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A comparison of standard and alternative measurement models for dealing with skewed data with applications to longitudinal data on the child psychopathy scale
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A comparison of standard and alternative measurement models for dealing with skewed data with applications to longitudinal data on the child psychopathy scale
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Content
A COMPARISON OF STANDARD AND ALTERNATIVE MEASUREMENT
MODELS FOR DEALING WITH SKEWED DATA WITH APPLICATIONS TO
LONGITUDINAL DATA ON THE CHILD PSYCHOPATHY SCALE
by
Leslie M. Owen
________________________________________________
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)
December 2010
Copyright 2010 Leslie M. Owen
ii
Dedication
I dedicate this work to my loving family: my parents, Margaret and Stuart Owen, my
brothers, Matthew and Robert Owen, and my future husband, Nikhil Pole. Thank you all
for your love, encouragement, and support through the years.
iii
Acknowledgments
I would like to first thank Jack McArdle for all of his guidance, wisdom, and support
through my first few years of graduate school and, in particular, through the process of
completing this project. I would also like to thank Laura Baker and Richard John for
their invaluable input.
iv
Table of Contents
Dedication ii
Acknowledgments iii
List of Tables vi
List of Figures vii
Abbreviations viii
Abstract ix
Chapter 1: Introduction 1
1.1 Factor Structure of Psychopathic Personality Traits 2
1.2 Longitudinal Factor Models – Standard Approach 6
1.3 Alternative Factor Models 8
1.4 Current Study 12
Chapter 2: Method 13
2.1 Participants 13
2.2 Measures 15
2.3 Procedure 16
2.4 Attrition 16
2.5 Data Screening 18
2.6 Initial CPS Factor Analysis 19
2.7 Evaluation of Alternative Models 20
2.8 Application of Alternative Models: CPS Longitudinal Factorial
Invariance 23
Chapter 3: Results 25
3.1 Attrition 25
3.2 Initial CPS Structural Factor Analysis 25
3.3 Evaluating Alternative Models 28
3.4 Analyzing the CPS Longitudinal Factor Model 34
Chapter 4: Discussion 40
4.1 Methodological Recommendations 40
4.2 Factor Structure of Psychopathic Personality Traits 42
4.3 Additional Measurement Issues of the CPS Data 44
4.4 Limitations & Further Considerations 45
Bibliography 48
v
Appendices
Appendix A: Descriptive Statistics for CPS Subscales Across Waves
for Group 1 52
Appendix B: Descriptive Statistics for Simulated Data 53
Appendix C: Histograms of CPS Subscales Across All Waves of
Assessment 54
Appendix D: CPS Two-Factor Model Figure 60
Technical Appendices
Technical Appendix A: SAS Program Code for Simulating One Factor
Model 61
Technical Appendix B: MPlus Program Code for Metric Invariance 62
Technical Appendix C: MPlus Program Code for Categorical
Longitudinal Factor Model 65
Technical Appendix D-1: SAS Program Code for Logistic Regressions
of Attrition 70
Technical Appendix D-2: Partial SAS Output for Full Logistic
Regressions of Attrition 71
Technical Appendix D-3: Partial SAS Output for Bivariate Logistic
Regressions of Attrition 73
Technical Appendix E: MPlus Program Code for Categorical Factor
Model 75
vi
List of Tables
Table 1: Demographic Descriptive Statistics for Sample 14
Table 2: Model Fit Indices for Initial CPS SFA 25
Table 3: Factor Loadings of Initial CPS SFA Using Three Methods of Analysis 27
Table 4: Estimated Factor Loadings from Simulated Data 31
Table 5: Model Fit for One Factor and Two Factor Models Using Simulated
Data 33
Table 6: Model Fit for CPS Longitudinal Factor Models 35
Table 7: Estimated Factor Loadings from CPS Longitudinal Two Factor Models 35
Table 8: Factor Means from CPS Longitudinal Two Factor Models 38
Table 9: CPS Descriptive Statistics for Group 1 52
Table 10: Descriptive Statistics for Simulated Data 53
vii
List of Figures
Figure 1: Penalty Function Plot for Initial CPS Structural Factor Models 27
Figure 2: Penalty Function Plot for CPS Longitudinal Factor Models 38
Figures 3-38: Histograms of CPS Subscales Across All Waves of Assessment 54
Figure 39: CPS Two-Factor Structure 60
viii
Abbreviations
SFA: Structural Factor Analysis
CPS: Child Psychopathy Scale
DSM: Diagnostic and Statistical Manual of Mental Disorders
PCL: Psychopathy Checklist
SCTS: Southern California Twin Study
SEM: Structural Equation Modeling
USC: University of Southern California
W1: Wave 1 of Assessment
W2: Wave 2 of Assessment
W3: Wave 3 of Assessment
ix
Abstract
The current study examined data from the Southern California Twin Study (Baker et al.,
2006), a longitudinal study assessing risk factors for antisocial behavior in a community
sample of N=1441 individuals who were twins, with specific focus on the Child
Psychopathy Scale (CPS; Lynam, 1997). These longitudinal data reflect self-reported
psychopathic personality traits assessed when individuals from a twin pair were 9-10
years, 11-14 years and 15-17 years of age. When analyzing severe emotional and
behavioral problems in samples such as this one, an inherent problem is the positively
skewed distributions of item responses. This type of data therefore becomes difficult to
accurately analyze using statistical methods of analysis such as structural equation
modeling (SEM), largely because one of the standard assumptions of SEM is the need to
have normally distributed residuals. Therefore, alternative models that deal with this type
of data distribution may be useful, and some of these were examined in the current study.
As a first step, we examined the factor structure of psychopathic personality traits as
measured by the CPS using three different methods: (1) ignoring the problem of
skewness, (2) using inverse transformations of the data in an attempt to normalize the
distributions, and (3) fitting the data using a categorical measurement model, which does
not assume any particular distributions of the data. We then simulated data for a one-
factor model with M = 9 indicators with both normal and non-normal distributions to
assess the effect of fitting skewed data to factor models. Results of the simulations
clearly indicated the categorical approach (3) to be the best method for analyzing
moderately skewed data. This conclusion was based on the accuracy of factor loadings
x
and the distinction between the correct and incorrect factor structure when the answer is
known. The categorical model was then applied to the CPS data to test for longitudinal
factorial invariance of psychopathic personality traits childhood to adolescence and the
numerical results were compared to those of the standard models.
1
Chapter 1: Introduction
Longitudinal data is key to understanding both the change and stability in
developmental processes over time (Baltes & Nesselroade, 1979). However, simply
using repeated measurements does not completely ensure that the factors and traits being
measured over time are in fact the same, even if the same instrument and the same people
are used at each time point (see McArdle, 2007). Therefore, in order to truly know if the
same constructs are being measured in the same way across time, it is necessary to assess
whether the same number of factors and factor patterns remain the same for a given trait.
Although the constructs may be expected to change, it is important to understand whether
these changes are due to the person or due to changes in measurement (Horn & McArdle,
1992; McArdle, 2007).
Data from the “Southern California Twin Study” (SCTS; Baker et al., 2006) were
collected in a longitudinal approach with the hope of assessing risk factors for antisocial
behavior in a community sample of twins. For the current study, these data were
analyzed with specific focus on the Child Psychopathy Scale (CPS; Lynam, 1997), one of
the measures of psychopathy in the SCTS. These data reflect self-reported psychopathic
personality traits collected from the agreeable participants at three different times from
childhood (~9-13 years of age) to adolescence (~14-17 years of age).
When analyzing this type of data in community samples with little or no
psychopathology, an inherent problem is the likelihood of a positively skewed sample
distribution (Micceri, 1989). As severe emotional and behavioral problems occur in
2
small proportions of the general population, the sample distributions tend to be positively
skewed with smaller proportions of people having higher levels of the traits and the
majority having minimal amounts. The resulting data becomes difficult to accurately
analyze using statistical methods of analysis such as structural equation modeling (SEM),
as one of the assumptions of SEM, as well as with other data analytic methods, is having
normally distributed residuals (Bollen, 1989; Kline, 2005; Tomarken & Waller, 2005).
Therefore, alternative models that can accurately analyze this type of data could be
useful. In the present study, this issue was investigated using a particular type of SEM,
the longitudinal factor model using the underlying factor structure of psychopathic
personality traits as measured in children and adolescents as an example.
1.1 Factor Structure of Psychopathic Personality Traits
Psychopathic personalities, exemplified by shallow emotions, lack of guilt and
empathy, and socially deviant behavior, have mostly been the ongoing focus of adult
research (Frick, O'Brien, Wootton, & McBurnett, 1994). However, many have felt that to
truly understand how this type of personality develops over time, its measurement should
begin at a young age (Frick et al.; Lynam, 1997). Part of understanding the development
of psychopathy involves understanding the underlying processes of the construct over
time and whether or not these processes remain stable or change. In the current study,
psychopathic personality traits were examined from childhood through early adolescence
to assess whether the same underlying personality structure that has been found to
represent psychopathy in adults exists prior to adulthood to aid in a better understanding
of the disorder and its antecedents.
3
In children, the focus of antisocial behavioral research has been on describing the
behavior itself and the study of Conduct Disorder rather than the personality traits
underlying the behavior (Lynam, 1997). This primary focus has led to many false
positives and problems in predictive validity (e.g., Robins, 1978; White, Moffitt, Earls,
Robin, & Silva, 1990), as it does not get at the root of what is motivating the behaviors.
For example, in a meta-analysis, Robins reported that most adolescents with antisocial
behaviors do not become antisocial adults, with fewer than 50% of severely antisocial
adolescents becoming antisocial adults. White and colleagues focused on just the
development of antisocial behavior in childhood with similar findings. They found that
85% of preschoolers who were predicted to have antisocial outcomes by age 11 based on
antisocial behavioral tendencies in preschool did not display antisocial behaviors at age
11 as predicted. These findings indicate studying the behaviors alone is not a viable
predictive tool. Still, although psychopathy is thought to occur only in adulthood
(according to the DSM-IV; APA, 2000), existing research has suggested that it can be
meaningfully assessed in adolescence (Forth, Kosson, & Hare, 2003). To further
understand the origins of psychopathy and the underlying mechanisms involved, we can
try to follow a developmental perspective. As a certain degree of disruptive and defiant
behavior is normative in children and adolescents, examining the manifestation of
specific psychopathic personality traits in childhood apart from the behaviors themselves
will provide insight to the development of the disorder.
The conceptualization of the factors composing psychopathy, however, is still
unresolved (Cooke & Michie, 2001). Most of the adult literature in psychopathy focuses
4
on two factors of psychopathic personality traits originally reported by Harpur, Hakstian,
and Hare (1988). The first factor represents the interpersonal and affective facets
involved in the psychopathic personality. The second factor represents social deviance or
facets of impulsivity and irresponsibility. These two factors were determined based on
adult manifestations of psychopathy, and have been the most widely used in research.
However, due to mixed support for the two-factor model, two other latent variable
models have been proposed based on confirmatory factor analysis (SFA) research for
both a three- and four-factor model (Cooke & Michie; Hare & Neumann, 2005). The
three-factor model separated the interpersonal/affective factor from the original two-
factor structure into interpersonal traits (such as lying and manipulation) and affective
traits (such as callousness and shallow affect). The four-factor model subsumed the
three-factor model and separated the two-factor structure even further by separating the
second, socially deviant lifestyle, factor into traits related to lifestyle (such as impulsivity
and irresponsibility) and antisocial traits (such as behavior problems and poor anger
control). Despite these suggestions of alternative factor structures, Lynam (1997)
suggested that the CPS might be best as a one-factor model, due to a high correlation (r =
.95) between the two factors representative of the two-factor structure proposed to reflect
adult psychopathic personalities.
Recently, these three different models were assessed for their applicability to the
youth version of the Psychopathy Checklist (PCL; Forth et al., 2003), which was created
to examine psychopathy from a developmental perspective. The findings of this study,
which focused on an incarcerated male adolescent sample (N = 505), indicated that the
5
four-factor model demonstrated a better fit to the data than the other models, even though
the three-factor model was still a good fit (Neumann, Kosson, Forth, & Hare, 2006).
