

10

0

60

00

.11

0.1

0.0

21

00

.04

.04

.02

10

00

.06

0.0

6.0

31

00

0.0

60

.06

.03

10

0

3,7

,1

2,1

8,

35

,2

0,5

10

00

.42

0.3

5.0

41

00

.10

.11

.05

10

00

.18

0.1

9.0

71

00

0.1

70

.18

.07

10

0

35

00

.16

0.1

8.0

21

00

.06

.06

.03

10

00

.09

0.1

0.0

41

00

0.0

80

.10

.04

10

0

60

00

.12

0.1

5.0

21

00

.04

.05

.02

10

00

.07

0.0

8.0

31

00

0.0

60

.08

.03

10

0

No

te:

Cat

.=n

um

ber

of

cate

go

ries

.

134



Tab

le3

(con

tin

ued

)

Ap

pro

ach

es

Ran

do

miz

edto

Fo

ur

Par

cels

Ran

do

miz

edto

Six

Par

cels

Fo

ur

Par

cels

of

Ad

jace

nt

Lo

adin

gs

Six

Par

cels

of

Ad

jace

nt

Lo

adin

gs

Fo

ur

Item

so

f

Div

erse

Lo

adin

gs

Fo

ur

Item

so

f

Hig

hL

oad

ing

s

g 1g 2

fn

g 1g 2

fn

g 1g 2

fn

g 1g 2

fn

g 1g 2

fn

g 1g 2

fn

1.3

91

.21

.00

10

00

.90

1.0

3.0

01

00

0.7

60

.74

.01

10

00

.08

0.8

1.0

11

00

1.0

22

.19

.11

82

.66

0.9

7.0

99

6

0.4

10

.40

.00

10

00

.46

0.3

9.0

01

00

0.3

10

.34

.00

10

00

.30

0.3

4.0

01

00

0.1

80

.60

.05

98

.18

0.1

8.0

51

00

0.3

20

.30

.00

10

00

.31

0.2

6.0

01

00

0.2

50

.23

.00

10

00

.25

0.2

2.0

01

00

0.1

90

.18

.05

10

0.1

40

.13

.04

10

0

1.2

92

.47

.00

10

01

.56

1.5

8.0

09

81

.57

1.6

8.0

09

91

.20

1.6

5.0

01

00

0.6

51

.87

.14

84

.65

1.3

2.1

48

3

0.6

40

.59

.00

10

00

.71

0.6

4.0

01

00

0.4

80

.50

.00

10

00

.49

0.5

0.0

01

00

0.4

90

.56

.06

98

.26

0.3

5.0

51

00

0.4

90

.51

.00

10

00

.56

0.4

9.0

01

00

0.3

50

.39

.00

10

00

.37

0.3

8.0

01

00

0.2

40

.24

.11

10

0.1

80

.17

.04

10

0

0.5

80

.65

.01

10

00

.59

0.5

1.0

11

00

0.4

10

.44

.02

10

00

.39

0.4

4.0

21

00

1.0

20

.76

.10

86

.43

0.9

6.0

89

8

0.2

40

.21

.01

10

00

.24

0.2

2.0

11

00

0.1

80

.19

.01

10

00

.18

0.1

9.0

11

00

0.2

00

.25

.04

10

0.1

70

.18

.03

10

0

0.1

80

.18

.00

10

00

.18

0.1

7.0

11

00

0.1

20

.15

.01

10

00

.13

0.1

5.0

11

00

0.1

20

.18

.03

10

0.1

30

.13

.03

10

0

0.7

40

.91

.01

