Transcript SEM and Longitudinal Data - University of Texas at Dallas

SEM and Longitudinal Data
Autoregressive models and
missing data
UTD
06.04.2006
1
1.1,1.2) Overview and Goals
• 1) Model specification for autoregressive, cross-lagged,
growth curves and ALT/hybrid models.
• 1a) Single group models;
• 1b) Multiple-group models.
• 1c) MIMIC models.
• 1d) Treatment of missing values.
• 2) Use of AMOS graphics to set up and test
autoregressive, cross-lagged, growth curves and
ALT/hybrid models.
• 3) Interpretation of parameters in the different types of
models in substantive examples (authoritarianism and
anomie in Germany).
• 4)
2 Comparison of the approaches.
Growth curves
models
Cross-lagged
models
Autoregressive
models (called
also quasisimplex or
markov models)
3
ALT/hybrid
models
Autoregressive Model
•each variable X at t2 is a function of its lagged measure at
t1 and residual (or any variable at t(n) is a function of a
variable at t(n-1) and not of any variable before (like a
variable at time t(n-2)).
•stability coefficients indicate degree of stability of
interindividual differences
res1
Xt1
Xt2
One has to differentiate between unstandardized stability
coefficients and standardized stability coefficients.
4
Cross-lagged autoregressive models
•each variable X at t2 function of its lagged measure at t1 and residual
•stability coefficients indicate degree of stability of inter-individual
differences
Xt1
a
Xt2
res
1
c
d
Yt1
5
b
Yt2
cross-lagged autoregressive model (Finkel 1995)
• cross-construct regression weights: X predicting Y,
controlling for former values of Y
res
2
latent growth curve models
•for analysing individual change processes using single/ multiple indicators
• assumption: a latent trajectory characterizing the sample
(or subgroups) can be found
• individual change as function of intercept and slope factors for each time
period
res
1
res
res
2
3
Xt2
Xt1
Xt3
1
1
0
1
Intercept
1
F
Slope
individual change as function of intercept and slope factors for each time
period
6
ALT/hybrid models
An ALT/hybrid model is a combination of both autoregressive and
growth curves models. It includes the stability coefficients of the
autoregressive model, and also the slope and intercept latent
variables.
res
1
res
res
2
3
Xt2
Xt1
Xt3
1
1
0
1
Intercept
7
1
Slope
F
1.3,1.4) Autoregressive models with t
waves on the latent level
Err2
b21
Xt1
Errn
bt t-1
Xt2
Xt
For the structural model the first order autoregressive structure
without intercepts is
Xit8 = bt t-1*Xit-1 + Err t ,
i=1,…
1.5) Interpretation of parameters:
• Stability is generally defined as a relation between
individuals in two subsequent times t-1 and t.
• Perfect stability means that the relation between
individuals does not change. For example, if we
have a sample of two individuals, and one individual
has a value twice as high as the second one has, then
we would expect this relation not to change in the
next time point under perfect stability. In such a
case, the standardized regression (stability)
coefficient between time1 and time 2 will be 1.
9
Exercise:
• 1) In a sample of 3 individuals we measure
attitude with one indicator. The values the
three respondents provide are:1,1,2. In the
second time point the first individual has a
value of 2, and the stability coefficient is 1.
What are the values that the two other
respondents provide in the second time
point?
10
• Positive stability (standardized coefficient close to +1):
the order remains the same between individuals.
• Negative stability (standardized coefficient close to –1):
the order of individuals inverts.
• Low stability (standardized coefficient close to zero):
the order just mixes, low become high, high become
low.
• No stability at all: standardized regression coefficient is
zero.
• Standardized and unstandardized regression coefficients
have different functions. For test propositions we can
compare only the unstandardized coefficients, just like
in other statistical tests. For interpretation, mostly
11
standardized
coefficients are used.
1.6) Parameterization of Autocorrelation:
Autoregressive models with three waves and multiple
indicators
e11
e21
e41
e31
e51
e71
e61
e81
e91
AU02W3R
AU03W3R
AU02W2R
AU03W2R AU01W3R
AU01W1R
AU02W1R
AU03W1RAU01W2R
f2c f3c
f2b f3b
1
1
1
f2a f3a
auto_02
a1
auto_03
a2
1
r1
A model with autocorrelations
12
auto_04
1
r2
Autoregressive model with method
factors
•
•
•
•
Every item is determined by:
1) a latent variable of interest;
2) measurement error;
3) specific/unique factor: each item has a specific meaning
that frequently will be invariant across time and groups.
• We interpret each item’s content as being constituted by two
facets: the common factor (the latent variable) and the
unique factor (the item meaning).
