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A comparison of classical methods and second order latent growth models for longitudinal data analysis

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A comparison of classical methods and second order latent growth models for longitudinal data analysis

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A COMPARISON OF CLASSICAL METHODS AND SECOND ORDER LATENT
GROWTH MODELS FOR LONGITUDINAL DATA ANALYSIS
by
Erin Dominique Shelton

A Thesis Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF ARTS
(PSYCHOLOGY)

August 2010

Copyright 2010 Erin Dominique Shelton

ii

Dedication
I dedicate this work to my wonderful parents, Howard and Greta Shelton, and to my
brothers Eriq and Evan. Thank you for your continued love and support.

iii

Acknowledgments
I would like to give a special thank you to Jack McArdle for his support and guidance
throughout this process. I would also like to say thank you to Rand Wilcox, Richard
John, Steve Miller, John Prindle, and Kevin Petway for their input.

iv

Table of Contents

Dedication ii
Acknowledgments iii
List of Figures vi
List of Tables vii
Abstract viii
Chapter 1: Introduction 1
Chapter 2: Methods 9
Chapter 3: Initial Simulation Studies 15
Chapter 4: Results from Multivariate Analysis of Variance Models 18
Chapter 5: Results from First Order Latent Growth Models 20
Chapter 6: Results from “Curve of Factors” Models 24
Chapter 7: Results from “Factor of Curves” Models 30
Chapter 8: Discussion 32
References 38
Appendices: 41
Appendix A: Algorithm for repeated measures multivariate analysis of
variance (MANOVA) 41

Appendix B: Algorithm for a first order latent growth model (LGM) 42
Appendix C: Algorithm for a Curve-of-Factors (CUFFS) second order
latent growth model 43

Appendix D: Algorithm for a Factor-of-Curves (FOCUS) second order
latent growth model 44

Appendix E: SAS Syntax for a Repeated Measures MANOVA 45

v

Appendix F: Mplus syntax for a one factor configural invariance
model of WISC data 46

Appendix G: Mplus syntax for a one factor partial metric/weak
invariance model of WISC data 49

Appenxix H : Mplus syntax for a one factor general metric/strong
invariance model of WISC data 50

Appendix I: Mplus syntax for a one factor full metric/strict invariance
model of WISC data 53

Appendix J: Mplus syntax for a one factor non linear CUFFS model
of WISC data 55

Appendix K: Mplus syntax for a one factor non linear FOCUS model
of WISC data 58

Appendix L: Mplus syntax for a two factor full CUFFS model of WISC
data 60

Appendix M: SAS syntax for simulating data for a one factor CUFFS
model and RAM path notation for fitting the CUFFS
and FOCUS models 64

Appendix N: Formula for Wilks’ lambda transformation to chi-square 69

vi

List of Figures
Figure 1: Curve-of-Factors (CUFFS) Path Diagram 5
Figure 2: Factor-of-Curves (FOCUS) Path Diagram 6
Figure 3: Trajectories of Information Scores over Grade at Time of Testing 10
Figure 4: Longitudinal trajectories of Mean Scores on the 8 WISC variables
at each Grade of Testing 12

Figure 5: First Order Latent Growth Model Path Diagram 14
Figure 6: Likelihood Ratio Tests and Degrees of Freedom for the
First Order LGMs 23

Figure 7: Likelihood Ratio Tests and Degrees of Freedom for the
CUFFS Models 29

vii

List of Tables
Table 1: Variance Distributions and Model Results for Initial Simulation
Analyses on CUFFS Generated Data. 16

Table 2: Model Descriptions for First and Second Order Latent Growth Models 22

Table 3: Fit Statistics for First Order Latent Growth Models 23

Table 4: Fit Statistics for One Factor CUFFS Second Order
Latent Growth Models. 25

Table 5: Fit Statistics for Two Factor CUFFS Second Order
Latent Growth Models. 28

Table 6: Fit Statistics for One Factor FOCUS Second Order
Latent Growth Models 31

Table 7: Fit Statistics for Verbal and Performance FOCUS Second Order
Latent Growth Models 32

viii

Abstract
In 1988, McArdle identified issues modeling multivariate growth using
what he termed “second order latent growth curve” models. Specifically, he raised
questions about which type of structural equation model to use for longitudinal data
analysis. For example, “Should the growth model be fit to common factors extracted
from the measured variables (CUFFS)?”; or “Should a growth model be fit to the
measured variables and then the intercepts and slopes of the common factors be
considered (FOCUS)?” Both SEMs can be fit to the same set of data, however, as
pointed ou in McArdle (1988), these two alternative models not strictly nested and
therefore statistical comparison becomes more difficult. Most recent works have utilized
only the CUFFS model (Ferrer, Balluerka, & Widaman, 2008; Grimm, Pianta, & Konold,
2009; Hancock, Kuo, & Lawrence, 2001; Lui, & Flay, 2009), however, because of the
close relationship of the FOUCS models to the newer dynamic multivariate modeling
issues (e.g., McArdle, 2008), it is an option worth further consideration.
In the examples presented by McArdle (1988), data from the Weschler
Intelligence Scale for Children (WISC) were used. The longitudinal WISC dataset was
completed in 1965 by R. T. Osborne (Osborne & Suddick, 1972). Children were
measured on 11 subscales of the Weschler Intelligence Scale for Children (WISC). The
selected sample consists of 204 children who were repeatedly measured under
standardized conditions at approximate ages of 6, 7, 9, and 11. In this research, the same
data and models are used but more focus is placed on the appropriate choice of SEM.
Classic and contemporary techniques for analyzing repeated measures date are compared.
The focus was on the specification of second order LGMs and the advantages of these

ix

methods over the more classic MANOVA and first order LGM techniques. The key
results and their implications are discussed.

1

Chapter 1: Introduction
Longitudinal data analysis techniques are intended to assess trajectories of data
over time. Several methods have been developed over the years to achieve this goal.
While many methods can be used to assess longitudinal trends, not all methods are
equally as helpful. There are a few classic methods, which are still extensively used, as
well as a few contemporary methods, that have proven to be very useful in the assessment
of longitudinal data. These methods essentially assess the same thing, but different
methods allow researchers to ask and answer different questions. Understanding the
capabilities of each method can provided researchers with the necessary tools to properly
analyze and explain trends in a set of longitudinal data.
Multivariate Analysis of Variance for Longitudinal Data
Multivariate analysis of variance (MANOVA) is a classic choice for assessing
longitudinal data. This method allows researchers to assess the effects of independent
variables on several dependent variables at the same time. There are several extensions
of the MANOVA analysis but here we focus on MANOVA for repeated measures
designs. MANOVA is often referred to as profile analysis in the repeated measures case
(see Tabachnick & Fidell, 2007; Stevens, 2009, Morrison, 1976; Finn, 1974; Timm,
1975). It assesses differences among groups of individuals on a set of dependent
variables which are measured at several different points in time. It is often preferred to
the more traditional Analysis of Variance (ANOVA) because it has a less restrictive form
(Hedeker & Gibbons, 2006). Repeated measures MANOVA allows for the testing of
several hypotheses all at once. Repeated measures forms of MANOVA typically assess

2

hypotheses of parallelism, flatness, and levels (Morrison, 1976). That is, it looks to
assess whether profiles for different groups are parallel over time, whether those
trajectories are flat between the groups, and they assess whether or not the groups are at
the same levels.
While MANOVA is a traditional method and is often the first choice for
researchers when attempting to analyze longitudinal data, it is limited in what it can tell
in a repeated measures analysis. One limitation of MANOVA is its inability to handle
data with missing cases. In this framework, subjects with missing data must be removed
from the analysis or data imputation techniques must be utilized. Because of this
constraint, results may be biased or may not generalize well to the population of choice.
Of course, MANOVA can be done as a multi-level model if appropriate (Goldstein,
1995) A second limitation of the MANOVA framework for repeated measures data is its
inability to assess subject specific growth curves. This analysis only compares group
means and group trajectories rather than comparing individual differences in trajectories
which can be done with more sophisticated analytical techniques. This limits the
complexity of possible hypothesis being tested. To test more complex hypotheses, more
complex analyses are required.
Latent Growth Curve Analysis
Latent growth models (LGMs) come from a family of techniques referred to as
structural equation models (SEM ; see Kline 2005; Duncan, Dunca, & Strycker 2006).
LGMs are typically used to assess growth or decline of a variable or construct over some
trajectory of time. Generally, researchers are interested in assessing intraindividual

3

changes over time as well as interindividual changes or changes between individuals’
trajectories. These models have been around for many years and are also very prominent
in the assessment of longitudinal data. Traditional models use a intercept and slope,
sometimes referred to as level and shape, to assess two important features of the change
process. The intercept or level assesses initial status at the beginning of the study and the
slope or shape assess the growth or decline of the construct over the repeated
measurement occasions. Using this method, researchers are able to deal with a wide
range of issues related to the assessment of change over time. Specifically, SEM
techniques offer and alternative framework which can address issues that arise when
variance assumptions are violated in ANOVA or when assumptions about covariates are
violated in ANCOVA (McArdle, 1988; Raykov & Marcoulides, 2006).
LGMs have several potential advantages over the MANOVA models. One major
advantage of LGM models over MANOVA is the ability to assess individual trajectories
over time rather than trajectories of group means. Using all of the available data rather
than group means allow for a more dynamic and detailed interpretation of how the
trajectories change over time. LGMs also allow researchers to get around issues with
assumptions that are generally present when using classic ANOVA frameworks.
Specifically, it is typically very difficult to meet assumptions of sphericity and
homogeneity of variance within the ANOVA framework (Raykov & Marcoulides, 2006).
This is not generally a problem when using SEM techniques and thus they are often
preferred over models utilizing the traditional ANOVA framework when analyzing
repeated measured data.