10

00

.78

0.7

7.0

11

00

0.5

00

.70

.02

10

00

.52

0.7

3.0

21

00

3.0

53

.48

.13

81

.82

1.0

0.0

99

6

0.3

40

.31

.00

10

00

.33

0.3

3.0

01

00

0.2

20

.25

.01

10

00

.22

0.2

5.0

11

00

0.3

40

.33

.05

99

.17

0.2

5.0

41

00

0.2

50

.22

.00

10

00

.26

0.2

2.0

01

00

0.1

80

.17

.01

10

00

.18

0.1

7.0

11

00

0.1

50

.17

.04

10

0.1

10

.13

.32

10

0

0.2

40

.26

.04

10

00

.28

0.2

1.0

41

00

0.1

90

.18

.05

10

00

.19

0.1

7.0

51

00

0.6

41

.48

.07

96

.04

0.3

6.0

61

00

0.1

20

.11

.02

10

00

.13

0.1

0.0

21

00

0.0

90

.09

.03

10

00

.09

0.0

9.0

31

00

0.1

70

.20

.04

10

0.1

30

.13

.03

10

0

0.0

90

.09

.01

10

00

.09

0.0

9.0

11

00

0.0

70

.07

.02

10

00

.07

0.0

7.0

21

00

0.1

40

.13

.03

10

0.0

90

.11

.02

10

0

0.2

10

.27

.03

10

00

.27

0.2

8.0

41

00

0.1

60

.22

.05

10

00

.16

0.2

2.0

51

00

0.4

80

.86

.08

96

.29

0.4

0.0

79

9

0.1

20

.12

.02

10

00

.13

0.1

2.0

21

00

0.0

90

.10

.03

10

00

.09

0.1

0.0

31

00

0.1

80

.17

.04

10

0.1

50

.13

.03

10

0

0.0

90

.10

.01

10

00

.10

0.1

0.0

11

00

0.0

70

.08

.02

10

00

.06

0.0

8.0

21

00

0.1

20

.15

.03

10

0.1

00

.10

.03

10

0

0.1

90

.20

.06

10

00

.21

0.2

2.0

61

00

0.1

40

.18

.09

10

00

.15

0.1

9.0

91

00

0.5

30

.59

.07

97

.30

0.3

9.0

69

8

0.1

00

.10

.03

10

00

.10

0.1

0.0

31

00

0.0

80

.08

.05

10

00

.08

0.0

8.0

51

00

0.1

50

.21

.04

10

0.1

20

.12

.04

10

0

0.0

70

.06

.03

10

00

.07

0.0

6.0

31

00

0.0

50

.05

.04

10

00

.05

0.0

5.0

51

00

0.1

10

.13

.03

10

0.1

00

.09

.02

10

0

0.1

80

.20

.06

10

00

.06

0.1

3.1

51

00

0.1

40

.17

.09

10

00

.15

0.1

7.0

91

00

0.5

30

.76

.08

96

.37

0.3

0.0

69

9

0.1

00

.11

.03

10

00

.10

0.1

0.0

41

00

0.0

70

.08

.05

10

00

.07

0.0

8.0

51

00

0.1

70

.21

.04

10

0.1

50

.13

.03

10

0

0.0

80

.07

.02

10

00

.08

0.0

8.0

31

00

0.0

60

.07

.04

10

00

.06

0.0

6.0

41

00

0.1

40

.16

.03

10

0.1

00

.11

.02

10

0

135



Tab

le3

(con

tin

ued

)