Although it is unclear whether this model would generalize to the broader construct of
psychopathy or to a nonincarcerated, community-based sample, the results suggest the
underlying factor structure may have a different manifestation prior to adulthood.
Lynam (1997) was the first to develop a measure of psychopathy to specifically
assess psychopathic personality traits in youth. The factor structure of this measure has
yet to be confirmed. In the first attempts at determining a factor structure, Lynam
rejected a two-factor model due to the high correlation between the factors. He based this
model on the one used to represent psychopathic traits in adulthood as measured by the
PCL (see Hare, 1991; Harpur et al., 1988). However, an unpublished study using the
same sample as the current study examined a different factor structure of this measure
based on reports from the children’s caregivers (Bezdjian, Raine, Baker, & Lynam,
2010). The findings of this study compared the two-factor model Lynam originally used
to the three-factor model proposed by Cooke and Michie (2001) and a second two-factor
model based on an exploratory factor analysis from a subsample of the SCTS CPS data.
Their findings suggested the optimal solution to be the alternative two-factor structure
(Callous/ Disinhibited and Manipulative/ Deceitful). The indicators of this two-factor
model exhibited a slightly different structure than those found using the PCL. The
current study examined this same factor structure phenotypically across the first three
waves of assessment of the SCTS. For a potentially more accurate reflection of the
children’s personality, the children’s self-reports were analyzed instead of the caregiver
6
reports. As the caregivers may not have accurately represented their children,
particularly as the children became adolescents, the self-reports may be a more valid of
the children’s personality.
As the Callous/ Disinhibited and Manipulative/ Deceitful two-factor structure
proposed by Bezdjian and colleagues (2010) has only been confirmed on the caregiver
report of the CPS in the first wave, invariance of the factor structure across each wave of
assessment was examined in the current study. This study is among the first to examine
if the same factor structure holds true measured across time and across raters.
1.2 Longitudinal Factor Models – Standard Approach
The primary test of whether the same factor structure is developmentally
consistent across time is to test if it fits invariantly across the three waves of assessment
from childhood through early adolescence. Therefore, the two-factor structure of
psychopathic personality traits was forced to be invariant from Wave 1 to Wave 3. This
step is important for future longitudinal studies of the data as one of the key assumptions
in longitudinal studies is that the constructs measured at different time points are the
same and are indicated by the same scores over time (Horn & McArdle, 1992; McArdle,
2007). For example, if the same factor structure is analyzed for change across time,
change cannot be accurately tested if the same construct is not being measured across
time. Therefore, by assuming the factors were invariant across each wave of assessment,
the longitudinal model of factorial invariance accounted for measurement error by
holding constant the same factor structure for each wave (Meredith, 1993).
To understand the underlying factor structure of a construct, such as psychopathic
7
personality traits, over time, there are multiple components to be addressed to determine
whether construct equivalence over time exists (described in detail in McArdle, 2007).
The first question is not only whether the construct has the same number of factors over
time, but also whether the factor structure or pattern remains the same over time. For
example, there may in fact be two factors underlying psychopathic personality traits from
childhood to adolescence, but the first factor may not always be identified by parasitic
lifestyle, untruthfulness, failure to accept responsibility, manipulativeness, glibness, and
lack of guilt. To determine whether the number of factors and factor structure is the same
at each repeated measurement, the number of factors and structure at each time point is
constrained to be equal (or invariant). If this model is not a good fit to the data, then the
same number of factors and factor structure is not the same over time. If, however, this
model is a good fit to the data, we can then impose more stringent constraints and
determine whether the factor loadings are the same at each time point. The last set of
model restrictions deals with the question of whether the factor score is the same over
time, tested by forcing the means of the indicators, or observed variables, to be invariant
over time. All of these questions deal with the same test of whether we are truly
measuring the same construct at each data assessment (McArdle, 2007; Meredith, 1993).
These questions tested by various levels of model restrictions have previously been
defined as configural and metric invariance (Horn & McArdle, 1980, 1992; McArdle,
2007). Configural Invariance exists when the factor patterns are set to be the same
across time. Partial Metric Invariance exists when the factor patterns as well as the
factor loadings of the subscales at each time point are constrained to be equal over time.
8
Metric Invariance exists when the factor patterns, factor loadings, and intercepts of the
subscales at each time point are constrained to be equal over time.
Determining longitudinal factorial invariance is important for future longitudinal
studies of the data as one of the key assumptions in longitudinal studies is that the
constructs measured at different time points are the same and are indicated by the same
scores over time (Horn & McArdle, 1992). However, as an inherent part of
psychopathology data is its tendency to be positively skewed with fewer numbers of
people experiencing high levels of mental health issues, such as psychopathy,
assumptions of the standard structural equation models may be unmet when fitting them
to this type of data. Specifically, one assumption of the standard models is that the
residuals be normally distributed, which could co-occur with severely skewed data.
Therefore, following the initial longitudinal confirmatory factor models done in the
standard way assuming interval scaling of the outcome variables, alternative models that
attempt to deal with data with non-normally distributed residuals were investigated and
applied to the data.
1.3 Alternative Factor Models
In factor analysis, maximum likelihood estimation of the model parameters
assumes multivariate normality of the residuals (Browne, 1984; Tomarken & Waller,
2005). However, when dealing with psychopathology data from a community sample,
the data will most likely not meet these criteria (Micceri, 1989). Due to this problem,
some statisticians in the field have suggested alternative analytic strategies for fitting data
with non-normally distributed residuals to a structural equation model (e.g., Bentler &
9
Yuan, 1999; Browne, 1982, 1984; Hamagami, 1998; Hilbe, 2008; King, 1998; Muthen &
Muthen, 2007; Sattora & Bentler, 1986, 1988, 1994; Yuan & Bentler, 1998). Others have
suggested simply transforming the data to attempt to normalize the distributions. One
problem with transformation, however, is that there are multiple types of transformations
available and one may need to try multiple different transformations before finding the
one that works for a particular distribution. Sometimes, as in the case of severely
nonnormal distributions, no transformations will work (Kline, 2005, p. 51). Additionally,
when transforming variables, the original scaling is clearly changed, and this can make
interpretations more difficult.
Some of the innovations in SEM have focused on the estimation methods. SEM
appears to rely on normal theory methods, such as maximum likelihood (MLE) and
generalized least squares (GLS) in order to estimate model parameters and test model fit.
Therefore, the ML test statistic tends to reject true models more often than the typically
set α=.05 rejection rate when using non-normal data (Curran, West, & Finch, 1996). In
other words, when fitting SEM with non-normal data, there is risk of an inflated Type I
error. Various solutions have been suggested to deal with violations of this assumption.
Some researchers have compared many of the methods suggested to overcome this
problem using various Monte Carlo simulations to compare the various modified test
statistics (e.g., Curran et al., 1996; Fouladi; 2000).
For example, some have suggested using an estimator with less restrictive
distributional assumptions, such as Browne’s asymptotic distribution free method (1982,
1984). However, this method seems to perform poorly with typical sample sizes (N<500;
10
Curran et al., 1996) and is computationally demanding (e.g., cannot be calculated with
>20 variables; Browne, 1984). Other alterations to the estimation techniques have
included scaling corrections to the ML estimator through use of alternative estimators,
such as the Sattorra-Bentler test statistic (SB; Sattora & Bentler, 1986, 1988, 1994) and
the Yuan-Bentler Residual Based Test Statistic (YB; Bentler & Yuan, 1999; Yuan &
Bentler, 1998). Although Fouladi highly recommended the SB-1986 test statistic based
on results from Monte Carlo simulations, the YB test statistic seems to perform better
when working with sample sizes less than 400 or for data with severely non-normal
distributions (see Boomsma & Hoogland, 2001 for details). However, in use (specifically
in programs such as Mplus; Muthen & Muthen, 2001) these scaling corrections only
apply to the ML test statistic and therefore seem most useful when determining model fit
or the detection of differences between competing models. Although this aspect is useful
and necessary for the interpretation of SEM, scaling corrections do not apply to
parameter estimates such as factor loadings or intercepts, which are important in
interpreting the meaning of a model (Muthen & Muthen, 2005; Tomarken & Waller,
2005).
Alternatives are using a different type of model, such as a categorical model based
on estimation of thresholds (e.g., Muthen & Muthen, 2005). The estimated thresholds
reflect differences between each category of the data, which are defined by the number of
points on an ordinal or categorical scale. The categorical model does not assume any
particular distributions of the data, but does presume the latent variables be normally
distributed (Hamagami, 1998; Muthen & Muthen). By using this alternative categorical
11
model, the estimation technique no longer presumes an equal interval scale. More
restrictive count response models, such as the Poisson and Negative Binomial model
(Hilbe, 2008; Muthen & Muthen) can also be used for analyzing data with skewed
distributions, especially when the reason for the skewness is due to the count nature of
the data. To apply these models, the data must be in the form of a count variable, an
integer value that must be greater than zero as it measures, for example, the number of
times a certain event occurs or whether or not a trait is endorsed. This type of scoring
results in a Poisson (King, 1988) or Negative Binomial distribution (Hilbe, 2008; Rider,
1955; Muthen & Muthen, 2007).
The Poisson Count and the Negative Binomial models are used for counting
conceptually rare events, such as having psychopathic personality traits. This type of
variable applies to the CPS scale, which counts the number of psychopathic personality
traits a person has, as the response format indicates either a presence (coded 1) or absence
(coded 0) of the trait. The problem with this type of data for standard statistical models,
such as Pearson Correlations, is that it can only indicate how psychopathic a person is
and does not measure the other side of the scale i.e., how non-psychopathic a person is).
Therefore, this data will most always result in a skewed distribution when measuring in a
community sample. Although these models may prove to be superior and optimal
estimators for analyzing the current measure of childhood psychopathy, we instead fit the
data to a categorical model, which has no distributional assumptions and is therefore
more inclusive of various distributions. This non-standard model is of potential benefit
12
for the current study as it accounts for the possibility of other types of distributions in
addition to the specific type of skewness seen in Poisson distributions, among others.
1.4 Current Study
In the current study, we examined standard and alternative measurement models
when dealing with skewed data. To determine the potential effects of ignoring the issue
of skewness completely, we simulated a one common factor model with skewed residuals
and fit the data using the correct one factor model. If ignoring the skewness is not a
problem, then this model should yield a good fit to the data and provide proper parameter
estimates. The simplest alternative methods and models discussed above, specifically
transformations of the data and the application of a categorical model, were then applied
to this data and the models were compared to determine the best method for fitting
longitudinal factor models to data with non-normally distributed residuals. The best
method (or methods) as determined by the simulations was then fit to the CPS
longitudinal factor model.
13
Chapter 2: Method
2.1 Participants
The participants for the current study were part of the ongoing University of
Southern California (USC) Twin Study. This longitudinal study has followed N = 1441
different pairs of twins since they were 9 to 10 years old to assess the various risk factors
for antisocial behavior (see Baker, Barton, Lozano, Raine, & Fowler, 2006; Baker,
Jacobson, Raine, Lozano, & Bezdjian, 2007 for a complete description). As the Twin
Study is currently in the fourth wave of assessment, the present study focused on the first
three waves of data assessment (Waves 1, 2, and 3) when the twins were aged 9-10 years,
11-14 years and 15-17 years old (see Table 1 for complete descriptive statistics).
14
Table 1. Demographic Descriptive Statistics for Sample
Percent
N Min Max Mean SD
Age Wave 1
624 8.42 11.09 9.60 0.59
Family SES
598 14.00 66.00 42.27 12.06
Age Wave 2
431 9.96 15.66 11.79 0.92
Age Wave 3
458 13.96 17.27 14.68 0.68
Male 49% 365
Female 51% 379
Caucasian 27% 161
Hispanic 38% 228
Black 14% 86
Asian 4% 27
Native American <1% 1
Group 1
Mixed 17% 103
Age Wave 1
1249 8.42 11.09 9.60 0.59
Family SES
1198 14.00 66.00 42.31 12.07
Age Wave 2
860 9.96 15.66 11.79 0.92
Age Wave 3 916 13.96 17.27 14.68 0.67
Male 49% 740
Female 51% 757
Caucasian 27% 326
Hispanic 38% 456
Black 14% 172
Asian 4% 54
Native American <1% 2
Total
Sample
Mixed 17% 207
For purposes of the current measurement analysis, the total sample was randomly
assigned to two separate groups, each a random sample of one twin per pair. The two
groups were formed using the SAS RANUNI procedure for random selection of
individual twins. The procedure was created such that no pair of twins was included in
the same group and each family was represented. The two groups were of near-equal size
(Group 1 n = 364, Group 2 n = 367) and each had similar breakdowns of gender and age.