4

While first order LGMs have advantages over the traditional MANOVA model, it
does have some limitations as well. One limitation that is discussed extensively in the
literature on more contemporary forms of LGMs is the inability to determine if the same
factor or construct is being measured at each time point. In these models, one can only
assume that the measured variables are representing the same construct over time (see
Sayer & Cumsille, 2001; Ferrer, Balluerka & Widaman, 2008). If the assumption of
factorial invariance over time is incorrect, then the results are invalid. A second
limitation of these LGMs is the use of composite scores. In order to assess the growth or
decline of a construct or factor over time, composite scores for each individual are
required. Using composite scores compounds measurement and time related variance
and provides less accurate estimates of the growth over time. In addition, this model is
restricted and lacks auto regressive terms which can be used in a less restricted latent
change score model. (see McArdle, 2009)
Second-Order Latent Growth Curve Analysis
An extension of the traditional first order LGMs is the second order LGM. Early
ideas of second order LGMs started in the 1980s, but received little attention until more
recently. McArdle (1988) identified issues modeling multivariate growth using second
order LGMs, and specified two possible models: the Curve-of-Factors (CUFFS), and the
Factor-of-Curves (FOCUS). In the CUFFS model, factors of measured variables are
extracted at the first level and a growth model is fit to the common factors at the second
level (see Figure 1). In the FOCUS model, growth models of each variable are fit at the
first level and the intercepts and slopes of the common factors are extracted at the second
level (see Figure 2). The question that arose is what should be fit at the first level.
5

6

7

One of the major advantages of these more contemporary LGMs is the ability to
test for factorial invariance of a measure over time. That is, these models allow
researchers to test whether a set of items measure the same construct at each
measurement occasion (Sayer & Cumsille, 2001; McArdle, 1988; Horn & McArdle,
1992; Ferrer et. al., 2008). The two types of invariance which are most commonly
mentioned in the literature are non-metric, also referred to as pattern invariance or
configural invariance, and metric invariance (Sayer & Cumislle, 2001; Horn & McArdle,
1992; Meredith & Horn, 2001; Ferrer et. al., 2008). Configural invariance implies that
the factor pattern is equivalent at each point in time. In other words, factors are made up
of the same indicators at each point in time regardless of the magnitude of the loadings at
those time points. This is to evaluate whether the factors present are represented the
same way each time. Metric invariance can be evaluated with several steps. I refer to the
first step as partial metric invariance, also known as weak factorial invariance ( Meredith,
1993). Partial metric invariance refers to invariance of the factor loadings of each
manifest variable over time. In this case, the factor loading of a given observed variable,
V
t
, is equated to be equal at each time of measurement. The second step builds upon that
of weak invariance and adds an equality constraint of the intercepts, or means of the
observed variables, over time. I refer to this step as general metric invariance but it is
also commonly referred to as strong factorial invariance. The final step I refer to as full
metric invariance, also known as strict factorial invariance, and it builds on the previous
two forms mentioned and requires an additional equality constraint to the unique error of
each observed variable over time (See Appendix I for model specification). Sayer and
Cumsille have argued that full metric invariance is not likely to hold when modeling

8

change because variances across time are typically not uniform, and have a tendency to
increase as means increase over time (Sayer & Cumsille, 2006). McArdle and Cattel,
however, have argued that full metric invariance is really essential and that lack of full
metric invariance could imply that the wrong factor pattern is being modeled (McArdle
and Cattel, 2004.)
A second advantage is the ability to separate out the variance in the observed
score. This model allows for the separation of measurement error and time specific error
which is confounded in the first order LGMs. By parsing out the measurement error in
the latent construct over time, more precise interpretations of the longitudinal changes
and relationships can be made. Essentially, second order LGMs make it possible avoid
using error saturated composites and instead creates a theoretically error free construct
for measuring growth (Sayer & Cumsille, 2006; Ferrer et. al., 2008). In addition to the
advantages already mentioned, second order LGMs also allow for the interpretation of
the individual measured variables which is not possible in the first order LGM case. In
the first order case, because composite scores are used, the model assumes that the
indicators are equally as important and therefore equally weighted. In the second order
LGMs, it is possible to test this assumption with a configural invariance model.
While the CUFFS and FOCUS LGMs appear to have several advantages over first
order models, it is still unclear under which conditions each of the models should be
preferred. The current literature which has utilized the concept of second order LGM has
focused largely on the CUFFS model and its ability to test factorial invariance over time
(See Ferrer et. al., 2008; Hancock & Lawrence, 2001; Grimm, Pianta, & Konold, 2009;
Liu & Flay, 2009; Sayer & Cumsille, 2006). The FOCUS model may have an advantage

9

over the CUFFS model in that it does not have the invariance restriction, which usually
makes it more difficult to fit the model and can result in poor model fit. Although the
direct comparison of these two models is difficult because they are not strictly nested, a
more in-depth look into the uses of these models is warranted.
Current Research Plan
The current study investigates the utility of several statistical techniques which
can be used for longitudinal data analysis. Initial simulation analyses will focus on the
utility of both second order LGM models, CUFFS and FOCUS, in order to examine the
optimal conditions under which each model may be preferred over another. Following
the simulation analyses, classical repeated measures MANOVA/ profile analysis and first
order LGMs will be run using the WISC data, followed by more contemporary second
order LGMs. The second order LGMs will follow the work of McArdle (1988) and will
consist of both CUFFS and FOCUS model specifications. McArdle (1988) specified the
CUFFS and FOCUS models only utilizing a slope at the second level. The current
analyses will go beyond what was previously done and utilize both an intercept and a
slope in both models. Results will be discussed.
Chapter 2: Methods
Participants
The dataset being used for this project was originally completed in 1965 by R. T.
Osborne. Children were measured on eleven subscales of the Weschler Intelligence
Scale for Children (WISC). The selected sample consists of 204 children who were
repeatedly measured under standardized conditions at approximate ages of 6, 7, 9, and

10

11. The children were originally tested in the spring prior to entering the first grade.
They were retested at the end of the first grade, again at the completion of the third grade
and the fifth grade. The mean age of the participants at the first time of testing was 6
years and one month (73 months; SD = 2 months). There were 109 girls, and 95 boys in
the sample, and the majority of the sample was white (58%; 42% = black; Osborne &
Suddick, 1972).
To permit direct comparisons with McArdle (1988), only eight of the eleven
subscales were utilized for the purposes of this study. The eight variables included in the
analyses were information (IN), Comprehension (CO), Similarities (SI), Vocabulary (V),
Picture Completion (PC), Block Design (BD), Picture Arrangement (PA), and Object
Assembly (OA). Trajectories of the information scores over grade at time of testing and
age at time of testing can be see below.

Figure 3: Trajectories of Information Scores over Grade at Time of Testing for a
Subset of N=20.

INFO[i]
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
GRADE AT TIME OF TESTING
1 2 3 4 5 6
INFO[i]
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
AGE IN MONTHS AT TIME OF TESTING
60 70 80 90 100 110 120 130 140

11

Simulation Analysis
Simulation analyses will be initially conducted to analyze further the accuracy
and utility three models – MANOVA, CUFFS, and FOCUS. The focus of these initial
analyses will be on how the three models perform when the data is generated for a
CUFFS model. Data will be generated for the CUFFS model using SAS statistical
software. Each of the three models, MANOVA, CUFFS, and FOCUS, will be fit to the
CUFFS generated data as well as the FOCUS generated data. The MANOVA model
should give information regarding how the means change over time and it is expected
that the CUFFS model will provide a nearly perfect fit to the CUFFS generated data. The
FOCUS model might provide a moderate fit to the data but it may not provide as good a
fit as the CUFFS model. Likewise, the FOCUS model should provide a nearly perfect fit
to the FOCUS generated data, however, because of the factorial invariance requirement
of the CUFFS model may provide a poor fit to the FOCUS generated data. This method
is generally used to assess whether these models work as they are thought to. If the
CUFFS model fails to fit the CUFFS generated data or the FOCUS model fails to fit the
FOCUS generated data, it may indicate that these models do not perform as they should
under general conditions and that the CUFFS or FOUCS models should not be preferred
over the first order LGMs.
Analyses
MANOVA
The MANOVA model for repeated measures designs assesses trajectories of
group means over time. A subject’s score is a function of the vector of means at each

12

timepoint, the vector group effects, and the vector of errors for that subject. A repeated
measures MANOVA or profile analyses of the WISC scores will be run first, as it is the
classical method. This model can be defined as a with-in subjects doubly multivariate
design (see Tabachnick & Fidell, 2001; Stevens, 2009 ). See Appendix A for model
algorithm. Each participant was measured on 8 WISC variables at 4 different time points
(approximate ages of 6, 7, 9, and 11). This model allows the use of DVs which are
measured on different metrics and avoids the assumption of sphericity by analyzing the
time effects and the dependent variables multivariately (Tabachnick & Fidell, 2001,
Stevens, 2009). This model will assess the trajectory of each dependent variable over
time using polynomial statements and will assess the relationship between the dependent
variables using canonical correlations. It will determine if scores on each of the
dependent variables are different at each grade level, and it will also determine whether
the trajectories of the variables are similar over time and what kind of trajectory they
follow. Figure 4 shows plots of the trajectories of mean WISC scores with grade.

Figure 4: Longitudinal Trajectories of Mean Scores on the 8 WISC Variables
at Each Grade of Testing

Note: Variables are listed in the order in which they were administered to participants.
0
5
10
15
20
25
30
35
40
INFO
COMP
SIMI
VOCA
PICC
PICA
BLOC
OBJE
0
5
10
15
20
25
30
35
40
INFO
COMP
SIMI
VOCA
PICC
PICA
BLOC
OBJE
Grade 1
Grade 2
Grade 4
Grade 6

13

First Order Latent Growth Models
First order LGMs will be run to look at the growth of the 8 scale scores over time,
assuming measurement invariance. In a first order model, a latent variable C is the sum
or average of a set of observed indicators T, U, and V. The variable score C at time t is a
function of the individual’s starting point at time 0, the systematic changes at time t, and
the random changes at time t. See Appendix B for algorithm.
Both one factor and a two factor first order LGMs will be fit and analyzed. For
the one factor model, composite WISC scores are calculated by using the average of the
scores on the 8 WISC variables for each individual, and the growth curve is fit to the
composite score variable/latent factor. The two factor model uses a verbal and
performance (V & P) split where the first four variables are thought to be strong
indicators of the verbal factor and the last four are thought to be strong indicators of the
performance factor (Horn & McArdle, 1992). For the two factor model, composite
scores are calculated for the four verbal indicators as well as the four performance
indicators to form a verbal and performance latent factor, and a growth model is fit to the
latent factors. Figure 5 is a path diagram of a one factor first order LGM.

