Developments in the Factor Analysis of
Individual Time Series
Michael W. Browne
Guangjian Zhang
The Ohio State University
Factor Analysis at 100: Historical Developments and Future Directions
L. L. Thurstone Psychometric Laboratory, University of North Carolina – Chapel Hill
May 13-15, 2004.
1
Outline
ñ Cattell's Original Contribution
ñ T.W. Anderson's Comments
ñ Two Dynamic Factor Analysis Models
ñ Two Approaches to Estimation
ñ Recent Developments
2
Origins of the P-Technique
ñ First carried out, named and promoted by Raymond
Cattell
ç Factor analysis is carried out on one person at a time.
♦ Repeated observations on tests over time on the same person
are analyzed in the same way as repeated observations on tests
over different persons at the same time.
♦ Correlated and factor analyzed in the usual manner.
♦ No special-purpose computational methodology required.
ñ Many applications since then. Reviews:
ç Luborsky, L., & Mintz, J. (1972)
ç Jones & Nesselroade (1990)
3
Initial Aims of P-Technique
ñ Cattell, Cattell & Rhymer (Psychometrika, 1947)
“P-Technique Demonstrated á ” (Self Ratings)
ç First empirical application
ñ Questions to be answered using P-technique:
ç Does a factor analysis on an individual yield a similar factor
pattern to that obtained from a large group of persons?
ç To what extent do factor patterns obtained from different
individuals differ.
ñ No mention was made at this stage of lagged correlation
or time series.
4
Early Reactions to P-technique
ñ T. W. Anderson (Psychometrika, 1963).
ç Pointed out that information about change over time is neglected
ç Two stage procedure: Factor analysis + Time series analysis of
factor score estimates.
ç Difficulty: Factor scores and factor variables do not have the same
time series.
ç Implied that factors represented a latent time series causing
concurrent and lagged correlations amongst manifest variables
ç Impetus to later work on factor analysis of individual time series
♦ Maximum likelihood estimation. No factor scores.
ñ Wayne Holtzman (1963, Harris volume)
ç Pointed out that lagged correlations and cross-correlations are
disregarded.
5
Influence on Cattell
ñ Cattell (1963, Harris volume on measurement of change)
ç Change over time is now given specific attention. Comments by
Anderson and Holtzman are discussed.
ç Rather than considering the effects of lagged shocks on current
variables which is usually done in time series, Cattell perceived
them as lead effects of current shocks on future variables.
♦ Mathematically equivalent when the time series is stationary, but
not conceptually equivalent.
♦ Stated that factors could be conceived of as being dependent or
independent: “á among factors some are dependent and some
are independent” (Cattell,1963, p. 188)
♦ Implied that factors could be latent shock variables influencing
future manifest variables at various leads: “The aim is now to
discover the interval between a change in the factor level and the
change in the variable level which it causes.” (Cattell,1963, p.
188)
6
Emergence of Dynamic Factor Analysis
ñ The Cattell (1963) chapter and Anderson (1963) article served as
impetus to the development of methodology for the factor analysis of
individual time series.
ñ Dynamic factor analysis: Factor analysis model with a few factors that
account for change in a larger number of manifest variables over time.
ñ Two main types of dynamic factor analysis (cf. Browne &
Nesselroade, McDonald Festschrift)
ç Process Factor Analysis Model: Factors follow a stationary vector
time series that induces a time series for manifest variables.
ç Shock Factor Analysis Model: Factors are random shocks that are
distributed independently across time but may be concurrently
correlated.
7
Process Factor Analysis (PFA) Model
ñ Influenced by Anderson's view of factors following a VARMA
time series.
