CONFIRMATORY BIFACTOR
MODELLING
Philip Hyland
philipehyland@gmail.com
University of Ulster
Lecture Outline
• Describe the nature of bifactor modelling
• Describe issues that arise in conceptualizing and
modelling multidimensionality,
• Outline confirmatory bifactor modelling,
• Differentiate between bifactor and second-order models,
and
• Describe strengths and limitations associated with bifactor
modelling.
DIMENSIONALITY
• Many psychological measures are constructed to
capture a single latent variable of interest (e.g.
depression, anxiety, self-esteem).
• Factor analytic studies often reveal evidence of both
a general factor and multidimensionality.
• When subjected to CFA, a scale almost always
displays a multidimensional structure.
DIMENSIONALITY
• Unsurprising given the challenges inherent in writing
a set of scale items that attempt to;
1. Measure a single latent variable of interest but are
not entirely redundant (i.e., the same question
asked over and over),
2. Are heterogeneous enough to validly represent the
diverse manifestations of the construct, and
3. Provide acceptable reliability.
DIMENSIONALITY
• Need to create a heterogeneous set of
items to appropriately capture the entire
breath of a given construct (e.g.
psychological and psychological
components of anxiety).
• Puts researchers in a difficult position:
• Measure one thing while simultaneously
measuring diverse aspects of this same
thing.
DIMENSIONALITY
• Not surprising to find conflicting
evidence of unidemsionality and
multidimensionality for certain
psychological measures.
• Leads to debates - measuring a single
construct or a variety of distinct, yet
related, constructs?
• Bifactor modelling in its simplest form
offers an appealing solution to these
kinds of problems.
UNIDIMENSIONAL MODEL
• Underlying structure of a measure are usually
assessed according to three general forms.
1. The unidemsional model
•
Each item is influenced by a single common factor
and a uniqueness term that reflects both
systematic and random error components.
UNIDIMENSIONAL MODEL
UNIDIMENSIONAL MODEL
• Often the hoped for structure.
• Summed scores on the individual indicators provide
a clear indication of individual differences on the
latent variable of interest (e.g. depression, selfesteem etc.).
• Variation in each item on the scale is determined by
levels of the general latent variable of interest.
UNIDIMENSIONAL MODEL
• Even when scales are constructed
to capture a single psychological
construct unidimensional solutions
are rarely identified.
• Necessary then to find alternative
multidimensional model structures.
• The second type of structural model
is the correlated traits model.
CORRELATED TRAITS MODEL
CORRELATED TRAITS MODEL
 Latent variable of interest is separated into its component
parts.
 Components usually considered to be related to varying
degrees.
 No attempt to measure a single common construct (e.g.
PTSD) – rather the various components of PTSD are being
measured.
 Multiple latent variables are correlated - no attempt made to
place a measurement structure on the correlated latent
factors.
HIGHER ORDER MODEL
• The third type of structural model normally
considered is a higher order model
HIGHER ORDER MODEL
 Includes a single common source of variation – unlike
correlated traits model
 Correlations between the first order factors are explained in
terms of a higher order factor
 Latent variable of interest is being modelled but at a more
abstract or distal level then in the unidimensional model.
• No direct relationship between the latent variable of interest
(PTSD) and each of the indicators of PTSD.

Rather there is an indirect pathway mediated by the first
order traits.
BIFACTOR MODEL
 An alternative model conceptualisation exists but is rarely
applied in the personality and psychopathology domains. This
type of model is the bifactor model.
BIFACTOR MODEL
• Variation among individual indicators is assumed to arise
for a number of sources.
• In a simple form - all indicators are modelled to load onto
a single common latent factor – the latent variable of
interest (PTSD).
• But, item covariation can also arise due to what are
called ‘group factors’ or in some models ‘nuisance
factors’.
BIFACTOR MODEL
• Bifactor models include two or more uncorrelated
(orthogonal) group/nuisance factors.
• Each indicator is allowed to load onto two distinct latent
variables: A general factor (e.g. PTSD) and one (and only
one!) group/nuisance factor (e.g. Intrusions).