13
Autoregressive models with three waves and
multiple indicators- an alternative approach
Ex2: autoregressive model of authoritarianism
with MTMM and Socratic effect
1
1
M1
M2
0
e1
e3
e2
1
1
1
AU03W 1R
AU04W 1R
1
1
aut1
e5
e4
e6
1
1
AU04W 3R
AU03W 2R
AU04W 2R AU03W 3R
1
1
1
1
aut3
aut2
1
res1
A model with uniqueness factors
14
1
1
res2
1.7) Socratic Effect
• H1) In addition to common factors, there are
specific factors that contribute significantly to the
autocorrelations of the indicators.
• The consistency process underlying the socratic
effect does not primarily change the relations
between items, but rather the relations between
items and their corresponding latent variables.
• This effect is postulated for short term panel
studies.
15
Socratic Effect (2)
• If there is a learning and a memory effect, then the
following hypothesis should hold:
• H2.1) The standardized coefficient linking an item
to its corresponding latent variable is significantly
lower in the first wave than in the second wave in
short-wave panel-studies (what is a short wave?…).
• We further hypothesize that the loading does not
increase beyond wave 2.
• H2.2) The standardized loading of the items on the
latent variable do not increase beyond the second
16
wave.
Socratic Effect (3)
• The same should hold for the uniqueness factors:
• H3.1) Standardized factor loadings of the items on
the uniqueness factors are significantly lower in the
first wave than in the second wave.
• H3.2) The standardized factor loading on the
uniqueness factor does not increase significantly
from the second wave on.
17
Socratic Effect (4)
• Logically we can derive from H2 and H3:
Because respondents are sensitized by the first
interview, their responses, in general, should be
more consistent.
H4.1) The variance of random measurement error is
greater in the first wave than in the second.
H4.2) The random measurement error does not
increase significantly from the second wave on.
18
Socratic Effect (5)
• H5) In a short-wave panel (for example only a few
weeks) the inter-temporal consistency of a latent variable
is nearly perfect, that is, the unstandardized and the
standardized stability coefficients are close to unity.
• Hypotheses H2-H5 form the core of the socratic effect.
• H7) The observed interwave correlation of an item
between t-1 and t is larger than the correlation between t1
and t2. Therefore we expect the stability from the second
time point to be higher than between the first and the
second.
• Empirical evidence was inconclusive.
• Saris/van der Putte proposed an alternative model which
had the same fit for the same data (but simpler) based on
a 19true score model (1988).
Cross-lagged effects: the issue of causality, standard and non-standard
model specification.
(1) Srole (1956, p. 716; see Scheepers et al. 1992):
•
anomic individuals choose authoritarian stances in order to
recover orientation
Anomia
Authoritarianism
(2) McClosky & Schaar (1965)
•
authoritarian individuals are hampered to interact effectively
•
less opportunities to escape from social isolation
•
resulting in anomia
Anomia
20
Authoritarianism
(3) reciprocal relationship: not necessarily unplausible
Anomia
Authoritarianism
Research questions for longitudinal analysis:
a)
• are authoritarian attitudes stable over time?
• are anomic attitudes stable over time?
• does anomia cause authoritarianism, does authoritarianism cause
anomia or do we get evidence for both processes?
b) Research questions which we will relate to tomorrow:
• if we get evidence for individual change of authoritarian and/ or
anomic attitudes: is there an increase or a decrease?
• do we get evidence for individual differences concerning such a
development?
• is there a relationship between the initial level of authoritarianism/
anomia and its dynamic?
21
Cross-lagged autoregressive models
autoregressive model
• each variable X at t2 function of its lagged measure at t1 and residual
• stability coefficients indicate degree of stability of interindividual
differences
Xt1
a
Xt2
res
1
c
d
Yt1
b
Yt2
cross-lagged autoregressive model (Finkel 1995)
• cross-construct regression weights: X predicting Y, controlling for
former values of Y
22
res
2
A Cross-lagged model with auto-correlations
e11
e21
e41
e31
e51
e71
e61
e81
e91
AU02W3R
AU03W3R
AU02W2R
AU03W2R AU01W3R
AU01W1R
AU02W1R
AU03W1RAU01W2R
f2c f3c
f2b f3b
1
1
f2a f3a
1
a1
auto_02
a2
auto_03
auto_04
1
1
r2
r1
aut_2
aut_1
b1
anom_02
1
1
f4a
AN01W1R AN02W1R
1
e10
23
1
e11
b2
anom_03
f4b
r3
AN01W2R
AN02W2R
1
e12
1
e13
anom_04
1
1
f4c
AN02W3R
AN01W3R
1
1
e14
e15
1
r4
A cross-lagged model with multiple indicators and
uniqueness factors
UAu2
UAu1
e11
e21
e41
e31
e51
UAu3
e71
e61
e81
e91
AU02W3R
AU03W3R
AU02W2R
AU03W2R AU01W3R
AU01W1R
AU02W1R
AU03W1RAU01W2R
f3c
f2c
f3b
f2b
1
1
f2a f3a
1
a1
auto_02
a2
auto_03
auto_04
1
1
r2
r1
aut_2
aut_1
b1
anom_02
1
1
f4a
AN01W1R AN02W1R
1
e10
1
e11
UAn1
24
b2
anom_03
f4b
1
r3
AN01W2R
AN02W2R
1
e12
anom_04
1
1
e13
AN02W3R
AN01W3R
1
1
e14
UAn2
f4c
e15
1
r4
1.9) Measurement invariance over time
• To establish equal meaning of the construct
over time and over groups, different criteria
for measurement invariance have been
proposed.