15
Although every family was represented, the groups were not exactly equal in size because
the complete twin pair did not always participate in every wave for some families. Group
1 was used for all subsequent CPS analyses in the present study.
2.2 Measures
Child Psychopathy Scale. The CPS (Lynam, 1997) was created based on the
adult measure of psychopathy, the Psychopathy Checklist (PCL; Hare, 1991), and is
intended to capture the same psychopathic constructs as the PCL in childhood. Lynam
attempted to identify items that could measure the constructs described in the PCL from
previously existing scales. However, not all constructs from the PCL were included, as
they have no childhood counterparts. Specifically, promiscuous sexual behavior, early
behavior problems, multiple short-lived marriages, and revocation of conditional release
were not included. Other constructs were not included because they did not correlate
with other items (grandiosity and lack of guilt). In total, 12 of the 20 PCL constructs
were used in the final 52-item CPS. Although Lynam did validate the scale through
comparisons to various other measures, he did not provide estimates of reliability of the
CPS. However, others conducted a reliability and validity study that incorporated a
modified version of the CPS and found it to have, what they termed, “a satisfactory
internal consistency” of α = .77 (Falkenbach, Poythress, & Heide, 2003, p. 797). This
measure of reliability was based on the total scale for a sample of arrested youth referred
to a court diversion program (Falkenbach et al.). In the same study, Falkenbach and
colleagues found the modified CPS to have good criterion-validity among youth
offenders. The only investigation of the distributional properties of the CPS, to the
16
author’s knowledge, has been from other research from the USC Twin Study. One such
study reported the CPS total scores to be positively skewed, ranging from skewness =
1.05 to 1.83 (Isen, Raine, Baker, Dawson, Bezdjian, & Lozano, 2010).
The 12 subscales of the CPS used in the current study (based on 52 yes or no
items) contain assessments of glibness, untruthfulness, callousness, impulsiveness,
boredom susceptibility, manipulation, poverty of affect, parasitic lifestyle, behavioral
dyscontrol, lack of planning, unreliability, and failure to accept responsibility. While the
CPS was administered to both the twins and to their primary caregivers, the current study
only examined the child self-report from the first three waves.
2.3 Procedure
Participants recruited from the Southern California Twin Register were invited to
USC on three separate occasions over the course of up to seven years to take part in the
Twin Study for a full day of assessments. The twins completed a variety of self-report
measures, and their primary caregivers (typically their biological mothers) completed
questionnaires and participated in interviews regarding each twins’ behavior. The
caregivers completed all measures and interviews on one twin before moving on to the
other. Complete details of the procedures and measures can be found in previous studies
(Baker et al., 2006; Baker et al., 2007).
2.4 Attrition
In longitudinal analyses, selective attrition can lead to incorrect results and bias in
the estimates (Heath, Madden, & Martin, 1998). Selective attrition may bias results even
if only the final wave of assessment was being measured, particularly if participants with
17
higher rates of psychopathy were those who did not return for assessment. Of the total
608 individuals from Group 1 who participated in the original sample in Wave 1, 415
(68%) continued participation until at least Wave 3 and 193 individuals (32%)
discontinued participation either by Wave 2 or 3. Of the 193 participants who dropped
out of the study, 16% refused to continue participation, 34% could no longer be found
using the original contact information or other contact resources available, and the
remaining participants were unable to be scheduled for follow up. Individuals who
joined the study either in Wave 2 or 3 were not included in the analysis.
Due to the relatively high number of participants who did not return for Wave 3,
reasons for attrition were investigated. Specifically of interest to the current study, the
total score on psychopathy assessed during Wave 1 was subjected to a logistic regression
to determine if the selection bias was associated with what the current study was
measuring. If so, the data would be missing at random (MAR) and may require
surrogates for the missing data to avoid bias in the estimates.
Additional logistic regression analyses were run to determine possible interactions
of psychopathy with various demographic variables. Demographic variables specifically
included family socioeconomic status (SES) as measured by the Hollingshead Four-
Factor index of Socio-Economic Status (Hollingshead, 1979), language of the interview
(English or Spanish), and the twin’s ethnicity (Caucasian, Hispanic, African American,
Asian, and Mixed Ethnicity).
18
2.5 Data Screening
Before conducting the longitudinal analyses, reasons for incomplete data and
attrition were investigated. Selection bias was examined to determine if any specific
variable was related to selection or specific reasons why participants did not return for the
current wave of assessment. Additionally, the distributions of the CPS subscales at each
wave of assessment were examined for skewness. As multivariate normality of the
residuals is an assumption of covariance structural models, skewed distributions of the
raw data could imply skewness of the residuals. To assess the particular distributional
properties of the subscales, we plotted histograms of each subscale over time (see Figures
3 – 38 in Appendix C). The majority of the distributions were positively skewed ranging
from skewness = .16 to 2.39 for Wave 1; .15 to 2.75 for Wave 2; and .04 to 1.55 for
Wave 3 (see Table 9 in Appendix A for skewness across all waves and subscales), with
skewness = 0 representing a normal distribution. Some distributions were closer to
normal, specifically, the subscale termed Boredom Susceptibility (the lower range of
skewness for Wave 3) and Poverty of Affect (the lower range of skewness for Waves 1
and 2). One difference was seen in the subscale Behavioral Dyscontrol, which had a
bimodal distribution at each wave of assessment indicating that the twins reported they
either had behavioral problems or did not, with fewer reporting moderate behavioral
problems.
Based on the skewness of the distributions, an inverse transform (1/x) could be
applied to all subscales in an attempt to normalize the distributions. This transformation
has been suggested when distributions differ severely from normal or are “J-shaped”
19
(Mosteller & Tukey, 1977). However, some observations were zero (indicating no
endorsement of the trait), so could not be transformed, as the calculation of these
observations would involve dividing by zero. Therefore, a constant of one was added to
every subscale (i.e., x + 1) prior to being transformed. The following inverse
transformation equation was applied to all subscales:
€
1
x +1
[1]
Although these transformations improved the skewness of most, but not all, of the
subscales (see Table 9 in Appendix A), no distributions transformed had a skewness
statistic of 0 and many were still not symmetrical.
Additionally, a reliability analysis of each factor over time was conducted for
Group 1 (the sample used for all CPS analyses in the current study). For the
Manipulative/Deceitful factor, Cronbach’s α was .66 at Wave 1 and .70 both at Wave 2
and Wave 3. For the Callous/Disinhibited factor, Cronbach’s α was .59 at Wave 1 and
.67 both at Wave 2 and Wave 3. Although the .70 reliability coefficient is considered
satisfactory (Cicchetti, 1994), these estimates demonstrate a low internal consistency of
the factors, which could lead to problems in the factor analysis.
2.6 Initial CPS Factor Analysis
The two-factor structure suggested by previous analyses of the CPS was first
examined phenotypically in a SFA using Maximum Likelihood (ML) estimation in the
MPlus modeling program (Muthen & Muthen, 2001) with a random sample of one twin
from each pair in Wave 3. The randomized sample was analyzed prior to the analyses to
20
ensure the percent of males and females was representative of the original sample. The
two-factor model consisted of two latent factors termed Callous/ Disinhibited and
Manipulative/ Deceitful. The Callous/ Disinhibited, factor consisted of the CPS
subscales measuring unreliability, poverty of affect, lack of planning, boredom
susceptibility, impulsivity, behavioral dyscontrol, and callousness. The Manipulative/
Deceitful factor consisted of the CPS subscales measuring manipulativeness, failure to
accept responsibility, glibness, parasitic lifestyle, and untruthfulness (see Figure 39 in
Appendix D). This factor structure was assessed in the standard method, assuming equal
intervals, using the raw data and the inverse transformed data, and was then assessed
using a categorical model.
In order to fit the CPS data to a categorical model, we transformed the original
means with possible boundaries of 0 and 1 (as all items only had a yes or no response
pattern) to a 10-point ordinal scale by changing, for example, a mean of .4 to 4 and a
mean of .6 to 6, etc.
All SFA models were fit under the assumption that the factors were correlated, or
had oblique factor structures. The goodness of fit of the confirmatory factor models was
directly compared through the difference in the chi-square statistic (Δχ
2
). In addition to
the chi-square statistic (χ
2
), all models were compared on the root mean squared error of
approximation (RMSEA) index (Browne & Cudeck, 1993: McDonald, 1989).
2.7 Evaluation of Alternative Models
To test the effect of non-normality of the residuals on measurement models and to
compare alternative methods for analyzing this type of data, we simulated data that had
21
similar distributions to that of the child self-reported CPS subscales from the USC Twin
Study. We first simulated data for a one factor model with M=9 indicators generated
from a normal distribution. Next, we simulated data for the same one factor model, but
skewed the residuals (e) using the g-and-h distribution, first introduced by Tukey (1977),
with the following equation:
€
e =
exp(gZ)−1
g
exp(hZ
2
/2),
[2]
where Z has a normal distribution and g and h are parameters that determine the level of
skewness and kurtosis (see Technical Appendix A for SAS code; Tukey, 1977; Wilcox,
2010). When g = h = 0, e can have a normal distribution. But as g gets large (g > 0), the
distributions become more positively skewed, and as h gets large (h > 0), the distributions
become more heavy-tailed. As the current study focused on the skewness of the
distributions, only g was manipulated to skew the data. We set h = 0 and g > 0 (tested g
= .5, g = 1, g = 1.5, and g = 2). When the value for g was equal to 1.75 or higher, the
exploratory factor analysis could not be performed as the communality was greater than
1. The skewness was still quite high with g = 1.5 (skewness = 18), so this was the value
chosen for the extreme skew and g = .5 was chosen for the moderate skew (skewness =
2). Additionally, three different sets of data were simulated for the three values of g (g =
0, g = .5, and g = 1.5) with low (λ = .3), moderate (λ = .5) and high (λ = .8) factor
loadings. Finally, to be able to use the data in categorical model, we then converted all
simulated data to integers using the INT function in SAS.
22
All sets of data were then fit to both the correct one factor model and to the
incorrect two-factor model to assess the effect of non-normally distributed residuals on
the estimated parameters and fit indices. We expected the model to be a good fit to the
data generated from a normal distribution and to be a poor fit to the data with the non-
normally distributed residuals. This data was then used to determine which methods
provide the most accurate factor loadings and correct number of factors when analyzing
non-normally distributed data.
We first fit each set of data to the standard, continuous model to assess the initial
effect of increasing levels of skewness and varying strengths of factor loadings on the fit
and estimates. Next, we transformed the skewed data in an attempt to normalize the
residuals and fit this data to the same model. We then transformed the data to be on an
ordinal scale of 10 categories (as one requirement of these models is that they can only fit
integers) using the following equation:
€
INT(10(
y +k
1
k
2
))
[3]
with k1 first added in order to produce positive integers, and varying values of k1 and k2
depending on the degree of skewness and strength of factor loadings. The ordinal data
was then fit to a categorical model with the same one-factor and two-factor structure as
the first two methods of analysis.
All models were compared to determine the best method (or methods). Based on
the results from the simulation study, the models that were shown to work best with non-
normal sample distributions were applied to the child self-reported CPS data. These
23
models were then compared to the standard structural models of longitudinal factorial
invariance fit to the raw data.