• Covariation is caused by a group of items tapping similar
aspects of the trait (e.g. items measuring symptoms of
Hyperarousal).
FACTOR ANALYSIS
• Appearances of multidimensionality can be the
result of heterogeneous item content
• A given set of indicator are primarily attempting
to measure the latent variable of interest but
they are also purposefully measuring another
factor (this is termed a ‘grouping factor’), or
• A given set of indicators are primarily
attempting to measure the latent variable of
interest but they inadvertently measuring
another factor (this is termed a ‘nuisance
factor),
BIFACTOR OR HIGHER ORDER
• Bifactor model allows researchers to retain the idea of a
single common construct (PTSD) while also recognising
multidimensionality (Intrusions etc.)
• A useful alternative to the traditionally employed higher
order model.
• In many ways the bifactor model and the higher-order
model are very similar but there are a few important
differences that are worth recognising.
BIFACTOR OR HIGHER ORDER
• Higher-order models are frequently used - bifactor models are
not.
• Major difference lies in how multidimensionality is
conceptualised.
• In “correlated traits” models assumption that the observed item
covariation is the result of two or more correlated primary
factors.
• Higher order model - latent variable of interest (PTSD) is
determined by what a set of primary latent variables have in
common not what observable indicators have in common.
BIFACTOR MODEL
• Bifactor models - latent variable(s) of interest explains
some proportion of covariation among observable
indicators
• Other group/nuisance factors explain additional
covariation among observable indicators.
• The latent variable of interest and group/nuisance
factors are therefore on equal conceptual footing and
compete for explaining item covariance—neither factor
“higher” or “lower” than the other.
BIFACTOR MODEL
• In contrast to the higher-order model conceptualisation,
in a bifactor model conceptualisation the latent
variable(s) of interest is measured by what is common
among the observable indicators.
• This means that variations in the intensity of the specific
latent variable of interest will directly influence levels of
the observable indicators rather than indirectly, as in the
case with the higher-order model.
APPLICATIONS
• A bifactor modelling approach can be used to address
important issues that routinely arise in psychometric
analysis of personality and psychopathology measures.
• Bifactor modelling is useful for:
1. Evaluating the plausibility of subscales,
2. Determining the degree to which sum scores reflect a
single factor, and
3. Evaluating the feasibility of applying a unidimensional
model structure to a measure with heterogeneous
indicators.
SUBSCALES
• Researchers often argue about the usefulness of
creating subscales.
• In many cases where a single general factor is broken
down into subscales these traits are highly correlated multicollinearity becomes a problem.
• Our ability to determine the unique contribution of each
of the subscales in predicting some important outcome is
severely compromised.
SUBSCALES
• A second reason often advanced for creating subscales
is that the subscales may have differential correlations
with external variables.
• Technically true but weak justification for “cutting up” a
measure.
• Any two items not perfectly correlated potentially have
different correlations with external variables.
• However it makes little sense to argue that one should
investigate external correlates for each item separately.
SUBSCALES
• A third and very seldom recognised problem with
creating subscales emerges from a bifactor modelling
perspective.
• Subscale scores reflect variation on both a general
factor and a more specific group/nuisance factor.
• The effect of this is that the subscale may appear to be
reliable, but in fact, that reliability is a function of the
general trait, not the specific group factor.
SUBSCALES
• Finally, it has also been noted that subscales are often
so unreliable compared to the composite score that the
composite score is actually a better predictor of an
individual’s true score on a subscale than is the subscale
score itself.
• Sinharay and Puhan (2007) have therefore argued that
subscale scores are seldom, if ever, empirically justified.
• How can bifactor models help?
SUBSCALES
• Provides a method for determining the usefulness of creating
subscales.
• Because the general and group/nuisance factors are
uncorrelated in a bifactor model, inspection of the factor
loadings on the general and group/nuisance factors is
informative.
• If factor loadings are high on the general factor and low on
the group/nuisance factors - makes little sense to create
subscales.
• If factor loadings are high for both the general factor and the
group/nuisance factors - subscale creation should be
considered.
APPLICATIONS
• Reise, Moore, and Haviland (2010) - main problem in
traditional psychometric evaluation of scales is that the
wrong default model is used.