ñ Key contributions: Engle & Watson (1981), Immink (1986),
Nesselroade, McArdle, Aggen & Meyers (2002)
ñ Data model:
ç Factor analysis model for manifest variables:
C> œ .  A0>  ?>
ç VARMAa:ß ; b time series model for factors:
0> œ "Ej 0>j  D>  "Fj D>j
:
;
jœ"
jœ"
8
PFA(1, 0) Model
ε11
ε22
ε33
ε11
ut,1
ut,2
ut,3
ut+1,1 ut+1,2 ut+1,3
1
1
1
yt,1
yt,2
yt,3
λ1
α
λ2
1
1
λ1
α
ψ1/2
ε33
ε11
λ2
1
1
1
λ1 λ2
α
ψ1/2
ε33
1
yt+2,1 yt+2,2 yt+2,3
λ3
ft+1
ε22
ut+2,1 ut+2,2 ut+2,3
yt+1,1 yt+1,2 yt+1,3
λ3
ft
ε22
λ3
ft+2
ψ1/2
zt
zt+1
zt+2
1
1
1
9
Shock Factor Analysis (SFA) Model
ñ Concordant with Cattell's view that shock factors influence
future manifest variables at various time points.
ñ Key contributions: Geweke & Singleton (1981), Molenaar
(1985)
ñ Data model WJ Ea; ‡ b
‡
C> œ .  "Aj D>j
 ?>
;
jœ!
Thus A! ß A" ß A# ß á ß A;‡ ß represent the effect of a shock
factor vector variableß D> ß on the manifest vector variable,
C> ß at the same time period, one time period in advance,
two time periods in advanceß á ; time periods in advance.
10
Shock Factor Analysis Model
ε11
ε22
ε33
ε11
ut,1
ut,2
ut,3
ut+1,1 ut+1,2 ut+1,3
1
1
1
yt,1
yt,2
yt,3
λ1(0)
λ3(0)
λ2(0)
ε22
1
1
ε33
λ2(1)
λ3(1)
λ1(2)
λ2(2)
ε22
ε33
ut+2,1 ut+2,2 ut+2,3
1
yt+1,1 yt+1,2 yt+1,3
λ1(1)
ε11
1
1
1
yt+2,1 yt+2,2 yt+2,3
λ3(2)
zt
zt+1
zt+2
1
1
1
11
Relationships Between PFA and SFA
ñ In PFA the emphasis is on a latent time series
representing a dynamic psychological process.
ñ In SFA the emphasis is on latent shocks driving a manifest
time series.
ñ Any PFAa:ß ; b model may be approximated arbitrarily
closely by a SFAa; ‡ b model if ; ‡ is chosen to be large
enough. The approximating model will have substantially
more parameters.
ñ It is straightforward to determine the SFAa; ‡ b model that
approximates a given PFAa:ß ; b model. The SFAa; ‡ b
model represents the implied effect of the random shocks
of the PFAa:ß ; b model on the manifest variables.
12
Relationship Between PFA and SFA
ε11
ε22
ε33
ε11
ut,1
ut,2
ut,3
ut+1,1 ut+1,2 ut+1,3
1
1
1
yt,1
yt,2
yt,3
λ 1 λ2
ε22
1
1
λ1
λ3
α
ft
ψ1/2
λ2
λ3(1)
ε11
1
λ2(2)
ψ1/2
1
λ1
1
λ2
λ3
λ3(2)
α
ft+1
ε33
yt+2,1 yt+2,2 yt+2,3
λ3
λ1(2)
ε22
ut+2,1 ut+2,2 ut+2,3
1
yt+1,1 yt+1,2 yt+1,3
(1)
(0) λ1
λ1(0)
λ
λ2(1)
λ2(0) 3
α
ε33
ft+2
ψ1/2
zt
zt+1
zt+2
1
1
1
13
Two Approaches to Estimation - I
There are two main approaches to estimation in dynamic
factor analysis.
ñ Full Information Maximum Likelihood. Engle & Watson
(1981), Immink (1986).
ç A likelihood function based on the raw data is maximized.
(Maximum Normal Likelihood)
♦ Estimators with known asymptotic properties are obtained.
♦ Computationally intensive. A large data set is stored and
manipulated by the program.
♦ No program is currently available.