2.8 Application of Alternative Models: CPS Longitudinal Factorial Invariance
Longitudinal factorial invariance was tested in a series of models with increasing
restrictions for the raw CPS data first fit to the standard model and then compared to the
best alternative determined by the simulations. The first model tested for Partial Metric
Invariance by forcing the previously described factor patterns (see Section 2.6) and
loadings to be invariant over time. Then, Metric Invariance was examined by adding an
additional restriction on the intercepts of both the subscales and the factors. This last
model held the factor pattern, the factor loadings of each subscale, and the intercepts of
each subscale equal over time (see Technical Appendix B). For the categorical model,
instead of forcing the intercepts to be invariant, the thresholds were forced to be invariant
(see Technical Appendix C). The factor means were also estimated for each model with
the factors at the first time period fixed at zero to get a general idea of whether the scores
increased or decreased over time. These models were assessed for goodness of fit using
the fit indices, and the factor loadings of the standard and alternative models were
compared. This series of models was repeated for a one-factor model and the difference
between the chi square relative to the degrees of freedom between the one-factor and
two-factor models was compared for both the standard and alternative models.
All sets of models, including the initial SFA, used MAR assumptions about time
(i.e., in MPlus (Muthen & Muthen, 2001, with analyses specified a TYPE = MISSING)
in order to include all participants in the study, regardless of having complete cases or
24
not, as not all of the participants were assessed for psychopathy at all three time points.
If the longitudinal analyses had been performed on complete cases only, instead of
having N=742 (complete data), it would have dropped to N=231 (only cases with scores
for all subscales across all three time points).
25
Chapter 3: Results
3.1 Attrition
To assess prediction of attrition, a logistic regression analysis was performed
(using SAS PROC LOGISTIC; see Technical Appendix D-1) to determine possible MAR
interpretation of the data. All participants from Wave 1 were included in the analysis. If
these same participants did not return for Wave 3, they were labeled as missing and those
who participated in both Wave 1 and Wave 3 were labeled as non-missing. This new
variable, labeled “missingness” was the outcome variable in the logistic regression
equation. To predict attrition, the variable under investigation, psychopathy (as measured
by the CPS), demographic data, and interactions of psychopathy and demographics were
included. All predictor variables were assessed at Wave 1. The full model with the total
CPS score, demographics, and interaction terms did not predict attrition (χ
2
(15) = 18, see
Technical Appendix D-2). Additionally, the variable under investigation, the total CPS
score, did not predict attrition in a simple bivariate logistic regression (χ
2
(1) = 3, see
Technical Appendix D-3). As these variables did not predict attrition, data for
individuals who did not return was not imputed for Wave 3 analyses.
3.2 Initial CPS Structural Factor Analysis
The factor structure of the CPS was evaluated by SFA (Structural Factor
Analysis; using Mplus; Muthen & Muthen, 2001). As a baseline comparison model, the
no-factor model was evaluated first. Alternative models included a one-factor model
followed by the theoretical two-factor model (see Figure 39 in Appendix D). We first fit
the raw data to the standard model, assuming equal intervals. Then we fit the inverse
26
transformed data to the same model. Finally, we fit the ordinal data to a categorical
model (see Technical Appendix E). For each of these three analyses, a no-factor, one-
factor, and two-factor model were compared.
For all three methods of analysis, the one-factor and two-factor models were both
improvements over the no-factor model, as indicated by the change in the chi-square (χ
2
)
statistic. Additionally, the two-factor models were improvements over the one-factor
models according to the chi square difference relative to the degrees of freedom,
indicating the two factor model to be a better fit to the data than the one factor model.
However, both the one-factor and two-factor models were fairly good fits to the data (see
Table 2 for all fit indices).
Table 2. Model Fit Indices for Initial CPS SFA
Cont
0F
Cont
1F
Cont
2F
Tran
0F
Tran
1F
Tran
2F
Cat
0F
Cat
1F
Cat
2F
N 442 442 442 442 442 442 442 442 442
χ
2
1008 177 147 707 110 95 2029 121 96
df 66 54 53 66 54 53 128 41 41
RMSEA .18 .07 .06 .15 .05 .04 .18 .07 .06
r
F1F2
- - .81 - - .83 - - .82
Δχ
2
- 831 30 - 597 15 - 1908 25
Δdf - 12 1 - 12 1 - 87 0
Δχ
2
/Δdf - 69 30 - 50 15 - 22 -
Model comparison can also be informed by a plot of the “penalty function,”
presented as a scatter plot of the χ
2
statistics as a function of their respective df for each
model (McArdle, 1988). In Figure 1, a plot of each of the initial factor models and their
relation to a regression line can be examined. The regression line runs through the origin
of χ
2
= 0 and df = 0 (out of the range of the pictured plot) with a regression coefficient of
27
B = 6.19. This regression line represents the “penalty line.” Models above this line are
considered to be a poor fit to the data, models clustering right next to the line cannot be
discerned from what would statistically be expected, and models below this line are
considered good fits to the data (the further below the line, the better the fit). This plot
indicates only one model, the no-factor model, to be a poor fit to the data for all three
methods of analysis. The one-factor and two-factor models from all three methods,
however, are all adequate fits if only the χ
2
/df ratio is considered. Based on this plot, as
well as the similar fit indices indicated in Table 2, there is no model that stands out as the
“best fit” to the data. Furthermore, each method of analysis yielded similar findings, so
the best method for analyzing this data is not clear.
Figure 1. Penalty Function Plot for Initial CPS Structural Factor Models
The loadings, however, did change depending on whether we fit the raw data or
the transformed data to the standard SFA or we fit the ordinal data to the categorical
model (see Table 3 for the three sets of loadings). Specifically, while the factor loadings
28
estimated from the transformed data were lower than those estimated from the raw data,
the factor loadings estimated from the categorical model were higher than those estimated
from the raw data. However, when analyzing real data, we cannot know which loadings
are correct. In order to know the right answer, we have to simulate data where the correct
model and factor loadings are known before fitting the data to alternative models to
assess how close the loadings are to those that were simulated.
Table 3. Factor Loadings of Initial CPS SFA Using Three Methods of Analysis
Factor Loadings
Subscale
Continuous
(Raw)
Transformed Categorical
Factor 1: Manipulative/Deceitful
Parasitic Lifestyle .69 .65 .79
Untruthfulness .63 .57 .67
Failure to Accept Responsibility .58 .53 .64
Manipulativeness .57 .52 .63
Glibness .39 .24 .39
Factor 2: Callous/Disinhibited
Impulsivity .62 .56 .66
Poverty of Affect .40 .33 .42
Boredom Susceptibility .57 .51 .60
Behavioral Dyscontrol .54 .51 .61
Unreliability .45 .43 .56
Callousness .37 .30 .40
Lack of Planning .38 .33 .42
3.3 Evaluating Alternative Models
To answer the question of which method yields the most accurate results when
analyzing skewed data, we simulated data with both normal and increasing levels of non-
normal distributions with low, moderate, and high factor loadings. We simulated data
with N=1001 individuals and M=9 observed variables with data for each observation at
29
each time point. The data was simulated for a one factor model indicated by the 9
observed variables first generated from a normal distribution. Three different sets of data
were simulated with g = 0 (no skewness) with low (λ = .3), moderate (λ = .5) and high (λ
= .8) factor loadings. We then simulated data so that the residuals had positively skewed
distributions. The simulated data was skewed using the g-and-h distributions, as
described by Wilcox (2010) and first introduced by Tukey (1977), with the previously
described equation (see Equation 3 in Section 2.7). As the current study focused on the
skewness of the distributions, only g was manipulated to skew the data. For the first
simulation with skewed distributions, we manipulated g until the level of skewness was
similar to that of the CPS distributions. The closest representation was for g = .5 (see
Table 10 in Appendix B for descriptive statistics of both the original and skewed data).
Similar to the normally distributed simulated data, three different sets of data were
simulated with g = .5 (skewness = 2) with low (λ = .3), moderate (λ = .5) and high (λ =
.8) factor loadings. For the final set of simulated data, g was manipulated so that the data
would be severely skewed. We set g = 1.5 (skewness = 18) and simulated three sets of
data with low (λ = .3), moderate (λ = .5) and high (λ = .8) factor loadings. A total of nine
sets of data were simulated to assess the effect of increasing levels of skewness and
strength of factor loadings on fit indices and factor loadings.
All nine sets of continuous data were first fit to a one-factor model (using a SFA
in MPlus). Next, we tried the different suggestions for what to do when fitting skewed
data to a structural equation model. First, we tried transforming the simulated data with
the moderately and severely skewed distributions in an attempt to normalize the residuals
30
using an inverse transformation and fit this data to the one factor model. Next, we used
another transformation on the data to make it ordinal with 10 or fewer categories (MPlus
only allows 10 categories) and fit this data to the categorical model. The estimated factor
loadings of the one factor model from the model using the raw data, the model using the
transformed data, and the categorical model were compared to one another, and the
estimated loadings from each model were compared to the simulated loadings. We then
fit the same 27 models to a two-factor model to assess the difference in fit between the
one factor and two factor models. All results were compared to determine the best
method (or methods) to use when analyzing skewed data with different degrees of factor
loadings.
We first assessed the differences in the factor loadings estimated from the three
methods for the one factor (correct model) only. The estimated factor loadings from all
simulations can be seen in Table 4. For the continuous, raw data, when the data were
normally distributed, the estimated factor loadings were the same as those we simulated.
However, for both levels of skewness, the loadings were lower than the simulated
loadings, whether they were low, moderate, or high. With the severely skewed data, the
factor loadings were extremely low and inaccurate compared to those that were
simulated.
The inverse transformations of the continuous data yielded much better results
than the raw data for both moderate to severe skewness. However, when the data was
normally distributed, transforming the data yielded somewhat lower loadings than what
had been simulated. Although the results from the transformations did yield loadings that
31
were closer to what was expected, for the moderately skewed data, the effect of the
transformations on the loadings was not consistent between the three values of factor
loadings. For the lowest value, the factor loadings were estimated to be slightly higher
than what was simulated, but for the moderate loadings, some of the factor loadings were
lower than what was simulated. For the highest loadings, some of the estimated factor
loadings were lower while some were higher. For the severely skewed data, the
transformations yielded much higher estimated factor loadings than the raw data;
however, about half of these loadings were lower than those that were simulated,
regardless of the degree of the simulated factor loadings.
The categorical models yielded the best results for the moderately skewed data
with moderate factor loadings. Also, changing the data to be ordinal in order to fit it to a
categorical model did not affect the results of the normally distributed data. However,
with the restriction in MPlus of limiting the data to 10 categories when fitting a
categorical model, we could not create categories out of severely skewed data and keep
the distribution. Furthermore, once the data were limited to 10 categories, all y scores
became linearly dependent.
32
Table 4. Estimated Factor Loadings from Simulated Data
No skew g = .5 g = 1.5 Sim’d
λ
Variable
Cont Tran Cat Cont Tran Cat Cont Tran Cat
y1 .28 .17 .28 .16 .28 .36 .03 .17 -
y2 .36 .41 .37 .06 .37 .31 .01 .20 -
y3 .31 .35 .32 .03 .44 .36 .03 .33 -
y4 .37 .29 .39 .37 .36 .44 .00 .28 -
y5 .34 .30 .35 .00 .41 .39 -.01 .33 -
y6 .32 .29 .34 .09 .39 .45 .01 .26 -
y7 .34 .32 .37 .17 .46 .52 -.49 .35 -
y8 .31 .29 .33 .11 .40 .32 -.02 .36 -
.3
y9 .29 .20 .29 .02 .34 .33 .01 .30 -
y1 .48 .28 .48 .22 .39 .52 .03 .21 -
y2 .54 .53 .56 .20 .43 .47 .01 .26 -
y3 .51 .50 .52 .22 .47 .48 .03 .50 -
y4 .54 .49 .55 .32 .48 .55 .00 .34 -
y5 .52 .46 .53 .17 .47 .50 -.01 .48 -
y6 .52 .48 .51 .22 .50 .56 .01 .32 -
y7 .52 .50 .53 .29 .50 .51 -.55 .48 -
y8 .50 .44 .50 .21 .45 .54 -.01 .50 -
.5
y9 .48 .42 .50 .18 .45 .50 .01 .45 -
y1 .80 .72 .81 .55 .55 .66 .61 .25 -
y2 .81 .75 .82 .46 .63 .65 -.02 .35 -
y3 .80 .70 .81 .52 .89 .66 -.01 .76 -
y4 .82 .79 .82 .58 .73 .71 .01 .48 -
y5 .80 .79 .81 .48 .89 .66 -.01 .75 -
y6 .80 .77 .81 .47 .67 .70 -.01 .38 -
y7 .81 .81 .81 .53 .89 .65 -.02 .76 -
y8 .80 .78 .81 .46 .79 .62 .02 .76 -
.8
y9 .79 .79 .81 .44 .80 .65 -.01 .74 -
33
Following the comparison of the estimated factor loadings from the one-factor
model, we fit the same 27 sets of data to a two-factor model to compare the ability to
detect a difference between a one-factor and a two-factor model when the correct answer
is known. These results were based on the difference in the chi square statistic between
these two models (see Table 5 for model fits).