• These criteria form prerequisites for a
comparison of latent means over time and
groups.
25
Measurement Invariance:
Equal factor loadings across groups and/or time
points
Group A
dA11
Item a
dA22
Item b
lA11=1
lA21
lA31
dA33
Group B
dB11
f 11
k A1
A
At1
dB22
Item a
Item b
lB11=1
lB21
lB31
dB33
Item c
Bt1
fB11
k B1
Item c
fB21
fA21
dA44
d
Item d
A
Item e
55
dA66
Item f
26
dB44
lA42=1
lA52
lA62

dB55
A
t2
fA22
k A2
dB66
Item d
Item e
lB42=1
lB52
lB62
Item f
Bt2
fB22
k B2
Steps in testing for Measurement Invariance
• Configural Invariance
• Metric/measurement Invariance
• Scalar Invariance
• Invariance of Factor Variances
• Invariance of Factor Covariances
• Invariance of latent Means
• Invariance of Unique Variances
27
Steps in testing for Measurement Invariance
• Configural Invariance
• Metric Invariance
• Equal factor loadings
• Same scale units in both groups/time points
• Presumption for the comparison of latent means
• Scalar Invariance
• Invariance of Factor Variances
• Invariance of Factor Covariances
• Invariance of latent Means
• Invariance of Unique Variances
28
Steps in testing for Measurement Invariance
• Configural Invariance
• Metric Invariance
• Scalar Invariance
• Equal item intercepts
• Same systematic biases in both groups/time points
• Presumption for comparison of latent means
• Invariance of Factor Variances
• Invariance of Factor Covariances
• Invariance of latent Means
29
• Invariance of Unique Variances
Steps in testing for Measurement Invariance
• Configural Invariance
• Metric Invariance
• Scalar Invariance
• Invariance of Factor Variances
• Invariance of Factor Covariances
• Invariance of latent Means
• Invariance of Unique Variances
30
Full vs. Partial Invariance
• Concept of ‘partial invariance’ introduced by
Byrne, Shavelson & Muthén (1989)
• Procedure
• Constrain complete matrix
• Use modification indices to find non-invariant
parameters and then relax the constraint
• Compare with the unrestricted model
• Steenkamp & Baumgartner (1998): Two indicators
with invariant loadings and intercepts are
sufficient for mean comparisons
31
• One of them can be the marker + one further
invariant item
1.10) Multi-group analysis of means and intercepts over time.
• ETA=Alpha(g) +Beta (g) *Eta (g) + Gamma (g) *Ksi (g)
+Zeta (g)
• X = constant + FA (g) + E (g)
• Where alpha and constant are vectors of constant
intercept terms. We assume that Zeta is uncorrelated
with Ksi, and E is uncorrelated with A. We also
assume as before that Ex(E)=Ex(Zeta)=0 (Ex is the
expected value operator), but it is not assumed that
Ex(A) is zero. The mean of A, Ex(A), will be a
parameter denoted by k.
32
Getting a model with means identified
• As Sörbom (1978) has shown, in order to estimate the
means, we must introduce some further restrictions:
• 1) setting the mean of the latent variable in one group-the
reference group- to zero. The estimation of the mean of the
latent variable in the other group is then the mean difference
with respect to the reference group.
• 2) setting the measurement models invariant across groups,
since it makes no sense to compare the means of constructs
having a different measurement model in the groups. One
intercept per construct has to be set to zero.
33
Getting a model with means identified (2)
• 3) invariant stabilities (not necessary for
identification).
• 4) When we have only one group: setting the
intercepts and factor loadings invariant over time as
we will do in the examples.
• 5) Two more new methods are dealt with by Little et
al. (2006).