14
Two Approaches to Estimation - II
ñ Two stage procedure. Calculate lagged correlation matrix
and apply standard structural equation modeling software
to obtain parameter estimates. Molenaar (1985),
Nesselroade, McArdle, Aggen & Meyers (2002).
ç Can be used in practice now!
ç Provides convenient residuals.
ç Maximum likelihood estimation with its associated statistical
theory is not available.
♦ Maximum Wishart likelihood is often used but the likelihood
function is inappropriate and no associated statistical properties
are known.
15
Stationarity Assumption
ñ Fundamental assumption for use of lagged correlation
coefficients
ç Stationarity
CovaC" ß C# b
œ CovaC# ß C$ b œ CovaC$ ß C% b œ â
CovaC" ß C"j b œ CovaC# ß C#j b œ CovaC$ ß C$j b œ â aj !
ç Implies 3"# œ 3#$ œ 3$% œ â
3"$ œ 3#% œ 3$& œ â
3"% œ 3#& œ 3$' œ â etc.
ñ For consistency of assumptions any time series model
fitted to lagged correlations should be constrained to be
stationary.
16
Path Diagram of a PFA(2,0) Model
ε11
ε22
ε33
ε11
ε22
ε33
ε11
ε22
ε33
E11
E21
E31
E12
E22
E32
E13
E23
E33
1.0
1.0
ACT1
LIV1
λ11
1.0
S1En
θ13 θ12
θ24
θ34
ACT2
ψ22
1.0
1.0
θ44
λ42
[Α2]11
ψ11
ψ22
Ζ21
1.0
λ52
[Α2]22
[Α1]22
Fa1
1.0
PEP2
ACT3
λ31
λ21
1.0
LIV3
λ11
[Α1]11
Ζ12
PEP3
λ31
λ21
[Α1]11
[Α2]11
1.0
[Α1]12
ψ12
ψ11
ψ22
Ζ22
Ζ13
[Α1]12
ψ12
Ζ23
1.0
1.0
λ52
1.0
En3
1.0
[Α1]22
Fa2
λ42
λ62
1.0
En2
ψ12
1.0
S1Fa
LIV2
λ11
[Α1]12
Z11
θ33
1.0
[Α1]11
1.0
ψ11
S2Fa
PEP1
En1
θ23
θ14
1.0
λ31
λ21
θ22
S2En
1.0
[Α1]22
Fa3
λ42
λ62
[Α2]22
λ52
λ62
SLG1
TRD1
WRY1
SLG2
TRD2
WRY2
SLG3
TRD3
WRY3
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
E41
E51
E61
E42
E52
E62
E43
E53
E63
ε44
ε55
ε66
ε44
ε55
ε66
ε44
ε55
ε66
17
Current Research Aims
ñ To develop special purpose methodology for process
factor analysis of lagged correlation matrices.
ñ To use this methodology to provide special purpose
software for fitting PFA models to lagged correlation
matrices.
ç This will improve on software developed primarily for other
purposes by À
♦ Ensuring stationarity of the fitted model
♦ Providing additional facilities - lagged factor correlations, show
whether or not autoregressive and moving average weights
satisfy invertibility conditions, breakdown of fit at different lags,
show indirect effects of shocks on manifest variables, exploratory
rotation of factors
18
The Correlation Structure
ñ The correlation structure employed is
T! œ AF! Aw  H%
Tj œ AFj Aw
where the 7 ‚ 7 lagged factor correlation matrix
Fj œ Cova0>j ß 0>w b is a function of the 7 ‚ 7
autoregressive weight matrices E" ß E# ß á , the moving
average weight matrices F" ß F# ß á , the random shock
covariance matrix, G and an initial state covariance matrix
@. Details are omitted to avoid presenting a mass of
algebra.