For the normally distributed data, all three methods of analysis clearly detected
the difference between one and two factors and supported the one-factor model no matter
the strength of the factor loadings. This difference, however, increased considerably as
the strength of the factor loadings increased. When the data was moderately skewed,
however, this difference was not detectable for the weak factor loadings, and was barely
detectable for the moderate factor loadings; however, for the strong factor loadings, the
gap between the fit of the two models was much larger. For the moderately skewed data,
both the transformations and categorical models yielded results much closer to those of
the normally distributed data, indicating both methods can detect a difference between a
one-factor and two-factor model when the correct answer is known.
34
Table 5. Model Fit for One Factor and Two Factor Models Using
Simulated Data
No Skew
Simulated
Loadings
Continuous Transform Categorical
1F χ
2
/df 23/27 25/27 31/25
2F χ
2
/df 172/27 136/27 267/25
.3
Δχ
2
149 111 236
1F χ
2
/df 22/27 41/27 19/25
2F χ
2
/df 534/27 424/27 1217/19
Δχ
2
512 383 1198
.5
Δχ
2
/Δdf - - 200
1F χ
2
/df 22/27 465/27 26/26
2F χ
2
/df 1665/27 1951/27 8912/9
.8
Δχ
2
1643 1486 8886
Moderate Skew (g = .5)
Simulated
Loadings
Continuous Transform Categorical
1F χ
2
/df 18/27 46/27 38/24
2F χ
2
/df 20/27 287/27 265/24
.3
Δχ
2
2 241 227
1F χ
2
/df 21/27 26/27 38/25
2F χ
2
/df 56/27 412/27 1215/22
Δχ
2
35 386 1180
.5
Δχ
2
/Δdf - - 393
1F χ
2
/df 21/27 784/27 60/24
2F χ
2
/df 472/27 2287/27 4040/16
.8
Δχ
2
451 1503 3980
3.4 Analyzing the CPS Longitudinal Factor Model
The results from the simulations indicated that the best method for fitting data
with moderate factor loadings and moderately skewed distributions to be the categorical
model. Based on these results, we fit the CPS data to a categorical factor model forced to
be invariant over the three waves of assessment (with invariant loadings and thresholds
35
over time) and again compared the findings to the standard factor models using the raw
data and inverse transformed data.
To test Partial Metric Invariance of the CPS factor structure over time, the two-
factor model of the child self-reported CPS (Lynam, 1997) was forced to be invariant
across the three waves of data assessment by constraining the factor patterns and the
factor loadings of each subscale to be the same at each time point. Next, to test for
Metric Invariance, an additional constraint was placed on the intercepts of the subscales
by forcing the intercepts of each subscale to be invariant across time, while estimating the
factor means of Waves 2 and 3 and fixing the factor means of Wave 1 at zero. These
models were first fit using the raw data, followed by the transformed data, and the two-
factor structure was compared to the one-factor structure. The ordinal data was then fit to
the categorical model with both constraints on the factor loadings and the thresholds,
which are estimated when fitting a categorical model. In this case, there were 9 possible
thresholds for the 10 categories of each CPS subscale, although not all 9 thresholds were
estimated for every subscale as the number of categories varied due to the different
number of items in the scales and the qualities of the traits themselves. The thresholds,
similar to the intercepts of the standard models, were forced to be invariant across time.
Based on the simulations, the categorical model was chosen as the best method of
analysis for data with moderate skewness and moderate factor loadings. Therefore, this
model was interpreted and treated as the most accurate.
The model fit indices associated with each model analyzed are summarized in
Table 6 below. Although all models demonstrated good fit as observed by the low
36
RMSEA values, the clearest distinction between the one-factor and two-factor model was
observed in the results of the categorical model, which had the largest difference in the χ
2
relative to the difference in df, demonstrating the two-factor model to be a better fit than
the one-factor model. Additionally, as seen in Table 6, the categorical model for the two-
factor structure of metric invariance yielded higher factor loadings than the continuous
models both with the raw and transformed data across all subscales (see Table 7 for
factor loadings). The factor loadings for both the raw and transformed data were almost
exactly the same.
Table 6. Model Fit for CPS Longitudinal Factor Models
Loadings
Equal
Continuous
Loadings +
Intercepts
Continuous
Loadings
Equal
Transformed
Loadings +
Intercepts
Transformed
Loadings +
Thresholds
Categorical
1F 2F 1F 2F 1F 2F 1F 2F 1F 2F
N 742 742 742 742 742 742 738 738 742 742
χ
2
1357 1233 1697 1567 1242 1128 1586 1439 751 675
df 613 599 635 623 613 599 635 623 201 201
RMSEA .04 .04 .05 .05 .04 .04 .05 .04 .06 .06
W1 r
F1F2
- .86 - .88 - .85 - .86 - .91
W2 r
F1F2
- .84 - .84 - .83 - .83 - .75
W3 r
F1F2
- .83 - .84 - .83 - .83 - .84
Δχ
2
- 124 - 130 - 114 - 147 - 76
Δdf - 14 - 12 - 14 - 12 - 0
Δχ
2
/Δdf - 9 - 11 - 8 - 12 - -
37
Table 7. Estimated Factor Loadings from CPS Longitudinal Two Factor Models
Factor Loadings
Subscale
Raw Transformed Categorical
Factor 1: Manipulative/Deceitful
Parasitic Lifestyle .64 .63 .78
Untruthfulness .66 .66 .80
Failure to Accept Responsibility .45 .42 .58
Manipulativeness .59 .55 .68
Glibness .38 .35 .45
Factor 2: Callous/Disinhibited
Impulsivity .59 .58 .69
Poverty of Affect .38 .37 .47
Boredom Susceptibility .49 .49 .63
Behavioral Dyscontrol .50 .51 .66
Unreliability .45 .44 .61
Callousness .49 .43 .51
Lack of Planning .28 .28 .30
These models were also compared by the “penalty function” chart of the χ
2
statistics as a function of their respective df for each model (McArdle, 1988). In Figure
2, a plot of each of the longitudinal factor models and their relation to a regression line
can be examined with the same properties as previously discussed (see Section 3.2). For
this set of models, none were above the penalty line, so none were poor fits to the data,
but there were models clustering right next to the line, indicating these models could not
be discerned from what would statistically be expected. These were the metric invariant
models fit using both the continuous and transformed data. Those furthest from the line
were the partial metric invariant models, again, fit using both the continuous and
transformed data. Those in the bottom left below the line, but not quite as far below the
line as the partial metric invariant models, were the categorical models. Although all
models below this line are considered good fits to the data, the further below the line, the
38
better the fit. Based on this plot alone, there is no distinction between the one-factor and
two-factor models for all three methods.
Figure 2. Penalty Line Plot
For the two factor models of metric invariance, we also assessed the change in the
two factor means over time. These means (and their changes over time) can be seen in
Table 8. Both the categorical model and the continuous model using the raw data
estimated similar mean changes, but comparably to the loadings, the categorical model
estimated slightly higher means for all factors except the Callous/Disinhibited factor at
Wave 3. However, the continuous model fit using the transformed data suggested a
different pattern, which may lend support to the problem of interpretation using
transformations. For both the continuous model using the raw data and the categorical
39
model, the means of the Manipulative/Deceitful factor increased slightly from Wave 1 to
Wave 2 and had a larger increase from Wave 2 to Wave 3. But, for the
Callous/Disinhibited factor, the mean decreased slightly from Wave 1 to Wave 2 and
increased from Wave 2 to Wave 3. These differing changes may provide further support
for the two-factor structure as they suggest possible evidence of different trajectories for
the two factors. Still, these changes were not evaluated for significance. We would need
to fit a latent growth model to fully understand these patterns of change over time.
Table 8. Factor Mean from CPS Longitudinal Two Factor Models
Factor
Change
Raw Transformed Categorical
Manipulative/Deceitful W1
.00 .00 .00
Manipulative/Deceitful W2
.12 -.08 .13
Manipulative/Deceitful W3
.80 -.85 .87
Manipulative/Deceitful W2-W1 .12 -.08 .13
W3-W2 .68 -.77 .74
W3-W1 .80 -.85 .87
Callous/Disinhibited W1
.00 .00 .00
Callous/Disinhibited W2
-.10 .18 -.19
Callous/Disinhibited W3
.37 -.32 .21
Callous/Disinhibited W2-W1 -.1 .18 -.19
W3-W2 .47 -.5 .4
W3-W1 .37 -.32 .21
40
Chapter 4: Discussion
The current study compared the effects of skewness on both the fit and estimated
factor loadings generated from standard and alternative factor models and determined the
best method for analyzing moderately skewed data. Results of the simulations indicated
the categorical model to be the best method for analyzing moderately skewed data with
moderate factor loadings and was therefore applied to a longitudinal factor model of
psychopathic personality traits self-reported from childhood to adolescence by
participants in the USC Twin Study.
We first examined the factor structure of psychopathic personality traits as
measured by the CPS (Lynam, 1997) using three different methods. As the sample
distributions were moderately skewed to the right, we compared two alternative methods
for analyzing skewed data to simply ignoring the problem. For the first alternative, we
transformed the data by computing the inverse of all scores and fit the transformed data
to the standard, continuous data model. For the second alternative, we transformed the
data to have an ordinal distribution and fit this data to a categorical model. Then, in order
to know which method was the most accurate when dealing with this type of data, we
simulated data with both normal and non-normal distributions and analyzed this data
using the same three methods.
4.1 Methodological Recommendations
In the evaluation of alternative models, the categorical methods yielded better
results than others, but this depended on the type of data being analyzed. The results of
41
the simulation indicated that a difference between a one-factor model and a two-factor
model is clearly detectable, when the correct model is known and the data are normally
distributed. Furthermore, this difference could still be detected when the data had
moderately skewed distributions for all strengths of factor loadings as long as the data
had been transformed to be more normally distributed or the data was fit to a categorical
model. The categorical model was chosen as a superior method of analysis to the
transformations when fitting data with moderate skewness and moderate loadings due to
the higher accuracy of estimated loadings in the categorical model compared to the
transformations. The categorical model can also be interpreted in an understandable way
to the original scale of measurement, whereas transformations lead to problems in
interpretability. However, the transformations estimated more accurate loadings when
the loadings were low (λ=.3) compared to the categorical model, which overestimated the
loadings in this case.
Despite the finding that the categorical model provided the most accurate results
of the simulated data representative of the CPS data, there are costs to fitting this type of
model. In order to transform the data to be ordinal, the categories have to be integers,
which is the first problem as it got rid of some of the accuracy when the data were
originally continuous with non-integer values. Additionally, using the current software
available for analyzing these models (Mplus; Muthen & Muthen, 2001), we are limited to
10 categories. Therefore, when there are more than 10 values in the data set (as there
were with the simulated skewed data in the current study), some values must be grouped
together, again, costing us additional accuracy.
42
One interesting finding from the simulations was seen in the models fit using the
normally distributed data. When we transformed the data without skewness and fit this
data to the correct, one factor model, the loadings were underestimated, particularly when
the simulated loadings were low. This finding emphasizes the need to always plot data
prior to analyzing it, particularly for those who think transformations will always help. If
you are lucky enough to have normally distributed data, transformations will give you the
wrong answer! On the other hand, if only distributional properties are known, in the case
of severe skewness, transformations are a far better option than simply ignoring the
problem, and are even better than categorical models because of certain limitations
involved in transforming data to have, at most, a 10-point ordinal scale.