14

Figure 5: first Order Latent Growth Model Path Diagram

Second order Latent Growth Curves
CUFFS. The CUFFS model adds an additional level the first order LGM. In this model,
manifest indicators form latent factors for each subject at each time point. A persons
factor score is a function of a their intercept score, slope, random changes, and
disturbance at time t. The growth of the factors over time is measured by an intercept and
slope at the second level (see Figure 1). See Appendix C for CUFFS algorithm. The one
factor model will use all 8 variables as indicators of a global WISC factor and the growth
model will be fit on top. The two factor model will use the V & P split mentioned earlier.
Prior to fitting the second order models, factorial invariance will be tested for both one
and two factor models as this is a requirement for the CUFFS models.
FOCUS. The FOCUS model also adds an additional level the first order LGM. In this
model, the growth of each manifest variable is measured at the first level. The growth of

15

a manifest variable for a given subject is a function of the persons intercept score, their
slope score for that variable, the random changes at time t, and the error that is
unexplained by their intercept and slope at time t. A factor of the intercepts and the
factor of the slopes are specified at the second level (see Figure 2). See Appendix D for
FOCUS algorithm. As with the CUFFS models, both one and two factor FOCUS
models will be specified. For the one factor model, a growth model will be fit to the 8
variables at each of the 4 time points and a factor of the intercepts and slopes will be fit
on top. In the two factor model, growth models will be fit for the 4 verbal variables and 4
performance variables separately and then factors of the intercepts and slopes will be fit
on top.
Chapter 3: Initial Simulation Studies
In order to assess the extent to which the CUFFS and FOCUS second order LGMs
work, data simulation was examined using statistical simulation (see Johnson, 1987;
Wang & McArdle, 2008; Parry & McArdle, 1991). In the first simulation, data was
generated to conform to a one factor CUFFS model. The model was generated to have 4
indicator variables which were measured at 4 time points, and a sample size of 1001 (see
Appendix M for data generation code and RAM path code). To assess the utility of these
models for longitudinal data, 7 different sets of data were generated based on the CUFFS
framework and assessed using the three data analysis techniques. The data sets had a
total variance of 1 and differed only in the proportion of variance coming from each
source, measurement variance (σ
u
2
), state or factor variance (σ
s
2
), intercept variance (σ
0
2
),
and slope variance (σ
i
2
). See Table 1 below for proportions of variance for each data set
generated.

16

Table 1: Variance Distributions and Model Results for Initial Simulation Analyses on
CUFFS Generated Data.
Total Variance = 1, Data = CUFFS CUFFS

σ
u
2
σ
s
2
σ
0
2
σ
1
2

χ
2
df p
A 0.25 0.25 0.25 0.25 101 105 0.60
B 1 0 0 0 82 105 0.95
C 0.5 0.16 0.17 0.17 96 105 0.72
D 0.75 0.083 0.083 0.083 89 105 0.87
E 0.25 0.25 0.5 0 86 105 0.91
F 0.25 0.25 0 0.5 96 105 0.73
G 0.25 0.25 0.375 0.125 102 105 0.56
Total Variance = 1, Data = CUFFS FOCUS

σ
u
2
σ
s
2
σ
0
2
σ
1
2

χ
2
df p
A 0.25 0.25 0.25 0.25 102 106 0.61
B 1 0 0 0 79 106 0.98
C 0.5 0.16 0.17 0.17 98 106 0.68
D 0.75 0.083 0.083 0.083 94 106 0.77
E 0.25 0.25 0.5 0 89 106 0.89
F 0.25 0.25 0 0.5 101 106 0.62
G 0.25 0.25 0.375 0.125 102 106 0.60

Total Variance = 1, Data = CUFFS MANOVA

σ
u
2
σ
s
2
σ
0
2
σ
1
2

Λ
w
F χ
2
p DF
A 0.25 0.25 0.25 0.25 0.0017 143,676 2,764 <.001 4/997
B 1 0 0 0 0.0003 921,663 3,516 <.001 4/997
C 0.5 0.16 0.17 0.17 0.0010 287,909 2,994 <.001 4/997
D 0.75 0.083 0.083 0.083 0.0003 719,841 3,516 <.001 4/997
E 0.25 0.25 0.5 0 0.0012 228,346 2,915 <.001 4/997
F 0.25 0.25 0 0.5 0.0056 43,911 2,247 <.001 4/997
G 0.25 0.25 0.375 0.125 0.0010 248,615 2,994 <.001 4/997

17

The first set of data (A) had an equal distribution of variance so each source of
variance received a value of .25. Both CUFFS and FOCUS models fit the data well and
MANOVA indicated significant changes in means over time. The second set of data (B)
attributed all variance to measurement. Both CUFFS and FOCUS models fit data set B
well and the MANOVA model again indicated significant mean changes over time. The
next set of data (C) attributed half of the variance to measurement and the remaining
variance was split between the three remaining sources. Again, Both CUFFS and
FOCUS models fit the data well and MANOVA indicated significant changes in means
over time. A fourth set of data (D) attributed 75% of the variance to measurement and
the remaining 25% was split equally between the remaining variance sources. Data set
number five (E) had half of the variance due to state/factor and measurement, and half
due to the intercept. There was no variance in the slope. Both CUFFS and FOCUS
models fit this data set very well and the MANOVA analysis indicated significant
changes in mean scores over time. Data set number six (F) had half of the variance due
to state/factor and measurement, and half due to the slope. There was no variance in the
intercept. Both CUFFS and FOCUS models fit this data set very well and the MANOVA
analysis indicated significant changes in mean scores over time. In the last model (G),
half of the variance was due to the state/factor and the measurement, and of the remaining
half was split between the intercept (37.5%) and slope (12.5%). Again, both CUFFS and
FOCUS models fit the data well and the MANOVA model again indicated significant
mean changes over time. See Table 1 for variance distributions and model results.
A second set of simulation analyses were done to assess alternative models fit to
dataset generated by a focus structure. In these analyses, the data was generated to have

18

4 variables measured at 4 time points with a non linear trend and a sample size of 1001.
This data also had a level and slope for each of the 4 variables as a factor of the levels
and a factor of the slopes. It is expected that the FOCUS model will fit the data perfectly
because the data is generated from that model, and the CUFFS model will provide a poor
fit to the FOCUS generated data. There were, however, complications with the FOCUS
simulations and thus the results are not presented here. Subsequent analyses will be done
to address the issues faced with the FOCUS simulations.
Chapter 4: Results From Multivariate Analysis of Variance Fitted to WISC Data
A multivariate repeated measures analysis of variance (MANOVA) was run with
SAS statistical software using Proc GLM. This model had 8 dependent variables at each
time of measurement, grades 1, 2, 4, and 6. The polynomial statement was specified in
order to assess the trajectory of the data, and the canonical statement was specified to
assess the relationship between the dependent variables (see Appendix E). The analysis
resulted in a high canonical correlation between the dependent variables, .988. This
indicates that the variables are very highly related. The overall effect of response was
significant, F(8, 196) = 964.41, p < .001. A transformation of the Wilks’ Lambda
statistic was used to obtain a chi-square likelihood value, Λ
W
= .0248, χ
2
= 320 (see
Appendix N for formulas). This indicates that the dependent variables differ from each
other at different grades, for example, information scores at time 1 are different than
those at time 2 or 3. This result is not surprising because we saw growth in the dependent
variables from grade 1 to 6 in figure 4. Follow up analyses indicate that all dependent
variables are different at different grade levels. This means that in general, children’s
scores on the dependent variables are significantly increasing from grade to grade. It is

19

not known from this analysis, however, if the variables are growing at the same rate or
whether they follow the same trajectories.
The next part of the analysis assesses the interaction between the dependent
variables. In other words, it assess the trajectories of the dependent variables. The
canonical correlation between grade and response (the dependent variables) was very
high, .985. This indicates a strong relationship between the dependent variables and each
grade level. The overall effect of the interaction was significant, F(24, 180) = 238.59, p <
.001, indicating that the trajectories of the dependent variables are not the same over
time. A transformation of the Wilks’ Lambda statistic was also used to obtain a chi-
square likelihood value, Λ
W
= .0305, χ
2
= 306 (see Appendix N for formulas). Follow up
analyses indicated that several variables followed similar trajectories, however, there
were a few that did not. All 8 variables were represented well by a linear trend. Only 7
of the dependent variables were represented well by a quadratic trend. The
Comprehension dv was not represented well by a quadratic trend, F(1, 203) = 2.88, p =
.09. Only 7 of the dependent variables were represented well by a cubic trend. The
dependent variable picture arrangement was not represented well by a quadratic trend,
F(1, 203) = 7.27, p = .17.
These results indicate that the variables are growing but that they do not follow a
strict linear trend. That is, they don’t change the same amount from grade 1 to grade 2 as
they do from grade 2 to grade 3 and so on. While the results indicate that the trajectories
are similar, it does not mean that each dependent variable is changing the same amount at
each time point. As can be seen in figure 4, some dependent variables grow more over
time than others. Some variables grow more in the early stages of development while

20

others grow more rapidly at the latter stages of development. Again, the MANOVA
analyses are limited in the information about the trajectories of the variables
Chapter 5: Results from First Order Latent Growth Models Fitted to WISC Data
A series of first order LGMs were fit to assess changes in 8 WISC scores over
time. For each of these models, composite WISC scores were created using the 8
measured variables and the LGMs were fit to the composite scores or latent factors.
Model fit was determined using the chi-squared (χ
2
) statistic, the comparative fit index
(CFI), the Tucker-Lewis index (TLI), Bayesian information criterion (BIC), and the root
mean squared error of approximation (RMSEA). The χ
2
is a function of the difference
between the proposed model and baseline model covariance matrices as well as the
sample size. Lower values indicate a better fitting model. The CFI assess increments in
model fit compared to the baseline model. It’s a function of the difference between the
chi-square and degrees of freedom for the baseline model minus the difference between
the chi-square and the degrees of freedom for the proposed model, all divided by the
difference between the chi-square and degrees of freedom for the baseline model, [{χ
2
-
df(baseline)} – {χ
2
- df(proposed)}] / [χ
2
-df(baseline)]. Values closer to one on the CFI
indicate better fit. The TLI is a function of the difference between the ratio of chi-square
to degrees of freedom for the baseline model and the proposed model, divided by the
ratio of chi-square to degrees of freedom for the baseline model minus 1, [χ
2
/df(baseline)
- χ
2
/df(proposed)] / [χ
2
/ df(baseline) – 1]. This index has a penalty for adding parameters.
Values closer to 1 indicate better fit. The BIC is a function of the chi-square value as
well as the number of free parameters (k(k - 1)/2 – df) and the sample size, χ
2
+ [k(k -
1)/2 - df]ln(N). It is not interpreted for a single model but rather used to compare two