• A unidimensional model is normally used as the default
model and is also frequently a researcher’s ideal
solution.
• Yet, item response data drawn from complex measures
are rarely, if ever, strictly unidimensional.
APPLICATIONS
• In a way it’s also not desirable to achieve such a solution
• To achieve unidimensionality one essentially has to write a
set of items with very narrow conceptual bandwidth (i.e., the
same item written over and over in slightly different ways)
• Results in poor predictive power or theoretical usefulness.
• Reise et al. (2010) argue that a bifactor model, which
contains a single common trait but also allows for
multidimensionality due to item content diversity, provides a
better foundational model for conceptualizing dimensionality.
STRENGTHS
• Ability to maintain a unidimensional conceptualisation while
also recognising issues of multidimensionality.
• What else?
• So far we have focused on the role on grouping factors.
• Factors that are present within a scale that arise from
intentionally attempting to capture complexities of a particular
psychological construct (e.g. the cognitive-affective and
somatic-performance components of depression).
• But, what about nuisance factors?
STRENGTHS
• Factors that are present within a scale which arise
unintentionally when attempting to capture the primary latent
variable(s) of interest.
• When unwanted nuisance factors are present within a scale
the fit of a theoretically derived model may be affected.
• Bifactor models allow researchers to model (control for) these
unwanted nuisance factors
• Allows for a clearer assessment of the adequacy of
theoretically derived models.
LIMITATIONS
• The general and group/nuisance factor must be specified to be
uncorrelated.
• What does it mean to propose the existence of a group factor
such as Intrusions that partially reflect PTSD but which also has
a specific component that is independent of PTSD?
• Many researchers are therefore sceptical that the model itself
makes any sense.
• Highly restrictive assumptions which need to satisfied in order
that group/nuisance factors will be identified.
• Data must be multidimensional but also that the
multidimensionality be well structured - each item measures a
general factor and one, and only one, group/nuisance factor.
CONCLUSION
• Many psychological scales are constructed to measure a
single latent variable of interest (e.g. depression, selfesteem).
• However due to item heterogeneity necessary to capture the
subtleties and complexities of a psych construct scales
invariably display signs of multidimensionality.
• Correlated trait models and higher-order models are
traditionally applied in order to model the underlying structure
of such scales.
CONCLUSION
• Bifactor models provide a plausible and useful alternative to
traditionally employed higher-order model in order to maintain
a unidimensional structure consistent with the theoretical
foundation upon which the measure was constructed while
also taking into account important item grouping factors.
• Bifactor modelling also allows researchers to model (control
for) unwanted/unintended/meaningless nuisance factors
which are preventing theoretically plausible models from
achieving appropriate model fit.
• A number of important strengths and limitations should be
considered when deciding whether or not to conduct a
bifactor analysis.
CONFIRMATORY BIFACTOR
MODELLING
HOW TO CARRY OUT IN
Mplus
EXAMPLE
• We will be further examining the factor structure of the
Measure of Criminal Social Identity (Boduszek et al., 2012).
• 8 Items designed to measure Criminal Social Identity –
Includes three grouping factors.
1. Cognitive Centrality
2. In-Group Affect
3. In Group Ties
SAVING DATA FOR USE IN MPLUS
• We will be using the data set entitled ‘Criminal Identity’
• Unlike SPSS, Mplus does not allow you to use drop-down
commands to estimate the model - you must write the syntax
yourself (don’t panic!).
• It is a good idea to create a shorter data set yourself for your
specific analysis in Mplus.
SAVING DATA FOR USE IN MPLUS
• Mplus cannot directly read an SPSS file.
• Mplus can easily read Tab delimited data, so we can save our
dataset as a .dat file. This can be done by choosing File,
Save as.
• Be sure to untick the box “Write Variable Names to
Spreadsheet”
• We will save the variable names quickly from SPSS by
copying them from the Variable View window and pasting
them into a new text editor or directly into an Mplus input file.
• Ready to open a new Mplus window and start writing syntax.