However, even when using transformations and categorical models for the CPS
data, minimal differences existed in the fit indices between the different factor models.
The first conclusion to a problem such as this may be that there is no correct model, and
all models may in fact be able to equally explain the underlying factors of psychopathy in
children. Therefore, one might say that the best model is the simplest model: the one-
factor model. This conclusion, however, may not be entirely accurate due to additional
problems in the measurement of the data beyond non-normally distributed residuals as
well as conceptual issues related to the theory of psychopathy.
4.2 Factor Structure of Psychopathic Personality Traits
In the simulations representative of the CPS data, it was clear that the one-factor
structure fit better than the two-factor structure; therefore, the lack of distinct clarity in
determining whether the one-factor or two-factor structure was the best fit to the CPS
43
data is indicative of further issues beyond skewness. Although the categorical two-factor
model was a better fit to the data than the one-factor model, the difference in the χ
2
statistics relative to the difference in df was not as clear a distinction as that found in the
simulations. The inability to decipher a clear difference between the one-factor and two-
factor structure of the CPS may imply additional measurement issues that were not
investigated in the current analysis, such as the skewness of the factor scores, the low
reliability of the subscales, or even related to the high correlation between the factors. It
may also be more of a conceptual issue in terms of the theory of psychopathy.
When validating this scale, Lynam (1997) confirmed that the psychopathic
personality factor structure in adulthood, as determined by the PCL (Hare, 1991),
adequately fit the CPS data; however, he noted a high correlation (r = .95) between the
two factors and therefore decided to only use the total score in further analyses. Despite
this report, others have continued to use the CPS two-factor structure because of their
differing trajectories and third variable correlates (Isen et al., 2010). For instance, Isen
and colleagues found that the two factors have different associations with respect to
internalizing problems and skin conductance reactivity. Results from the current study
also suggested a possible difference in the trajectories of the two factors with the
Manipulative/Deceitful factor increasing from childhood to adolescence and the
Callous/Disinhibited first decreasing from age 9 to age 12 and then increasing from age
12 to 15. Therefore, in attempts to understand the psychopathic personality in relation to
other variables and its change over time, it makes sense to keep the two-factor structure
to get a clearer understanding of the disorder’s development.
44
Similar to the original validation of the CPS (Lynam, 1997), in the current analysis,
the correlation between the factors was also quite high, and sometimes estimated to be a
near perfect correlation (r = .9 for Wave 1). However, while both the one-factor and two-
factor models fit the data well, the two-factor models fit slightly better, evidenced by all
methods used. Along with the conceptual issues regarding the factor’s differing
associations, it may continue to make sense to keep a two-factor structure.
4.3 Additional Measurement Issues of the CPS Data
In addition to the conceptual issues, other measurement issues exist in the CPS data
beyond skewness that could contribute to the difficulty in determining a clear factor
structure. For one, the reliability of the factors in the current sample is particularly low
(see Section 2.5). As the correlations among the observed variables are the basis for
computing both reliability and factor loadings (Reuterberg & Gustafsson, 1992), it is not
surprising that scales with low reliability also have lower factor loadings. The additional
measurement issues of low reliability and high correlation between the two factor scores
in the CPS data that were noted in the current study could be affecting the results as well
as the non-normal sample distributions. Further measurement issues that were not
investigated in the current analysis could also contribute to the parameter estimates and
fit indices, such as the skewness of the factor scores or low reliability of the subscales.
Continued evaluations of these different measurement issues could help clarify their
specific effects when fitting data with these properties to factor models. The simulation
of data with these properties as well as other types of models should be investigated for
better methods of analyzing the factor structure of the CPS. If the psychopathic
45
personality can be understood in its early stages of development, research could lead to
future implementation of quality early interventions and possible prevention of antisocial
behaviors that may result from psychopathic tendencies.
4.4 Limitations & Further Considerations
Adding in the longitudinal component to the final CPS analysis without an
accompanying simulation study raised additional questions regarding which of the three
methods yielded the most accurate result. Specifically, the factor means changed in
differing ways for the model applied to the transformed data compared to the model
applied to the raw data and the categorical model. Further simulations of longitudinal
data could help clarify the most accurate method.
To continue to understand the effect of fitting data with non-normally distributed
residuals to these models, future analyses should also investigate how these effects may
change depending on variation in sample size, as the findings from the current study only
apply to fairly large sample sizes (N=1,001). Furthermore, attrition and missing data
could influence these results, particularly if those who discontinued participation reported
higher levels of the trait under investigation than those who continued.
Finally, although the data for the current study consisted of twin pairs, we focused
on phenotypic analyses of one randomly selected twin, as the purpose of the current study
was primarily measurement oriented. When using both twins in a pair, we would
partition the variance of the latent factors into additive genetic (A), shared environmental
(C), and non-shared environmental (E) influences (Evans, Gillespie, & Martin, 2002).
By adding this biometric structure to the longitudinal factor model, we could then assess
46
the additional effect of skewness on the estimates of these three additional latent
variables. Even without studying this effect using a simulation, we could first assess
whether using a categorical model instead of the standard models changes the estimates
of the genetic and environmental influences in any way.
As the estimation of the genetic components also assumes normally distributed
data (Evans et al., 2002), failure to uphold this assumption could result in similar
problems to those found in the current study. Furthermore, future analysis of the data
should continue with assessing factorial invariance over time by partitioning the variance
of the two factors into genetic and environmental influences from a substantive
perspective. To further disentangle the genetic and environmental influences of
psychopathic personality traits from mid-childhood to adolescence, a biometric model
should be fit to the longitudinally invariant factor structure to evaluate these contributions
to the part of the CPS that was commonly measured at each wave. This final analysis can
be used to further develop the longitudinal dynamic trajectory of psychopathic
personality traits as measured by the CPS. Additionally, this analysis would further
elucidate whether the influences on psychopathic personality traits do remain the same
over time or change in their importance to etiological contributions.
If the factor structure of childhood psychopathic personality traits can be untangled,
future longitudinal analyses of the current sample should be conducted. Specifically,
Latent Curve analyses and biometric models in combination (McArdle, 2006) would both
be valuable tools to understanding how psychopathy develops and changes over time.
The genetic analyses incorporated in these types of models may also further illuminate
47
the underlying factor structure of early inklings of the psychopathic personality.
48
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Appendix A
Table 9. Descriptive Statistics for CPS Subscales Across Waves for Group 1
N Min Max Mean SD Skew SkewT
Wave 1
Glibness 603 0 1.00 0.27 0.20 0.73 0.10
Untruthfulness 603 0 1.00 0.13 0.19 1.59 -1.03
Boredom Susceptibility 603 0 1.00 0.28 0.19 0.35 0.35
Manipulativeness 603 0 1.00 0.14 0.24 1.76 -1.25
Poverty of Affect 603 0 0.90 0.35 0.16 0.16 0.49
Callousness 603 0 1.00 0.17 0.18 1.25 -0.48
Parasitic Lifestyle 603 0 1.00 0.08 0.17 2.39 -1.76
Behavioral Dyscontrol 603 0 1.00 0.33 0.36 0.59 -0.20
Lack of Planning 603 0 1.00 0.30 0.31 0.66 -0.15
Impulsiveness 603 0 1.00 0.31 0.29 0.69 -0.02
Unreliability 603 0 1.00 0.13 0.21 1.54 -1.07
Fail to Accept Responsibility 603 0 1.00 0.38 0.27 0.30 0.46
Wave 2
Glibness 396 0 1.00 0.28 0.24 0.94 -0.12