Ap

pro

ach

es

Fo

ur

Item

so

f

Med

ium

Lo

adin

gs

Fo

ur

Item

so

f

Lo

wL

oad

ing

s

Six

Item

so

f

Div

erse

Lo

adin

gs

Six

Item

so

f

Hig

hL

oad

ing

s

Six

Item

so

f

Med

ium

Lo

adin

gs

Six

Item

so

f

Lo

wL

oad

ing

s

g 1g 2

fn

g 1g 2

fn

g 1g 2

fn

g 1g 2

fn

g 1g 2

fn

g 1g 2

fn

1.4

40

.93

.09

78

1.9

12

.46

.09

61

0.8

20

.89

.09

95

.47

0.6

5.0

81

00

1.3

82

.47

.09

88

1.7

92

.01

.08

78

0.2

60

.33

.05

99

1.2

50

.97

.04

94

0.1

70

.25

.05

10

0.1

30

.17

.04

10

00

.19

0.2

4.0

41

00

0.6

00

.60

.04

10

0

0.1

80

.20

.04

10

00

.34

0.3

5.0

39

90

.14

0.1

4.0

41

00

.12

0.1

1.0

31

00

0.1

60

.18

.03

10

00

.23

0.3

2.0

31

00

1.4

71

.41

.15

74

1.6

21

.3.1

25

31

.31

1.1

1.1

29

1.4

41

.19

.12

91

1.1

71

.56

.10

79

1.1

11

.42

.11

67

0.4

10

.66

.05

99

0.8

91

.56

.05

87

0.2

20

.28

.06

10

0.1

90

.20

.04

10

00

.26

0.4

5.0

51

00

0.7

91

.35

.05

95

0.2

40

.41

.04

10

00

.74

0.5

8.0

49

90

.21

0.1

9.0

41

00

.14

0.1

4.0

31

00

0.1

80

.22

.03

10

00

.44

0.3

4.0

31

00

0.9

61

.37

.09

92

1.9

1.2

.07

69

0.6

10

.92

.08

97

.31

0.3

9.0

79

80

.97

0.9

9.0

89

71

.48

1.8

9.0

78

4

0.2

20

.36

.03

10

00

.93

0.4

8.0

39

90

.16

0.1

6.0

41

00

.14

0.1

3.0

31

00

0.1

90

.27

.03

10

00

.39

0.3

4.0

31

00

0.1

30

.18

.03

10

00

.38

0.3

9.0

31

00

0.1

20

.14

.03

10

0.1

20

.12

.03

10

00

.12

0.1

5.0

21

00

0.2

50

.28

.02

10

0

0.9

91

.16

.07

86

2.5

61

.78

.09

76

0.6

60

.80

.10

97

.43

0.7

3.0

99

90

.91

0.6

9.0

89

54

.42

17

.5.0

88

6

0.2

90

.33

.04

10

00

.68

0.6

9.0

49

60

.15

0.2

0.0

41

00

.14

0.1

5.0

41

00

0.2

00

.21

.04

10

00

.71

0.9

2.0

49

8

0.1

60

.22

.03

10

00

.32

0.3

6.0

31

00

0.1

30

.13

.03

10

0.1

00

.10

.03

10

00

.14

0.1

8.0

31

00

0.2

10

.23

.03

10

0

0.8

41

.06

.06

96

1.1

71

.64

.06

86

0.6

00

.50

.06

10

0.2

70

.27

.06

10

00

.37

0.4

7.0

59

91

.41

.64

.05

95

0.2

10

.20

.03

10

00

.34

0.4

4.0

39

90

.16

0.1

3.0

31

00

.12

0.1

3.0

31

00

0.1

70

.17

.03

10

00

.23

0.3

5.0

31

00

0.1

20

.13

.02

10

00

.25

0.2

3.0

21

00

0.1

10

.10

.02

10

0.0

80

.10

.02

10

00

.12

0.1

2.0

21

00

0.1

70

.21

.02

10

0

0.5

40

.43

.07

10

01

.25

1.3

3.0

78

70

.46

0.6

1.0

79

9.2

30

.30

.06

10

00

.45

0.3

3.0

61

00

1.4

71

.66

.06

95

0.1

60

.19

.04

10

00

.39

0.6

3.0

31

00

0.1

40

.18

.03

10

0.1

20

.11

.03

10

00

.15

0.1

8.0

31

00

0.2

40

.39

.03

10

0

0.1

10

.16

.02

10

00

.31

0.3

8.0

21

00

0.1

00

.12

.02

10

0.1

00

.09

.02

10

00

.11

0.1

5.0

21

00

0.1

70

.27

.02

10

0

0.3

90

.57

.06

96

1.1

21

.48

.06

93

0.3

50

.38

.05

10

0.3

00

.43

.05

10

00

.58

1.1

9.0

61

00

12

.94

.92

.05

96

0.1

90

.23

.03

10

00

.38

0.4

0.0

31

00

0.1

50

.16

.03

10

0.1

10

.11

.04

10

00

.17

0.1

7.0

31

00

0.2

80

.37

.02

10

0

0.1

30

.15

.02

10

00

.25

0.2

4.0

21

00

0.1

00

.10

.03

10

0.0

90

.08

.02

10

00

.11

0.1

2.0

21

00

0.1

70

.18

.02

10

0

0.5

90

.70

.07

96

1.7

92

.19

.07

84

0.5

00

.74

.06

10

0.3

30

.26

.05

10

00

.46

0.5

2.0

51

00

0.6

11

.16

.05

90

0.1

90

.22

.03

10

00

.29

0.5

0.0

31

00

0.1

30

.16

.03

10

0.1

30

.13

.03

10

00

.18

0.1

8.0

31

00

0.2

60

.43

.03

10

0

0.1

40

.13

.02

10

00

.29

0.2

5.0

21

00

0.1

10

.13

.02

10

0.0

90

.10

.02

10

00

.12

0.1

2.0

21

00

0.1

90

.24

.02

10

0

136



the latent scoring approach performed well with large samples and when binary indicators

were not extremely skewed.