21

nested models. It increases penalties as the sample size increases, and lower values
indicate better models . The RMSEA is a function of the ratio of chi-square to degrees of
freedom as well as the sample size, √[([χ
2
/df] - 1)/(N - 1)]. Values closer to zero indicate
better fit (Kenny, 2010). Nested models were compared using the difference between the
likelihood ratios (Δχ
2
) and the difference between the degrees of freedom (Δdf) for the
two models being compared. If the change in the likelihood ratio in relation to the
change in df is significant, it is interpreted that one model is significantly better or worse
than the other and the better model is preferred.
The first two models specified were one factor models. Composite scores for
these models were created by averaging each participant’s scores on the 8 WISC
variables. This resulted in each participant having one WISC score at each of the 4 time
points. Model 1(LGM
1
) was the linear model in which the slope was fixed with a linear
trajectory over the time points. This model resulted in a poor fit to the data. Model 2
(LGM
2
) freed the linearity constraint and allowed the trajectory to deviate from linearity.
This model resulted in a significantly better fit to the data, Δχ
2
= 53.53, Δdf = 2, p <.001.
The next two models, LGM
3
and LGM
4
, were two factor LGMs. With these
models, the first 4 variables were averaged to form verbal composite scores and the last 4
were averaged to form performance composite scores. In these models, a growth curve is
fit to the verbal and performance scores separately and the intercepts and slopes are
allowed to correlate. Model 3 (LGM
3
) is a linear model where the slopes of the verbal
and performance models are forced to a linear trajectory. This model provided a poor fit
to the data. Model 4 (LGM
4
) was a non linear model where the verbal and performance
slopes were anchored at 0 and 5, and the values were freely estimated in between

22

allowing deviations from linearity. LGM
4
was significantly better than LGM
3
, Δχ
2
=
111.5, Δdf = 4, p <.001. See Table 1 for a description of each model and Table 2 for
model fit statistics.
Table 2: Model Descriptions for First and Second Order Latent Growth Models.
Model Description
LGM
1
1 Factor linear model
LGM
2
1 Factor non-linear model
LGM
3
2 Factor linear model
LGM
4
2 Factor non-linear model

CUFFS
1
1 Factor correlated longitudinal model (configural invariance)
CUFFS
2
1 Factor partial metric invariance (equal loadings; λ
x1
= λ
x2
)
CUFFS
3
1 Factor metric invariance (equal loadings and equal means; λ
x1
= λ
x2
, μ
x1
= μ
x2

CUFFS
4
1 Factor non linear CUFFS model
CUFFS
5
1Factor linear CUFFS model
CUFFS
6
2 Factor correlated longitudinal model (configural invariance)
CUFFS
7
2 Factor partial metric invariance (equal loadings; λ
x1
= λ
x2
)
CUFFS
8
2 Factor metric invariance (equal loadings and equal means; λ
x1
= λ
x2
, μ
x1
= μ
x2

CUFFS
9
2 Factor non linear CUFFS model
CUFFS
10
2Factor linear CUFFS model
CUFFS
11
2 Factor full CUFFS model with a non linear trend
CUFFS
12
2 Factor full CUFFS model with a linear trend

FOCUS
1
1 Factor correlated longitudinal model
FOCUS
2
1 Factor non linear FOCUS model
FOCUS
3
1 Factor linear FOCUS model
FOCUS
4
Verbal correlated longitudinal model
FOCUS
5
Verbal non linear FOCUS model
FOCUS
6
Verbal linear FOUCS model
FOCUS
7
Performance correlated longitudinal model

23

Table 3. Fit Statistics for First Order Latent Growth Models.
Model X
2
df CFI TLI BIC RMSEA
90%
CI
LGM
1
60 5 0.94 0.92 3469 0.23 .18, .29
LGM
2
6 3 1.00 0.99 3426 0.08 .00, .16
LGM
3
142 22 0.92 0.90 7535 0.16 .14, .19
LGM
4
31 18 0.99 0.99 7444 0.06 .02, .09

After assessing the fit and comparing nested models, a plot was produced to assess which
model was the best out of first order LGMs. Figure 6 is a plot of the chi- square (χ
2
)
likelihood ratio tests in relation to their respective degrees for each of the first order
LGMs. A regression line is fit through the points. Points above the line have relatively
high LRT for the number of degrees of freedom, therefore, these models are considered
poor. The points below the line have relatively low LRT for the number of degrees of
freedom, therefore, these models are preferred. According to Figure 6, LGM
4
is the best
model for the given data, χ
2
= 30, df = 18.

Figure 6. Likelihood Ratio Tests and Degrees of Freedom for the first Order LGMs.

Chi-square Likelihood Ratio Test (LRT)
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
Degrees of Freedom (DF)
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
1
2
3
4

24

Chapter 6: Results From “Curve of Factors” Models Fitted to WISC Data
A series of Curve of Factor Score (CUFFS) models were fit and analyzed. Fit
was again determined using the χ
2
likelihood ratio test, the comparative fit index (CFI),
the Tucker-Lewis index (TLI), the Bayesian information criterion (BIC), and the root
mean squared error of approximation (RMSEA). Nested models will again be compared
using the model difference statistics (Δχ
2
and Δdf).
The first set of models run assumed one common factor. As mentioned earlier,
invariance tests must be conducted before running the actual CUFFS model. The first
invariance model (CUFFS
1
) is sometimes referred to as the configural model of
invariance (Sayer & Cumislle, 2001; Horn & McArdle, 1992; Meredith & Horn, 2001;
Ferrer et. al., 2008) or a correlated longitudinal model. CUFFS
1
tests the invariance of
the structure of the data over the 4 time points. This is modeled at the first order level.
Measured variables are correlated at each time point and the factors are correlated over
time (see Appendix F for additional details of model specification). CUFFS
1
fit the data
moderately well. The second model (CUFFS
2
) adds an additional invariance constraint
to the loadings of each measured variable on its respective factor. This level of
invariance has been referred to as weak (Sayer & Cumislle, 2001; Horn & McArdle,
1992; Meredith & Horn, 2001; Ferrer et. al., 2008) or partial metric invariance. For
example, the loading for the measured variable information (IN) is constrained to be
equal at each time point on each of the 4 factors (see Appendix G for additional detail).
CUFFS
2
resulted in a worse fit to the data. The decrease in model fit is expected because
of the invariance restriction, however, the model fit is still acceptable and the analyses
were continued. The third model (CUFFS
3
) adds another invariance restriction in

25

addition to the restrictions in the first two models. This level of invariance has been
referred to as strong invariance or metric invariance (Sayer & Cumislle, 2001; Horn &
McArdle, 1992; Meredith & Horn, 2001; Ferrer et. al., 2008). In addition to the
constrained structure and factor loadings, the intercepts or means of each measured
variable are equated over time. For example, the mean of the measured variable IN is
estimated but constrained to be equal at each of the four time points (see Appendix H for
additional detail). Again, as expected, CUFFS
3
was worse than the previous models due
to the added constraints. Although CUFFS
3
was worse, the model fit was still at an
acceptable level so analyses were continued. See Table 1 for a description of each model
and Table 3 below for model fit statistics.

Table 4. Fit Statistics for One Factor CUFFS Second Order
Latent Growth Models.
Model X
2
df CFI TLI BIC RMSEA
90%
CI
CUFFS
1
698 410 0.93 0.91 33795 0.06 .05, .07
CUFFS
2
892 431 0.88 0.86 33877 0.07 .07, .08
CUFFS
3
1153 450 0.82 0.80 34037 0.09 .08, .09
CUFFS
4
1167 456 0.82 0.80 34091 0.09 .08, .09

Invariance restrictions must be met in order to move on to the second order
models because the more complex models do not make sense without the invariance
constraint. Since the invariance restrictions were met, the one factor CUFFS models
were fit next. These models add an intercept and a slope at the second level. The first

26

CUFFS model (CUFFS
4
) was fit with a non linear slope. In other words, the slope was
anchored at 0 and 5 and the values in between were allowed to deviate from linearity
rather than being fixed to a linear trend (See Appendix J for additional details). CUFFS
4
provided a moderate fit to data. The addition of the second order level and slope did not
change the a fit statistics by much. The fifth model (CUFFS
5
) was a linear extension of
CUFFS
4
. The slope values were anchored at 0 and 5 as with Model 4, however, the
values in between were also fixed to force a linear trend to the slope. CUFFS
5
provided a
significantly worse fit to the data than did CUFFS
4
, Δχ
2
= 51.79, Δdf = 2, p <.001. The
previous result indicates that the variables do not follow a pure linear trend over time.
See Table 1 for a description of each model and Table 3 for model fit statistics.
In addition to one factor CUFFS models, a two factor extension of the CUFFS
models were fit and assessed. As mentioned earlier, the two factor models spit the 8
measured variables into a verbal and a performance factor, with the first four variables
loading on the verbal factor and the remaining four loading on the performance factor.
As with the one factor models, 3 invariance models were tested for the two factor models.
The first two factor invariance model, CUFFS
6
, mimicked that of the one factor
configural, or correlated longitudinal model with the addition of a second factor.
CUFFS
6
fit the data quite well and provided a significantly better fit to the data than did
CUFFS
1
, Δχ
2
= 165.91, Δdf = 22, p <.001. The next model, CUFFS
7
, was a two factor
partial metric invariance model which adds the constraint of equal loadings across time
points. This model was slightly worse than CUFFS
6
as expected but was also provided a
significantly better fit to the data than the one factor partial metric model (CUFFS
2
), Δχ
2

= 172.71, Δdf = 25, p <.001. The third two factor invariance model, CUFFS
8
, was a full

27

metric invariance model, which added a constraint of equal intercepts for the measured
variables. Fit for this model was worse than its predecessor, however, it was also a
significantly better fit than the similar one factor full metric model (CUFFS
3
), Δχ
2
=
254.8, Δdf = 28, p <.001. These results would suggest that the data is better represented
with two factors than with one as believed in previous research (Horn & McArdle, 1992;
Osborne & Suddick, 1972). See Table 1 for a description of each model and Table 4 for
model fit statistics.
The next set of models added the second level to the invariance models. The
ninth model, CUFFS
9
, was a two factor non linear CUFFS model. As with the one factor
models, the slope for the verbal and performance variables in this model were anchored
at 0 and 5 and the two values in between were allowed to deviate from normality on both
factors. This model provided a moderate fit to the data, χ
2
= 959.65, df = 446, CFI =
.866, TLI = .851, RMSEA = .075. This model provided a significantly better fit to the
data than the one factor non linear model (CUFFS
4
), Δχ
2
= 207.56, Δdf = 10, p <.001.
The next model, CUFFS
10
, was a two factor linear model. In this model, the slopes of the
verbal and performance factors were fixed to a linear trend, as was done with the one
factor linear model (CUFFS
5
). This model had a significantly worse fit to the data than
the two factor non linear model (CUFFS
9
), Δχ
2
= 93.14, Δdf = 4, p <.001. This result
suggests that the trajectory of the variables over time is non linear. See Table 1 for a
description of each model and Table 4 for model fit statistics.