Untruthfulness 396 0 1.00 0.18 0.25 1.44 -0.86
Boredom Susceptibility 396 0 1.00 0.31 0.25 0.68 0.11
Manipulativeness 394 0 1.00 0.14 0.24 1.81 -1.30
Poverty of Affect 396 0 0.83 0.31 0.17 0.15 0.44
Callousness 394 0 0.80 0.12 0.17 1.39 -0.89
Parasitic Lifestyle 395 0 1.00 0.08 0.17 2.75 -1.95
Behavioral Dyscontrol 389 0 1.00 0.37 0.41 0.54 -0.26
Lack of Planning 396 0 1.00 0.16 0.24 1.49 -0.97
Impulsiveness 391 0 1.00 0.29 0.30 0.85 -0.18
Unreliability 393 0 1.00 0.14 0.22 1.51 -1.04
Fail to Accept Responsibility 394 0 1.00 0.37 0.30 0.52 0.21
Wave 3
Glibness 438 0 1.00 0.34 0.24 0.46 0.33
Untruthfulness 438 0 1.00 0.32 0.26 0.31 0.18
Boredom Susceptibility 438 0 1.00 0.45 0.26 0.04 0.66
Manipulativeness 438 0 1.00 0.23 0.29 1.01 -0.54
Poverty of Affect 438 0 0.90 0.33 0.17 0.12 0.49
Callousness 438 0 0.80 0.14 0.19 1.31 -0.77
Parasitic Lifestyle 438 0 0.75 0.14 0.19 1.22 -0.79
Behavioral Dyscontrol 438 0 1.00 0.46 0.41 0.17 0.14
Lack of Planning 438 0 1.00 0.18 0.28 1.32 -0.90
Impulsiveness 437 0 1.00 0.37 0.31 0.44 0.21
Unreliability 438 0 1.00 0.14 0.23 1.55 -1.04
Fail to Accept Responsibility 438 0 1.00 0.43 0.29 0.14 0.57
53
Appendix B
Table 10. Descriptive Statistics for Simulated Data
No skew g = .5 g = 1.5 Sim’d
λ
Vrbl
Mean SD Skew Mean SD Skew Mean SD Skew
y1 -0.04 1.65 -0.07 1.05 3.42 4.04 26.86 206.55 18.19
y2 0.09 1.77 0.15 1.46 4.68 5.33 70.55 648.33 13.88
y3 0.01 1.80 -0.03 1.29 3.90 4.00 37.07 253.96 11.99
y4 -0.03 1.70 0.10 1.08 3.67 6.38 39.70 616.79 28.43
y5 -0.11 1.84 -0.02 1.23 4.22 5.23 51.25 678.59 28.84
y6 -0.11 1.80 0.05 1.16 4.68 9.22 101.44 2125.34 30.73
y7 0.13 1.82 0.00 1.56 4.40 4.03 51.51 342.86 10.95
y8 -0.04 1.85 0.10 1.44 4.74 5.23 72.19 769.60 19.01
.3
y9 0.01 1.73 0.17 1.38 4.84 8.05 101.62 1848.97 29.72
y1 -0.04 1.85 -0.06 1.01 3.53 3.79 26.80 206.54 18.19
y2 0.08 1.98 0.04 1.42 4.80 5.08 70.50 648.36 13.88
y3 -0.02 2.03 -0.05 1.26 4.02 3.69 37.03 253.99 11.99
y4 -0.05 1.92 0.06 1.05 3.79 5.93 39.66 616.83 28.43
y5 -0.13 2.08 -0.10 1.17 4.32 4.88 51.19 678.60 28.84
y6 -0.13 2.00 0.10 1.10 4.76 8.60 101.38 2125.31 30.73
y7 0.15 2.05 0.00 1.53 4.51 3.77 51.47 342.87 10.95
y8 -0.04 2.07 0.02 1.39 4.82 4.87 72.14 769.60 19.01
.5
y9 0.01 1.95 0.21 1.32 4.95 7.65 101.59 1848.98 29.72
y1 -0.08 2.89 0.06 0.97 4.16 2.24 26.77 206.54 18.20
y2 0.04 3.05 0.05 1.39 5.34 3.83 70.48 648.42 13.88
y3 -0.05 3.07 -0.05 1.20 4.66 2.46 36.98 254.01 11.99
y4 -0.09 3.00 0.04 1.01 4.44 3.82 39.61 616.86 28.43
y5 -0.17 3.13 -0.11 1.14 4.93 3.35 51.14 678.61 28.84
y6 -0.17 3.03 0.09 1.05 5.29 6.18 101.33 2125.28 30.73
y7 0.09 3.12 -0.04 1.47 5.13 2.66 51.44 342.92 10.94
y8 -0.06 3.12 -0.02 1.33 5.33 3.45 72.12 769.57 19.01
.8
y9 -0.02 2.98 0.09 1.27 5.47 5.85 101.52 1848.99 29.72
54
Appendix C
Figure 3. Glibness Histogram at W1 Figure 4. Untruthfulness Histogram at W1
Figure 5. Boredom Susceptibility Histogram at W1 Figure 6. Manipulativeness Histogram at W1
Figure 7. Poverty of Affect Histogram at W1 Figure 8. Callousness Histogram at W1
55
Figure 9. Parasitic Lifestyle Histogram at W1 Figure 10. Behavioral Dyscontrol Histogram at W1
Figure 11. Lack of Planning Histogram at W1 Figure 12. Impulsivity Histogram at W1
Figure 13. Unreliability Histogram at W1 Figure 14. Fail to Accept Responsibility Histogram at W1
56
Figure 15. Glibness Histogram at W2 Figure 16. Untruthfulness Histogram at W2
Figure 17. Boredom Susceptibility Histogram at W2 Figure 18. Manipulativeness Histogram at W2
Figure 19. Poverty of Affect Histogram at W2 Figure 20. Callousness Histogram at W2
57
Figure 21. Parasitic Lifestyle Histogram at W2 Figure 22. Behavioral Dyscontrol Histogram at W2
Figure 23. Lack of Planning Histogram at W2 Figure 24. Impulsivity Histogram at W2
Figure 25. Unreliability Histogram at W2 Figure 26. Fail to Accept Responsibility Histogram at W2
58
Figure 27. Glibness Histogram at W3 Figure 28. Untruthfulness Histogram at W3
Figure 29. Boredom Susceptibility Histogram at W3 Figure 30. Manipulativeness Histogram at W3
Figure 31. Poverty of Affect Histogram at W3 Figure 32. Callousness Histogram at W3
59
Figure 33. Parasitic Lifestyle Histogram at W3 Figure 34. Behavioral Dyscontrol Histogram at W3
Figure 35. Lack of Planning Histogram at W3 Figure 36. Impulsivity Histogram at W3
Figure 37. Unreliability Histogram at W3 Figure 38. Fail to Accept Responsibility Histogram at W3
60
Appendix D
Figure 39. CPS Two-Factor Structure
61
Technical Appendix A
Title
Title 'Simulation of Factor Model';
Title2 'Generating Simulation Data with Moderate Skew (g=.5) and
Moderate Loadings (L=.5)';
Title3 'Correct model is 1 factor model';
DATA G1L5;
*setting mathematical parameters;
*these top parameters preset to limit random estimation;
mu_f = 0; sigma_f = 1.2;
mu_u = 0; sigma_u = 2; *sigma_f^2 + sigma_u^2 = total variance;
L = 1 ;
mu_y = 0;
Beta = 2;
g=.5;
*setting statistical parameters;
N = 1001; seed = 20100907;
*Generating raw data for each person;
DO n1 = 1 TO N;
f1 = mu_f + (sigma_f*RANNOR(seed));
*Arrays for more variables;
ARRAY u_score{9} u1-u9;
ARRAY y_score{9} y1-y9;
ARRAY us_score{9} us1-us9;
ARRAY ys_score{9} ys1-ys9;
*creating items and associated uniquenesses;
DO m = 1 TO 9;
u_score{m} = mu_u + (sigma_u*RANNOR(seed));
y_score{m} = mu_y + L*f1 + u_score{m};
*creating items and associated uniquenesses with
skewness;
us_score{m} = (exp(g*u_score{m})-1)/g;
ys_score{m} = mu_y + L*f1 + us_score{m};
END;
KEEP n1 y1--y9 u1--u9 ys1--ys9 us1--us9 f1;
OUTPUT;
END;
62
Technical Appendix B
TITLE: Factorial Invariance Over Time – Equal Loadings and Intercepts –
2-Factor Model
DATA: FILE = y123cpsG1t.dat;
VARIABLE: NAMES = token
y1glib y1untr y1bore y1mani y1pova y1call y1para
y1behd y1lapl y1impu y1unre y1fail y2glib
y2untr y2bore y2mani y2pova y2call y2para y2behd
y2lapl y2impu y2unre y2fail y3glib y3untr y3bore
y3mani y3pova y3call y3para y3behd y3lapl y3impu
y3unre y3fail;
USEVAR = y1glib y1untr y1bore y1mani y1pova y1call y1para
y1behd y1lapl y1impu y1unre y1fail y2glib
y2untr y2bore y2mani y2pova y2call y2para y2behd
y2lapl y2impu y2unre y2fail y3glib y3untr y3bore
y3mani y3pova y3call y3para y3behd y3lapl y3impu
y3unre y3fail;
MISSING = all(-99);
ANALYSIS: TYPE = MEANSTRUCTURE MISSING;
MODEL:
!Intercepts invariance across waves
[y1glib y2glib y3glib] (i1);
[y1untr y2untr y3untr] (i2);
[y1bore y2bore y3bore] (i3);
[y1mani y2mani y3mani] (i4);
[y1pova y2pova y3pova] (i5);
[y1call y2call y3call] (i6);
[y1para y2para y3para] (i7);
[y1behd y2behd y3behd] (i8);
[y1lapl y2lapl y3lapl] (i9);
[y1impu y2impu y3impu] (i10);
[y1unre y2unre y3unre] (i11);
[y1fail y2fail y3fail] (i12);
!Factor loadings invariant across waves
FACTOR11 by y1para@1 (L1);
FACTOR11 by y1untr*.1 (L2);
63
FACTOR11 by y1fail*.1 (L3);
FACTOR11 by y1mani*.1 (L4);
FACTOR11 by y1glib*.1 (L5);
FACTOR21 by y1impu@1 (L7);
FACTOR21 by y1pova*.1 (L8);
FACTOR21 by y1bore*.1 (L9);
FACTOR21 by y1behd*.1 (L10);
FACTOR21 by y1unre*.1 (L11);
FACTOR21 by y1call*.1 (L12);
FACTOR21 by y1lapl*.1 (L13);
FACTOR11 WITH FACTOR21 (r1);
FACTOR12 by y2para@1 (L1);
FACTOR12 by y2untr*.1 (L2);
FACTOR12 by y2fail*.1 (L3);
FACTOR12 by y2mani*.1 (L4);
FACTOR12 by y2glib*.1 (L5);
FACTOR22 by y2impu@1 (L7);
FACTOR22 by y2pova*.1 (L8);
FACTOR22 by y2bore*.1 (L9);
FACTOR22 by y2behd*.1 (L10);
FACTOR22 by y2unre*.1 (L11);
FACTOR22 by y2call*.1 (L12);
FACTOR22 by y2lapl*.1 (L13);
FACTOR12 WITH FACTOR22 (r2);
FACTOR13 by y3para@1 (L1);
FACTOR13 by y3untr*.1 (L2);
FACTOR13 by y3fail*.1 (L3);
FACTOR13 by y3mani*.1 (L4);
FACTOR13 by y3glib*.1 (L5);
FACTOR23 by y3impu@1 (L7);
FACTOR23 by y3pova*.1 (L8);
FACTOR23 by y3bore*.1 (L9);
FACTOR23 by y3behd*.1 (L10);
FACTOR23 by y3unre*.1 (L11);
FACTOR23 by y3call*.1 (L12);
FACTOR23 by y3lapl*.1 (L13);
FACTOR13 WITH FACTOR23 (r3);
64
[FACTOR11-FACTOR21@0];
[FACTOR12-FACTOR23];
OUTPUT: SAMP STAND RES TECH1;
65
Technical Appendix C
TITLE: Categorical Longitudinal Factor Model 2F
DATA: FILE = y123cpsg1NB.txt;
VARIABLE: NAMES = token fid twin gender
y1glib y1untr y1bore y1mani y1nogu y1pova y1call y1para
y1behd y1lapl y1impu y1unre y1fail y1gran y2glib
y2untr y2bore y2mani y2nogu y2pova y2call y2para y2behd
y2lapl y2impu y2unre y2fail y2gran y3glib y3untr y3bore
y3mani y3nogu y3pova y3call y3para y3behd y3lapl y3impu
y3unre y3fail y3gran twrand;
USEVAR = y1glib y1untr y1bore
y1mani y1pova y1call y1para y1behd
y1lapl y1impu y1unre y1fail
y2glib y2untr y2bore
y2mani y2pova y2call
y2para y2behd y2lapl y2impu
y2unre y2fail
y3glib y3untr y3bore
y3mani y3pova y3call y3para y3behd
y3lapl y3impu y3unre y3fail;
MISSING = all(-99);
CATEGORICAL = y1glib y1untr y1bore
y1mani y1pova y1call y1para y1behd
y1lapl y1impu y1unre y1fail
y2glib y2untr y2bore
y2mani y2pova y2call
y2para y2behd y2lapl y2impu
y2unre y2fail
y3glib y3untr y3bore
y3mani y3pova y3call y3para y3behd
y3lapl y3impu y3unre y3fail;
ANALYSIS: TYPE=MEANSTRUCTURE MISSING ;
ESTIMATOR = WLSMV; ITERATIONS = 10000;
CONVERGENCE = 0.00005; COVERAGE = 0.01;
66
PARAMETERIZATION=THETA;
MODEL:
[y1glib$1 y2glib$1 y3glib$1] (T11);
[y1glib$2 y2glib$2 y3glib$2] (T12);
[y1glib$3 y2glib$3 y3glib$3] (T13);
[y1glib$4 y2glib$4 y3glib$4] (T14);
[y1glib$5 y2glib$5 y3glib$5] (T15);
[y1glib$6 y2glib$6 y3glib$6] (T16);
[y2glib$7 y3glib$7] (T17);
[y2glib$8] (T18);
[y1untr$1 y2untr$1 y3untr$1] (T11);
[y1untr$2 y2untr$2 y3untr$2] (T12);