Shortened Scales

To compare and simplify the information concerning the effects of shortening scales, a

repeated measures analysis of variance was used to compare the biases of shortened-scale

approaches, including four ways of using four indicators and four ways of selecting six

indicators. This analysis yielded a significant within-subject effect (shortening strategies)

on g1, F(7, 147)= 13.98, p< .01; g2, F(7, 147)= 10.10, p< .01; and f, F(7, 147)=476.89, p< .01; and a significant between-subject effect of sample size on g1, F(1,

21)= 3.95, p< .05; g2, F(2, 21)= 8.99, p< .05; and f, F(1, 21)= 7.76, p< .01. These

effects suggest that various shortening strategies and sample sizes yielded different biases

in the three parameters. The estimated marginal means showed that selecting four or six

items with diverse or high factor loadings for SEM biased all three parameters within the

maximum acceptable range of 0.10. The raw means in Table 2 show that biases could

exceed 0.20 with small sample sizes. The estimated marginal means also show that select-

ing four or six items with medium factor loadings for the model biased the partial covar-

iances maximally by 0.08 but underestimated the covariance of F1 and F2 by more than

0.10 on average. Low factor loadings in the four or six items biased g1 minimally by 0.07

on average and the other two parameters (g1 and f) minimally by 0.13. In sum, the opti-

mal conditions for using four or six indicators selected from large scales were when indi-

vidual items had diverse or high factor loadings and sample sizes were relatively large.

Findings From Empirical Data

The selected approaches produced mixed results when applied to the empirical data. A

measurement model of aggression as a latent construct with 20 categorical indicators was

estimated and found to fit the data modestly from each wave. Standardized factor loadings

for each wave, their reliabilities, and model fit indices are listed in Table 4. Latent scores

were obtained from estimating each measurement model. The goodness of fit indices and

the factor loadings suggest that these items measured a unidimensional construct. Only

three kinds of parceling (single parcel, two odd–even split parcels, and four random par-

cels) were applied for the measurement of each wave, because the factor loadings of this

longitudinal data did not display the same pattern specified for the population model in

the simulation. A linear growth model was estimated using the latent scores or single par-

cels as the observed variables, two odd–even split parcels or four random parcels as indi-

cators of the aggression construct, or six individual items as indicators of the aggression

construct. The six individual items were selected to have the highest mean loadings over

time, and are marked with P in Table 4. All the models fit the data acceptably, with

CFI= .92 to .95, TLI= .94 to .98, and RMSEA= .06 to .07. The means of the intercept

(ai) and slope (as) and the covariance (fis) of the intercept and slope factors differed in

their magnitude and statistical significance across different approaches to using the scale.

The models of single parcels and odd–even split parcels as indicators of the latent