ç F! is a symmetric matrix with unit diagonals
ç Fj ß 6 œ "ß #ß á are nonsymmetric square matrices
19
Summary of Parameter Matrices
ñ Free parameters
ç
ç
ç
ç
A
H%
E3
F4
ç G
5 ‚ 7 Factor matrix
Diagonal 5 ‚ 5 error covariance matrix
7 ‚ 7 Autoregressive weight matrix
7 ‚ 7 Moving average weight matrix
7 ‚ 7 symmetric random shock covariance matrix
ñ Number of identification conditions œ 7# as in standard factor
analysis.
ñ Nonlinear matrix functions of free parameters
ç Fj
ç @
j œ !ß "ß á ß P À 7 ‚ 7 lagged factor correlation matrices.
Nonlinear constraints are applied to Fj to yield unit
diagonals.
=7 ‚ =7 covariance matrix of the = initial state vectors.
(= œ maxa:ß ; b)
20
Two Sets of Regression Weights in PFA
ñ In Classical Factor Analysis there is a single system of
regression equations with the factor loadings in A as
regression weights.
ñ In Process Factor Analysis there are two systems of
regression equations
ç Factor Analysis model with factor loadings in A as regression
weights
ç Autoregression with elements of E" ß E# ß á as regression
weights plus moving average weights F" ß F# á
ñ Two latent variable covariance matrices: G, F
ñ In exploratory PFA the matrices Aß E" ß E# ß á ß
F" ß F# ß á ß Gß F are all affected by rotation of factors.
21
Ordinary Least Squares Estimation
ñ Lagged correlation matrices (5 ‚ 5Ñ À V! ß V" ß á ß VP
ñ Implied covariance matrices (5 ‚ 5Ñ À T! ß T" ß á ß TP
ñ Discrepancy function to be minimized. Overall residual
sum of squares,
J œ "Jj
P
jœ!
Jj œ ""cVj  Tj d#34
5
where
5
3œ" 4œ"
is the residual sum of squares at lag j and P : œ # is
the maximum lag considered. (e.g. P œ # or $ or %)
22
Starting Values
ñ Convergence of the Gauss-Newton algorithm can be
problematical if poor starting values are used.
ñ Algorithm used for obtaining starting values:
~
s%
ç OLS factor analysis of V! to yield A and H
s and F
s!
ç Oblique rotation to yield A
s j ß j œ "ß #ß á by ordinary least squares.
ç Obtain estimates F
ç Estimate the E3 , F4 , and G by applying a generalization to
multivariate time series of methods suggested by Box &
s j.
Jenkins (1976, Appendix 6.2) to the F
ñ Excellent approximation when ; œ !
23
An Example
ñ Data from Lebo & Nesselroade (1978).
ñ Each of four female subjects rated her mood on six
adjective rating scales daily for X œ "!$ days.
ñ Scales were Active, Lively, Peppy to define an Energy
factor and Sluggish, Tired, Weary to define a Fatigue
factor.
ñ Subject #4 is employed. For illustrative purposes an
ARMA(1,1) model is used for the latent time series.
ç NoÞ of variables, 5 œ '
ç No. of factors, 7 œ #
ç Maximum lag employed, P œ $
24
Lebo #4 Example: Exploratory PFA(1,1)
Starting values for parameter estimates
s
A
!Þ*#
!Þ)"
!Þ(#
!Þ"%
!Þ!!
!Þ"%
!Þ!!
!Þ"(
!Þ#*
!Þ("
!Þ*$
!Þ((
s%
!Þ"'
!Þ"%
!Þ"$
!Þ$&
!Þ"%
!Þ#'
s"
E
!Þ%(
 !Þ#*
!Þ%%
!Þ)(
k-max k œ !Þ($
s"
F
 !Þ"&
!Þ&$
 !Þ$&  !Þ&$
k-max k œ !Þ&"
Shock correlations and
s
sd's obtained from G
"Þ!!  !Þ($
 !Þ($
"Þ!!
Sd !Þ*&
!Þ)*
Figures in red represent parameter values fixed for identification
purposes.