28

Table 5. Fit Statistics for Two Factor CUFFS Second Order
Latent Growth Models.
Model X
2
df CFI TLI BIC RMSEA
90%
CI
CUFFS
5
1219 458 0.80 0.79 34061 0.09 .08, .10
CUFFS
6
532 388 0.96 0.95 33746 0.04 .03, .05
CUFFS
7
719 406 0.92 0.90 33837 0.06 .05, .07
CUFFS
8
898 422 0.88 0.85 33931 0.07 .07, .08
CUFFS
9
960 446 0.87 0.85 33865 0.08 .07, .08
CUFFS
10
1053 450 0.84 0.83 33937 0.08 .08, .09
CUFFS
11
1096 445 0.83 0.81 34007 0.09 .08, .09
CUFFS
12
1156 450 0.82 0.80 34040 0.09 .08, .09

The last two models fit were linear and non linear two factor full models. In these
models, the 8 variables were not split and fixed to be on the verbal or the performance
variable, but rather each variable was allowed to be estimated on each factor at each time
point. In other words, all 8 measured variables were estimated on the verbal factor at
time point 1 and all 8 measured variables were estimated on the performance factor at
time point one. The measured variables were not forced to load only on one factor (see
Appendix L for model specification). This model was suggested by Horn and McArdle
(1992). The two factor full non linear model (CUFFS
11
) provided a significantly worse
fit to the data than the CUFFS
9
(two factor linear), Δχ
2
= 136.35, Δdf = 1, p <.001. The
final CUFFS model, CUFFS
12
, was a two factor full linear model. As with all other
linear models, the values of slope for each factor were fixed with a linear trend. This
model provided a significantly worse fit to the data than the previous model, CUFFS
11
,
Δχ
2
= 60.09, Δdf = 5, p <.001. These results suggest that the variables should be

29

represented as two distinct factors rather than two factors of cross loading variables. See
Table 1 for a description of each model and Table 4 for model fit statistics.
To assess which of the CUFFS models provided the best fit for the current set of
data, another LRT by DF plot was utilized (Figure 7). Figure 7 is a plot of the LRT by
DF for the CUFFS models, excluding the 6 invariance models. So it includes the one
factor non linear and linear models (CUFFS
4
and CUFFS
5
, respectively), the two factor
non linear and linear models (CUFFS
9
and CUFFS
10
), respectively, and the two factor
full non linear and linear models (CUFFS
11
and CUFFS
12
, respectively). According to
this figure, CUFFS
9
and CUFFS
10
are the best models to represent the data. As stated
earlier, CUFFS
9
is significantly better than CUFFS
10
so it seems that the two factor non
linear model (CUFFS
9
) is the best representation of the data.

Figure 7. Likelihood Ratio Tests and Degrees of Freedom for the CUFFS Models.

Chi-square Likelihood Ratio Test (LRT)
900
1000
1100
1200
1300
Degrees of Freedom (DF)
445 446 447 448 449 450 451 452 453 454 455 456 457 458
4
5
9
10
11
12

30

Chapter 7: Results From “Factor of Curves” Models Fitted to the WISC Data
A series of Factor of Curves (FOCUS) models were fit and analyzed. Fit was
again determined using the χ
2
likelihood ratio test, the comparative fit index (CFI), the
Tucker-Lewis index (TLI), the Bayesian information criterion (BIC), and the root mean
squared error of approximation (RMSEA). Nested models will again be compared using
the model difference statistics, Δχ
2
and Δdf.
The first set of FOCUS models were based on one global factor. At the
first level, growth curves were fit to each of the 8 variables. At the second level, a factor
of the 8 levels and a factor of the 8 slopes were added. A Longitudinal growth model
was fit and assessed first. In this model (FOCUS
1
) the intercepts and slopes of the 8
growth models are correlated with each other and it is. This model fit the data fairly well
so analyses were continued. The next model (FOCUS
2
) was a non linear one factor
model. In this model, the second order factors were added and the slopes of the first
order growth models were allowed to deviate from linearity. This model provided a
moderate fit to the data,
2
= 879.41, df = 463, CFI = .892, TLI = .884, RMSEA = .066.
In addition to the previous one factor FOCUS models mentioned, a linear one factor
model was fit (FOCUS
3
). In this model, the slopes of the growth models are forced to
follow a linear trend (see Appendix K for model specification). This model provided a
significantly worse fit the data than the non linear model, Δχ
2
= 221.82, Δdf = 14, p
<.001. See Table 1 for a description of each model and Table 5 for model fit statistics.

31

Table 6. Fit Statistics for One Factor FOCUS Second Order
Latent Growth Models
Model X
2
df CFI TLI BIC RMSEA
90%
CI
FOCUS
1
550 360 0.95 0.93 33913 0.05 .04, .06
FOCUS
2
879 463 0.89 0.88 33694 0.07 .06, .07
FOCUS
3
1101 477 0.84 0.83 33842 0.08 .07, .09

A verbal and performance split was also assessed using the FOCUS models
however these models were assessed in separate runs for purposes which will be
mentioned later. As stated earlier, the verbal factor is made up of the first four variables
(IN, CO, SI, and VO). A longitudinal growth model was fit for the verbal indicators
(FOCUS
4
). This model provided a moderate fit to the data. A subsequent non linear
second order model was fit to the verbal indicators (FOCUS
5
). The slopes of the growth
models were anchored at 0 and 5, and the values in between were allowed to deviate from
normality. This model provided a moderate fit to the data as well,
2
= 194.11, df = 103,
CFI = .954, TLI = .947, RMSEA = .066. The final verbal model was a linear second
order model, in which the slopes of the growth models were fixed to a linear trend
(FOCUS
6
). This model provided a significantly worse to the data than the non linear
model did, Δχ
2
= 112.54, Δdf = 8, p <.001. Again, these results provided evidence that
the growth of the verbal indicators is not strictly linear. See Table 1 for a description of
each model and Table 6 for model fit statistics.

32

Table 7. Fit Statistics for Verbal and Performance FOCUS Second Order
Latent Growth Models
Model X
2
df CFI TLI BIC RMSEA
90%
CI
FOCUS
4
154 84 0.97 0.95 15392 0.06 .05, .08
FOCUS
5
194 103 0.95 0.95 15331 0.07 .05, .08
FOCUS
6
307 111 0.90 0.89 15401 0.09 .08, .11
FOCUS
7
163 84 0.95 0.93 18432 0.07 .05, .08

The performance factor is made up of the last four variables mentioned (PC, PA,
OA, BD). A longitudinal growth model was fit to the performance variables (FOCUS
7
).
The slopes for this model remained non linear and the intercepts and slopes of the
variables were correlated over time. This longitudinal model fit the data fairly well,
2
=
163.18, df = 84, CFI = .951, TLI = .930, RMSEA = .068. Following the longitudinal
model, linear and non linear second order models were fit to the data. The FOCUS
model was not a good representation of the performance indicators. These two models
had serious issues and did not converge. Part of the reason for the non convergence had
to deal with issues of correlations between variables which were above 1. Based on these
result, it appears that the FOCUS model does not represent the data well as a two factor
verbal and performance model. See Table 1 for a description of each model and Table 2
for additional model fit statistics.
Chapter 8: Discussion
Classical and contemporary methods for assessing longitudinal data were
explored. More specifically, methods for assessing growth over time were chosen. The

33

first of the methods was the classical multivariate analysis of variance (MANOVA). The
MANOVA results revealed that the 8 dependent variables differed at each grade level.
That is, each variable grew from grade1 through grade 6. Analyses also revealed that the
trajectories of dependent variables differed over time. The trajectories for some variables
were similar to each other, however, they did not all follow exactly the same trajectory
from time 1 to time 4. This indicates that while scores on the dependent variables grow,
they grow at different rates. Some variables have steeper trajectories than others as can
be seen in figure 5.
Because of the limitations of the MANOVA analysis, structural equation models
were also run and analyzed. The classic first order latent growth model (LGM) was run
and assessed first. In this model, composite scores of the 8WISC variables were
computed using the average and a latent construct of global intelligence was formed. The
growth model was fit to the latent variables formed at each time point. Both one and two
factor models were fit. Results revealed that the two factor verbal and performance
model was a better fit to the data than the one global factor model. Results also indicated
that the trend of the variables was not linear. Essentially, kids scores on the verbal and
performance factors of the WISC change at different rates from ages 6 to 11 and grades 1
to 6. While first order factor model have advantages over the MANOVA model, it still
has limitations and thus more contemporary second order LGMs were fit and assessed.
Curve of Factor Scores (CUFFS) second order models were fit first. As with the
first order models, a two factor verbal and performance model fit the data best.
Invariance of verbal and performance factor structure was assessed prior to fitting the two
factor CUFFS model. Results indicated that the factor structure was reasonably

34

consistent over time and the same construct was being assessed at each time of
measurement. That is, the 8 measured variables were good indicators of verbal
intelligence and performance intelligence for children at each time of measurement.
Additional analyses also revealed that the verbal and performance factors did not follow
linear trends over time and changed differently from grade to grade.
The final longitudinal models run were FOCUS second order LGMs. The one
factor model provided a moderate fit to the data, however, based on previous research as
well as the previous models that were run. The models were fit separately to the verbal
and performance factors. The longitudinal verbal model fit the data well as did the verbal
FOCUS model. This analysis also indicated that the verbal indicators did not change at
the same rate at each gap in time, meaning they did not follow a linear trend. Next, the
performance factor was analyzed. A longitudinal performance model fit the data well,
however, there were several warnings when running the FOCUS model on the
performance variables. This model had issues with convergence. One reason is due to
the fact that some of the measured variables had correlations above 1. Because of this, it
was not possible or reasonable to fit the FOCUS model to the performance variables and
a two factor FOCUS model was not analyzed.
In addition to the repeated measures MANOVA and growth models of the WISC
data, simulation analyses were run to gain further understanding of the second order
LGMs. In these initial simulation analyses, data was generated based on a one factor
CUFFS model as well as a 4 variable FOCUS model. CUFFS and FOCUS models were
fit to several simulated data sets as well as a repeated measures MANOVA model.
Results indicated that the CUFFS and FOCUS models both represent CUFFS generated

35

data very well under general conditions and low levels of variance. The MANOVA
model was also able to indicate significant changes in means over time with each data set
generated. These analyses indicate that the CUFFS model performs as it should under
general conditions and that the FOCUS model also fits CUFFS generated data well. Due
to complications, the FOCUS simulation analyses were not presented here thought tt is
thought that while the FOCUS model does well when data are generated from a CUFFS
framework, the same may not be true when the analyses are reversed. That is, the
CUFFS model may not represent data generated from a FOCUS framework well. Future
analyses will continue to assess these models when fitted to FOCUS generated data.
Based on all of the analyses, it appears that a two factor nonlinear Curve-of-
Factors second order growth model (CUFFS
9
) best represents the WISC data. As
mentioned earlier, the use of second order LGMs can have advantages over the more
classic methods used to assess longitudinal data. The use of composite factor scores in
the first order LGMs requires assumptions to be made about the measurement properties
of a factor over time. Inaccurate assumptions of the factorial assumptions can lead to
very inaccurate interpretations of results. Being able to test the invariance of a factor
over time within the model can lead to more accurate interpretations of trends in data or
trends of factors over time. The main difference between the CUFFS and FOCUS second
order LGMs seems to be meeting the assumptions of factorial invariance. With a better
understanding the usefulness of the FOCUS model, it is possible to get more accurate
results by eliminating composite factor scores and the invariance restrictions are not
necessary. These two second order LGMs can provide researchers with better tools to
analyze longitudinal data.