34
Ex6a: estimation of means and intercepts
0;
0;
0;
e1
e2
1
1
In_Au3
0;
e3
1
0
0;
e4
In_Au3
1
e5
1
0
0;
e6
In_Au3
1
0
AU04W3R
AU04W2RAU03W3R
AU03W1R
AU04W1R AU03W2R
1
1
aut1
1
1
1
aut3
aut2
1
0;
res1
35
1
1
0;
res2
Ex6a: estimation of means and intercepts
,00
,21
,23
,04
,22
0; ,45
0; ,46
0; ,30
e1
e2
1
1
,51
e3
1
0
,01
0; ,32
0; ,42
e4
,51
1
e5
1
0
0; ,29
e6
1
,51
0
AU04W3R
AU04W2RAU03W3R
AU03W1R
AU04W1R AU03W2R
aut1
2,15; ,43
1,001,00
1,00 1,00
1,00 1,00
,84
,96
aut3
aut2
,38
1
0; ,08 ,08
res1
36
1
0; ,07
res2
More about stability
• A structural stability refers to longitudinal
consistency in multi-dimensional measurement
properties, and provides a basis for comparison of
factor means and relations over time (Meredith
1993). That is, it refers to measurement invariance
between the two time points.
• Differential stability represents the correlation of
individual latent scores assessed at two separate
occasions (corresponds to our earlier stability
concept).
37
More about stability (2)
• Latent mean stability describes changes in
the latent construct means over time.
38
1.12) MIMIC: autoregressive models
and cross-lagged effects
• Uses:
• Predicting exogenous variables in the autoregressive
models but possibly also on other endogenous variables.
• Auxiliary variables may improve FIML in case they
have no missing data, as they can help predicting
missing values in the model.
• Characteristics: the newly introduced exogenous
auxiliary/background/instrumental variables may be
dichotomous or interval. Some of them must be
observable.
• Distributional assumptions are not violated when
background
variables are dummies and used as
39
exogenuous variables.
e11
e21
e41
e31
e51
e71
e61
e81
e91
AU02W3R
AU03W3
AU02W2R
AU03W2R AU01W3R
AU01W1R
AU02W1R
AU03W1RAU01W2R
f3c
f2c
f2b f3b
1
1
1
1
r0
Age
40
f2a f3a
auto_02
a1
auto_03
a2
1
r1
Education East/West
auto_04
1
r2
1.13) Robustness and non-normality with multiplegroup analyses in SEM models (Satorra 2001)
• Satorra studied three chi square test statistics with
non-normal data and structural equation models with
multiple-groups.
• The normal theory-NT (ML) approach proposed by
Joereskog 1971 has been complemented by Satorra
with robust statistics and results on asymptotic
robustness. It was built in in LISREL.
• His conclusion: for non-normal data the various test
statistics like chi square and CR (critical ratio) have
different powers (power is type 2 error).
41
Robustness and non-normality(2)
• Non normality is not a problem with large sample
sizes (1000 and more), and results are robust.
However, performance of chi square statistics in any
SEM model in small and medium samples with nonnormal data is rather poor.
• Differences in power do not arise when asymptotic
robustness (non-bias) holds.
42
Stability – different definitions
Type of Stability
Phenomenon
Coefficient
Interpretation
1.
Positional Stability
The relative position of
individuals within a
sample
AR coefficientsstandardized and
unstandardized
Perfect positional stability
means coefficients are all
1. No stability:0. medium:
around 0.5
2.
Absolute Stability
The absolute mean
In AR model the mean
over time.
In LGC model coefficient
from latent slope across
time.
Whether level or mean or
coefficients change. May
change when positional
stability is 1. However, if
there is absolute stability,
there is also positional
stability.
3.
Dynamic Stability
Equilibrium of a dynamic
model
Eigenvalues are all nonzero (positive or negative)
A dynamic stability means
long-term growth or
decline.
43
2.11 Beyond SEM: General
Latent Variable Modeling/
Muthen 2002
• The generalized approach: it views all kinds of latent
variable models as specific cases of one general
model.
• Its goals:
• 1) better integration of Psychometric modeling into
mainstream statistics.
• 2) to show how many statistical analyses implicitly
utilize the idea of latent variables in the form of
random effects, components of variation, missing
44
data,
mixture components and clusters.
• The generality of the model is achieved by considering not only
continuous latent variables, but also categorical latent variables.
• This makes it possible to unify and to extend the following analyses:
classical SEM, growth curve modeling, multi-level modeling, missing
data modeling, finite mixture modeling, latent class modeling and
survival modeling.
• The general framework is represented by a square D, and is a
combination of 3 special cases (ellipses):
• A) continuous latent variables: includes measurement error, measureme
invariance in conventional SEM, latent variables in growth modeling an
variance components in multi-level modeling.
• B) categorical latent variables: includes latent class analysis and latent
class growth analysis.
• C) latent profile models and models that combine continuous and
categorical latent variables such as growth mixture modeling.
• D) new types of models including modeling with missing data on a
categorical latent variable in randomized trial (like modeling under
MNAR
).
45