25
Lebo #4 Example: Exploratory PFA(1,1)
Functions of starting values
s!
F
"Þ!! !Þ'&
"Þ!!
!Þ'&
s#
F
!Þ"$ !Þ"&
!Þ!"
!Þ#*
s"
F
!Þ#! !Þ!*
!Þ""
!Þ$)
s$
F
!Þ!' !Þ"'
!Þ!&
!Þ")
RSS
s ! !Þ!!
J
s " !Þ"$
J
s # !Þ"%
J
s $ !Þ$!
J
s X !Þ&(
J
Figures in red represent function values that are constrained for
identification purposes.
26
Lebo #4 Example: Exploratory PFA(1,1)
OLS solution
s
A
!Þ*%
!Þ'*
!Þ(!
!Þ"'
!Þ!!
!Þ!#
!Þ!!
!Þ#*
!Þ$$
!Þ'%
!Þ*!
!Þ*%
s%
!Þ""
!Þ")
!Þ""
!Þ%$
!Þ"*
!Þ"&
s"
E
!Þ($
!Þ"#
!Þ!'
!Þ)&
k-max k œ !Þ)*
s"
F
!Þ'% !Þ##
!Þ"! !Þ%)
k-max k œ !Þ&)
Shock correlations and
s
sd's obtained from G
"Þ!!
!Þ(%
Sd !Þ*(
!Þ(%
"Þ!!
!Þ)*
27
Lebo #4 Example: Exploratory PFA(1,1)
OLS solution
s!
F
"Þ!! !Þ'&
"Þ!!
!Þ'&
s#
F
!Þ"# !Þ!&
!Þ!(
!Þ$!
s"
F
!Þ") !Þ"#
!Þ!*
!Þ$'
s$
F
!Þ!)
!Þ!!
!Þ!&
!Þ#'
s!
J
s"
J
s#
J
s$
J
sX
J
RSS
OLS Start
!Þ!# !Þ!!
!Þ"! !Þ"$
!Þ") !Þ"%
!Þ!' !Þ$!
!Þ$' !Þ&(
28
Lebo #4 Example: Exploratory PFA(1,1)
Oblique CF-Varimax Rotation
s
A
!Þ*'
!Þ($
!Þ(%
!Þ#$
!Þ!*
!Þ!(
!Þ!$
!Þ#&
!Þ#*
!Þ&*
!Þ)%
!Þ))
s%
!Þ""
!Þ")
!Þ""
!Þ%$
!Þ"*
!Þ"&
s"
E
!Þ("
!Þ"!
!Þ!&
!Þ)'
k-max k œ !Þ)*
s"
F
!Þ'$ !Þ#"
!Þ"! !Þ&!
k-max k œ !Þ&)
Shock Corr.
"Þ!! !Þ(!
!Þ(!
"Þ!!
Sd
!Þ*(
!Þ))
29
Lebo #4 Example: Exploratory PFA(1,1)
Oblique CF-Varimax Rotation
s!
F
"Þ!! !Þ'"
"Þ!!
!Þ'"
s#
F
!Þ"# !Þ!%
!Þ!(
!Þ$$
s"
F
!Þ") !Þ"#
!Þ!*
!Þ$*
s$
F
!Þ!)
!Þ!!
!Þ!&
!Þ#)
RSS
s ! !Þ!#
J
s " !Þ"!
J
s # !Þ")
J
s $ !Þ!'
J
s X !Þ$'
J
30
Lebo #4 Example: Exploratory PFA(1,1)
Alternative representation of ARMA process: MA Weights
are replaced by shocks that are correlated across time.
Lagged shock correlations and standard
s and F
s".
deviations obtained from G
Lag 0
Lag 1
Sd
"Þ!!
!Þ')
!Þ')
"Þ!!
!Þ$*
!Þ$&
!Þ#!
 !Þ%$
"Þ!*
"Þ!#
31
Lebo #4 Example: Exploratory PFA(1,1)
Shock Factor analysis representation of PFA(1,1)
‡
s
A!