36

Limitations
While the analyses highlighted in this paper have provided new insights into the
use of the CUFFS and FOCUS second order models, there are several limitations of this
study. The simulation analyses presented were limited. They provided insight into the
capabilities of the CUFFS and FOCUS LGMs, however, there is still a lot that can be
learned about these models. A more extensive simulation investigation of these models is
warranted in order to fully understand where these models succeed and where they fail.
Further analyses should also be done to determine under what circumstances one model
should be preferred over another. For example, are there certain types of data or specific
circumstances where the FOCUS model is a better choice than the CUFFS model, and
vice versa? Another limitation of this study is that only one set of data was used to
examine the strengths and weaknesses of these analytical techniques. Examining these
models with several sets of data can also increase knowledge about the proper uses and
capabilities associated with them. Further investigation into the FOCUS model is
needed. While the current analyses shed some light on the FOCUS model, the issues that
arose when specifying the more complicated two factor model limited the understanding
gained from these models. More specifically, information about the specification of the
FOCUS models as well as the interpretations were obtained, however, it was not possible
to compare one and two factor FOCUS models which is vital in this case considering that
research suggests that two factors underlie the WISC variables.

37

Future Directions
It is important to continue investigations into the uses of these more contemporary
longitudinal data analysis techniques. Future analyses should explore further the
complexities and utilities of second order LGMs. It is important to extend the literature
on the FOUCS model to determine its usefulness as a tool for researchers in addition to
the CUFFS model. Future analyses should be conducted to understand these models in
more depth. Because of the limited literature on the FOCUS model, little is known about
the practicality of this model in analyzing longitudinal data. Analyses should focus on
the specification of the FOCUS model in a variety of situations to determine if it
performs as it should under more complex conditions. Future analyses should also focus
on the specification of the CUFFS model in comparison to the FOCUS model in more
complex situations, such as with non normal distributions or with very small sample
sizes. Continued investigation into second

order LGMs is warranted.

38

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41

Appendix A
Algorithm for repeated measures multivariate analysis of variance (MANOVA)

A simple repeated measures MANOVA can be specified as follows:
y
gi
=μ + γ
g
+ ε
gi

where μ is the vector of time point means, γ
g
is the vector effect for the population from
which the gth group of subjects was drawn, and ε
gi
is the vector of errors for subject i in
group g (see Hedeker and Gibbons, 2006).

42

Appendix B
Algorithm for a first order latent growth model (LGM)
A first order LGM can be specified as follows:
C[t]
n
= I
n
+ A[t] S
n
+ u[t]
n
,
where C is the score on the latent variable at time t for person n, I
n
is the intercept score
for person n, S
n
is the slope score for person n, A[t] are the random changes, and u[t]
n
is
the error that is unexplained by the intercept and slope for person n at time t.

43

Appendix C
Algorithm for a Curve-of-Factors (CUFFS) second order latent growth model

The CUFFS Model can be specified as follows:

T[t]
n
= λ
T
[t] β
n
+ e
T
[t]
n
,
U[t]
n
= λ
U
[t] β
n
+ e
U
[t]
n
,
V[t]
n
= λ
V
[t] β
n
+ e
V
[t]
n
,
F[t]
n
= I
n
+ A[t] S
n
+ E[t]
n
,
I
F
= β
0F
+ Aβ
sF
+ D
x

S
F
= β
0F
+ Aβ
sF
+D
x

where T, U, and V are manifest indicators of the variable X for person n at time t, λ
represents the variable loading on latent factor F, e represents the unique factor score, I
n
is the intercept score for person n, S
n
is the slope score for person n, A[t] are the random
changes, and E[t]
n
is a disturbance term at time t for factor F. As seen in Figure 1, X[t] is
made up of T[t], U[t], and V[t] and the growth of F over time is modeled by a second
order intercept and slope, I
X
and S
X
.

44

Appendix D
Algorithm for a Factor-of-Curves (FOCUS) second order latent growth model

The FOCUS model can be specified as follows:
T[t]
n
= I
n
+ A
T
[t] S
n
+ E
T
[t]
n,
U[t]
n
= I
n
+ A
U
[t] S
n
+ E
U
[t]
n,
V[t]
n
= I
n
+ A
V
[t] S
n
+ E
V
[t]
n,
f
i
= Λ
i
β
i
+ ε
i
f
s
= Λ
s
β
s
+ ε
s
where I
n
is the intercept score for person n on each variable, S
n
is the slope score for
person n on each variable, A[t] are the random changes for each person on each measured
variable at time t, E[t]
n
is the error that is unexplained by the intercept and slope for
person n at time t on each variable, and f
i
and f
s
represent the factor of the intercepts and
the factor of the slopes fit at the second level, respectively.

45

Appendix E
SAS Syntax for a Repeated Measures MANOVA

proc glm data=Wisc8;

model info6 comp6 simi6 voca6 picc6 pica6 bloc6 obje6
info7 comp7 simi7 voca7 picc7 pica7 bloc7 obje7
info9 comp9 simi9 voca9 picc9 pica9 bloc9 obje9
info11 comp11 simi11 voca11 picc11 pica11 bloc11 obje11 = / nouni int;
repeated Grade 4 (1 2 4 6 )polynomial, Response 8 identity/short summary canonical;
run;

46

Appendix F
Mplus syntax for a 1 factor configural invariance model of WISC data
INPUT INSTRUCTIONS

Title:
1 Factor CUFFS;
Data:

File is "C:\WISC.dat ";
Format is (1F5.0, 14F2.0, 1F12.0, 1F1.0, 1F2.0/
1F7.0, 13F2.0, 1F10.0/
1F7.0, 13F2.0, 1F10.0/
1F7.0, 13F2.0, 1F8.0);

USEVARIABLES ARE
info6 comp6 simi6 voca6 picc6 pica6 bloc6 obje6
info7 comp7 simi7 voca7 picc7 pica7 bloc7 obje7
info9 comp9 simi9 voca9 picc9 pica9 bloc9 obje9
info11 comp11 simi11 voca11 picc11 pica11 bloc11 obje11;

ANALYSIS: TYPE= MEANSTRUCTURE;
ITERATIONS=100000;
MODEL:
!First-Order Factor Model
WISC1 BY info6@1
comp6
simi6
voca6
picc6
pica6
bloc6
obje6;
WISC2 BY info7@1
comp7
simi7
voca7
picc7
pica7
bloc7
obje7;
WISC3 BY info9@1
comp9
simi9
voca9

47

picc9
pica9
bloc9
obje9;
WISC4 BY info11@1
comp11
simi11
voca11
picc11
pica11
bloc11
obje11;

[info6 comp6 simi6 voca6 picc6 pica6 bloc6 obje6]
[info7 comp7 simi7 voca7 picc7 pica7 bloc7 obje7]
[info9 comp9 simi9 voca9 picc9 pica9 bloc9 obje9]
[info11 comp11 simi11 voca11 picc11 pica11 bloc11 obje11];

info6 comp6 simi6 voca6 picc6 pica6 bloc6 obje6
info7 comp7 simi7 voca7 picc7 pica7 bloc7 obje7
info9 comp9 simi9 voca9 picc9 pica9 bloc9 obje9
info11 comp11 simi11 voca11 picc11 pica11 bloc11 obje11;

!residual correlations
info6 WITH info7 info9 info11;
info7 WITH info9 info11;
info9 WITH info11;

comp6 WITH comp7 info9 comp11;
comp7 WITH comp9 comp11;
comp9 WITh comp11;

simi6 WITH simi7 simi9 simi11;
simi7 WITH simi9 simi11;
simi9 WITH simi11;

voca6 WITH voca7 voca9 voca11;
voca7 WITH voca9 voca11;
voca9 WITH voca11;

picc6 WITH picc7 picc9 picc11;
picc7 WITH picc9 picc11;
picc9 WITH picc11;

pica6 WITH pica7 pica9 pica11;
pica7 WITH pica9 pica11;

48

pica9 WITH pica11;

bloc6 WITH bloc7 bloc9 bloc11;
bloc7 WITH bloc9 bloc11;
bloc9 WITH bloc11;

obje6 WITH obje7 obje9 obje11;
obje7 WITH obje9 obje11;
obje9 WITh obje11;

OUTPUT: STANDARDIZED SAMPSTAT TECH4 TECH1;

49

Appendix G
Mplus syntax for a 1 factor partial metric/weak invariance model of WISC data
MODEL:
!First-Order Factor Model – Equal factor loadings across time
WISC1 BY info6@1
comp6(L2)
simi6(L3)
voca6(L4)
picc6(L5)
pica6(L6)
bloc6(L7)
obje6(L8);
WISC2 BY info7@1
comp7(L2)
simi7(L3)
voca7(L4)
picc7(L5)
pica7(L6)
bloc7(L7)
obje7(L8);
WISC3 BY info9@1
comp9(L2)
simi9(L3)
voca9(L4)
picc9(L5)
pica9(L6)
bloc9(L7)
obje9(L8);
WISC4 BY info11@1
comp11(L2)
simi11(L3)
voca11(L4)
picc11(L5)
pica11(L6)
bloc11(L7)
obje11(L8);

[info6 comp6 simi6 voca6 picc6 pica6 bloc6 obje6]
[info7 comp7 simi7 voca7 picc7 pica7 bloc7 obje7]
[info9 comp9 simi9 voca9 picc9 pica9 bloc9 obje9]
[info11 comp11 simi11 voca11 picc11 pica11 bloc11 obje11];

info6 comp6 simi6 voca6 picc6 pica6 bloc6 obje6
info7 comp7 simi7 voca7 picc7 pica7 bloc7 obje7

50

info9 comp9 simi9 voca9 picc9 pica9 bloc9 obje9
info11 comp11 simi11 voca11 picc11 pica11 bloc11 obje11;