[y1untr$3 y2untr$3 y3untr$3] (T13);
[y1untr$4 y2untr$4 y3untr$4] (T14);
[y1untr$5 y2untr$5 y3untr$5] (T15);
[y1untr$6 y2untr$6 y3untr$6] (T16);
[y1untr$7 y2untr$7 y3untr$7] (T17);
[y2untr$8] (T18);
[y1bore$1 y2bore$1 y3bore$1] (T11);
[y1bore$2 y2bore$2 y3bore$2] (T12);
[y1bore$3 y2bore$3 y3bore$3] (T13);
[y2bore$4 y3bore$4] (T14);
[y1bore$6 y2bore$6 y3bore$6] (T16);
[y1bore$7 y2bore$7 y3bore$7] (T17);
[y2bore$8] (T18);
[y1mani$1 y2mani$1 y3mani$1] (T11);
[y1mani$2 y2mani$2 y3mani$2] (T12);
[y1mani$3 y2mani$3 y3mani$3] (T13);
[y2mani$4 y3mani$4] (T14);
[y1pova$1 y2pova$1 y3pova$1] (T11);
[y1pova$2 y2pova$2 y3pova$2] (T12);
[y1pova$3 y2pova$3 y3pova$3] (T13);
[y1pova$4 y2pova$4 y3pova$4] (T14);
[y1pova$5 y2pova$5 y3pova$5] (T15);
[y1pova$6 y2pova$6 y3pova$6] (T16);
[y1pova$7 y2pova$7 y3pova$7] (T17);
[y1pova$8 y2pova$8 y3pova$8] (T18);
[y1pova$9] (T19);
67
[y1call$1 y2call$1 y3call$1] (T11);
[y1call$2 y2call$2 y3call$2] (T12);
[y1call$3 y2call$3 y3call$3] (T13);
[y1call$4 y2call$4 y3call$4] (T14);
[y1call$5 y2call$5 y3call$5] (T15);
[y1call$6 y2call$6] (T16);
[y2call$7] (T17);
[y1para$1 y2para$1 y3para$1] (T11);
[y1para$2 y2para$2 y3para$2] (T12);
[y1para$3 y2para$3 y3para$3] (T13);
[y1para$4 y2para$4] (T14);
[y2para$5] (T15);
[y1behd$1 y2behd$1 y3behd$1] (T11);
[y1behd$2 y2behd$2 y3behd$2] (T12);
[y1behd$3 y2behd$3 y3behd$3] (T13);
[y1behd$4 y2behd$4] (T14);
[y1lapl$1 y2lapl$1 y3lapl$1] (T11);
[y1lapl$2 y2lapl$2 y3lapl$2] (T12);
[y1lapl$3 y2lapl$3 y3lapl$3] (T13);
[y1lapl$4 y2lapl$4 y3lapl$4] (T14);
[y1impu$1 y2impu$1 y3impu$1] (T11);
[y1impu$2 y2impu$2 y3impu$2] (T12);
[y1impu$3 y2impu$3 y3impu$3] (T13);
[y1impu$4 y2impu$4 y3impu$4] (T14);
[y1unre$1 y2unre$1 y3unre$1] (T11);
[y1unre$2 y2unre$2 y3unre$2] (T12);
[y1unre$3 y2unre$3 y3unre$3] (T13);
[y1unre$4 y2unre$4 y3unre$4] (T14);
[y1fail$1 y2fail$1 y3fail$1] (T11);
[y1fail$2 y2fail$2 y3fail$2] (T12);
[y1fail$3 y2fail$3 y3fail$3] (T13);
[y1fail$4 y2fail$4 y3fail$4] (T14);
!Factor loadings invariant across waves
FACTOR11 by y1para@1 (L1);
FACTOR11 by y1untr*.1 (L2);
FACTOR11 by y1fail*.1 (L3);
FACTOR11 by y1mani*.1 (L4);
68
FACTOR11 by y1glib*.1 (L5);
FACTOR21 by y1impu@1 (L7);
FACTOR21 by y1pova*.1 (L8);
FACTOR21 by y1bore*.1 (L9);
FACTOR21 by y1behd*.1 (L10);
FACTOR21 by y1unre*.1 (L11);
FACTOR21 by y1call*.1 (L12);
FACTOR21 by y1lapl*.1 (L13);
FACTOR11 WITH FACTOR21 (r1);
FACTOR12 by y2para@1 (L1);
FACTOR12 by y2untr*.1 (L2);
FACTOR12 by y2fail*.1 (L3);
FACTOR12 by y2mani*.1 (L4);
FACTOR12 by y2glib*.1 (L5);
FACTOR22 by y2impu@1 (L7);
FACTOR22 by y2pova*.1 (L8);
FACTOR22 by y2bore*.1 (L9);
FACTOR22 by y2behd*.1 (L10);
FACTOR22 by y2unre*.1 (L11);
FACTOR22 by y2call*.1 (L12);
FACTOR22 by y2lapl*.1 (L13);
FACTOR12 WITH FACTOR22 (r2);
FACTOR13 by y3para@1 (L1);
FACTOR13 by y3untr*.1 (L2);
FACTOR13 by y3fail*.1 (L3);
FACTOR13 by y3mani*.1 (L4);
FACTOR13 by y3glib*.1 (L5);
FACTOR23 by y3impu@1 (L7);
FACTOR23 by y3pova*.1 (L8);
FACTOR23 by y3bore*.1 (L9);
FACTOR23 by y3behd*.1 (L10);
FACTOR23 by y3unre*.1 (L11);
FACTOR23 by y3call*.1 (L12);
FACTOR23 by y3lapl*.1 (L13);
FACTOR13 WITH FACTOR23 (r3);
[FACTOR11-FACTOR21@0];
69
[FACTOR12-FACTOR23];
OUTPUT: PATTERNS SAMPSTAT STANDARDIZED RESIDUAL TECH1;
70
Technical Appendix D-1
TITLE1 "Attrition Analysis";
LIBNAME Data "\\vmware-host\Shared Folders\Leslie\Desktop\Second Year
Project\SAS";
DATA Attrition;
SET Data.attritionG1;
RUN;
proc logistic;
model dropout = CPST Nfamses sex_dum Hispanic Black Asian Mixed
INTLANG YCPSses ycpssex ycpslang YCPSxHisp YCPSxBlack YCPSxAsian
YCPSxMixed/risklimits;
run;
proc logistic;
model dropout = CPST/risklimits;
run;
71
Technical Appendix D-2
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 17.6506 15 0.2815
Score 17.7303 15 0.2771
Wald 17.0730 15 0.3145
The LOGISTIC Procedure
Analysis of Maximum Likelihood Estimates
Standard Wald
Parameter DF Estimate Error Chi-Square Pr>ChiSq
Intercept 1 1.7360 0.5273 10.8405 0.0010
CPS 1 -4.0736 1.9203 4.4998 0.0339
SES 1 -0.1661 0.2930 0.3215 0.5707
Sex 1 -0.0067 0.4560 0.0002 0.9882
Hispanic 1 -0.4870 0.7682 0.4019 0.5261
Black 1 -1.8842 0.7025 7.1936 0.0073
Asian 1 0.7795 1.4134 0.3042 0.5813
Mixed 1 -1.0695 0.7130 2.2496 0.1336
InterviewLang 1 -0.2584 0.7990 0.1046 0.7464
CPSxSES 1 1.0749 1.1182 0.9241 0.3364
CPSxSex 1 1.0306 1.6892 0.3723 0.5418
CPSxLang 1 0.1812 2.7890 0.0042 0.9482
CPSxHisp 1 2.2949 2.8097 0.6671 0.4141
CPSxBlack 1 5.8628 2.4984 5.5066 0.0189
CPSxAsian 1 -3.5760 6.1905 0.3337 0.5635
CPSxMixed 1 4.3071 2.7264 2.4958 0.1141
Odds Ratio Estimates
Point 95% Wald
Effect Estimate Confidence Limits
CPS 0.017 <0.001 0.734
SES 0.847 0.477 1.504
Sex 0.993 0.406 2.428
Hispanic 0.614 0.136 2.769
Black 0.152 0.038 0.602
Asian 2.180 0.137 34.799
72
Mixed 0.343 0.085 1.388
InterviewLang 0.772 0.161 3.698
CPSxSES 2.930 0.327 26.220
CPSxSex 2.803 0.102 76.811
CPSxLang 1.199 0.005 283.610
CPSxHisp 9.924 0.040 >999.999
CPSxBlack 351.696 2.627 >999.999
CPSxAsian 0.028 <0.001 >999.999
CPSxMixed 74.229 0.355 >999.999
73
Technical Appendix D-3
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 2.7186 1 0.0992
Score 2.7591 1 0.0967
Wald 2.7377 1 0.0980
The LOGISTIC Procedure
Analysis of Maximum Likelihood Estimates
Standard Wald
Parameter DF Estimate Error Chi-Square Pr>ChiSq
Intercept 1 1.0886 0.2140 25.8697 <.0001
CPS 1 -1.2804 0.7739 2.7377 0.0980
Odds Ratio Estimates
Point 95% Wald
Effect Estimate Confidence Limits
CPST 0.278 0.061 1.267
74
Technical Appendix E
TITLE: Categorical Two-Factor Model at One Time Point
DATA: FILE = y123cpsg1NB.txt;
VARIABLE: NAMES = token fid twin gender
y1glib y1untr y1bore y1mani y1nogu y1pova y1call y1para
y1behd y1lapl y1impu y1unre y1fail y1gran y2glib
y2untr y2bore y2mani y2nogu y2pova y2call y2para y2behd
y2lapl y2impu y2unre y2fail y2gran y3glib y3untr y3bore
y3mani y3nogu y3pova y3call y3para y3behd y3lapl y3impu
y3unre y3fail y3gran twrand;
USEVAR = y3glib y3untr y3bore
y3mani y3pova y3call y3para y3behd
y3lapl y3impu y3unre y3fail;
MISSING = all(-99);
CATEGORICAL = y3glib
y3untr y3bore
y3mani y3pova y3call y3para y3behd
y3lapl y3impu y3unre y3fail;
ANALYSIS: TYPE=MEANSTRUCTURE MISSING ;
ESTIMATOR = WLSMV; ITERATIONS = 10000;
CONVERGENCE = 0.00005; COVERAGE = 0.01;
PARAMETERIZATION=THETA;
MODEL:
[y3glib$1] (T11);
[y3glib$2] (T12);
[y3glib$3] (T13);
[y3glib$4] (T14);
[y3glib$5] (T15);
[y3glib$6] (T16);
[y3glib$7] (T17);
[y3untr$1] (T21);
[y3untr$2] (T22);
[y3untr$3] (T23);
[y3untr$4] (T24);
[y3untr$5] (T25);
75
[y3untr$6] (T26);
[y3untr$7] (T27);
[y3bore$1] (T31);
[y3bore$2] (T32);
[y3bore$3] (T33);
[y3bore$4] (T34);
[y3bore$6] (T36);
[y3bore$7] (T37);
[y3mani$1] (T41);
[y3mani$2] (T42);
[y3mani$3] (T43);
[y3mani$4] (T44);
[y3pova$1] (T51);
[y3pova$2] (T52);
[y3pova$3] (T53);
[y3pova$4] (T54);
[y3pova$5] (T55);
[y3pova$6] (T56);
[y3pova$7] (T57);
[y3pova$8] (T58);
[y3call$1] (T61);
[y3call$2] (T62);
[y3call$3] (T63);
[y3call$4] (T64);
[y3call$5] (T65);
[y3para$1] (T71);
[y3para$2] (T72);
[y3para$3] (T73);
[y3behd$1] (T81);
[y3behd$2] (T82);
[y3behd$3] (T83);
[y3lapl$1] (T91);
[y3lapl$2] (T92);
[y3lapl$3] (T93);
[y3lapl$4] (T94);
[y3impu$1] (T101);
[y3impu$2] (T102);
76
[y3impu$3] (T103);
[y3impu$4] (T104);
[y3unre$1] (T111);
[y3unre$2] (T112);
[y3unre$3] (T113);
[y3unre$4] (T114);
[y3fail$1] (T121);
[y3fail$2] (T122);
[y3fail$3] (T123);
[y3fail$4] (T124);
FACTOR1 by y3para@1 (L1);
FACTOR1 by y3untr*.2 (L2);
FACTOR1 by y3fail*.2 (L3);
FACTOR1 by y3mani*.2 (L4);
FACTOR1 by y3glib*.2 (L5);
FACTOR2 by y3impu@1 (L6);
FACTOR2 by y3pova*.2 (L7);
FACTOR2 by y3bore*.2 (L8);
FACTOR2 by y3behd*.2 (L9);
FACTOR2 by y3unre*.2 (L10);
FACTOR2 by y3call*.2 (L11);
FACTOR2 by y3lapl*.2 (L12);
OUTPUT: PATTERNS SAMPSTAT STANDARDIZED RESIDUAL TECH1;
Abstract (if available)
Abstract
The current study examined data from the Southern California Twin Study (Baker et al., 2006), a longitudinal study assessing risk factors for antisocial behavior in a community sample of N=1441 individuals who were twins, with specific focus on the Child Psychopathy Scale (CPS
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Asset Metadata
Creator
Owen, Leslie M.
(author)
Core Title
A comparison of standard and alternative measurement models for dealing with skewed data with applications to longitudinal data on the child psychopathy scale
School
College of Letters, Arts and Sciences
Degree
Master of Arts
Degree Program
Psychology
Publication Date
11/22/2010
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
child psychopathy scale,factor analysis,factorial invariance,longitudinal data,nonnormal distributions,OAI-PMH Harvest,simulation study,skewed data
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
McArdle, John J. (
committee chair
), Baker, Laura A. (
committee member
), John, Richard S. (
committee member
)
Creator Email
lesliemowen@gmail.com,leslieow@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m3553
Unique identifier
UC1309259
Identifier
etd-Owen-4208 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-410147 (legacy record id),usctheses-m3553 (legacy record id)
Legacy Identifier
etd-Owen-4208.pdf
Dmrecord
410147
Document Type
Thesis
Rights
Owen, Leslie M.
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
Tags
child psychopathy scale
factor analysis
factorial invariance
longitudinal data
nonnormal distributions
simulation study
skewed data