!Þ*$
!Þ!$
!Þ(" !Þ##
!Þ(# !Þ#&
!Þ##
!Þ&#
!Þ!)
!Þ(%
!Þ!(
!Þ((
‡
s
A"
!Þ!) !Þ!)
!Þ!$ !Þ"&
!Þ!# !Þ"'
!Þ!(
!Þ#"
!Þ"#
!Þ#)
!Þ"#
!Þ#*
‡
s
A#
!Þ!) !Þ!#
!Þ!# !Þ!*
!Þ!# !Þ"!
!Þ!'
!Þ"(
!Þ"!
!Þ#$
!Þ""
!Þ#%
‡
s
A$
!Þ!(
!Þ!"
!Þ!# !Þ!'
!Þ!# !Þ!'
!Þ!&
!Þ"%
!Þ!*
!Þ#!
!Þ"!
!Þ#"
Note that the contemporaneous correlation of the two factors
is !Þ(!
The largest absolute difference between process- and shockimplied correlations at ; ‡ œ $ is !Þ!'
32
Lebo #4 Example: Exploratory PFA(1,1)
s
A
Alternative identification conditions
s"
E
s%
!Þ"! !Þ*%
!Þ"# !Þ*!
!Þ"& !Þ*$
!Þ%#
!Þ'$
!Þ'#
!Þ''
!Þ'%
!Þ'(
!Þ""
!Þ")
!Þ""
!Þ%$
!Þ"*
!Þ"&
!Þ*#
!Þ"#
!Þ!' !Þ'&
k-max k œ !Þ)*
s
G
!Þ'#
!Þ!!
!Þ!!
!Þ*&
s"
F
!Þ'$ !Þ"$
!Þ"& !Þ&!
k-max k œ !Þ&)
s!
F
"Þ!!
!Þ!!
!Þ!!
"Þ!!
Note poor simple structure
33
Lebo #4 Example: Exploratory PFA(1,1)
Alternative identification conditions
s!
s#
F
F
RSS
s ! !Þ!#
"Þ!!
!Þ!!
!Þ&! !Þ!#
J
s " !Þ"!
"Þ!!
!Þ!" !Þ"#
J
!Þ!!
s # !Þ")
J
s"
s$
s $ !Þ!'
F
F
J
s X !Þ$'
!Þ&$ !Þ!!
!Þ%' !Þ!$
J
!Þ!%
!Þ")
!Þ!$ !Þ!)
34
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35
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36
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Abstract
The development of methodology for the factor analysis of data from a single subject over
repeated observations on a battery of variables arose out of the P-technique recommended by
Cattell. In this approach the data are treated as if the repeated observations are independent and
subjected to a standard factor analysis. Comments by T. W. Anderson that the P-technique
should be supplemented by the modeling of time dependence in the data were influential on later
modifications of Cattell's approach.
Work on the P-technique is first reviewed and associated developments are presented. It is seen
that two types of model have been used. In one, the process factor analysis model, the factors
follow a VARMA process. This model has a single factor matrix and additional time series
parameter matrices. In the other, the shock factor analysis model, the factors are independently
distributed random shock vectors or, equivalently, white noise. This model has several factor
matrices, each representing the effect of the latent shock vector on the vector of manifest
variables at one of a number of lags. It has no time series parameter matrices. Different
perspectives on the same situation are provided by the two models and it is instructive to employ
them concurrently.
Two classes of estimation methodology are also considered. One uses full information maximum
likelihood based on the original observations. The other is a moment-based approach. It involves
the initial computation of lagged correlation matrices, followed by the fitting of appropriate
correlation structures using a minimum discrepancy approach.
A modern development is then described. The process factor analysis model is estimated using a
moment-based approach. Both confirmatory, and exploratory variants with rotation of factors, are
available. In addition, the shock factor representation for any process factor analysis is provided.
Examples are employed to illustrate capabilities of the model.
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