OUTPUT: STANDARDIZED SAMPSTAT TECH4 TECH1;

51

Appendix H
Mplus syntax for a 1 factor metric/strong invariance model of WISC data
!First-Order Factor Model
WISC1 BY info6@1
comp6(L2)
simi6(L3)
voca6(L4)
picc6(L5)
pica6(L6)
bloc6(L7)
obje6(L8);
WISC2 BY info7@1
comp7(L2)
simi7(L3)
voca7(L4)
picc7(L5)
pica7(L6)
bloc7(L7)
obje7(L8);
WISC3 BY info9@1
comp9(L2)
simi9(L3)
voca9(L4)
picc9(L5)
pica9(L6)
bloc9(L7)
obje9(L8);
WISC4 BY info11@1
comp11(L2)
simi11(L3)
voca11(L4)
picc11(L5)
pica11(L6)
bloc11(L7)
obje11(L8);

!intercepts – equal means over time
[info6@0];
[comp6] (M2)
[simi6] (M3)
[voca6] (M4)
[picc6] (M5)
[pica6] (M6)
[bloc6] (M7)

52

[obje6] (M8)
[info7@0];
[comp7] (M2)
[simi7] (M3)
[picc7] (M5)
[pica7] (M6)
[bloc7] (M7)
[obje7] (M8)
[info9@0];
[simi9] (M3)
[voca9] (M4)
[picc9] (M5)
[pica9] (M6)
[bloc9] (M7)
[obje9] (M8)
[info11@0];
[comp11] (M2)
[simi11] (M3)
[voca11] (M4)
[picc11] (M5)
[pica11] (M6)
[bloc11] (M7)
[obje11] (M8);

info6 comp6 simi6 voca6 picc6 pica6 bloc6 obje6
info7 comp7 simi7 voca7 picc7 pica7 bloc7 obje7
info9 comp9 simi9 voca9 picc9 pica9 bloc9 obje9
info11 comp11 simi11 voca11 picc11 pica11 bloc11 obje11;

! Means of factors relaxed
[wisc1-wisc4];

OUTPUT: STANDARDIZED SAMPSTAT TECH4 TECH1;

53

Appendix I
Mplus syntax for a 1 factor full metric/strict invariance model of WISC data
!First-Order Factor Model
WISC1 BY info6@1
comp6(L2)
simi6(L3)
voca6(L4)
picc6(L5)
pica6(L6)
bloc6(L7)
obje6(L8);
WISC2 BY info7@1
comp7(L2)
simi7(L3)
voca7(L4)
picc7(L5)
pica7(L6)
bloc7(L7)
obje7(L8);
WISC3 BY info9@1
comp9(L2)
simi9(L3)
voca9(L4)
picc9(L5)
pica9(L6)
bloc9(L7)
obje9(L8);
WISC4 BY info11@1
comp11(L2)
simi11(L3)
voca11(L4)
picc11(L5)
pica11(L6)
bloc11(L7)
obje11(L8);

!intercepts – equal means over time
[info6@0];
[comp6] (M2)
[simi6] (M3)
[voca6] (M4)
[picc6] (M5)
[pica6] (M6)
[bloc6] (M7)

54

[obje6] (M8)
[info7@0];
[comp7] (M2)
[simi7] (M3)
[picc7] (M5)
[pica7] (M6)
[bloc7] (M7)
[obje7] (M8)
[info9@0];
[simi9] (M3)
[voca9] (M4)
[picc9] (M5)
[pica9] (M6)
[bloc9] (M7)
[obje9] (M8)
[info11@0];
[comp11] (M2)
[simi11] (M3)
[voca11] (M4)
[picc11] (M5)
[pica11] (M6)
[bloc11] (M7)
[obje11] (M8);

!Invavariant uniqunesses
info6 info7 info9 info11 (U2_in);
comp6 comp7 comp9 comp11 (U2_co);
simi6 simi7 simi9 simi11 (U2_si);
voca6 voca7 voca9 voca11 (U2_vo);
picc6 picc7 picc9 picc11 (U2_pi);
pica6 pica7 pica9 pica11 (U2_pc);
bloc6 bloc7 bloc9 bloc11 (U2_bl);
obje6 obje7 obje9 obje11 (U2_ob);

! Means of factors relaxed
[wisc1-wisc4];

OUTPUT: STANDARDIZED SAMPSTAT TECH4 TECH1;

55

Appendix J
Mplus syntax for a 1 factor non linear CUFFS model of WISC data

!First-Order Factor Model
WISC1 BY info6@1
comp6(L2)
simi6(L3)
voca6(L4)
picc6(L5)
pica6(L6)
bloc6(L7)
obje6(L8);
WISC2 BY info7@1
comp7(L2)
simi7(L3)
voca7(L4)
picc7(L5)
pica7(L6)
bloc7(L7)
obje7(L8);
WISC3 BY info9@1
comp9(L2)
simi9(L3)
voca9(L4)
picc9(L5)
pica9(L6)
bloc9(L7)
obje9(L8);
WISC4 BY info11@1
comp11(L2)
simi11(L3)
voca11(L4)
picc11(L5)
pica11(L6)
bloc11(L7)
obje11(L8);

!intercepts
[info6](M1)
[comp6] (M2)
[simi6] (M3)
[voca6] (M4)
[picc6] (M5)
[pica6] (M6)

56

[bloc6] (M7)
[obje6] (M8)
[info7] (M1)
[comp7] (M2)
[simi7] (M3)
[picc7] (M5)
[pica7] (M6)
[bloc7] (M7)
[obje7] (M8)
[info9] (M1)
[simi9] (M3)
[voca9] (M4)
[picc9] (M5)
[pica9] (M6)
[bloc9] (M7)
[obje9] (M8)
[info11](M1)
[comp11] (M2)
[simi11] (M3)
[voca11] (M4)
[picc11] (M5)
[pica11] (M6)
[bloc11] (M7)
[obje11] (M8);

!variances
info6 comp6 simi6 voca6 picc6 pica6 bloc6 obje6
info7 comp7 simi7 voca7 picc7 pica7 bloc7 obje7
info9 comp9 simi9 voca9 picc9 pica9 bloc9 obje9
info11 comp11 simi11 voca11 picc11 pica11 bloc11 obje11;

WISC1 WISC2 WISC3 WISC4

! Means of factors relaxed
[WISC1@0 WISC2@0 WISC3@0 WISC4@0];

!residual correlations
info6 WITH info7 info9 info11;
info7 WITH info9 info11;
info9 WITH info11;

comp6 WITH comp7 info9 comp11;
comp7 WITH comp9 comp11;
comp9 WITh comp11;

simi6 WITH simi7 simi9 simi11;

57

simi7 WITH simi9 simi11;
simi9 WITH simi11;

voca6 WITH voca7 voca9 voca11;
voca7 WITH voca9 voca11;
voca9 WITH voca11;

picc6 WITH picc7 picc9 picc11;
picc7 WITH picc9 picc11;
picc9 WITH picc11;

pica6 WITH pica7 pica9 pica11;
pica7 WITH pica9 pica11;
pica9 WITH pica11;

bloc6 WITH bloc7 bloc9 bloc11;
bloc7 WITH bloc9 bloc11;
bloc9 WITH bloc11;

obje6 WITH obje7 obje9 obje11;
obje7 WITH obje9 obje11;
obje9 WITh obje11;

!Growth Model
I BY WISC1@1 WISC2@1 WISC3@1 WISC4@1;
S BY WISC1@0
WISC2*1
WISC3*3
WISC4@5;
[I@0 S* ];
I S;

OUTPUT: STANDARDIZED SAMPSTAT TECH4 TECH1;

58

Appendix K
Mplus syntax for a 1 factor linear FOCUS model of WISC data
MODEL:
!FOCUS Model
INFOL BY info6@1 info7@1 info9@1 info11@1;
INFOS BY info6@0 info7*1 info9@3 info11@5;
[info6@0 info7@0 info9@0 info11@0];
info6 info7 info9 info11;

COMPL BY comp6@1 comp7@1 comp9@1 comp11@1;
COMPS BY comp6@0 comp7@1 comp9@3 comp11@5;
[comp6@0 comp7@0 comp9@0 comp11@0];
comp6 comp7 comp9 comp11;

SIMIL BY simi6@1 simi7@1 simi9@1 simi11@1;
SIMIS BY simi6@0 simi7@1 simi9@3 simi11@5;
[simi6@0 simi7@0 simi9@0 simi11@0];
simi6 simi7 simi9 simi11;

VOCAL BY voca6@1 voca7@1 voca9@1 voca11@1;
VOCAS BY voca6@0 voca7@1 voca9@3 voca11@5;
[voca6@0 voca7@0 voca9@0 voca11@0];
voca6 voca7 voca9 voca11 ;

PICCL BY picc6@1 picc7@1 picc9@1 picc11@1;
PICCS BY picc6@0 picc7@1 picc9@3 picc11@5;
[picc6@0 picc7@0 picc9@0 picc11@0];
picc6 picc7 picc9 picc11;

PICAL BY pica6@1 pica7@1 pica9@1 pica11@1;
PICAS BY pica6@0 pica7@1 pica9@3 pica11@5;
[pica6@0 pica7@0 pica9@0 pica11@0];
pica6 pica7 pica9 pica11;

BLOCL BY bloc6@1 bloc7@1 bloc9@1 bloc11@1;
BLOCS BY bloc6@0 bloc7@1 bloc9@3 bloc11@5;
[bloc6@0 bloc7@0 bloc9@0 bloc11@0];
bloc6 bloc7 bloc9 bloc11;

OBJEL BY obje6@1 obje7@1 obje9@1 obje11@1;
OBJES BY obje6@0 obje7@1 obje9@3 obje11@5;
[obje6@0 obje7@0 obje9@0 obje11@0];
obje6 obje7 obje9 obje11;

59

[INFOL COMPL SIMIL VOCAL PICCL PICAL BLOCL OBJEL F_of_L@0];
[INFOS COMPS SIMIS VOCAS PICCS PICAS BLOCS OBJES F_of_S@0];

F_of_L BY INFOL COMPL SIMIL VOCAL PICCL PICAL BLOCL OBJEL;
F_of_S BY INFOS COMPS SIMIS VOCAS PICCS PICAS BLOCS OBJES;

OUTPUT: STANDARDIZED SAMPSTAT TECH1 TECH4;

60

Appendix L
Mplus syntax for a 2 factor full CUFFS model of
MODEL:
!First-Order Factor Model
V6 BY info6@1
comp6*1(L2)
simi6*1(L3)
voca6*1(L4)
picc6*.5(L5)
pica6*.5(L6)
Bloc6*.5(L7)
obje6@0;