Tab

le4

Sta

nd

ard

ized

Fact

or

Load

ings

of

the

Late

nt

Con

stru

ctof

Aggre

ssio

nM

easu

red

Fro

mK

ind

ergart

ento

Gra

de

8

Item

and

Co

nte

nts

Kin

der

gar

ten

Gra

de

1G

rad

e2

Gra

de

3G

rad

e4

Gra

de

5G

rad

e6

Gra

de

7G

rad

e8

3.A

rgu

es.8

6.8

8.9

3.9

1.9

2.8

7.8

8.8

7.9

1

7.B

rag

s.7

5.7

3.7

0.7

5.7

0.8

0.7

5.7

3.8

1

16

.C

ruel

too

ther

sP

.88

.94

.92

.94

.93

.92

.89

.87

.94

19

.D

eman

ds

atte

nti

on

s.7

6.8

1.8

4.8

0.7

9.8

1.8

2.7

4.8

9

20

.D

estr

oy

so

wn

thin

gs

.80

.76

.78

.80

.79

.82

.76

.63

.87

21

.D

iffi

cult

yfo

llo

win

g

dir

ecti

on

s

.84

.84

.85

.90

.84

.86

.90

.84

.92

23

.D

iso

bed

ien

tat

sch

oo

lP

.87

.85

.91

.91

.90

.91

.88

.87

.90

27

.E

asil

yje

alo

us

.69

.82

.81

.69

.73

.71

.74

.81

.82

37

.F

igh

tsP

.90

.89

.91

.92

.89

.83

.85

.87

.88

43

.L

ies

.79

.75

.87

.79

.84

.74

.82

.72

.83

57

.P

hy

sica

lly

atta

cksP

.87

.81

.89

.91

.88

.91

.87

.92

.88

68

.S

crea

ms

.82

.69

.89

.90

.83

.82

.88

.81

.93

74

.S

ho

ws

off

.82

.82

.80

.77

.75

.79

.81

.84

.87

86

.S

tub

bo

rn.8

2.8

5.8

7.8

4.8

6.8

5.8

6.8

4.7

6

87

.M

oo

dy

.68

.78

.82

.79

.81

.81

.87

.82

.76

93

.T

alk

sto

om

uch

.78

.77

.80

.75

.82

.80

.81

.83

.88

94

.T

ease

s.7

8.8

8.8

2.8

3.8

5.8

1.8

5.8

6.8

6

95

.H

ot

tem

per

P.8

5.8

8.8

6.8

9.8

9.8

9.8

9.9

1.8

8

97

.T

hre

aten

sP

.90

.85

.89

.91

.91

.92

.88

.93

.94

10

4.

Lo

ud

.87

.86

.83

.84

.87

.78

.88

.89

.93

Mea

sure

men

tm

od

elfi

tw2

=2

58

.11

,

df=

50

,

p<

.01

,

CF

I=

.94

,

TL

I=

.98

,

RM

SE

A=

.08

w2=

24

5.3

1,

df=

48

,

p<

.01

,

CF

I=

.94

,

TL

I=

.98

,

RM

SE

A=

.08

w2=

19

1.0

4,

df=

50

,

p<

.01

,

CF

I=

.96

,

TL

I=

.99

,

RM

SE

A=

.07

w2=

18

8.4

0,

df=

53

,

p<

.01

,

CF

I=

.97

,

TL

I=

.99

,

RM

SE

A=

.07

w2=

18

0.8

1,

df=

42

,

p<

.01

,

CF

I=

.96

,

TL

I=

.98

,

RM

SE

A=

.08

w2=

20

4.3

5,

df=

52

,

p<

.01

,

CF

I=

.94

,

TL

I=

.98

,

RM

SE

A=

.08

w2=

17

9.0

2,

df=

43

,

p<

.01

,

CF

I=

.95

,

TL

I=

.98

,

RM

SE

A=

.08

w2=

13

1.7

9,

df=

.32

,

p<

.01

,

CF

I=

.96

,

TL

I=

.98

,

RM

SE

A=

.08

w2=

15

0.4

5,

df=

37

,

p<

.01

,

CF

I=

.97

,

TL

I=

.99

,

RM

SE

A=

.08

No

te:

CF

I=

com

par

ativ

efi

tin

dex

;T

LI=

Tu

cker

–L

ewis

ind

ex;

RM

SE

A=

roo

tm

ean

squ

are

erro

rap

pro

xim

atio

n.

138



construct found significant upward linear growth, as = .06, z= 3.13, p< .01, and as = .05,

z= 2.96, p< .01, whereas no growth was found by modeling latent scores, as = .05,

z= 1.33, p> .05, an aggression construct with four random parcels as indicators, as = :01,

z= 0.76, p> .05, or six individual items as indicators of the aggression construct,

as = .06, z= 0.72, p> .05.

Discussion

This study was designed to evaluate the effectiveness of three approaches to incorporat-

ing lengthy ordinal scales of varied measurement qualities into SEM, with a focus on

biases in estimates of partial variances and the covariance between exogenous constructs.

Three approaches were similarly effective under their own appropriate conditions in this

study. First, parceling approximately reflected the true population parameters when the

indicators had five response categories and parcels were further used as indicators of latent

constructs. Desirable parceling procedures included odd–even splitting into two parcels,

balancing item discrimination functions (factor loadings) to form three parcels, and ran-

dom parceling into four or six parcels. An ineffective procedure under the conditions of

this study was forming four or six parcels of items with contiguous factor loadings. Sec-

ond, latent scoring best reflected the true parameters in most data conditions of this study

except in the case of binary indicators with small samples or extremely skewed binary

indicators even with large samples. Third, scale length could be shortened to four or six

items with diverse or high factor loadings to adequately recover the population parameters

with medium to large samples. The best approach depended on the available data

conditions.

In contrast to previous findings that used continuous data, identical measurement of

both the exogenous and endogenous constructs, or a measurement model rather than a full

SEM, biases of parceling in the two partial covariance parameters were found to be

severely upward when the indicators had two or three response categories but slightly

downward when the indicators had seven categories. Besides differences in data, model,

and identical parceling of indicators of both exogenous and endogenous constructs, these

inconsistencies could also be attributed to the fact that parceling changed the original non-

linear functions between categorical indicators and latent constructs into linear functions,

resulting in transformation errors. These errors became large with binary indicators,

whose variances and means are dependent on each other and are typically bound into a

range from 0 to 1 (Ferrando, 2009). Parceling of trichotomous indicators could result in

distorted variance and covariance, because the range of the parcel was limited to the origi-

nal scale, as opposed to that of the latent construct it reflected, which is theoretically

infinite. Parceling under these conditions would lose information on variances and covar-

iances of the items and thus bias the estimates of partial covariances more severely.

Consistent with previous findings based on continuous data, optimal conditions for parcel-

ing in this study were found to be when parcels were created from items having five categories

and used as indicators of the latent constructs, regardless of parceling strategies and degree of

nonnormality (e.g., Sass & Smith, 2006). Parceling of five-category indicators could increase

the range of the parcel to approximately 4 (± 2) standard deviations of a normal distribution.




As a linear transformation of the categorical variables into continuous ones, parceling did not

lose much information on variances and covariances and thus yielded estimates that approxi-

mated the population partial covariance parameters (Ferrando, 2009).

Small sample size appeared to be disadvantageous to both latent scoring and shortened-

scale approaches to SEM. The latent scoring approach produced trivial biases in the esti-

mates of effects, except when the measurement model for obtaining the latent scores had

binary indicators and was estimated with small samples, or when the binary indicators

were extremely skewed even with large samples. As expected, a small sample typically

could not offer sufficient information for any model to describe its population with accu-

rate parameters. This problem could be complicated with binary indicators, whose means

and variances are dependent on each other. Simply increasing the sample size did not

improve the information that severely skewed binary indicators could provide about the

population, partly because certain information might have been lost when multivariate

continuous data were categorized into these severely skewed indicators. In spite of these

limitations, biases of latent scores in the covariance (f) and the two partial covariances

(g1 and g2) were not as far from the population parameters as those caused by using par-

cels. With large samples, shortened scales with either good or diverse measurement quali-

ties also became robust in recovering the population parameters.

Findings from the simulation study could shed some light on results of linear growth

modeling of the empirical data that adopted several approaches to the lengthy aggression

scale. Although discrimination functions of the aggression scale did not exactly match

those of the simulated data either in magnitude or pattern, no increase in aggression was

consistently detected by modeling the latent scores or factors with six individual indica-

tors of good measurement qualities. These two approaches had performed equivalently

well in recovering the population parameters with the simulated data under favorable sam-

ple sizes. In addition, the linear growth model showed no increase of factors with four

random parcels as indicators. Four random parcels as indicators of the aggression con-

struct should have provided more reliable measurement than a single parcel or two parcels

as indicators of the construct (Kenny, 1979). These consistent findings suggested that

aggression might not have increased over time.

In contrast, a significant increase in aggression over time was detected with linear

growth modeling of single parcels or a latent construct with two odd–even split parcels as

indicators. This finding corresponded to the overestimation of the partial covariances

when parcels of trichotomous indicators were used as indicators in the simulation study.

Therefore, using findings from the simulation study as guidelines, the efficient approaches

to the empirical data problem were (a) latent growth modeling of factors reflected by a

shortened scale with the best six indicators and (b) modeling of latent scores of the full

scale, which led to the inference that aggression did not increase over time in this sample.

Conclusions

If all items in a lengthy ordinal scale or testlets with various item discrimination func-

tions are to be included for SEM, parceling is one desirable option given the following

conditions: Items have five response categories and are parceled by odd–even splitting of




the scale; item discrimination functions are balanced or randomized; and two to six par-

cels obtained through parceling are used as indicators of a construct. Another option is to

obtain latent scores through preliminary item response modeling or CFA with categorical

indicators based on medium or large samples. However, binary items should not be extre-

mely skewed. If some items of a lengthy scale are not desirable, four or six items having

good, diverse measurement properties may be selected to provide robust indicators of a

construct, preferably based on a medium or large sample.

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