MPLUS SYNTAX FOR BIFACTOR
SAVING DATA FOR USE IN MPLUS
• First we have to provide a TITLE for our analysis (Criminal
Social Identity Bifactor Model)
• To read our DATA we indicate the location of the .dat file we
saved.
• Under the VARIABLE heading after ‘names are’ you paste in
your variable names from your SPSS data set.
• In the next line, we indicate which values should be
considered missing in each variable. In our example missing
are all (-9).
SAVING DATA FOR USE IN MPLUS
• In USEVAR enter those variables which are to be used for the
current analysis.
• The CATEGORICAL option is used to specify which variables
are treated as binary or ordered categorical (ordinal)
variables in the model and its estimation.
• Under the ANALYSIS heading we must indicate what
ESTIMATOR we will be using.
SAVING DATA FOR USE IN MPLUS
• Because our observed variables are measured on 5point Likert scale we will use Robust Maximum
Likelihood (MLR) estimation.
• If your observed variables are categorical use Estimator
= WLSMV
SAVING DATA FOR USE IN MPLUS
• Because our observed variables are measured on 5-point
Likert scale we will use Robust Maximum Likelihood (MLR)
estimation.
• If your observed variables are categorical use Estimator =
WLSMV
• Under the MODEL command we must specify the bifactor
model
• This is the place where you have to create your latent
variables (seven factors in this example – four general factors
and three nuisance factors).
• In bifactor modelling, like CFA, we use the command “by” to
create latent variables.
SAVING DATA FOR USE IN MPLUS
• The latent variable “CSI” is measured by all 8 items CI1C*
CI2C CI3C CI4IA CI5IA CI6IT CI7IT CI8IT
• For each latent variable created the first indicator must be
followed by an astrix (*)
• This sets the variance of each Latent Variable to be 1 –
Allows for model identification and latent variable scaling.
• We then must indicate....
• CSI@1; C@1; A@1; T@1;
SAVING DATA FOR USE IN MPLUS
• The group factors must all be uncorrelated with the general
factors.
• We need to provide Mplus with this command
• CSI with C-T@0;
• The grouping factors must also be uncorrelated with each
other
• C with A-T@0; A with T@0;
• If you have more than 1 latent variable of interest these are
allowed to correlate.
SAVING DATA FOR USE IN MPLUS
• Once you have created syntax for confirmatory factor
analysis press
to run the model.
• Save this as an input file under some name e.g., CSI bifactor
model.inp in the same folder as the criminal identity.dat file.
• This produces a text output (.out) file stored in the working
directory with the results.
• For this model the output file looks like the following:
MPLUS OUTPUT BIFACTOR
MPLUS OUTPUT BIFACTOR
• The first part of the output provides a summary of the
analysis including:
• The number of groups (1)
• The number of observations (participants included in the
analysis, N = 303)
• The number of items included in the confirmatory model
(number of dependent variables = 8)
• The number of latent variables (4).
• Furthermore, Mplus gives more info which you do not need to
report except what Estimator was used (in this example it was
MLR= robust maximum likelihood).
MPLUS OUTPUT BIFACTOR
• The next step is to investigate how well the model fits the
data.
• This model of CSI was specified and estimated in Mplus as a
restricted bifactor solution.
• Before we look at the factor structure we have to assess the
fit between the data and pre-established theoretical model.
• Goodness-of-fit indices are used to assess model fit – Same
as used in EFA and CFA.
MPLUS OUTPUT BIFACTOR
MPLUS OUTPUT BIFACTOR
• This bifactor model represents a superior fitting model
compared to all other tested models
• We are interested in more details about this model.
• Mplus output provides lots of information however you are
interested only in few of them.
• Unstandardized Factor Loadings and Standard Errors
• Standardized Factor Loadings and Significance Levels
• Factor Correlations.
UNSTANDARDIZED/S.E.
STANDARDIZED LOADINGS 1
MPLUS OUTPUT BIFACTOR
• You should inspect the strength of the factor loadings for the
psychological factors relative to the grouping factors.
• If strong on grouping factors – consider creating subscales.
• If weak or weaker than the general factor – consider
unidimensionality.
• Also you would be interested in the factor correlations for the
psychological factors – if 2 or more.