P6 BY info6@0
comp6*.5(L2)
simi6*.5(L3)
voca6*.5(L4)
picc6*1(L5)
pica6*1(L6)
bloc6*1(L7)
obje6@1;

V7 BY info7@1
comp7*1(L2)
simi7*1(L3)
voca7*1(L4)
picc7*.5(L5)
pica7*.5(L6)
bloc7*.5(L7)
obje7@0;

P7 BY info7@0
comp7*.5(L2)
simi7*.5(L3)
voca7*.5(L4)
picc7*1(L5)
pica7*1(L6)
bloc7*1(L7)
obje7@1;

V9 BY info9@1
comp9*1(L2)
simi9*1(L3)
voca9*1(L4)

61

picc9*.5(L5)
pica9*.5(L6)
bloc9*.5(L7)
obje9@0;

P9 BY info9@0
comp9*.5(L2)
simi9*.5(L3)
voca9*.5(L4)
picc9*1(L5)
pica9*1(L6)
bloc9*1(L7)
obje9@1;

V11 BY info11@1
comp11*.5(L2)
simi11*.5(L3)
voca11*.5(L4)
picc11*1(L5)
pica11*1(L6)
bloc11*1(L7)
obje11@0;

P11 BY info11@0
comp11*.5(L2)
simi11*.5(L3)
voca11*.5(L4)
picc11*1(L5)
pica11*1(L6)
bloc11*1(L7)
obje11@1;

info6 comp6 simi6 voca6 picc6 pica6 bloc6 obje6
info7 comp7 simi7 voca7 picc7 pica7 bloc7 obje7
info9 comp9 simi9 voca9 picc9 pica9 bloc9 obje9
info11 comp11 simi11 voca11 picc11 pica11 bloc11 obje11;
!intercepts
[info6] (M1)
[comp6] (M2)
[simi6] (M3)
[voca6] (M4)
[picc6] (M5)
[pica6] (M6)
[bloc6] (M7)
[obje6] (M8)
[info7] (M1)

62

[comp7] (M2)
[simi7] (M3)
[picc7] (M5)
[pica7] (M6)
[bloc7] (M7)
[obje7] (M8)
[info9] (M1)
[simi9] (M3)
[voca9] (M4)
[picc9] (M5)
[pica9] (M6)
[bloc9] (M7)
[obje9] (M8)
[info11] (M1)
[comp11] (M2)
[simi11] (M3)
[voca11] (M4)
[picc11] (M5)
[pica11] (M6)
[bloc11] (M7)
[obje11] (M8);

!Means of Factors relaxed
[V6@0 V7@0 V9@0 V11@0];
[P6@0 P7@0 P9@0 P11@0];

!Growth Model
IV BY V6@1 V7@1 V9@1 V11@1;
IP BY P6@1 P7@1 P9@1 P11@1;

SV BY V6@0
V7@1
V9@3
V11@5;
SP BY P6@0
P7@1
P9@3
P11@5;

[IV@0 IP@0 SV* SP*];
IV IP SV SP;
IV WITH SV;
IP WITH SP;
IV WITH SP;
IV WITH IP;

63

SV WITH SP;
SV WITH IP

OUTPUT: STANDARDIZED SAMPSTAT TECH4 TECH1;

64

Appendix M
SAS syntax for simulating data for a 1 factor CUFFS model and RAM path notation
for fitting the CUFFS and FOCUS models

Title 'Simulation of CUFFS LGM';
options nodate linesize=79 pagesize=59;
***************************************;
Title2 'Generating Simulation Data';
Title3 'Correct model is 1 factor CUFFS';
DATA CUFFS;
*setting mathematical parameters;
*these top parameters preset to limit random estimation;
mu_int = 10; sigma_int = 3; *mean fixed @0;
mu_slope = 5; sigma_slope = 1;
sigma_z = 1; *disturbance/residual is same for all four factors;
mu_u = 0; sigma_u = 1;
L = 1 ;
*setting statistical parameters;
N = 1001; seed = 201003;
*Arrays for more variables;
ARRAY u_score{16} u1-u16;
ARRAY Y_score{16} y1-y16;
ARRAY y_mu{4} (5 6 8 9);
ARRAY Beta{4} (0 1 3 5);
ARRAY f_score{4} f1-f4;

*Generating raw data;
DO _N_ = 1 TO N;
*defining intercept and slope;
f_intercept = mu_int + sigma_int * RANNOR(seed);
f_slope = mu_slope + sigma_slope * RANNOR(seed);
*creating items and associated uniquenesses;
DO m = 1 TO 16;
k=INT((m-1)/4)+1; /* range between 1 and 4 for m=1 to 16 */
f_score{k} = f_intercept + (Beta{k}*f_slope) + sigma_z;
u_score{m} = mu_u + sigma_u * RANNOR(seed);
y_score{m} = y_mu{k}+ f_score{k} + (u_score{m});
END;
KEEP y1--y16 u1--u16 f1-f4 f_intercept f_slope;
OUTPUT;
END;
run;
proc means data=CUFFS mean var std;

65

var y1-y16 f1-f4 f_intercept f_slope;
run;

PROC CALIS
COV DATA=CUFFS;
VAR y1--y16;
RAM
1 1 17 1,
1 2 17 .1 L2,
1 3 17 .1 L3,
1 4 17 .1 L4,

1 5 18 1,
1 6 18 .1 L2,
1 7 18 .1 L3,
1 8 18 .1 L4,

1 9 19 1,
1 10 19 .1 L2,
1 11 19 .1 L3,
1 12 19 .1 L4,

1 13 20 1,
1 14 20 .1 L2,
1 15 20 .1 L3,
1 16 20 .1 L4,

1 17 21 1,
1 18 21 1,
1 19 21 1,
1 20 21 1,

1 17 22 0,
1 18 22 1 S2,
1 19 22 3 S3,
1 20 22 5,

2 1 1 .1 U1,
2 2 2 .1 U2,
2 3 3 .1 U3,
2 4 4 .1 U4,
2 5 5 .1 U5,
2 6 6 .1 U6,
2 7 7 .1 U7,
2 8 8 .1 U8,
2 9 9 .1 U9,

66

2 10 10 .1 U10,
2 11 11 .1 U11,
2 12 12 .1 U12,
2 13 13 .1 U13,
2 14 14 .1 U14,
2 15 15 .1 U15,
2 16 16 .1 U16,
2 17 17 .1 U17,
2 18 18 .1 U18,
2 19 19 .1 U19,
2 20 20 .1 U20,
2 21 21 .1 V5,
2 22 22 .1 V6,
1 1 23 .1 I1,
1 5 23 .1 I1,
1 9 23 .1 I1,
1 13 23 .1 I1,

1 2 23 .1 I2,
1 6 23 .1 I2,
1 10 23 .1 I2,
1 14 23 .1 I2,

1 3 23 .1 I3,
1 7 23 .1 I3,
1 11 23 .1 I3,
1 15 23 .1 I3,

1 4 23 .1 I4,
1 8 23 .1 I4,
1 12 23 .1 I4,
1 16 23 .1 I4,

2 23 23 1;
RUN;

PROC CALIS
COV DATA=CUFFS; /*FOCUS Model*/
VAR y1--y16;
RAM
1 1 17 1,
1 5 17 1,
1 9 17 1,
1 13 17 1,
1 1 18 0,
1 5 18 .1 S2,

67

1 9 18 .1 S3,
1 13 18 1,

1 2 19 1,
1 6 19 1,
1 10 19 1,
1 14 19 1,
1 2 20 0,
1 6 20 .1 S6,
1 10 20 .1 S7,
1 14 20 1,

1 3 21 1,
1 7 21 1,
1 11 21 1,
1 15 21 1,
1 3 22 0,
1 7 22 .1 S10,
1 11 22 .1 S11,
1 15 22 1,

1 4 23 1,
1 8 23 1,
1 12 23 1,
1 16 23 1,
1 4 24 0,
1 8 24 .1 S14,
1 12 24 .1 S15,
1 16 24 1,

1 17 25 11 FI1,
1 19 25 15 FI2,
1 21 25 25 FI3,
1 23 25 35 FI4,

1 18 26 11 FS1,
1 20 26 15 FS2,
1 22 26 25 FS3,
1 24 26 35 FS4,

2 17 17 1 v1,
2 18 18 1 v2,
2 19 19 1 v3,
2 20 20 1 v4,
2 21 21 1 v5,
2 22 22 1 v6,

68

2 23 23 1 v7,
2 24 24 1 v8,

2 1 1 0 U1,
2 2 2 0 U2,
2 3 3 0 U3,
2 4 4 0 U4,
2 5 5 0 U1,
2 6 6 0 U2,
2 7 7 0 U3,
2 8 8 0 U4,
2 9 9 0 U1,
2 10 10 0 U2,
2 11 11 0 U3,
2 12 12 0 U4,
2 13 13 0 U1,
2 14 14 0 U2,
2 15 15 0 U3,
2 16 16 0 U4,

2 25 25 .1 E1,
2 26 26 .1 E2;
run;

69

Appendix N
Formula for Wilks’ lambda transformation to chi-square

Wilks’ lambda can be transformed to a chi-square distribution using the following
formula:
[(P-n+1)/2)-m]*Log Λ(P,n,m)

Where P is the number of dimensions, n is the degrees of freedom for the hypothesis, and
m is the degrees of freedom for the error.

Abstract (if available)

Abstract In 1988, McArdle identified issues modeling multivariate growth using what he termed “second order latent growth curve” models. Specifically, he raised questions about which type of structural equation model to use for longitudinal data analysis. For example, “Should the growth model be fit to common factors extracted from the measured variables (CUFFS)?”

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Asset Metadata

Creator Shelton, Erin Dominique (author)

Core Title A comparison of classical methods and second order latent growth models for longitudinal data analysis

School College of Letters, Arts and Sciences

Degree Master of Arts

Degree Program Psychology

Publication Date 08/05/2010

Defense Date 06/23/2010

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Growth,growth curves,growth modeling,latent growth modeling,longitudinal analysis,OAI-PMH Harvest

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor McArdle, John J. (committee chair), John, Richard S. (committee member), Wilcox, Rand R. (committee member)

Creator Email erinshelton2@gmail.com,eshelton@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3292

Unique identifier UC1326728

Identifier etd-Shelton-3877 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-367789 (legacy record id),usctheses-m3292 (legacy record id)

Legacy Identifier etd-Shelton-3877.pdf

Dmrecord 367789

Document Type Thesis

Rights Shelton, Erin Dominique

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu